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% that were lost in the 2022-08 Great Delete Disaster.
@article{lia-zha-lej-zha-liz-22-aa-sensit,
author = {Xiaobing Liao and Min Zhang and Jian Le and Lina Zhang and Zicheng Li}
title = {Global Sensitivity Analysis of Static Voltage Stability Based on Extended Affine Model}
fullhash = {3657877d001c8db56f9a12d3dd0119e5}
journal = {Electric Power Systems Research}
month = {6}
volume = {208}
year = {2022}
pages = {Article 107872}
numpages = {-1}
doi = {10.1016/j.epsr.2022.107872}
abstract = {Large scale centralized grid-connection of renewable energy significantly increases the uncertainty of a power system, and accurate analysis and quantitative evaluation of the impact of this uncertainty on the static voltage stability of a power system is the important premise and basis for improving the power system operation security. This paper studies a global sensitivity analysis method of the static voltage stability based on extended affine model. A static voltage stability interval evaluation model based on L index is constructed according to the extended affine arithmetic, and the global sensitivity analysis method based on variance decomposition is introduced to evaluate the importance of the input interval variables on the static voltage stability. The analysis of the simulation results on the IEEE systems shows that the proposed method can suppress the effect of interval expansion more effectively than the traditional affine algorithm, and the global sensitivity analysis method based on analytic variance decomposition can identify the importance of input interval variables more effectively.}
}
@inproceedings{riv-fra-pus-22-aa-compile,
author = {Joao Rivera and Franz Franchetti and Markus P{\"{u}}schel}
title = {A Compiler for Sound Floating-Point Computations using Affine Arithmetic}
fullhash = {67813303cebcc8db4fb8652f26959d0b}
booktitle = {Poc. International Symposium on Code Generation and Optimization (CGO 2022)}
month = {4}
year = {2022}
pages = {66-78}
numpages = {13}
url = {https://acl.inf.ethz.ch/publications/cgo22-p57-preprint.pdf}
abstract = {Floating-point arithmetic is extensively used in scientific and engineering applications to approximate real arithmetic. Unfortunately, floating-point arithmetic is not a sound implementation of real arithmetic, i.e., it may produce different results, does not provide error guarantees, and the errors can become arbitrarily large. In this paper, we introduce SafeGen, a source-to-source compiler that rewrites a given C program using floating-point arithmetic to an efficient C program performing the same computation soundly, i.e., it returns an error bound that is guaranteed to contain the correct result of the program if it had been executed in real arithmetic. Equivalently, it gives a precision certificate on the number of correct bits in the result. SafeGen uses affine arithmetic (AA) that keeps accuracy high compared to interval arithmetic by preserving linear correlations between variables. To mitigate its high cost, SafeGen combines a novel form of static analysis to identify these correlations with a flexible policy-based approach for their selection. SafeGen supports SIMD intrinsics in the input and can output SIMDoptimized code. Our results show that SafeGen-generated code is 30--70 times faster than manually rewritten code using AA libraries. Equivalently, SafeGen can offer many more bits of certified accuracy within a reduced time budget.}
}
@article{zhe-wan-sha-zha-liy-22-aa-powflow,
author = {Wendi Zheng and Xiangjie Wang and Zhenguo Shao and Min Zhang and Yixin Li}
title = {A Modified Affine Arithmetic-Based Interval Optimization for Integrated Energy System with Multiple Uncertainties}
fullhash = {392b84117dc0bc51396867b1f12a29a4}
journal = {Journal of Renewable and Sustainable Energy}
number = {1}
volume = {14}
year = {2022}
pages = {016302}
numpages = {1}
doi = {10.1063/5.0079306}
abstract = {In integrated energy systems, uncertainties cause day-ahead scheduling plans to be uncertain rather than an exact solution. However, traditional interval optimization models ignore the correlation between variables, resulting in an over-conservative range of solutions. To overcome this limitation, the present paper proposes a modified affine arithmetic to establish an interval optimal scheduling model based on deterministic optimization. The correlation between different variables is tracked by sharing noise terms, and the solution in affine form is transformed into a more intuitive interval solution by a modified affine approximation method and linear relaxation method. Finally, with the minimum midpoint and radii of daily operating cost interval as the objective function, the multi-objective function is converted into a single-objective function through Lagrange multiplication to solve the model. In a case study, the proposed method is compared with the Monte Carlo simulation method and the traditional interval optimization method without affine arithmetic. The simulation results show that the radii of the interval solutions obtained by the proposed method is 9.39{\%} less than the traditional interval arithmetic at the 15{\%} level of uncertainty.}
}
@article{pep-car-vac-djo-22-aa-markov,
author = {Antonio Pepiciello and Fabrizio {De Caro} and Alfredo Vaccaro and Sasa Djokic}
title = {Affine Arithmetic-Based Reliable Estimation of Transition State Boundaries for Uncertain Markov Chains}
fullhash = {4f9feffffcc582a83d0e2dd8b6663aea}
journal = {Electric Power Systems Research}
volume = {204}
year = {2022}
pages = {107711}
numpages = {1}
doi = {10.1016/j.epsr.2021.107711}
abstract = {Markov Chain-based models have been extensively used in numerous power system applications, ranging from decision support systems, to reliability and resilience analysis. The state transition probabilities, which determine the overall model dynamics, have been traditionally described by deterministic values. This common assumption might not be suitable for analyzing modern power systems, where large number of complex and correlated uncertainties could significantly affect the correct evaluation and estimation of both model parameters and model outputs. In addressing that problem, this paper advocates the role of affine arithmetic, by proposing a reliable framework for Markov Chain transient analysis in the presence of data uncertainties. The main idea is to represent the uncertainties in state transition probabilities through affine forms, which then allow to apply affine arithmetic-based operators and to identify the corresponding transient state probabilities. Two case studies are presented and discussed, in order to demonstrate the effectiveness of the proposed methodology in practical power system problems and applications.}
}
@article{zha-son-che-cho-22-aa-battery,
author = {Xuehan Zhang and Yongju Son and Taesu Cheong and Sungyun Choi}
title = {Affine-Arithmetic-Based Microgrid Interval Optimization Considering Uncertainty and Battery Energy Storage System Degradation}
fullhash = {b05026bdcd41d76b33b533117025ab56}
journal = {Energy}
volume = {242}
year = {2022}
pages = {123015}
numpages = {1}
doi = {0.1016/j.energy.2021.123015}
abstract = {Microgrids can effectively integrate renewable energy sources (RESs) and provide power for local customers. However, uncertainties of RESs and loads pose challenges to microgrid operation. The traditional point optimization method is unrealistic, and the widely used stochastic optimization (SO) method is time-consuming. Besides, battery energy storage systems (BESSs) are critical dispatchable devices to alleviate adverse effects of uncertainty, so an accurate nonlinear degradation cost model of BESSs should also be proposed. To handle such problems, the paper proposes an affine--arithmetic (AA)-based microgrid interval optimization (IO) method considering uncertainty and BESS degradation. First, the AA theory is introduced to model the RES and load variation ranges as intervals and calculate the interval uncertainty. Then, a nonlinear BESS degradation cost model is proposed, which can assess battery degradation costs considering different charging and discharging behaviors. The nondominated sorting genetic algorithm-II (NSGA-II) is employed to solve the proposed microgrid IO framework. For validation, the proposed IO method was compared with the point optimization method and SO method under various uncertainty realizations in a modified IEEE 33 bus system. The simulation results indicated the effectiveness of the proposed IO method in terms of an equilibrium between the simulation time and optimization performance.}
}
@incollection{cha-21-aa-struct,
author = {S. Rout and Snehashish Chakraverty}
title = {Affine approach in solving linear structural dynamic problems with uncertain parameters}
publisher = {Academic Press}
fullhash = {9d8ed3323911ffb7b0429aadd9162bf0}
booktitle = {New Paradigms in Computational Modeling and Its Applications}
chapter = {8}
isbn = {978-0-12-822133-4}
year = {2021}
pages = {97-121}
numpages = {25}
doi = {10.1016/B978-0-12-822133-4.00003-7}
abstract = {The present chapter proposes an affine approach for computing the linear structural dynamic problems having uncertain and nonprobabilistic model parameters with the help of affine arithmetic operations. The solutions of linear dynamic problems of structures lead to eigenvalue problems, viz., generalized and standard eigenvalue problems. When there is a deficiency of large data, the material and geometric properties of the said problems may be considered as intervals due to uncertainty. The error explosion problem occurs in interval arithmetic, which leads to wide-ranging results. In this case, affine arithmetic may be developed to handle the uncertainty with ease. Using this affine arithmetic, we have included a new procedure for solving standard as well as generalized interval eigenvalue problems having uncertainty in the form of closed intervals. Several numerical examples related to various applications of structural dynamic problems have been worked out to illustrate the reliability and efficiency of the present approach.}
}
@article{ngo-wuw-21-aa-reldist,
author = {Vietcuong Ngo and Wenchuan Wu}
title = {Interval Distribution Power Flow With Relative-Distance-Measure Arithmetic}
fullhash = {50a341970f3410235a001c6cb9e99736}
journal = {IEEE Transactions on Smart Grid}
month = {9}
number = {5}
volume = {12}
year = {2021}
pages = {3858-3867}
numpages = {10}
doi = {10.1109/TSG.2021.3074018}
abstract = {Interval distribution power flow can analyze the uncertainties incurred by the random fluctuations of distributed renewable generation and load demand in distribution networks. The conventional interval arithmetic (IA) and affine arithmetic (AA) cannot fully apply the characteristics of general algebra: associativity, inverse elements, distributive law, so the IA or AA based interval power flows (IPFs) are either very conservative or unreliable. Relative-Distance-Measure (RDM) arithmetic can overcome these defects in the conventional arithmetic for interval analysis and therefore a RDM based IPF is proposed in this paper. Furthermore, a linear programming contractor (LPC) is developed to solve this RDM based IPF, which can iteratively transform the IPF problem into serial linear optimization models and its results are reliable. Simulation results of IEEE 123-bus test systems show that the proposed IPF can obtain much better results than those of baseline methods in terms of accuracy and reliability.}
}
@article{wan-don-zha-zha-21-aa-bilevel,
author = {Shouxiang Wang and Yichao Dong and Qianyu Zhao and Xu Zhang}
title = {Bi-Level Multi-Objective Joint Planning of Distribution Networks Considering Uncertainties}
fullhash = {00060049efd81666fc33963b24c73300}
journal = {Journal of Modern Power Systems and Clean Energy}
year = {2021}
pages = {1-13}
numpages = {13}
doi = {10.35833/MPCE.2020.000930}
abstract = {With the increasing penetration of photovoltaics in distribution networks, the adaptability of distribution network under uncertainties needs to be considered in the planning of distribution systems. In this paper, the interval arithmetic and affine arithmetic are applied to deal with uncertainties, and an affine arithmetic based bi-level multi-objective joint planning model is built, which can obtain the planning schemes with low constraint violation risk, high reliability and strong adaptability. On this basis, a bi-level multi-objective solution methodology using affine arithmetic based non-dominated sorting genetic algorithm II is proposed, and the planning schemes that simultaneously meet economy and adaptability goals under uncertainties can be obtained. To further eliminate bad solutions and improve the solution qualities, an affine arithmetic based dominance relation weakening criteria and a deviation distance based modification method are proposed. A 24-bus test system and a Chinese 10kV distribution system are used for case studies. Different uncertainty levels are compared, and a sensitivity analysis of key parameters is conducted to explore their impacts on final planning schemes. The simulation results verify the advantages of proposed affine arithmetic based planning method.}
}
@article{liu-wan-zha-wan-21-aa-powflow,
author = {Qi Liu and Shouxiang Wang and Qianyu Zhao and Kai Wang}
title = {Interval Power Flow Calculation Algorithm for Multi-Terminal {DC} Distribution Networks Considering Distributed Generation Output Uncertainties}
fullhash = {4335fbff8da9b529e4025fe61d2e90cb}
journal = {IET Generation, Transmission {\&} Distribution}
number = {5}
volume = {15}
year = {2021}
pages = {986-996}
numpages = {11}
doi = {10.1049/gtd2.12074}
abstract = {Abstract An interval power flow calculation (PFC) algorithm for multi-terminal DC distribution networks is proposed to handle the uncertainties of distributed generation output powers and loads. Firstly, an equivalent resistance is introduced to represent the effect of droop control in DC distribution networks. The power losses of the equivalent resistance are offset by adding the injection powers to droop nodes, which guarantees the precision of the equivalent resistance model. The Newton--Raphson method is used for deterministic PFC based on the equivalent resistance, and it is applicable to DC distribution networks with different kinds of network topologies and voltage source converter control modes. Secondly, based on the deterministic PFC algorithm and affine arithmetic, an interval PFC algorithm is presented to handle the uncertainties of distributed generation output powers and loads. The affine arithmetic based interval PFC algorithm not only can determine power flow intervals quickly and accurately, but also can analyse the influences of distributed generation outputs to nodal voltages. Finally, a series of test systems are used to validate the proposed algorithms. Simulation results illustrate the accuracy and efficiency of the deterministic PFC algorithm and the interval PFC algorithm for multi-terminal DC distribution networks.}
}
@article{dor-21-aa-routhdelgam,
author = {H. Mallesam Dora}
title = {Model Order Reduction of Discrete Uncertain System Using Modified $\delta$-$\gamma$ {Routh} Approximation and Affine Arithmetic Method}
fullhash = {c06269ffa547e3ba7115c6c880ff8725}
journal = {Journal of Advanced Research in Electrical Engineering and Technology}
note = {Published in 2021-10-03}
number = {1}
volume = {3}
year = {2020}
pages = {22-25}
numpages = {4}
url = {http://www.thejournalshouse.com/index.php/electrical-engg-technology/article/view/464}
abstract = {This paper deals with a new technique for order reduction of higher order discrete uncertain system into lower order discrete time interval system. Using modified Routh Approximation and Affine Arithmetic developed to obtained numerator and denominator of reduced order model. The numerator and denominator of reduced order model is evaluated using modified $\delta$- and $gamma$-table respectively. Which is calculated based on Dolgin {\&} Zehab's concepts [3]. For understanding the effectiveness of the proposed method, it has been applied on two standard numerical examples, it is also found that the proposed method gives less integral square error and better approximation to the original higher order system.}
}
@article{dor-21-aa-routh,
author = {H. Mallesam Dora}
title = {New Aspect in Order Reduction of Large-scale Uncertain {MIMO} System Using Modified Routh Approximation and Affine Arithmetic}
fullhash = {c06269ffa547e3ba7115c6c880ff8725}
journal = {Journal of Advanced Research in Electrical Engineering and Technology}
note = {Published in 2021-10-03}
number = {1}
volume = {2}
year = {2020}
pages = {1-5}
numpages = {5}
urlraw = {http://www.thejournalshouse.com/index.php/SignalProcessing-Applications/issue/view/108}
url = {http://www.thejournalshouse.com/index.php/SignalProcessing-Applications/issue/view/108}
abstract = {This paper presents the Modified Routh Approximation (MRA) and Affine Arithmetic (AA) methods for obtaining the reduced order model (ROM) of higher order model (HOM).The reduced order model of a large-scale Uncertain MIMO System is required to simplify analysis of any system. The proposed method is compared with different MIMO uncertain systems in the literature. It is analyze that the proposed method is nearly identical in behavior to that of with original higher order systems. Observed numerical examples are considered to illustrate the effectiveness of the proposed method.}
}
@inproceedings{wan-zhe-21-aa-mumicro,
author = {Xiangjie Wang and Wendi Zheng}
title = {Optimal Dispatch of Multi-Microgrids System Based On Affine Arithmetic}
fullhash = {b6868e4c66f1d7e8a4ddfe1e0c17279d}
booktitle = {Proc. 2nd IEEE China International Youth Conference on Electrical Engineering (CIYCEE 2021)}
month = {12}
year = {2021}
pages = {1-6}
numpages = {6}
doi = {10.1109/CIYCEE53554.2021.9676774}
abstract = {With the development of the integrated energy system, a distribution area is usually composed of multiple microgrid systems, forming the multi-microgrid system of electricity, gas and heat combined supply. In order to maximize the interests of energy service companies, this paper establishes an interval optimization scheduling model of multi-microgrid integrated energy systems considering multi-energy complementarity and energy interaction between multi-micro grid. In addition, in order to solve the disadvantages of traditional interval optimization methods that are too conservative, affine arithmetic is used to represent the uncertainty of prediction variables, and the correlation between prediction variables of different microgrids is traced by the noise terms in affine arithmetic. Moreover, the effectiveness and applicability of this method are analyzed by comparing with monte Carlo simulation method and interval optimization method. Simulation results show that when the uncertainty is 15{\%}, the width of the interval solution obtained by this method is 16.25{\%} smaller than that of the interval arithmetic.}
}
@inproceedings{ras-sch-21-aa-powflex,
author = {Sasan J. Rasti and Peter Schegner}
title = {A Novel Approach to Describe and Aggregate Multi-Energy Flexibility in Cellular Energy Systems Using Affine Arithmetic}
fullhash = {2f4b674c793c4b35b9864d7ea645decb}
booktitle = {Proc. Power Engineering Society (ETG) Congress 2021}
month = {3}
year = {2021}
pages = {1-6}
numpages = {6}
abstract = {Flexibilities in Energy Systems are required to integrate renewable energy sources such as Photovoltaic and Wind Power. Cellular Energy Systems are discussed as a method to integrate and manage flexibilities such as Electrical Vehicles, Batteries and Controllable Loads on different levels of the Energy System. However, it is lacking a holistic flexibility model when extending the approach to Multi-Energy Systems (Electrical and Thermal Storages, Heat Pumps, CHPs, Fuel Cells, Electrolyzers etc.). This paper presents a flexibility model based on Affine Arithmetic which specifies allowed power and energy ranges while considering relations between time steps and energy forms.}
}
@inproceedings{mor-pen-lop-fel-21-aa-3phase,
author = {John Pe{\~{n}}aloza Mor{\'{a}}n and Julio C. L{\'{o}}pez and Antonio Padilha Feltrin}
title = {Three-Phase Optimal Power Flow based on Affine Arithmetic}
fullhash = {de43fb6fa6eadedd8f47a93c41ac1730}
booktitle = {IEEE PES Innovative Smart Grid Technologies Conference - Latin America (ISGT-LA 2021)}
month = {9}
year = {2021}
pages = {1-5}
numpages = {5}
doi = {10.1109/ISGTLatinAmerica52371.2021.9543033}
abstract = {This paper presents a non-linear stochastic mathematical formulation for the three-phase optimal power flow problem, which addresses the uncertainties of both the load and the renewable generation using the self-validation method called Affine Arithmetic. The affine shapes for tree-phase variables was performed using the theoretical approach of Reduced Affine Arithmetic together with the Chebyshev approximation method for non-affine operations. The stochastic mathematical model is formulated into as an equivalent optimization problem through a set of affine operators to later be solved by a two-stage stochastic optimization problem. Two three-phase distribution systems 19-bus and 25-bus are used to show the effectiveness of the proposed methodology.}
}
@misc{imm-21-aa-descr,
author = {Fabian Immler}
title = {Affine Arithmetic}
fullhash = {d0b0722da31cffb37f836c379a269a38}
howpublished = {Online dicument}
month = {12}
year = {2021}
urlraw = {https://www.isa-afp.org/browser_info/current/AFP/Affine_Arithmetic/document.pdf}
url = {https://www.isa-afp.org/browser_info/current/AFP/Affine_Arithmetic/document.pdf}
abstract = {We give a formalization of affine forms [1, 2] as abstract representations of zonotopes. We provide affine operations as well as overapproximations of some non-affine operations like multiplication and division. Expressions involving those operations can automatically be turned into (executable) functions approximating the original expression in affine arithmetic. Moreover we give a verified implementation of a functional algorithm to compute the intersection of a zonotope with a hyperplane, as described in the paper [3].}
}
@article{adu-raj-adu-21-aa-capac,
author = {Bala Surendra Adusumilli and Vinod Raj and Vijaya Bhaskar Adusumilli}
title = {Capacitor Placement and Sizing to Minimize Losses in a Radial Distribution Network Considering Uncertainty Using Modified Affine Arithmetic Division}
fullhash = {632eec80e8bb20ea0a0896ea61736d5d}
journal = {Sustainable Energy, Grids and Networks}
volume = {27}
year = {2021}
pages = {100492}
numpages = {1}
doi = {10.1016/j.segan.2021.100492}
abstract = {In a distribution system, load and distributed generation connected to a feeder are subject to uncertainty. Conventional capacitor placement and sizing methods do not consider uncertainty in power injections, due to which the results may be erroneous. In literature, affine arithmetic (AA) is one of the tools used for incorporating uncertainty in power system analysis. However, conventional division operation in AA gives rise to extra noise terms in the resulting affine form, which are not due to the actual uncertainty sources but are due to the nonaffine operations. The present work incorporates modified AA division which does not generate any additional noises, thereby improving the accuracy. Buses in the distribution network are ranked in decreasing order of rate of change of active power loss in line with respect to the change in the effective reactive power flow in that line. The top three ranked buses are chosen as candidate buses for capacitor connection. The required affine reactive current injection at the candidate buses is calculated, and the corresponding affine capacitive kVAR required is obtained through modified AA-based backward/forward sweep (BFS) power flow analysis. The intervals for the cost incurred in losses, voltage profiles with and without capacitor are calculated, and savings for capacitor connected system is calculated. Switchable capacitors are used for reactive power injection. The proposed method is tested on 15, 33 and 118 bus radial distribution systems. The results show that the proposed modified AA method is accurate than existing interval arithmetic-based method.}
}
@article{zho-zho-hao-zha-21-aa-bridge,
author = {Yun Zhou and Sai Zhou and Guanwang Hao and Jian Zhang}
title = {Bridge influence line identification based on big data and interval analysis with affine arithmetic}
fullhash = {8da1bc068847dc241aa797e291ac664f}
journal = {Measurement}
volume = {183}
year = {2021}
pages = {109807}
numpages = {1}
doi = {10.1016/j.measurement.2021.109807}
abstract = {On-site bridge influence line (IL) calibration traditionally relies upon attached sensors and the pre-weighing of trucks, which limits its popularity for engineering applications. This paper proposes a novel technology for bridge IL identification based on big data of pre-defined statistical vehicle information and interval analysis using the affine arithmetic (AA) method. By employing computer vision techniques, the individual axle-weight intervals were determined. The AA algorithm was used to obtain the interval solutions due to its high computation efficiency (only consumes 1 s). The binary classification algorithm for the support vector machine was adopted to extract the certain IL from the massive amount of IL interval data. The proposed method was verified by employing vehicle-bridge coupled numerical simulation and laboratory tests. Calculations results indicated that the errors of the distinguished IL obtained by the proposed method ranged from 7.43{\%} to 15.16{\%}. The ILs were successfully identified in multiple cases.}
}
@article{xuf-yan-wan-sun-21-aa-ballist,
author = {Fengjie Xu and Guolai Yang and Liqun Wang and Quanzhao Sun}
title = {Interval Uncertain Optimization for Interior Ballistics Based on {Chebyshev} Surrogate Model and Affine Arithmetic}
fullhash = {9cb83c947575f01c6732a6ee7f8ebc3d}
journal = {Engineering Optimization}
publisher = {Taylor {\&} Francis}
number = {8}
volume = {53}
year = {2021}
pages = {1331-1348}
numpages = {18}
doi = {10.1080/0305215X.2020.1790551}
abstract = {This article proposes an interior ballistic interval optimization method with the consideration of parameter uncertainty. Interior ballistic parameters such as charge mass, web thickness, powder aperture and chamber volume are considered as design variables and described by interval number. A one-dimensional two-phase interior ballistic model is constructed, and the MacCormack scheme is used to calculate the interior ballistic parameters. A Chebyshev surrogate model is constructed to replace the one-dimensional two-phase interior ballistic model and applied to the optimization process. For each set of design variables, affine arithmetic is introduced to calculate the interval bounds of the performance index of the interior ballistic, avoiding nested double-loop optimization and improving computational efficiency. The Pareto-optimal solution is searched for by an improved non-dominant sorting genetic algorithm. An example is given to demonstrate the effectiveness of the proposed method.}
}
@article{son-mij-che-bai-che-20-aa-reliab,
author = {Yufei Song and Jinhua Mi and Yuhua Cheng and Libing Bai and Kai Chen}
title = {A Dependency Bounds Analysis Method for Reliability Assessment of Complex System with Hybrid Uncertainty}
fullhash = {2bb670b6e3e4f3a32c482ea4bdde8545}
journal = {Reliability Engineering {\&} System Safety}
volume = {204}
year = {2020}
pages = {article 107119}
numpages = {-1}
doi = {10.1016/j.ress.2020.107119}
abstract = {In reliability assessment, a difficulty is to handle a complex system with hybrid uncertainty (aleatory and epistemic uncertainty) and dependency problem. Probability-box is a general model to represent hybrid uncertainty. Arithmetic rules on the structure are mostly used between independent random variables. However, in practice, dependency problems are also common in reliability assessment. In addition, in most real applications, there is some prior information on the dependency of components, but the available information may be not enough to determine dependent parameters. The issue is named non-deterministic dependency problem in the paper. Affine arithmetic is hence used to produce dependent interval estimates. The arithmetic sometimes has a better effect than probability-box arithmetic (interval arithmetic) in dealing with dependency problem. Bayesian network is a commonly used model in reliability assessment. Under Bayesian network framework, this paper proposes a dependency bounds analysis method that combines affine arithmetic and probability-box method to handle hybrid uncertainty and non-deterministic dependency. For the sake of illustration, this method is applied to two real systems. To show the advantages of the proposed method, the proposed method is compared with the Frechet inequalities and 2-stage Monte Carlo method in the second case study.}
}
@article{xie-lin-lia-yan-tan-son-liu-20-aa-powflow,
author = {Yuquan Xie and Shunjiang Lin and Weikun Liang and Yuerong Yang and Zhiqiang Tang and Yunong Song and Mingbo Liu}
title = {Interval Probabilistic Energy Flow Calculation of {CCHP} Campus Microgrid Considering Interval Uncertainties of Distribution Parameters}
fullhash = {7a6bb23cce478a3813f07053a45c9702}
journal = {IEEE Access}
volume = {8}
year = {2020}
pages = {141358-141372}
numpages = {15}
doi = {10.1109/ACCESS.2020.3013151}
abstract = {Due to the absence of historical data and the errors of measurement instruments, there may be uncertainties in the distribution parameters of the random variables describing the uncertain fluctuations of node power including renewable energy station output and load power in the combined cooling heating and power (CCHP) campus microgrid. In this paper, intervals are used to describe the uncertainties of distribution parameters of the random variables, and an interval probabilistic energy flow (IPEF) calculation model of the CCHP campus microgrid is established. Introducing the interval arithmetic (IA) into the cumulant method, an IA-based IPEF algorithm is proposed to obtain the analytical expressions of probability density function or cumulative distribution function intervals of the state variables. Moreover, affine arithmetic (AA) is introduced to address the interval extension problem in the calculation, and an AA{\&}IA-based IPEF algorithm is proposed. By constructing the correlation transformation matrixes, the correlation among different node power is considered in the IPEF calculation. A case study on a CCHP campus microgrid demonstrates that the results of the AA{\&}IA-based IPEF algorithm are more accurate than those of the IA-based IPEF algorithm by using the results of the double-layer Monte Carlo method as a reference. Moreover, the proposed algorithms are more efficient than the double-layer Monte Carlo method.}
}
@inbook{vac-20-aa-powflowsurv,
author = {Alfredo Vaccaro}
title = {The Role of Affine Arithmetic in Robust Optimal Power Flow Analysis}
publisher = {Springer}
fullhash = {40325c9a3d98c5800e14f986b3783b17}
booktitle = {Decision Making under Constraints}
isbn = {978-3-030-40814-5}
year = {2020}
pages = {189-196}
numpages = {8}
doi = {10.1007/978-3-030-40814-5_24}
urlraw = {https://doi.org/10.1007/978-3-030-40814-5_24}
url = {https://doi.org/10.1007/978-3-030-40814-5_24}
abstract = {Optimal Power Flow (OPF) analysis represents the mathematical foundation of many power engineering applications. For the most common formalization of the OPF problem, all input data are specified using deterministic variables, and the corresponding solutions are deemed representative of the limited set of system conditions. Hence, reliable algorithms aimed at representing the effect of data uncertainties in OPF analyses are required in order to allow analysts to estimate both the data and solution tolerance, providing, therefore, insight into the level of confidence of OPF solutions. To address this issue, this Chapter outline the role of novel solution methodologies based on the use of Affine Arithmetic.}
}
@inproceedings{est-lop-reg-20-aa-fixpt,
author = {Luis Esteban and Juan Antonio L\'{o}pez Mart\'{\i}n and Alberto Regad\'{\i}o}
title = {Round-off noise estimation of fixed-point algorithms using Modified Affine Arithmetic and Legendre Polynomials}
fullhash = {afa48a3f8eb8e447bdfa91bc9d0bd793}
booktitle = {Proc. XXXV Conference on Design of Circuits and Integrated Systems (DCIS 2020)}
month = {11}
year = {2020}
pages = {1-6}
numpages = {6}
doi = {10.1109/DCIS51330.2020.9268668}
abstract = {The implementation of algorithms in fixed-point format causes the apparition of Round-Off Noise which propagates through the different functional units of the system. This issue causes the Signal-to-Noise Ratio of the outputs is degraded. Given an algorithm, it is essential to estimate the integer and fractional bit-widths of all the variables and operations to comply with the Signal-to-Noise Ratio requirements. In this context, Affine Arithmetic can obtain fast and accurate estimations of the bit-widths for linear systems. However, for non-linear systems, Affine Arithmetic loses the temporal correlation of the variables. Other existing frameworks are either time consuming or lead to inaccurate bound estimations. In this paper, a Modified Affine Arithmetic framework with Legendre polynomials is used to obtain fast and accurate bound estimations also for non-linear systems. Moreover, the approach proposed in this paper obtains speedups in the range of 7 to 100 compared to Monte-Carlo simulations.}
}
@mastersthesis{jaa-20-aa-hask-thesis,
author = {Joosep J{\"{a}}{\"{a}}ger}
title = {Implementation of affine arithmetic in {Haskell}}
institution = {University of Tartu}
fullhash = {d2235044c1898fc54e3e8ad6ed52738c}
month = {3}
note = {Bachelors Thesis; supervisors A. Pankova, D. Unruh}
year = {2020}
abstract = {Interval arithmetic and affine arithmetic are methods in numerical analysis that deal with ranges of numerical values. Affine arithmetic is often used instead of interval arithmetic since it can result in smaller errors. The result of this thesis is an affine arithmetic library written in Haskell. This library is written in a way that makes it more difficult to make errors when using it. The library was tested using certain mathematical properties of affine arithmetic.}
}
@inproceedings{atc-rah-cec-20-aa-tpowtr,
author = {Forest Atchison and Mahbubur Rahman and Valentina Cecchi}
title = {Temperature-Dependent Power Transfer Capability of Transmission Systems Under Uncertainty: {An} Affine-Arithmetic Approach}
fullhash = {b6005184297d0783f1ee5441d732811c}
booktitle = {IEEE/PES Transmission and Distribution Conference and Exposition (T{\&}D 2020)}
month = {10}
year = {2020}
pages = {1-5}
numpages = {5}
doi = {10.1109/TD39804.2020.9299664}
abstract = {Traditionally used continuation power flow methods do not consider the variation in line conductor temperature with increase in line loading. In this paper, a temperature-dependent continuation power flow (TD-CPF) approach is used and considers both thermal and voltage stability limits to determine the maximum transfer capacity of a transmission system. Moreover, the TD-CPF method is enhanced via use of affine arithmetic to account for the presence of uncertainties in weather parameters, overhead line conductor properties, and regional pollution levels. Considering these uncertainties, the voltage stability is assessed and compared to the line thermal limits in order to obtain a non-deterministic solution of the maximum power transfer capability of the system. The methodology is tested on a 4-bus and a 30-bus test cases to demonstrate the contribution of the proposed method.}
}
@article{len-liu-ran-che-zha-20-aa-powflow,
author = {Shipeng Leng and Kaipei Liu and Xiaohong Ran and Shuyao Chen and Xunyue Zhang}
title = {An Affine Arithmetic-Based Model of Interval Power Flow With the Correlated Uncertainties in Distribution System}
fullhash = {239d84f00bb0d98404a0c4bb0b73a5c0}
journal = {IEEE Access}
volume = {8}
year = {2020}
pages = {60293-60304}
numpages = {12}
doi = {10.1109/ACCESS.2020.2982928}
abstract = {An interval power flow (IPF) method that considers the interval correlations of input random variables is proposed to improve the calculation accuracy of IPF, i.e., as correlated distributed generations (DGs) and some correlated DGs-loads are integrated into distribution system. The interval correlation for input variables is described by parallelogram model (PM), whose shape and size are determined by the interval correlated level. Based on affine arithmetic (AA) method, the IPF is solved through nonlinear optimization instead of traditional interval iterative computations. The optimization model of IPF is established, and the interval correlations of input variables (DGs-DGs and DGs-loads) are added into the IPF optimization problem in the form of additional constraint, to make the power flow solutions, i.e., bus voltage magnitude, voltage angle, active and reactive power of branches, more accurate. Finally, several cases, i.e., numerical case, IEEE33-bus, PG{\&}E69-bus and IEEE118-bus distribution system, not only demonstrate the effectiveness of the proposed method, but also indicate that the IPF results are affected by uncertainty level, and the widths of IPF increase with the increasing of uncertainty level.}
}
@inproceedings{raj-kum-20-aa-ptswarm,
author = {Vinod Raj and Boddeti Kalyan Kumar}
title = {An Improved Affine Arithmetic-Based Optimal {DG} Sizing and Placement Algorithm Using {PSO} for Radial Distribution Networks with Uncertainty}
fullhash = {4182eb4e343e0073025e478ed1032211}
booktitle = {21st National Power Systems Conference (NPSC 2020)}
month = {12}
year = {2020}
pages = {1-6}
numpages = {6}
doi = {10.1109/NPSC49263.2020.9331886}
abstract = {In the present work, Particle Swarm optimization (PSO) based optimal Distributed Generation (DG) placement and sizing for radial distribution systems using Affine Arithmetic (AA), considering uncertainty in load and generation, is presented to reduce real power losses and improve voltage profile with minimum investment cost. The PSO algorithm incorporates both affine as well as crisp discrete variables for uncertain DG generation and bus locations respectively to arrive at the optimal solution. An affine expression for real power loss is used which has sensitivity coefficients of real power losses in distribution lines corresponding to power injections at different buses. The resulting affine loss intervals are further tightened using an optimization procedure. The proposed method is implemented on IEEE 33 and 119 bus radial distribution systems. The optimal loss intervals obtained by the proposed method are compared with existing AA based optimal DG allocation method. The simulation results show that the proposed AA-PSO method gives more optimal DG size and location, for IEEE 33 and 119 bus radial distribution systems with uncertainty when compared to existing method which uses AA.}
}
@article{rui-her-jur-20-aa-harmonic,
author = {F.J. Ruiz-Rodriguez and J.C. Hernandez and F. Jurado}
title = {Iterative Harmonic Load Flow by using the Point-Estimate Method and Complex Affine Arithmetic for Radial Distribution Systems with Photovoltaic Uncertainties}
fullhash = {4db0591961a2db2c1adc594d11fd93ef}
journal = {International Journal of Electrical Power {\&} Energy Systems}
volume = {118}
year = {2020}
pages = {105765}
numpages = {1}
doi = {10.1016/j.ijepes.2019.105765}
abstract = {Load and generation variations and the random nature of harmonics in non-linear devices (NLDs) are the source of multiple uncertainties that can be handled in harmonic load flows (HLFs) by probabilistic or interval formulations. The paper presents a new combined analytical technique (CAT) for iterative HLFs in the presence of correlated input uncertainties from photovoltaic (PV) systems in radial distribution systems (RDSs). This technique merges the point-estimate method (PEM), a probabilistic formulation, and complex affine arithmetic (AA), an interval formulation. It then uses the information derived in Legendre series approximation (LGSA) to approximate harmonic voltage distributions. Unlike other methods, this CAT includes iterative harmonic penetration (IHP), which provides a way to deal with the interaction of background harmonic voltage on PV harmonic current. The CAT was examined in a real ENDE 100 RDS system. Thanks to PEM and AA, the CAT significantly reduced the computational burden, an evident improvement over the Monte-Carlo simulation (MCS). Furthermore, the simulation results showed that it accurately reconstructed the harmonic voltage distributions (magnitude and phase angle). The iterative approach also underlines the relevance of background harmonic interaction. The CAT outperformed the incomplete CAT (ICAT), which was based solely on a probabilistic HLF formulation.}
}
@inproceedings{adu-kum-20-aa-bachforw,
author = {Bala Surendra Adusumilii and Boddeti Kalyan Kumar}
title = {Backward/Forward Sweep based Power Flow Analysis of Distribution Systems under Uncertainty using New Affine Arithmetic Division}
fullhash = {0d8f5f8917523704b5446d66ce039665}
booktitle = {Proc. 2020 IEEE Power Energy Society Innovative Smart Grid Technologies Conference (ISGT)}
year = {2020}
pages = {1-5}
numpages = {5}
doi = {10.1109/ISGT45199.2020.9087718}
abstract = {Power flow analysis is a very important study for planning and operation of power systems. Conventional power flow analysis techniques will fail when uncertainty is present in power injections. Adequate analytical techniques are required for power flow analysis under uncertainty. Affine Arithmetic (AA) is one of the effective tools to handle uncertainty. However, existing AA method lead to generation of extra noise terms in each division operation. The error in solution bounds increases because of these extra noise terms. In this work, backward/forward sweep (BFS) based power flow analysis, which do not generate any extra noise terms, is proposed. The proposed method is tested on 33 and 118 bus radial distribution systems. Simulation results are compared with Monte Carlo simulations based as well as existing AA-BFS based power flow analysis. Simulation results show that the proposed method is more accurate than existing AA-BFS based power flow analysis.}
}
@article{lia-liu-lej-zhu-hua-lib-zha-20-aa-powflow,
author = {Xiaobing Liao and Kaipei Liu and Jian Le and Shu Zhu and Qing Huai and Ben Li and Yantian Zhang}
title = {Extended Affine Arithmetic-Based Global Sensitivity Analysis for Power Flow with Uncertainties}
fullhash = {32478f932beb88c3db7b20417cd8c841}
journal = {International Journal of Electrical Power {\&} Energy Systems}
volume = {115}
year = {2020}
pages = {105440}
numpages = {1}
doi = {10.1016/j.ijepes.2019.105440}
abstract = {In order to quantitatively assess the impacts of input uncertainties on power flow solutions, a novel analytical variance-based global sensitivity analysis method using extended affine arithmetic was proposed in this paper. With input uncertain variables described as intervals, the power flow output models based on extended affine arithmetic were originally derived. These models are normally expressed by second-order interval response surface model, and thus the total variance of the models can be calculated analytically. Finally, we proposed a novel framework of variance decomposition based global sensitivity analysis method to clarify the components of total variance contributions. The tests on the IEEE14-bus, IEEE300--bus, 2383wp system demonstrate that extended affine arithmetic-based global sensitivity analysis method can acquire maximum relative error of total sensitivity indices is less than 4.55{\%} and max relative error of main sensitivity indices is less than 9.1{\%} when comparing to Monte Carlo simulation method, which is able to greatly improve the evaluation efficiency relative to Monte Carlo simulation method.}
}
@article{sch-sch-20-aa-lovolt,
author = {Maximilian Schmidt and Peter Schegner}
title = {Deriving power uncertainty intervals for low voltage grid state estimation using affine arithmetic}
fullhash = {355550ca207135c9aa8a0752536fce5a}
journal = {Electric Power Systems Research}
volume = {189}
year = {2020}
pages = {article 106703}
numpages = {-1}
doi = {10.1016/j.epsr.2020.106703}
abstract = {Pseudo-measurements are usually applied as substitutes for measurement data to obtain observability even in case of sparse measurement configurations. The quality of pseudo-measurements strongly affects the accuracy of state estimation results. Generating pseudo-measurements is not a straightforward task in particular for customers with highly stochastic behavior. This paper proposes a novel approach to obtain verified boundaries of pseudo-measurements, which are consistent to currently available measurement information. These boundaries allow to verify generated pseudo-measurements, whether they are compliant to current system conditions. Additionally, tight boundaries are suitable to generate pseudo-measurements on-line regardless of the customer's structure. The proposed algorithm applies Affine Arithmetic - a self-validated numerical model - to the state estimation problem and treats pseudo-measurements as main source of uncertainty. The key properties of the algorithm are rigorous, iterative, linearizing and large-scale. These properties are demonstrated by numerical simulation of two benchmark grid models of different complexity. Computational and implementation issues are addressed in detail.}
}
@book{cha-rou-20-aa-book,
author = {Snehashish Chakraverty and Saudamini Rout}
title = {Affine Arithmetic Based Solution of Uncertain Static and Dynamic Problems}
publisher = {Morgan {\&} Claypool}
fullhash = {5e5f4810aeaa0194cdbfc0b83485801a}
month = {3}
series = {Synthesis Lectures on Mathematics and Statistics}
year = {2020}
pages = {170}
numpages = {1}
doi = {10.2200/S00983ED1V01Y202001MAS032}
abstract = {Uncertainty is an inseparable component of almost every measurement and occurrence when dealing with real-world problems. Finding solutions to real-life problems in an uncertain environment is a difficult and challenging task. As such, this book addresses the solution of uncertain static and dynamic problems based on affine arithmetic approaches. Affine arithmetic is one of the recent developments designed to handle such uncertainties in a different manner which may be useful for overcoming the dependency problem and may compute better enclosures of the solutions. Further, uncertain static and dynamic problems turn into interval and/or fuzzy linear/nonlinear systems of equations and eigenvalue problems, respectively. Accordingly, this book includes newly developed efficient methods to handle the said problems based on the affine and interval/fuzzy approach. Various illustrative examples concerning static and dynamic problems of structures have been investigated in order to show the reliability and efficacy of the developed approaches.}
}
@article{tan-don-zhu-son-20-aa-powflow,
author = {Kunjie Tang and Shufeng Dong and Chengzhi Zhu and Yonghua Song}
title = {Affine Arithmetic-Based Coordinated Interval Power Flow of Integrated Transmission and Distribution Networks}
fullhash = {c43d78308648281e8ae2ab5ec8a62b36}
journal = {IEEE Transactions on Smart Grid}
number = {5}
volume = {11}
year = {2020}
pages = {4116-4132}
numpages = {17}
doi = {10.1109/TSG.2020.2991210}
abstract = {Considering the increasing coupling between transmission and distribution networks, an affine arithmetic-based interval power flow algorithm of integrated transmission and distribution (I-T{\&}D) networks is proposed. First, a novel two-step approximation method for sine and cosine functions is proposed to reduce the number of noise variables in the affine form of the injected power, as the basis of the proposed algorithm. Then, this algorithm is divided into four stages, which are implemented alternatively between a transmission system operator (TSO) and multiple distribution system operators (DSOs) by exchanging parameters at TSO-DSO boundaries. Three-phase unbalanced I-T{\&}D networks and boundary tap changer action are also discussed. In addition, an asynchronous power flow method is proposed to mitigate the influences of communication networks on the proposed interval power flow algorithm. Numerical experiments demonstrate the effectiveness of the proposed two-step approximation method, as well as the conservative property, fairly good approximation effect, high efficiency, and low data exchange amount of the proposed algorithm in various circumstances, including single-phase and three-phase networks, varying degrees of unbalance, various smart inverter controls, etc.}
}
@article{ran-len-liu-20-aa-triang,
author = {Xiaohong Ran and Shipeng Leng and Kaipei Liu}
title = {A Novel Affine Arithmetic Method With Missed the Triangular Domain With Uncertainties}
fullhash = {a3e30f7ee48796c5c5d29e289c406fa1}
journal = {IEEE Transactions on Smart Grid}
number = {2}
volume = {11}
year = {2020}
pages = {1430-1439}
numpages = {10}
doi = {10.1109/TSG.2019.2938080}
abstract = {A novel affine arithmetic (AA) method with missed triangular domain (remaining area is convex polygons) is proposed in this paper for solving interval power flow (IPF) problem with uncertainties. An interval correlation model of random variables is established, and the original correlated interval variables are transformed into the standard space using interval arithmetic (IA) to reduce overestimation problem because of interval operation. To alleviate the errors caused by inaccurate enveloping of different intervals, an interval model with missed triangular domain is proposed and integrated into optimization model of IPF in form of additional constraints, which is developed in this paper through AA. IPF results, i.e., voltage magnitude, voltage angles and line flows are much narrower because it enhances restriction of correlated variables. Case studies on various test systems, i.e., IEEE 33-bus, 69-bus, and 118-bus system, indicate effectiveness of proposed technique compared with the other methods.}
}
@inproceedings{gie-xam-gan-mit-19-aa-missign,
author = {Georges Gielen and Nektar Xama and Karthik Ganesan and Subhasish Mitra}
title = {Review of Methodologies for Pre- and Post-Silicon Analog Verification in Mixed-Signal {SOCs}}
fullhash = {7196c28b41dbdb929172276247fb861b}
booktitle = {Proc. Design, Automation {\&} Test in Europe Conference Exhibition (DATE 2019)}
month = {3}
year = {2019}
pages = {1006-1009}
numpages = {4}
doi = {10.23919/DATE.2019.8714828}
abstract = {The integration of increasingly more complex and heterogeneous SOCs results in ever more complicated demands for the verification of the system and its underlying subsystems. Pre-silicon design validation as well as post-silicon test generation of the analog and mixed-signal (AMS) subsystems within SOCs proves extremely challenging as these subsystems do not share the formal description potential of their digital counterparts. Several methods have been developed to cope with this lack of formalization during AMS pre-silicon validation, including model checkers, affine arithmetic formalisms and equivalence checkers. However, contrary to the industrial practice for digital circuits of using formal verification and ATPG tools, common industry practice for analog circuits still largely defaults to simulation-based validation and test generation. A new formal digital-inspired technique, called AMS-QED, can potentially solve these issues in analog and mixed-signal verification.}
}
@inproceedings{yul-kon-che-19-aa-intsta,
author = {Li Yu and Xiangyu Kong and Ying Chen}
title = {Internal State Estimation for Distribution Network with {PMU} Measurement Information}
fullhash = {602653a6558cda8d87057b69fd232204}
booktitle = {Proc. IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia 2019)}
month = {5}
year = {2019}
pages = {1149-1153}
numpages = {5}
doi = {10.1109/ISGT-Asia.2019.8881090}
abstract = {Due to more and more intelligent devices such as phasor measurement unit (PMU) access to the distribution network, state estimation for distribution network need to consider more and more uncertainty and the great difference in accuracy of different measurement information may lead to non-convergence of weighted least square (WLS) methods. Aiming at the situation of only knowing the upper and lower limits of uncertainties, an interval state estimation for distribution system is proposed, which describes these uncertainties of system measurements are quantitatively d by the interval numbers and affine arithmetic. In order to accurately obtain the results of DSSE, an algorithm based on the combination of interval arithmetic and affine arithmetic is proposed. In this method, the interval numbers and affine arithmetic are transformed into each other in the iterative process to reducing the conservatism of interval state estimation. The estimation results in interval form are used to provide upper and lower bound information of DSSE results for dispatchers more intuitively.}
}
@inproceedings{rou-cha-19-aa-eigen,
author = {S. Rout and S. Chakraverty}
title = {Affine Approach to Solve Nonlinear Eigenvalue Problems of Structures with Uncertain Parameters}
publisher = {Springer}
fullhash = {c1f2f44d89e9efc8e61b4f68df1f4673}
booktitle = {Recent Trends in Wave Mechanics and Vibrations - Select Proc. of 8th National Conference on Wave Mechanics and Vibrations (WMVC 2018)}
isbn = {978-981-15-0286-6}
year = {2020}
pages = {407-425}
numpages = {19}
abstract = {Various science and engineering problems involve uncertainty with respect to parameters due to different causes. The uncertain parameters may be contemplated as closed intervals. Uncertain material parameters of structural vibration problems may produce interval mass matrices and interval stiffness matrices. In general, dynamic problems with interval uncertainty lead to generalized interval eigenvalue problems. Further, the inclusion of damping factor may transform the problem to a nonlinear interval eigenvalue problems, viz., quadratic and/or cubic eigenvalue problems. In this respect, affine arithmetic may be used to handle the uncertainties due to the overestimation problem occurred in some of the cases of interval arithmetic. Accordingly, this manuscript aims to deal with solving the nonlinear eigenvalue problems with interval parameters using affine arithmetic. Numerical examples have been worked out to illustrate the reliability and efficiency of the present approach.}
}
@inproceedings{jac-ved-19-aa-divmet,
author = {Maxime Jacquemin and Franck V{\'{e}}drine}
title = {A Dividing Method Minimizing the Linearization Term in Affine Arithmetic}
fullhash = {a57df743c529eb0fb65932adbee21706}
booktitle = {Proc. 8th International Workshop on Numerical and Symbolic Abstract Domains (NSAD 2019)}
location = {Porto, Portugal}
year = {2019}
pages = {article 5 (3 pages)}
numpages = {-1}
abstract = {Affine arithmetic is a well known tool to derive first order guaranteed approximations of general formulas over real numbers. It is also a useful tool in static analysis to keep track of linear correlations between program variables. However non-linear operations, like multiplications or divisions, introduce compensation terms to over-approximate the residue of the linearization process. Tight estimations of those terms are of utmost importance to obtain precise abstractions and thus a useful analysis. In this paper, we propose a new and simple technique to compute precise compensation terms for divisions.}
}
@inproceedings{mah-zha-zha-wan-zha-19-aa-powramp,
author = {Huan Ma and Kang Zhao and Bing Zhang and Liang Wang and Pengfei Zhang}
title = {An Early Warning Method Based on Improved Affine Arithmetic for Wind Power Ramp Events in Power System}
publisher = {IOP}
fullhash = {ba12ac2418ca2895c77b8c15a9ffd992}
booktitle = {Proc 2019 International Conference on Oil {\&} Gas Engineering and Geological Sciences}
location = {Dalian, China}
month = {9}
series = {Earth and Environmental Science Conferences}
volume = {384}
year = {2019}
pages = {article 012199}
numpages = {-1}
doi = {10.1088/1755-1315/384/1/012199}
abstract = {With more and more wind power integrated into power system, increasing attention is paid to wind power ramp events (WPREs) due to the significant impact on the power balance of power system. Considering the impact of large volatility of WPREs on power control, the permissible intervals of ramping amplitude are analysed to determine the boundaries between different warning stages, which are corresponding to different power control measures to keep the power balance with Area Control Error (ACE) deviation maintained within the allowable range. Moreover, interval analysis is introduced to deal with the strong uncertainty, and an improved affine arithmetic is proposed to deal with the conservation brought by interval analysis by considering the temporal correlation of WPREs prediction error. A multi-stage early warning method for WPREs is established in further, by which the warning information, including boundaries and probability distributions, can be provided reflecting the degree of potential harm inflicted by WPREs. Simulation results demonstrate that, the provided waring information have clear physical meanings to provide guidance for the operators, and the proposed warning method has simple and rapid characteristics, which reflects the verify the effectiveness of the proposed method.}
}
@inproceedings{adu-kum-19-aa-contin,
author = {Bala Surendra Adusumilii and Boddeti Kalyan Kumar}
title = {Continuation Power Flow Analysis of Distribution Systems under Uncertainty using Modified Affine Arithmetic}
fullhash = {0d8f5f8917523704b5446d66ce039665}
booktitle = {Proc. IEEE Milan PowerTech 2019}
month = {6}
year = {2019}
pages = {1-6}
numpages = {6}
doi = {10.1109/PTC.2019.8810841}
abstract = {Conventional continuation power flow analysis (CPF) gives the maximum loadability from voltage versus power curves (V-P curves) for a deterministic set of loads and generations. But, in presence of uncertainties in power generations and loads conventional analysis may give wrong results. In this work, a range arithmetic analysis known as Affine Arithmetic (AA) with modifications is used to incorporate the uncertainties associated with loads {\&} generations in CPF analysis of distribution systems. Newton-Raphson method based CPF with modified AA is used to determine the bounds of maximum loadability from V-P curves. The proposed modified AA based CPF analysis under uncertainty can be applied to radial as well as mesh connected distribution systems and is tested on 69 bus radial and 119 bus radial {\&} mesh distribution systems. Simulation results with the proposed method are compared with existing AA based CPF and Monte Carlo simulations based CPF analyses.}
}
@phdthesis{kra-19-aa-svm-thesis,
author = {Anna Krause}
title = {Variation-Aware Behavioural Modelling Using Support Vector Machines and Affine Arithmetic}
institution = {Gottfried Wilhelm Leibniz University, Hanover}
fullhash = {8561fb992a15ec151a7d5d18f3f55cc9}
year = {2019}
doi = {10.15488/9165}
abstract = {Generalised Interval Arithmetic Simulator (AGIAS) is a specialised simulator which uses affine arithmetic to model parameter variations. It uses a specialised root-finding algorithm to simulate analogue circuits with parameter variations in one single simulation run. This is a significant speed-up compared to the multiple runs needed by industrialised solutions such as Monte-Carlo (MC) or Worst-Case Analysis (WCA). Currently, AGIAS can simulate analogue circuits only under very specific conditions. In many cases, circuits can only be simulated for certain operating points. If the circuits is to be evaluated in other operating points, the solver becomes numerically unstable and simulation fails. In these cases, interval widths approach infinity. Behavioural modelling of analogue circuits was introduced by researchers working around limitations of simulators. Most early approaches require expert knowledge and insight into the circuit which is modelled. In recent years, Machine Learning techniques for automatic generation of behavioural models have made their way into the field. This thesis combines Machine Learning techniques with affine arithmetic to include the effects of parameter variations into models. Support Vector Machines (SVMs) train two sets of parameters: one slope parameter and one offset parameter. These parameters are replaced by affine forms. Using these two parameters allows affine SVMs to model effects of parameter variations with varying widths. Training requires additional information about maximum and minimum values in addition to the nominal values in the data set. Based on these changes, affine $\varepsilon$ Support Vector Machine ($\varepsilon$-SVR) and $\nu$ Support Vector Machine ($\nu$-SVR) algorithms for regression are presented. To train the affine parameters directly and profit from the Sequential Minimal Optimisation algorithm (SMO)'s selectivity, the SMO is extended to handle the new, larger optimisation problems. The new affine SVMs are tested on analogue circuits that have been chosen based on whether they could be simulated with AGIAS and how strongly non-linear their characteristic function is.}
}
@inproceedings{che-qin-guo-liu-wei-19-aa-powdistr,
author = {Lin Cheng and Ning Qi and Yanfei Guo and Nan Liu and Wei Wei}
title = {Potential Evaluation of Distributed Energy Resources with Affine Arithmetic}
fullhash = {8451b3323b10ff5aec81e8bebd2368ab}
booktitle = {IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia 2019)}
month = {5}
year = {2019}
pages = {4334-4339}
numpages = {6}
doi = {10.1109/ISGT-Asia.2019.8881198}
abstract = {With the rapid development of Energy Internet, the penetration of distributed energy resources is gradually increasing. New methods for potential evaluation of DERs is becoming more and more important for planning, operation, and control. In this paper, the traditional affine arithmetic-based forward-backward power flow calculation method is introduced. Then, modified method is proposed mainly considers arithmetic correction, weekly looped structure, and different types of DG. Moreover, comprehensive potential evaluation structure for DERs is proposed based on affine arithmetic which involves three parts. Finally, through numerical analysis, the applicability and error of the affine arithmetic method are analyzed.}
}
@inproceedings{raj-kum-20-aa-weakpow,
author = {Vinod Raj and Boddeti Kalyan Kumar}
title = {A New Affine Arithmetic-Based Power Flow Analysis for Weakly Meshed Distribution Systems under Uncertainty}
fullhash = {4182eb4e343e0073025e478ed1032211}
booktitle = {Proc. IEEE International Conference on Environment and Electrical Engineering and IEEE Industrial and Commercial Power Systems Europe (EEEIC/I{\&}CPS Europe 2020)}
month = {6}
year = {2020}
pages = {1-6}
numpages = {6}
doi = {10.1109/EEEIC/ICPSEurope49358.2020.9160489}
abstract = {An affine arithmetic based load flow for weakly meshed distribution systems with uncertainty using optimized solution intervals is proposed in the present work. Uncertainty is considered in loads and Distributed Generation (DG) connected at buses. Affine Arithmetic (AA) is used to represent uncertainty in distribution load flow. Bus Injection to Branch Current (BIBC) and Branch Current to Bus Voltage (BCBV) matrices are used in the load flow. Noise intervals for the initial affine solution are optimised with respect to the specified uncertainty in power injections. The proposed method, existing AA and Monte Carlo simulations based power flow analyses are tested on weakly meshed 33 and 118 bus distribution systems.}
}
@article{raj-kum-19-aa-binswarm,
author = {Vinod Raj and Boddeti Kalyan Kumar}
title = {A New Affine Arithmetic-Based Optimal Network Reconfiguration to Minimize Losses in a Distribution System Considering Uncertainty Using Binary Particle Swarm Optimization}
fullhash = {4182eb4e343e0073025e478ed1032211}
journal = {Electric Power Components and Systems}
publisher = {Taylor {\&} Francis}
number = {6-7}
volume = {48}
year = {2020}
pages = {628-639}
numpages = {12}
doi = {10.1080/15325008.2020.1797940}
abstract = {In the present work, Binary Particle Swarm Optimization (BPSO) based optimal re-configuration for balanced and unbalanced radial distribution networks using Affine Arithmetic (AA), with uncertainty in generation and load, is proposed to minimize the system losses. An expression for three phase real affine power loss is derived with partial deviations of real power loss in lines with respect to power injections in other buses and also with respect to power injections in other phases in case of unbalanced distribution systems. The major contribution of the present work is the application of AA based optimal network reconfiguration, to both balanced and unbalanced radial distribution networks with uncertainty. The proposed method is tested on IEEE 16, 33, 85 and 119 bus balanced distribution systems and an unbalanced 123 bus system with Distributed Generation (DG) connected at some buses. The optimal loss intervals obtained by the proposed method are compared with that obtained by Interval Arithmetic (IA) and Monte Carlo (MC) simulations based methods. The simulation results show that proposed AA based analysis gives an optimal reconfiguration, for both balanced and unbalanced radial distribution systems with uncertainty as compared to existing IA based method.}
}
@article{luf-dup-liu-liu-19-aa-hybrid,
author = {Fang Lu and Pingjing Du and Hongda Liu and Fanming Liu}
title = {Affine Arithmetic-Based Power Flow Algorithm Considering Uncertainty for Hybrid {AC/DC} Grids Incorporating {VSCs}}
fullhash = {8b64f88cde5101ed85632f4a4e942a3a}
journal = {IET Generation, Transmission {\&} Distribution}
number = {10}
volume = {13}
year = {2019}
pages = {1856-1864}
numpages = {9}
doi = {10.1049/iet-gtd.2018.6095}
abstract = {The randomness of new energy generation and variation in load demands bring uncertainty to large AC/DC grids. Traditional deterministic power flow equations have difficulty coping with the uncertainties of new energy generation and load. Here, an affine arithmetic (AA)-based uncertainty power flow algorithm for hybrid AC/DC grids incorporating voltage source converters (VSCs) is presented. VSC station and DC grid models are established using AA. Diverse VSC control modes are considered in the power flow iterations. A comparison using the Monte--Carlo method shows that the proposed algorithm is able to obtain a feasible solution. The proposed method can be used by power system operators and planners to monitor and control AC/DC grids under various uncertainties.}
}
@article{col-vac-vil-19-aa-powflow,
author = {Guido Coletta and Alfredo Vaccaro and Domenico Villacci}
title = {Fast and Reliable Uncertain Power Flow Analysis by Affine Arithmetic}
fullhash = {ae3b05d9505849d825be448db2821a38}
journal = {Electric Power Systems Research}
volume = {175}
year = {2019}
pages = {105860}
numpages = {1}
doi = {10.1016/j.epsr.2019.04.038}
abstract = {The massive penetration of renewable power generators in modern power systems is pushing the research in developing reliable computing techniques aimed at addressing the issues caused by the increasing uncertainty sources induced by their intermittent power profiles and the corresponding power transactions. In this domain affine arithmetic-based power flow analysis has been recognized as one of the most promising research directions, since it allows to address many important power systems operation functions by directly keeping track of uncertainties propagation as part of the computation process. Anyway, AA-based computing requires the deployment of specific software modules implementing the main AA-based mathematical operators, which could increase the overall complexity of the solution algorithms. To reduce the complexity of AA-based power flow analysis, this paper proposes a new formulation of the uncertain power flow equations, which allows to explicitly compute the Jacobian matrix and apply a traditional Newton--Raphson based algorithm to solve the overall problem. Detailed results obtained on realistic power systems will be presented and discussed in order to evaluate the performances of the proposed method compared to other traditional AA-based methodologies.}
}
@article{wan-wan-wul-wan-19-aa-polar,
author = {Shouxiang Wang and Kai Wang and Lei Wu and Chengshan Wang}
title = {Polar Affine Arithmetic: Optimal Affine Approximation and Operation Development for Computation in Polar Form Under Uncertainty}
fullhash = {d67741435fcbc13ff509bc48d9a66f06}
journal = {ACM Transactions on Mathematical Software}
month = {2}
number = {1}
volume = {45}
year = {2019}
doi = {10.1145/3274659}
urlraw = {https://doi.org/10.1145/3274659}
url = {https://doi.org/10.1145/3274659}
abstract = {Uncertainties practically arise from numerous factors, such as ambiguous information, inaccurate model, and environment disturbance. Interval arithmetic has emerged to solve problems with uncertain parameters, especially in the computational process where only the upper and lower bounds of parameters can be ascertained. In rectangular coordinate systems, the basic interval operations and improved interval algorithms have been developed in the numerical analysis. However, in polar coordinate systems, interval arithmetic still suffers from issues of complex computation and overestimation. This article defines a polar affine variable and develops a polar affine arithmetic (PAA) that extends affine arithmetic to the polar coordinate systems, which performs better in many aspects than the corresponding polar interval arithmetic (PIA). Basic arithmetic operations are developed based on the complex affine arithmetic. The Chebyshev approximation theory and the min-range approximation theory are used to identify the best affine approximation. PAA can accurately keep track of the interdependency among multiple variables throughout the calculation procedure, which prominently reduces the solution conservativeness. Numerical examples implemented in MATLAB programs show that, compared with benchmark results from the Monte Carlo method, the proposed PAA ensures completeness of the exact solution and presents a more compact solution region than PIA when dependency exists in the calculation process. Meanwhile, a comparison of affine arithmetic in polar and rectangular coordinates is presented. An application of PAA in circuit analysis is quantitatively presented and potential applications in other research fields involving complex variables in polar form will be gradually developed.}
}
@article{bel-lam-bel-tag-bel-19-aa-polyev,
author = {Rima Bellal and El-sedik Lamini and Hac\`{e}ne Belbachir and Samir Tagzout and Adel Belouchrani}
title = {Improved Affine Arithmetic-Based Precision Analysis for Polynomial Function Evaluation}
fullhash = {e0c80f56bbe2754c71e6d372259b92f0}
journal = {IEEE Transactions on Computers}
number = {5}
volume = {68}
year = {2019}
pages = {702-712}
numpages = {11}
doi = {10.1109/TC.2018.2882537}
abstract = {Word-length allocation is the most important design phase to optimize hardware resources while guaranteeing a determined accuracy for circuits with fixed-point numbers. This paper presents an enhanced precision analysis for degree-n polynomial Horner's rule. It is based on affine arithmetic and introduces an error propagating formula for a degree-n polynomial Horner's rule. It takes into account quantization error of all the circuit's connections including the inputs. Furthermore, a tighter upper bound error is defined, exploiting the dependencies between intermediate connections. Hardware implementations show that the proposed upper bound results in an area reduction that reaches 70 percent.}
}
@inproceedings{vac-pet-ber-19-aa-sched,
author = {Alfredo Vaccaro and Marina Petrelli and Alberto Berizzi}
title = {Robust Optimization and Affine Arithmetic for Microgrid Scheduling under Uncertainty}
fullhash = {dce1a6af2936ec2d86602ea01a2768e5}
booktitle = {Proc. 2019 IEEE International Conference on Environment and Electrical Engineering (EEEIC/I{\&}CPS Europe)}
year = {2019}
pages = {1-6}
numpages = {6}
doi = {10.1109/EEEIC.2019.8783572}
abstract = {Optimization analyses are commonly used in microgrids to identify the most efficient and reliable operation of the available energy resources. Unfortunately, most of the times these programming problems rely on input parameters which are not accurately known. In this context, advanced computing paradigms for solving uncertainty optimization problems represent the most promising enabling methodology. These techniques may show their effectiveness during both the dispatch and the pre-dispatch phase, when operators need to solve the unit-commitment and the economic dispatch problems. To this aim, this paper discusses and compares experimentally some promising existing alternatives to deterministic methods to deal with the solution of optimization problems in the presence of data uncertainty.}
}
@article{pep-vac-man-19-aa-enerhub,
author = {Antonio Pepiciello and Alfredo Vaccaro and Mario Ma{\~{n}}ana}
title = {Robust Optimization of Energy Hubs Operation Based on Extended Affine Arithmetic}
fullhash = {b28953fc8c2f454c79ead20da091e484}
journal = {Energies}
number = {12}
volume = {12}
year = {2019}
pages = {article 2420}
numpages = {-1}
doi = {10.3390/en12122420}
abstract = {Traditional energy systems were planned and operated independently, but the diffusion of distributed and renewable energy systems led to the development of new modeling concepts, such as the energy hub. The energy hub is an integrated paradigm, based on the challenging idea of multi-carrier energy systems, in which multiple inputs are conditioned, converted and stored in order to satisfy different types of energy demand. To solve the energy hub optimal scheduling problem, uncertainty sources, such as renewable energy production, price volatility and load demand, must be properly considered. This paper proposes a novel methodology, based on extended Affine Arithmetic, which enables the solving of the optimal scheduling problem in the presence of multiple and heterogeneous uncertainty sources. Realistic case studies are presented and discussed in order to show the effectiveness of the proposed methodology.}
}
@article{fan-zou-col-vac-djo-19-aa-probopf,
author = {Duo Fang and Mingzhe Zou and Guido Coletta and Alfredo Vaccaro and Sasa Z. Djokic}
title = {Handling Uncertainties with Affine Arithmetic and Probabilistic {OPF} for Increased Utilisation of Overhead Transmission Lines}
fullhash = {a41e655870797393ab41a66c336c206f}
journal = {Electric Power Systems Research}
volume = {170}
year = {2019}
pages = {364-377}
numpages = {14}
doi = {10.1016/j.epsr.2019.01.027}
abstract = {Large-scale integration of variable and unpredictable renewable-based generation systems poses significant challenges to the secure and reliable operation of transmission networks. Application of dynamic thermal rating (DTR) allows for a higher utilisation of transmission lines and effectively avoids high-cost upgrading and/or reinforcing of transmission system infrastructure. In order to efficiently handle ranges of uncertainties introduced by the variations of both wind energy sources and system loads, this paper introduces a novel optimization model, which combines affine arithmetic (AA) and probabilistic optimal power flow (P-OPF) for DTR-based analysis of transmission networks. The proposed method allows for the improved analysis of underlying uncertainties on the supply, transmission and demand sides, which are expressed in the form of probability distributions (e.g. for wind speeds, wind directions, wind power generation and demand variations) and related interval values. The paper presents a combined AA-P-OPF method, which can provide important information to transmission system operators for evaluating the trade-off between security and costs at a planning stage, as well as for selecting optimal controls at operational stage. The AA-P-OPF methodology is illustrated for a day-ahead planning, using a case study of a real transmission network and a medium size test distribution network.}
}
@article{ziv-gri-olb-sch-bar-19-aa-analog,
author = {Carna Zivkovic and Christoph Grimm and Markus Olbrich and Oliver Scharf and Erich Barke}
title = {Hierarchical Verification of {AMS} Systems With Affine Arithmetic Decision Diagrams}
fullhash = {bb838b86683f7bac7ee4990cf3ec93bf}
journal = {IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems}
number = {10}
volume = {38}
year = {2019}
pages = {1785-1798}
numpages = {14}
doi = {10.1109/TCAD.2018.2864238}
abstract = {Formal methods are a promising alternative to simulation-based verification of mixed-signal systems. However, in practice, such methods fail to scale with heterogeneity and complexity of today's analog/mixed-signal systems. Furthermore, it is unclear how they can be integrated into existing verification flows. This paper shows a path to overcome these obstacles. The idea is to use a hierarchical verification flow, in which components can be verified by formal methods or by multirun simulation. To transport verification results across hierarchies, we represent parameters and properties by affine arithmetic decision diagrams. We study to which extent this approach fulfills the needs of practical application by the verification of a phase-locked loop of an IEEE 802.15.4 transceiver system.}
}
@article{rom-gar-19-aa-heatpow,
author = {David Romero-Quete and Javier Rosero Garcia}
title = {An Affine Arithmetic-Model Predictive Control Approach for Optimal Economic Dispatch of Combined Heat and Power Microgrids}
fullhash = {8551e7df458b9dd7476c4443fd3dd539}
journal = {Applied Energy}
volume = {242}
year = {2019}
pages = {1436-1447}
numpages = {12}
doi = {10.1016/j.apenergy.2019.03.159}
abstract = {This paper presents a novel approach for the optimal economic dispatch of Combined Heat and Power (CHP) Microgrids (MGs), which incorporates an Affine Arithmetic-based Economic Dispatch (AAED) problem into a Model Predictive Control (MPC) framework. The proposed algorithm solves an AAED problem each $\Delta t$ minutes (e.g. 15 min) with time steps of $\Delta t$ minutes over a time horizon $T$ (e.g. 24 h). It uses the available forecast and the current state of the system to provide the schedule and the affine forms that represent the operation intervals of the generators and Energy Storage Systems (ESS) for the next time interval $[t,t+\Delta t]$. Online set points for generators and ESS are then obtained by computing the noise symbols values of the affine forms, based on the most updated information of electricity and heat demands and available renewable energy power. A theoretical CHP-based MG, comprising PVs, a gas boiler, a CHP unit, a battery, and a thermal tank, is used to assess the performance of the AA-MPC approach in both connected and isolated operation modes. The method is also compared with a deterministic MPC approach. Results show the ability of the method to better address forecasting errors, resulting in more cost-effective solutions without considerably affecting the computation performance.}
}
@article{col-vac-vil-fan-djo-19-aa-weathpow,
author = {Guido Coletta and Alfredo Vaccaro and Domenico Villacci and Duo Fang and Sasa Z. Djokic}
title = {Affine Arithmetic for Efficient and Reliable Resolution of Weather-Based Uncertainties in Optimal Power Flow Problems}
fullhash = {ac5109f85d7aa47b1939d7044c62224e}
journal = {International Journal of Electrical Power {\&} Energy Systems}
volume = {110}
year = {2019}
pages = {713-724}
numpages = {12}
doi = {10.1016/j.ijepes.2019.03.022}
abstract = {The massive diffusion of renewable power generators in existing power grids introduces large uncertainties in power system operation, hindering their hosting capacity, and introducing several critical issues in network management. To address these challenging issues, weather-based optimal power flow has been recognized as one of the most promising enabling methodology for increasing the system flexibility by exploiting the real power components loadability. Anyway, the deployment of this technique in a real operation scenario could be seriously compromised due to the effects of data uncertainty, which could sensibly affect both the generated/demanded power profiles, and the components thermal modeling. In this context, the research for reliable techniques aimed at representing and managing these uncertainties represents one of the most relevant problem to solve. Armed with such a vision, this paper advocates the role of Affine Arithmetic in reliable solving weather-based OPF problems in the presence of multiple and correlated uncertainties. Experimental results obtained on a real case study, which is based on a congested portion of a transmission system characterized by a massive pervasion of wind generators, will be presented and discussed in order to assess the benefits deriving by the application of the proposed method.}
}
@article{raj-kum-19-aa-powflow,
author = {Vinod Raj and Boddeti Kalyan Kumar}
title = {A Modified Affine Arithmetic-Based Power Flow Analysis for Radial Distribution System with Uncertainty}
fullhash = {4182eb4e343e0073025e478ed1032211}
journal = {International Journal of Electrical Power {\&} Energy Systems}
volume = {107}
year = {2019}
pages = {395-402}
numpages = {8}
doi = {10.1016/j.ijepes.2018.12.006}
abstract = {In a distribution system, the loads and distributed generators connected at a bus have lot of uncertainty due to which the power injections at that bus become uncertain. These uncertainties need to be incorporated in distribution system power flow analysis to get an idea of the uncertainty in the power flow solution. One of the methods used for finding solution intervals for power flow is Monte Carlo simulations. However, Monte Carlo simulations approach suffers from the drawback of large time consumption due to repeated power flows. To overcome this limitation, affine arithmetic based power flow analysis is introduced in literature which is capable of generating solution intervals to the interval power flow problem without the need of repeated power flow analysis. Multiplication operation between two affine terms in conventional affine arithmetic generates an extra noise term in the affine product. In affine based power flow analysis for distribution system, multiplication operation is used repeatedly since backward/forward sweep algorithm is iterative in nature. This generates a large number of noise terms in certain affine parameters which ultimately leads to larger solution intervals for the problem. In the present work, a new formulation of complex affine multiplication is introduced for the affine arithmetic based distribution power flow analysis which does not generate any extra noise term for the resulting complex affine product. The proposed modified affine arithmetic based distribution power flow analysis is tested on IEEE 33, 69, 202 and 874 bus radial distribution systems. All the four test systems are assumed to have distributed generation. The solution intervals obtained are compared with the solution intervals of conventional affine arithmetic based distribution power flow analysis. Results show that the solution intervals of the proposed method are closer to Monte Carlo solution intervals as compared to conventional affine arithmetic based distribution power flow analysis.}
}
@article{rom-can-19-aa-enerman,
author = {David Romero-Quete and Claudio A. Ca{\~{n}}izares}
title = {An Affine Arithmetic-Based Energy Management System for Isolated Microgrids}
fullhash = {0cea52991fb342bbb8d659fabf54ea92}
journal = {IEEE Transactions on Smart Grid}
number = {3}
volume = {10}
year = {2019}
pages = {2989-2998}
numpages = {10}
doi = {10.1109/TSG.2018.2816403}
abstract = {This paper presents a mathematical formulation of an energy management system (EMS) for isolated microgrids, which addresses uncertainty using the affine arithmetic (AA) method. The proposed EMS algorithm is based on an AA unit commitment (AAUC) problem for day-ahead dispatch, using uncertainty intervals of both load and renewable energy (RE) to provide robust commitment and dispatch solutions in AA form, which are feasible for all the possible realizations within the predetermined uncertainty bounds. A real-time dispatch solution is then found by the proposed algorithm, which computes the noise symbols values of the affine forms obtained by the AAUC, based on the current and actual load and RE power levels and available reserves. If the actual forecast error is outside the uncertainly bounds considered in the AAUC solution process, leading to possible load and/or RE curtailment, the AAUC is recalculated with updated forecast information. The proposed AA-based EMS is tested on a modified CIGRE microgrid benchmark and is compared against day-ahead deterministic, model predictive control (MPC), stochastic optimization, and stochastic-MPC approaches. The simulation results show that the proposed EMS provides robust and adequate cost-effective solutions, without the need of frequent re-calculations as with MPC-based approaches, or assumptions regarding statistical characteristics of the uncertainties as in the case of stochastic optimization.}
}
@article{zha-wan-wan-hua-19-aa-multobj,
author = {Qianyu Zhao and Shouxiang Wang and Kai Wang and Bibin Huang}
title = {Multi-Objective Optimal Allocation of Distributed Generations under Uncertainty Based on {D-S} Evidence Theory and Affine Arithmetic}
fullhash = {24f7807c3d07961fb1c4162e6bd14795}
journal = {International Journal of Electrical Power {\&} Energy Systems}
volume = {112}
year = {2019}
pages = {70-82}
numpages = {13}
doi = {10.1016/j.ijepes.2019.04.044}
abstract = {Many efforts on distributed generation (DG) allocation are based on deterministic methods in previous studies. However, the influence of stochastic fluctuation characteristics of DGs and loads is enormous. Therefore, affine arithmetic (AA) is used to represent the uncertainty, and a multi-objective uncertainty optimization model for DG allocation with minimum investment cost, highest income, lowest environmental cost, and minimal network loss is built in this paper. Then multi-objective interval decision-making and bi-layer optimization method based on D-S evidence theory (ET) and genetic algorithm (GA) is proposed to achieve the optimal allocation of DGs. In the proposed method, GA is used as an outer layer optimization for multi-point searches to generate different allocation schemes. Moreover, the ET method is employed as the inner layer to evaluate the candidate allocation schemes which from the outer layer optimization. Also, the ET method guides the evolution direction of GA in the outer layer. For validating the effectiveness and performance, the proposed method is applied to a typical 33-bus distribution system. The comparison with the existing TOPSIS method demonstrates the advantages when handling the uncertainty of the proposed method and achieves a more robust optimal solution.}
}
@article{luo-shi-niy-18-aa-windpow,
author = {Jinqing Luo and Libao Shi and Yixin Ni}
title = {A Solution of Optimal Power Flow Incorporating Wind Generation and Power Grid Uncertainties}
fullhash = {363dcb54ced659c3d9f6356c50f7b92a}
journal = {IEEE Access}
volume = {6}
year = {2018}
pages = {19681-19690}
numpages = {10}
doi = {10.1109/ACCESS.2018.2823982}
abstract = {This paper proposes a novel approach for the solution of optimal power flow with consideration of uncertainties caused by wind generation and various factors in the power grid. Regarding the uncertainties studied here, multiple types of uncertainty modeling techniques are applied during research. Evidence theory and extended affine arithmetic are employed and mixed as the framework of uncertainty propagation to fuse probability distributions, possibility distributions, and intervals so as to obtain the best possible probability bounds, and the dependence among variables is handled by copula theory and affine arithmetic. Moreover, the uncertainty of wind farm active power and the characteristic of wind farm reactive power are modeled and integrated into the power flow calculation. An enhanced particle swarm optimization algorithm with introduction of fitness comparison and constraint handling techniques under the evidence theory framework is applied to the solution of this problem. The proposed model and method are tested on the IEEE 30-bus standard test system and a real-sized 183-bus power system to demonstrate the validity and effectiveness.}
}
@book{ska-18-aa-survbook,
author = {Iwona Skalna}
title = {Parametric Interval Algebraic Systems}
publisher = {Springer}
fullhash = {79a5c772124ee18ed172632c5bcc2850}
isbn = {978-3-319-75186-3}
series = {Studies in Computational Intelligence book series (SCI)}
volume = {766}
year = {2018}
pages = {191}
numpages = {1}
doi = {10.1007/978-3-319-75187-0}
abstract = {This self-contained book presents a framework for solving a general class of linear systems with coefficients being continuous functions of parameters varying within prescribed intervals. It also provides a comprehensive overview of the theory related to solving parametric interval linear systems and the basic properties of parametric interval matrices. In particular, it develops several new algorithms delivering sharp rigorous bounds for the solutions of such systems with full mathematical rigor. The framework employs the arithmetic of revised affine forms that enables the readers to handle dependent data. The book is intended not only for researchers interested in developing rigorous methods of numerical linear algebra, but also for engineers dealing with problems involving uncertain data. The theory discussed is also useful in various other fields of numerical analysis, in computer graphics, economics, computational geometry, computer-aided design, computer-assisted proofs, computer graphics, control theory, solving constraint satisfaction problems, and global optimization.}
}
@inbook{ska-18-aa-altern,
author = {Iwona Skalna}
title = {Alternative Arithmetic}
publisher = {Springer}
fullhash = {79a5c772124ee18ed172632c5bcc2850}
booktitle = {Parametric Interval Algebraic Systems}
chapter = {2}
isbn = {978-3-319-75187-0}
year = {2018}
pages = {25-50}
numpages = {26}
doi = {10.1007/978-3-319-75187-0_2}
abstract = {Over the years, a lot of effort has been put into the development of self-validated computational (SVC) models that will be able to overcome the ``memoryless nature'' of interval arithmetic, i.e., to take into account the dependencies between variables involved in a computation and/or reduce the so-called wrapping effect. This effort has resulted in several such models worth mentioning: ellipsoid calculus (Chernousko, Izv Akad Nauk SSSR, Tekh Kibern 3:3--11; 4:3--11; 5:5--11, 1980, [27]), V{\'{a}}lyi, Ellipsoidal calculus for estimation and control, Birkh{\"{a}}user, Boston, 1997, [269]), constrained interval arithmetic (Lodwick, Constrained interval arithmetic, 1999, [131]), Hansen's generalized interval arithmetic (Hansen, A generalized interval arithmetic, Springer, Berlin, 1975, [72]), affine arithmetic (de Figueiredo, Stolfi, Self-validated numerical methods and applications, 1997, [33]), reduced affine arithmetic (Messine, New affine forms in interval branch and bound algorithms, 1999, [139]), and revised affine arithmetic (Vu, Sam-Haroud, Faltings, A generic scheme for combining multiple inclusion representations in numerical constraint propagation, 2004, [271]).}
}
@inbook{ska-18-aa-matrices,
author = {Iwona Skalna}
title = {Interval and Parametric Interval Matrices}
publisher = {Springer}
fullhash = {79a5c772124ee18ed172632c5bcc2850}
booktitle = {Parametric Interval Algebraic Systems}
chapter = {3}
isbn = {978-3-319-75187-0}
year = {2018}
pages = {51-83}
numpages = {33}
doi = {10.1007/978-3-319-75187-0_3}
abstract = {We start this chapter from an overview of the preliminary theory on interval matrices (cf. Neumaier, Interval methods for systems of equations. Cambridge University Press, Cambridge (1990) [160]), since it lays a basis for our main theoretical considerations. The remaining part of the chapter is devoted to parametric interval vectors and matrices.}
}
@inbook{ska-18-aa-linsys,
author = {Iwona Skalna}
title = {Linear Systems}
publisher = {Springer}
fullhash = {79a5c772124ee18ed172632c5bcc2850}
booktitle = {Parametric Interval Algebraic Systems}
chapter = {4}
isbn = {978-3-319-75187-0}
year = {2018}
pages = {85-98}
numpages = {14}
doi = {10.1007/978-3-319-75187-0_4}
abstract = {In this chapter we define parametric interval linear systems and characterize various types of parametric solution sets. We provided as well several visualizations of the most general (united) solution set (in 2D and 3D) that aim to facilitate the exploration of the properties of this solution set. We start with a brief introduction to interval linear systems (Neumaier, Interval methods for systems of equations. Cambridge University Press, Cambridge, 1990) [160], (Rohn, Interval linear systems. Freiburger Intervall-Ber. 84/7. Universit{\"{a}}t Freiburg, Freiburg, 1984) [206] from which parametric interval linear systems originate.}
}
@inbook{ska-18-aa-linsolv,
author = {Iwona Skalna}
title = {Methods for Solving Parametric Interval Linear Systems}
publisher = {Springer}
fullhash = {79a5c772124ee18ed172632c5bcc2850}
booktitle = {Parametric Interval Algebraic Systems}
chapter = {5}
isbn = {978-3-319-75187-0}
year = {2018}
pages = {99-163}
numpages = {65}
doi = {10.1007/978-3-319-75187-0_5}
abstract = {In this chapter we present various methods for solving parametric interval linear systems with general dependencies. Furthermore, we propose several modifications and improvements of selected methods that aim at extending their applicability and accuracy.}
}
@inbook{ska-18-aa-linprog,
author = {Iwona Skalna}
title = {Parametric Interval Linear Programming Problem}
publisher = {Springer}
fullhash = {79a5c772124ee18ed172632c5bcc2850}
booktitle = {Parametric Interval Algebraic Systems}
chapter = {6}
isbn = {978-3-319-75187-0}
year = {2018}
pages = {165-172}
numpages = {8}
doi = {10.1007/978-3-319-75187-0_6}
abstract = {We shall now switch to the following parametric interval linear programming (PILP) problem (Kolev, L. (2014). Applied Mathematics and Computation, 229--246.) (cf. [113]): given parametric objective function.}
}
@article{eve-18-aa-mtcarlo,
author = {Richard G. Everitt}
title = {Efficient Importance Sampling in Low Dimensions Using Affine Arithmetic}
fullhash = {e81f39c296c2c64e45eb9b29a6a4349f}
journal = {Computational Statistics}
volume = {33}
year = {2018}
pages = {1-29}
numpages = {29}
abstract = {Despite the development of sophisticated techniques such as sequential Monte Carlo (Del Moral et al. in J R Stat Soc Ser B 68(3):411--436, 2006), importance sampling (IS) remains an important Monte Carlo method for low dimensional target distributions (Chopin and Ridgway in Leave Pima Indians alone: binary regression as a benchmark for Bayesian computation, 32:64--87, 2017). This paper describes a new technique for constructing proposal distributions for IS, using affine arithmetic (de Figueiredo and Stolfi in Numer Algorithms 37(1--4):147--158, 2004). This work builds on the Moore rejection sampler (Sainudiin in Machine interval experiments, Cornell University, Ithaca, 2005; Sainudiin and York in Algorithms Mol Biol 4(1):1, 2009) to which we provide a comparison.}
}
@article{luf-yan-liu-liu-18-aa-cpowflow,
author = {Fang Lu and Liang Yan and Hongda Liu and Fanming Liu}
title = {Complex Affine Arithmetic-Based Power Flow Analysis for Zonal Medium Voltage Direct Current Shipboard Power Systems in the Presence of Power Variation}
fullhash = {908f914875e2d388bc2b0afc2879f39a}
journal = {Energies}
number = {7}
volume = {11}
year = {2018}
pages = {article 1697}
numpages = {-1}
doi = {10.3390/en11071697}
abstract = {The medium-voltage direct-current (MVDC) ship power system has been extensively investigated and discussed in recent years. This paper presents an alternating current/direct-current (AC/DC) power flow algorithm based on complex affine arithmetic for the zonal MVDC shipboard power system in the presence of power variation. The power converter effect is considered, and an affine power converter model is proposed in the power flow model. An affine-analysis-based sequential method is adopted to solve AC/DC power flows. With the proposed algorithm, the bounds of bus voltages can be effectively obtained. A relative influence factor is defined to quantitatively assess the effect of power variation on voltage profile. Four cases are studied in detail to demonstrate the applicability of the proposed algorithm. The proposed algorithm is useful in network architecture design, planning, and online operations of MVDC power systems when such decisions are subject to power variation.}
}
@inproceedings{col-vac-vil-zol-18-aa-powflow,
author = {Guido Coletta and Alfredo Vaccaro and Domenico Villacci and Andrea Zollo}
title = {Solving Uncertain Power Flow Problem by Affine Arithmetic}
fullhash = {a68b02221b9ca1dc67b7a09d2c2ea799}
booktitle = {2018 AEIT International Annual Conference}
month = {10}
year = {2018}
pages = {1-6}
numpages = {6}
doi = {10.23919/AEIT.2018.8577403}
abstract = {Uncertainty management is becoming a very challenging tool in operation scheduling of networks interested by massive distributed generators' penetration. From this point of view, self-validated computing techniques are very useful tools, allowing to intrinsically track data uncertainty effects into power system operation procedures. In this field, the use of Affine Arithmetic has been demonstrated to be one of the most promising research direction, since it prevents the typical error explosion phenomena affecting the standard range arithmetic-based computing frameworks. Starting from these considerations, the aim of this paper is to investigate the Affine Arithmetic-based Power Flow problem, which, by mean of the uncertainty bounds characterization, can provide additional information to the system operators in their decision making processes. The methodology will be tested on the 30-bus IEEE test network and the results will be validated by application of traditional sampling-based techniques.}
}
@article{shi-nin-18-aa-safpow,
author = {Rashmi Shivarudraswamy and Shankaraiah Ningaiah}
title = {An Affine Arithmetic Approach to Model and Estimate the Safety Parameters of {AC} Transmission Lines}
fullhash = {67d1c34b29f8643a5254ee67641a98c4}
journal = {International Journal of Image, Graphics and Signal Processing}
month = {1}
volume = {1}
year = {2018}
pages = {11-22}
numpages = {12}
doi = {10.5815/ijigsp.2018.01.02}
abstract = {With an increase in population of the country day by day and with high growing speed of geographical residential plots, the demand by the public for new set up of electrical power transmission system has become a common mandate. Therefore it is the responsibility of the concerned authority to protect the interests of common man in a smart manner and develop solutions for the growth of country in an intelligent way keeping safety of public as prime importance. This paper proposes an Affine Arithmetic approach of mathematical modelling to estimate the safety parameters of AC transmission line leading to sag, like, temperature, wind loading, ice loading, weight of the conductor, stress, tension, pressure etc., taking into account the uncertainty conditions so that the solutions developed address in real time. The proposed model is executed in MATLAB integrated development environment and gives the complete behavior of sag in transmission lines with respect to each of the safety parameters individually and closely ascertains the safety threshold limits considering 50{\%} factor of safety and 5m ground clearance. The critical safety threshold limit for wind loading is found to be about 3.02 kg/m and the typical value is about 1.51 kg/m. Similarly, the critical safety threshold limit for weight of ice is found to be about 1.45 kg/m, whereas its typical value is about 0.7 kg/m. Extending further, the critical safety threshold limit for conductor weight is found to be about 3.07 kg/m, whereas its typical value is about 1.6 kg/m, which is a close approximation to the typical weight per unit length of industry-grade conductors like Aluminum Conductor Steel Reinforced (ACSR). The critical safety threshold limit for tension in the conductor and the span lengths are found to be 1850 kg and 230m respectively}
}
@article{che-che-shi-zha-sun-hua-18-aa-windpow,
author = {Weijie Cheng and Renli Cheng and Jun Shi and Cong Zhang and Gaoxing Sun and Dong Hua}
title = {Interval Power Flow Analysis Considering Interval Output of Wind Farms through Affine Arithmetic and Optimizing-Scenarios Method}
fullhash = {a7a130ab1396e753ff7d281a57f8b7f5}
journal = {Energies}
number = {11}
volume = {11}
year = {2018}
pages = {article 3176}
numpages = {-1}
doi = {10.3390/en11113176}
abstract = {Wind power belongs to sustainable and clean energy sources which play a vital role of reducing environment pollution and addressing energy crisis. However, wind power outputs are quite difficult to predict because they are derived from wind speeds, which vary irregularly and greatly all the time. The uncertainty of wind power causes variation of the variables of power grids, which threatens the power grids' operating security. Therefore, it is significant to provide the accurate ranges of power grids' variables, which can be used by the operators to guarantee the power grid's operating security. To achieve this goal, the present paper puts forward the interval power flow with wind farms model, where the generation power outputs of wind farms are expressed by intervals and three types of control modes are considered for imitating the operation features of wind farms. To solve the proposed model, the affine arithmetic-based method and optimizing-scenarios method are modified and employed, where three types of constraints of wind control modes are considered in their solution process. The former expresses the interval variables as affine arithmetic forms, and constructs optimization models to contract the affine arithmetic forms to obtain the accurate intervals of power flow variables. The latter regards active power outputs of the wind farms as variables, which vary in their corresponding intervals, and accordingly builds the minimum and maximum programming models for estimating the intervals of the power flow variables. The proposed methods are applied to two case studies, where the acquired results are compared with those acquired by the Monte Carlo simulation, which is a traditional method for handling interval uncertainty. The simulation results validate the advantages, effectiveness and the applicability of the two methods.}
}
@article{liu-cai-jia-han-zha-18-aa-intinv,
author = {Jie Liu and Heng Cai and Chao Jiang and Xu Han and Zheng Zhang}
title = {An Interval Inverse Method Based on High Dimensional Model Representation and Affine Arithmetic}
fullhash = {531100bd36d7ea7d2493a563232f8b8e}
journal = {Applied Mathematical Modelling}
volume = {63}
year = {2018}
pages = {732-743}
numpages = {12}
doi = {10.1016/j.apm.2018.07.009}
abstract = {This paper proposes an interval inverse method through a high dimensional model representation and affine arithmetic, which can effectively solve inverse problems with interval uncertainty. Firstly, when only the bounds of responses can be obtained from a limited number of experimental measurements, an interval model can be employed to describe the uncertainty of the measured responses and identified parameters. Secondly, in order to reasonably estimate the degree of the closeness between the measured and calculated responses, an error interval and corresponding optimization model are constructed for the interval inverse problem. Thirdly, a high dimensional model representation is utilized to approximate the original system model, and an affine arithmetic is adopted to efficiently calculate the response bounds. Finally, the optimization model for interval inverse problem is solved using genetic algorithm to identify the upper and lower bounds of the system parameters. Three examples are studied to demonstrate the correctness and effectiveness of the proposed method.}
}
@article{adu-kum-18-aa-voltst,
author = {Bala Surendra Adusumilli and Boddeti Kalyan Kumar}
title = {Modified Affine Arithmetic Based Continuation Power Flow Analysis for Voltage Stability Assessment under Uncertainty}
fullhash = {6d4a6944b6a9dc4c54e4e83716557c48}
journal = {IET Generation, Transmission {\&} Distribution}
number = {18}
volume = {12}
year = {2018}
pages = {4225-4232}
numpages = {8}
doi = {10.1049/iet-gtd.2018.5479}
abstract = {Continuation power flow (CPF) analysis has been used in the literature to determine the voltage collapse point from active power versus voltage curves (PV curves) for steady-state voltage stability assessment. Affine arithmetic-based (AA) CPF analysis to determine PV curve bounds under uncertainty in power generation was introduced in the literature to overcome the problem of large computational time with Monte Carlo (MC) simulations, by getting a faster solution with a reasonably good accuracy. However, AA operations lead to more noise terms and hence overestimation of bounds. In the present work, a modified AA (modAA)-based CPF analysis is proposed to determine PV curve bounds by considering uncertainties associated with active and reactive power injections at all buses in the system. The proposed method reduces the overestimation caused by the AA operations and gives more accurate solution bounds. The proposed modAA-based CPF analysis is tested on 5-bus test case, IEEE 57, European 1354 and Polish 2383-bus systems. The simulation results with the proposed method are compared with MC simulations and AA-based CPF analysis to show the efficacy of the proposed method.}
}
@article{luo-shi-niy-18-aa-upowflow,
author = {Jinqing Luo and Libao Shi and Yixin Ni}
title = {Uncertain Power Flow Analysis Based on Evidence Theory and Affine Arithmetic}
fullhash = {363dcb54ced659c3d9f6356c50f7b92a}
journal = {IEEE Transactions on Power Systems}
number = {1}
volume = {33}
year = {2018}
pages = {1113-1115}
numpages = {3}
doi = {10.1109/TPWRS.2017.2691539}
abstract = {A novel uncertain power flow analysis method is proposed in this letter. It utilizes evidence theory to fuse probabilistic, possibilistic, and interval uncertainties. Besides, the extended affine arithmetic is applied to handle variable dependencies existed in power flow. The final analysis results are represented as probability boxes (p-boxes). The comparisons with Monte Carlo simulation are given to demonstrate the validity of the proposed method.}
}
@article{wan-wan-ten-str-wul-18-aa-multobj,
author = {Shouxiang Wang and Kai Wang and Fei Teng and Goran Strbac and Lei Wu}
title = {An Affine Arithmetic-Based Multi-Objective Optimization Method for Energy Storage Systems Operating in Active Distribution Networks with Uncertainties}
fullhash = {bfc5082af2b86ecd471ddd9d34b7223b}
journal = {Applied Energy}
volume = {223}
year = {2018}
pages = {215-228}
numpages = {14}
doi = {10.1016/j.apenergy.2018.04.037}
abstract = {Considering uncertain power outputs of distributed generations (DGs) and load fluctuations, energy storage system (ESS) represents a valuable asset to provide support for the smooth operation of active distribution networks. This paper proposes an affine arithmetic-based multi-objective optimization method for the optimal operation of ESSs in active distribution networks with uncertainties. Affine arithmetic is applied to the optimization model for handling uncertainties of DGs and loads. Two objectives are formulated with affine parameters including the minimization of total active power losses and the minimization of system voltage deviations. The affine arithmetic-based forward-backward sweep power flow is first improved by the proposed pruning strategy of noisy symbols. Then, the affine arithmetic-based non-dominated sorting genetic algorithm II (AA-NSGAII) is used to solve the multi-objective optimization problem for ESSs operation under uncertain environment. Furthermore, three types of indices with respect to convergence, diversity, and uncertainty are defined for performance analysis. Numerical studies on a modified IEEE 33-bus system with embedded DGs and ESSs show the effectiveness and superiority of the proposed method. The optimization results demonstrate that the obtained Pareto front has better convergence and lower conservativeness in comparison to the interval arithmetic-based NSGA-II. A multi-period case considering seasonality of DGs and loads is further simulated to show the applicability in real applications.}
}
@article{adu-rak-bod-18-aa-powflow,
author = {Bala Surendra Adusumilli and Vinod Raj and Kalyan Kumar Boddeti}
title = {Modified Affine Arithmetic-Based Power Flow Analysis with Uncertainty}
fullhash = {98d3b53cf96b45404d8428ca10681353}
journal = {Electric Power Components and Systems}
number = {6}
volume = {46}
year = {2018}
pages = {728-737}
numpages = {10}
doi = {10.1080/15325008.2018.1465143}
abstract = {Power flow studies are very important and are essential for fault analysis and stability analysis. Conventional Newton--Raphson-based power flow analysis is for fixed loads and generations. However, load and generation powers are uncertain due to factors like forecasting error, sudden outages, etc. Monte Carlo was widely used for power flow analysis with uncertainty. Due to excessive computational time involved in Monte Carlo, affine arithmetic-based power flow analysis was introduced. In the present work, a modified affine arithmetic-based power flow analysis is proposed. The proposed method needs less computational memory and gives tighter bounds on voltage magnitudes and angles of the system buses for a given uncertainty range of load and generation powers as compared to existing affine arithmetic-based power flow analysis. The proposed method is implemented on IEEE 30 and IEEE 118 bus systems and detailed simulation results are presented. The results are compared with Monte Carlo simulations as well as existing affine arithmetic-based power flow analysis. The results show that the proposed modified affine arithmetic-based power flow analysis yields more accurate solution intervals for voltage magnitude, voltage angles, real, and reactive power flows as compared to existing affine arithmetic-based power flow analysis.}
}
@article{coe-vuy-cha-fri-17-aa-control,
author = {Adrien Le Co{\"{e}}nt and Florian De Vuyst and Ludovic Chamoin and Laurent Fribourg}
title = {Control Synthesis of Nonlinear Sampled Switched Systems using {Euler's} Method}
fullhash = {cb77ada78ba81dc3ba6d75b431a3cfc0}
journal = {Electronic Proceedings in Theoretical Computer Science}
publisher = {Open Publishing Association}
month = {4}
volume = {247}
year = {2017}
pages = {18-33}
numpages = {16}
doi = {10.4204/eptcs.247.2}
abstract = {In this paper, we propose a symbolic control synthesis method for nonlinear sampled switched systems whose vector fields are one-sided Lipschitz. The main idea is to use an approximate model obtained from the forward Euler method to build a guaranteed control. The benefit of this method is that the error introduced by symbolic modeling is bounded by choosing suitable time and space discretizations. The method is implemented in the interpreted language Octave. Several examples of the literature are performed and the results are compared with results obtained with a previous method based on the Runge-Kutta integration method.}
}
@article{abe-rao-nai-17-aa-powflow,
author = {Yoseph Mekonnen Abebe and P Mallikarjuna Rao and M Gopichand Naik}
title = {Load Flow Analysis of an Uncertain System in the Presence of Renewable Energy Sources Using Complex Affine Arithmetic}
fullhash = {93283eb7c7835323f2f32e58978edbd6}
journal = {Journal on Electrical Engineering}
publisher = {i-Manager}
month = {6}
number = {4}
volume = {10}
year = {2017}
pages = {28-40}
numpages = {13}
abstract = {The limitation of fossil fuel resource and its effects on global warming leads to the increased usage of Renewable Energy Sources (RES). Even though the Renewable Energy Sources (RES) notably wind and solar energy are advantageous in many aspects their intermittent nature is a great deal of concern in power system control. The penetration of such renewable energy source makes the generation uncertain and leads to uncertainty based steady state analysis. In this paper, a complex Affine Arithmetic (AA) based load flow analysis in the presence of generation and load uncertainty is proposed. Vectorial representation is applied to denote the partial deviation values of the variables. The proposed approach is tested on a conventional IEEE test bus systems and the result is compared with the Monte Carlo Approach. In terms of fast convergence, less memory usage, and conservatism, the proposed approach is superior than the traditional random number based Monte Carlo approach.}
}
@inproceedings{hua-zha-sha-17-aa-newton,
author = {Weida Huang and Yan Zhang and Zhenguo Shao}
title = {{Newton}-{Raphson} Interval Harmonic Power Flow Based on Affine Arithmetic}
fullhash = {af4580a87be6dbaeb41f514a9e1072b6}
booktitle = {Proc. 43rd Annual Conference of the IEEE Industrial Electronics Society (IECON 2017)}
month = {10}
year = {2017}
pages = {6093-6098}
numpages = {6}
doi = {10.1109/IECON.2017.8217058}
abstract = {Interval harmonic power flow can take into account the uncertainty of distributed generation, and obtain the interval results including all possible harmonic current solutions. In this paper, affine arithmetic is used to reduce the conservatism, which is a shortcoming of interval arithmetic, and a Newton-Raphson harmonic current iterative equation is established in affine form. By the equal value at both ends of the equation, the coefficients of noise element in affine correction quantities are solved one by one, and in this case, the complex performance of Interval affine matrix inversion can be avoided. The interval arithmetic method proposed in this paper is proved to be complete enough and less conservative compared to the Monte Carlo method with IEEE-30 node example.}
}
@inproceedings{liu-han-gua-wug-sha-17-aa-windpow,
author = {Yixuan Liu and Zhongkuan Han and Mingfeng Guan and Guochang Wu and Zhenguo Shao}
title = {Optimizing the SVG Capacity for Wind Farm Based on Affine Arithmetic}
fullhash = {115ea0750016dfab00f042d17f835195}
booktitle = {Proc. International Conference on Smart Grid and Electrical Automation (ICSGEA 2017)}
month = {5}
year = {2017}
pages = {509-513}
numpages = {5}
doi = {10.1109/ICSGEA.2017.20}
abstract = {With the ever-increasing penetration level of wind power, its impacts on power system reliability and power quality are becoming more and more significant mainly due to the randomness and uncertainty of wind resource. For accurate express of the model about input wind speed of wind turbines, some influence factors of wind speed will be considered as noise variables, and then a model about the uncertainty of input wind speed in wind farms will be set up by affine arithmetic method. So the output power affine model of wind turbines can be calculated from the model above, attaching the requirements of power factor by the state grid, its easy to gain the reactive power affine model. Thereupon, with the relative parts of voltage fluctuation, SVG capacity in wind farms can be reasonably optimizing and analyzing after knowing the reactive power compensation.}
}
@inproceedings{abe-mal-gop-18-aa-initvar,
author = {Yoseph Mekonnen Abebe and P. Mallikarjuna Rao and M. Gopichand Naik}
title = {Finding the Initial Variables for Affine Arithmetic-Based Power Flow Analysis}
publisher = {Springer}
fullhash = {623ea460a7cb0c29a04d449ea1b44c82}
booktitle = {Advances in Power Systems and Energy Management: Proc. ETAEERE-2016}
isbn = {978-981-10-4394-9}
year = {2018}
pages = {147-156}
numpages = {10}
doi = {10.1007/978-981-10-4394-9_15}
abstract = {Power flow analysis has been playing a major role in providing information about the power system network. Due to the advancement of variable generation (VG) sources and weather variation, the punctual power flow analysis cannot give any information about the worst case scenario. The introduction of affine arithmetic (AA) is gaining high recognition due to its ability to consider all forms of uncertainty than interval arithmetic (IA). The first work in applying AA for power flow analysis is finding the initial affine forms in order to start the analysis. This paper focuses on finding the initial voltage and angle affine forms for polar coordinates and the initial real and reactive voltage forms for rectangular coordinate system.}
}
@article{ska-hlad-17-aa-optmul,
author = {Iwona Skalna and Milan Hlad\'{\i}k}
title = {A New Algorithm for {Chebyshev} Minimum-Error Multiplication of Reduced Affine Forms}
fullhash = {9abc94c7847236ba8a830704c5d21c24}
journal = {Numerical Algorithms}
volume = {76}
year = {2017}
pages = {1131-1152}
numpages = {22}
abstract = {Reduced affine arithmetic (RAA) eliminates the main deficiency of the standard affine arithmetic (AA), i.e. a gradual increase of the number of noise symbols, which makes AA inefficient in a long computation chain. To further reduce overestimation in RAA computation, a new algorithm for the Chebyshev minimum-error multiplication of reduced affine forms is proposed. The algorithm yields the minimum Chebyshev-type bounds and works in linear time, which is asymptotically optimal. We also propose a simplified O(nlogn) version of the algorithm, which performs better for low dimensional problems. Illustrative examples show that the presented approach significantly improves solutions of many numerical problems, such as the problem of solving parametric interval linear systems or parametric linear programming, and also improves the efficiency of interval global optimisation.}
}
@inproceedings{rom-can-17-aa-unitcom,
author = {David Fernando {Romero-Quete} and Claudio A. Canizares}
title = {Affine Arithmetic Formulation of the Unit Commitment Problem Under Uncertainty}
fullhash = {ed1c572da3ce1da16ec7907060a98c3e}
booktitle = {Proc. Bulk Power Systems Dynamics and Control Symposium (IREP 2017)}
location = {Espinho, Portugal}
month = {8}
year = {2017}
abstract = {This paper proposes a method based on affine arithmetic (AA) to solve the unit commitment (UC) problem, considering load and renewable energy (RE) uncertainties. The main idea is to formulate an AA-based constrained multi-objective problem that not only provides a robust commitment solution, but also provides confidence intervals for active powers and operating costs, as well as dispatch solutions for all possible load and RE generation realizations for the predetermined uncertainty bounds. Moreover, the AA-based UC (AAUC) approach allows to explore solutions where the impact of the re-dispatch cost in the total operation cost can be reduced by adjusting the objective function weight values. The proposed approach can be used to better estimate day-ahead energy prices and explore more costefficient solutions under uncertainty, as well as for real-time dispatching. The AAUC approach is tested and compared against stochastic optimization UC (SOUC), interval optimization UC (IOUC), and Monte Carlo simulation (MCS) approaches, on a sixbus system and a modified IEEE 118-bus system. The simulations results show that the proposed approach provides more accurate confidence intervals for active power and operating costs than IOUC, using MCS results as the benchmark, and has a better computational performance than SOUC}
}
@inproceedings{din-zha-tri-sti-can-17-aa-wceletr,
author = {Tongyu Ding and Liang Zhang and Riccardo Trinchero and Igor S. Stievano and Flavio G. Canavero}
title = {Worst-Case Analysis of Electrical and Electronic Equipment via Affine Arithmetic}
fullhash = {a099ecae0c47bfe54bf18efd44ae1682}
booktitle = {Proc. International Conference on Electromagnetics in Advanced Applications (ICEAA 2017)}
month = {9}
year = {2017}
pages = {991-993}
numpages = {3}
doi = {10.1109/ICEAA.2017.8065425}
abstract = {This paper proposes to generate a smart tool that can inherently and effectively capture the results of parameter variations on the system responses of lumped and distributed electrical circuits. This methodology leverages the so-called affine arithmetic and represents parameter-dependent responses in terms of a multivariate polynomial. The affine representation is propagated from input parameters to circuit responses through a suitable redefinition of the basic operations, such as addition, multiplication or matrix inversion, that are involved in the circuit solution. The proposed framework is applied to the frequency-domain analysis of switching converters, and it turns out to be accurate and more efficient than traditional solutions based on Monte Carlo analysis.}
}
@article{abe-rao-nak-17-aa-powflow,
author = {Yoseph Mekonnen Abebe and P. Mallikarjuna Rao and M. Gopichand Nak}
title = {Load Flow Analysis of a Power System Network in the Presence of Uncertainty using Complex Affine Arithmetic}
fullhash = {48d966609290ff11b5c17038c380dd6a}
journal = {International Journal of Engineering and Manufacturing}
month = {9}
volume = {5}
year = {2017}
pages = {48-64}
numpages = {17}
doi = {10.5815/ijem.2017.05.05}
abstract = {The depletion of fossil fuel is driving the world towards the application of renewable energy sources. However, their intermittent nature, in addition to load variation and transmission line sag-tension change due to temperature, is a great deal of problems for reliable power delivery. Without a reliable power delivery, power generation is just a waste of resources. A reliable power delivery can be achieved when the best and the worst case steady state information of a power system network is known to plan and control accordingly. If a system is affected by the presence of uncertainty, a deterministic load flow analysis fails to provide the worst-case load flow result in a single analysis. As a result, a load flow analysis considering the presence of uncertainty is mandatory. On this paper, a novel complex affine arithmetic (AA) based load flow analysis in the presence of generation and load uncertainties is proposed. The proposed approach is tested on an IEEE bus systems and compared with Monte Carlo approach. The proposed approach convergence faster than the Monte Carlo based method and it is slightly conservative.}
}
@article{bos-pel-sol-17-aa-projsche,
author = {Gaetano Bosurgi and Orazio Pellegrino and Giuseppe Sollazzo}
title = {Project Duration Evaluated Using Affine Arithmetic}
fullhash = {e8b2fc9cbd776ffc90ff143304114fd3}
journal = {Periodica Polytechnica Civil Engineering}
number = {3}
volume = {61}
year = {2017}
pages = {412-420}
numpages = {9}
doi = {10.3311/PPci.8972}
abstract = {A civil engineering work can be performed by organizing the available resources (manpower, equipment and materials) in many different ways. Each different configuration results in a realization time and a cost that a building company has to bear. To produce reliable duration forecasts and money savings, it is essential to take into account all the uncertainties involved in the project operations. Generally, since it is impractical to process numerous uncertain variables - also undefined from a statistical point of view -, traditional probabilistic methods involve application difficulties for complex environments such as construction sites. To properly handle this issue, the authors propose in this paper the application of the Affine Arithmetic technique. This method treats the variables as intervals and returns reliable results, even when the variables are mutually dependent. The numerical example presented in the paper proves the efficiency of the procedure, even if some analytical complications are included in the analysis (dependency between variables, non-linear functions, etc.). Comparisons with Interval Analysis and traditional procedures are also provided. Adopting Affine Arithmetic, the results are reported in terms of intervals, avoiding the definition of unrealistic deterministic values that can strongly affect the operation organization. Furthermore, without increasing the problem complexity, the model admits continuous modifications (interval amplitudes, new variable dependencies, etc.) to correct and optimize the durations.}
}
@article{luo-guw-wan-che-17-aa-unsch,
author = {Lizi Luo and Wei Gu and Yonghui Wang and Chunxi Chen}
title = {An Affine Arithmetic-Based Power Flow Algorithm Considering the Regional Control of Unscheduled Power Fluctuation}
fullhash = {fa53f6faf6d483cbbcd1d46b47fba5dd}
journal = {Energies}
number = {11}
volume = {10}
year = {2017}
pages = {article 1794}
numpages = {-1}
doi = {10.3390/en10111794}
abstract = {With consideration of the intermittency of renewable generation and uncertain load, a regional control strategy is presented to smooth the unscheduled power fluctuation in this letter. Then, an affine arithmetic-based modeling method is proposed to describe the unscheduled power tracking characteristic of dispatchable generation resources (DGRs), based on which interval power flow solutions with narrower ranges can be obtained. Finally, the proposed algorithm is applied to a modified IEEE 33-bus distribution system to demonstrate its effectiveness.}
}
@inproceedings{rah-kie-cec-wat-17-aa-weathpow,
author = {Mahbubur Rahman and Mareike Kiesau and Valentina Cecchi and Bobby Watkins}
title = {Investigating effects of weather parameter uncertainty on transmission line power handling capabilities using affine arithmetic}
fullhash = {d058d6ed58ebed4b7dcb96a215f2c1e0}
booktitle = {Proc. 2017 IEEE Power {\&} Energy Society General Meeting}
location = {Chicago}
year = {2017}
pages = {1-5}
numpages = {5}
doi = {10.1109/PESGM.2017.8274733}
abstract = {This paper presents an approach to deal with uncertainty in weather parameters when determining transmission line power handling capabilities, in terms of both line thermal rating and voltage stability limits. Weather parameters affect conductor temperature, which in return affects line electrical parameters. Wind speed and direction, for example, have a significant cooling effect, which can result in higher allowable currents through the line before reaching the maximum conductor temperature. At the same time, the line electrical parameters, particularly resistance, are a function of temperature and would therefore affect the line electrical model and its maximum power transfer point before voltage collapse. Weather conditions are variable and uncertain; in order to account for the uncertainty in weather parameters, this paper presents an approach based on the affine form of interval arithmetic. The results of the affine arithmetic-based approach are then compared with static line ratings. The approach is tested on a single-line 2-bus system and on a 14-bus transmission network.}
}
@article{wan-wuz-dou-hum-xuy-17-aa-powflow,
author = {Yang Wang and Zaijun Wu and Xiaobo Dou and Minqiang Hu and Yiyue Xu}
title = {Interval power flow analysis via multi-stage affine arithmetic for unbalanced distribution network}
fullhash = {d250c3fa77ab14d3d264035c723fcf88}
journal = {Electric Power Systems Research}
volume = {142}
year = {2017}
pages = {1-8}
numpages = {8}
doi = {10.1016/j.epsr.2016.08.024}
abstract = {Integrating large numbers of intermittent energy resources, such as photovoltaic generations and wind turbines, brings huge uncertainty to the operation of distribution network. In this paper, an interval power flow method via multi-stage affine arithmetic is proposed to address the impact of distributed generation output power and load uncertainty on the power flow solution for the unbalanced distribution network, which acquires affine expression of bus voltage by solving the fixed component, first-order perturbation and non-linear perturbation of the bus complex voltage respectively. Multi-numerical results based on modified IEEE 123 nodes test feeder are presented and discussed, demonstrating that the proposed method achieves much better performance than existing ones in both precision and computation aspects.}
}
@article{vac-can-17-aa-uncpow,
author = {Alfredo Vaccaro and Claudio A. Ca{\~{n}}izares}
title = {An Affine Arithmetic-Based Framework for Uncertain Power Flow and Optimal Power Flow Studies}
fullhash = {f3c52580fa856a293dd258d79678cb74}
journal = {IEEE Transactions on Power Systems}
number = {1}
volume = {32}
year = {2017}
pages = {274-288}
numpages = {15}
doi = {10.1109/TPWRS.2016.2565563}
abstract = {This paper proposes a unified framework based on affine arithmetic for computing reliable enclosures of uncertain power flow (PF) and optimal power flow (OPF) solutions. The main idea is to formulate a generic mathematical programming problem under uncertainty by means of equivalent deterministic problems, and to identify the affine forms describing the data uncertainty by means of a signal processing technique based on principal components analysis. Compared to existing solution algorithms, this formulation presents greater flexibility, as it allows to find feasible solutions and inclusion of multiple equality and inequality constraints, and reduce the approximation errors to obtain better PF and OPF solution enclosures. Detailed numerical results are presented and discussed using a variety of realistic test systems, demonstrating the effectiveness of the proposed methodologies and comparing it to existing techniques for uncertain PF and OPF analysis.}
}
@article{sun-luo-tan-huy-zho-16-aa-reliab,
author = {Jianping Sun and Yiping Luo and Zhaoping Tang and Yutao Hu and Xinjian Zhou}
title = {An Improved Affine Algorithm for the Non-Probabilistic Reliability Index of Interval Model}
fullhash = {4e4c0543564999b25940e84c9783afc0}
journal = {International Journal of Control and Automation}
number = {4}
volume = {9}
year = {2016}
pages = {333-346}
numpages = {14}
urlraw = {https://www.earticle.net/Article/A272700}
url = {https://www.earticle.net/Article/A272700}
abstract = {Aimed at that standard interval arithmetic is easy to cause error explosion, as well as traditional affine arithmetic is easy to cause calculation error due to not considering the correlation among the first degree item, quadratic term, even the multiple-item and the nonlinear term expressed by the same noise, this paper proposed an improved algorithm which took affine algorithm operation as the main body. When there are those correlation items, the algorithm calculates the upper and lower bound of their total value by interval arithmetic, then uses the bound information and introduces the new noise affine form to instead of those terms. The method improved the conventional affine operation by combining interval arithmetic. When it applied to calculate the nonprobabilistic reliability index, the calculation results has higher accuracy than the interval arithmetic and the conventional affine algorithm, and it has strong practical significance.}
}
@inproceedings{wan-wan-cai-16-aa-storage,
author = {Kai Wang and Shouxiang Wang and Shengxia Cai}
title = {Quantitative Evaluation of Uncertainty Mitigation by {ESS} via Complex Affine Distribution Power Flow}
fullhash = {d5566266d1910e95cffe0ac147e33431}
booktitle = {Proc. IEEE Power and Energy Society General Meeting (PESGM 2016)}
month = {7}
year = {2016}
pages = {1-5}
numpages = {5}
doi = {10.1109/PESGM.2016.7741556}
abstract = {This paper quantitatively evaluates the uncertainty mitigation by energy storage system (ESS) via complex affine arithmetic-based distribution power flow. First, the affine models of distributed generations (DGs) and ESS under uncertain situations are explicitly established. Then, a forward-backward sweep power flow based on complex affine arithmetic is proposed in distribution systems with DGs and ESS. Third, an index of uncertainty mitigation by ESS to quantify the impacts on the power flow results is defined. Finally, the IEEE 13-bus system case is used to test the proposed evaluation method. The uncertainty mitigation index of ESS is calculated. Test results demonstrate that the presented index can quantitatively reflect the uncertainty mitigation impacts of ESS on separate bus voltage of distribution systems.}
}
@phdthesis{rad-16-aa-thesis,
author = {{\v{C}}arna Radoji{\v{c}}i{\'{c}}}
title = {Symbolic Simulation of Mixed-Signal Systems with Extended Ane Arithmetic}
institution = {University of Kaiserslautern}
fullhash = {f55aa79ad640d35399b8b9650d049fb8}
month = {9}
note = {Ph.~D. Thesis; Advisor: Christoph Grimm}
year = {2016}
abstract = {Mixed-signal systems combine analog circuits with digital hardware and software systems. A particular challenge is the sensitivity of analog parts to even small deviations in parameters, or inputs. Parameters of circuits and systems such as process, voltage, and temperature are never accurate; we hence model them as uncertain values (`uncertainties'). Uncertain parameters and inputs can modify the dynamic behavior and lead to properties of the system that are not in specified ranges. For verification of mixed signal systems, the analysis of the impact of uncertainties on the dynamical behavior plays a central role. Verification of mixed-signal systems is usually done by numerical simulation. A single numerical simulation run allows designers to verify single parameter values out of often ranges of uncertain values. Multi-run simulation techniques such as Monte Carlo Simulation, Corner Case simulation, and enhanced techniques such as Importance Sampling or Design-of-Experiments allow to verify ranges -- at the cost of a high number of simulation runs, and with the risk of not finding potential errors. Formal and symbolic approaches are an interesting alternative. Such methods allow a comprehensive verification. However, formal methods do not scale well with heterogeneity and complexity. Also, formal methods do not support existing and established modeling languages. This fact complicates its integration in industrial design flows. In previous work on verification of Mixed-Signal systems, Affine Arithmetic is used for symbolic simulation. This allows combining the high coverage of formal methods with the ease-of use and applicability of simulation. Ane Arithmetic computes the propagation of uncertainties through mostly linear analog circuits and DSP methods in an accurate way. However, Affine Arithmetic is currently only able to compute with contiguous regions, but does not permit the representation of and computation with discrete behavior, e.g. introduced by software. This is a serious limitation: in mixed-signal systems, uncertainties in the analog part are often compensated by embedded software; hence, verification of system properties must consider both analog circuits and embedded software. The objective of this work is to provide an extension to Affine Arithmetic that allows symbolic computation also for digital hardware and software systems, and to demonstrate its applicability and scalability. Compared with related work and state of the art, this thesis provides the following achievements: 1. The thesis introduces extended Affine Arithmetic Forms (XAAF) for the representation of branch and merge operations. 2. The thesis describes arithmetic and relational operations on XAAF, and reduces over-approximation by using an LP solver. 3. The thesis shows and discusses ways to integrate this XAAF into existing modeling languages, in particular SystemC. This way, breaks in the design flow can be avoided. The applicability and scalability of the approach is demonstrated by symbolic simulation of a $\Sigma$-$\Delta$ Modulator and a PLL circuit of an IEEE 802.15.4 transceiver system}
}
@inproceedings{hei-ahm-ali-ahm-16-aa-comperr,
author = {Azar Heidari and Arash Ahmadi and Shahpour Alirezaee and Majid Ahmadi}
title = {A Modified Affine Arithmetic Method for Computational Error Analysis}
fullhash = {01c5de6bad09fa852135ba6314733e22}
booktitle = {Proc. International Conference on Informatics, Management Engineering and Industrial Application (IMEIA 2016)}
isbn = {978-1-60595-345-8}
year = {2016}
abstract = {Computational precision always has been a concern in digital systems design and optimization. This study presents a modified Affine Arithmetic method to calculate uncertainty range in the output of arithmetic units. We have considered five case studies, namely: quadratic equation, RGB to CrYCb system, fourth-order equation, multivariate polynomial functions and low pass filter, by which the proposed method is evaluated. Presented results indicate the modified affine arithmetic can outperform the traditional methods.}
}
@misc{kas-16-aa-nitsui,
author = {Masahide Kashiwagi}
title = {About Affine Arithmetic}
fullhash = {043579a540ab675a207a5254db9be693}
howpublished = {Online document}
note = {In Japanese.}
year = {2016}
pages = {15}
numpages = {1}
url = {http://verifiedby.me/kv/affine/affine.pdf}
}
@inproceedings{mat-ued-16-aa-simul,
author = {Shota Matsumoto and Kazunori Ueda}
title = {Symbolic Simulation of Parametrized Hybrid Systems with Affine Arithmetic}
fullhash = {b6d1c8113de0465b19995c94f1feed98}
booktitle = {23rd International Symposium on Temporal Representation and Reasoning (TIME 2016)}
month = {10}
year = {2016}
pages = {4-11}
numpages = {8}
doi = {10.1109/TIME.2016.8}
abstract = {The purpose of this research is to develop a highly reliable simulator of hybrid systems, i.e., systems involving both discrete change and continuous evolution. In particular, we aim at rigorous simulation of parametrized hybrid systems, which enables not only the analysis of model's possible behavior but also the design of parameters that realize desired properties. Simulators with interval arithmetic can reliably compute a reachable set of states, but preserving the dependency of uncertain quantities in models is still challenging. In this paper, we discuss a simulation method that is based on symbolic computation and cooperates with the interval Newton method and affine arithmetic, which is able to preserve first-order dependency of uncertain quantities. We implemented the algorithm on the symbolic simulator we have been developing and evaluated the performance of the method with example models.}
}
@inproceedings{zha-che-guo-lei-leh-wan-16-aa-mixpol,
author = {Cong Zhang and Haoyong Chen and Manlan Guo and Jia Lei and Hailin Ge and Chulin Wan}
title = {A Mixed Interval Power Flow Analysis Based on Affine Arithmetic}
fullhash = {bb97ac60406e251fb25eeb401ef7dd1e}
booktitle = {Proc. IEEE PES Asia-Pacific Power and Energy Engineering Conference (APPEEC 2016)}
month = {10}
year = {2016}
pages = {758-762}
numpages = {5}
doi = {10.1109/APPEEC.2016.7779597}
abstract = {A rectangular interval power flow (RIPF) method is proposed to evaluate solutions' intervals of power flow including uncertain loads and output of generators. It is based on affine arithmetic which is a more accurate method to compute intervals, and it obtains intervals through optimization methods to enhance efficiency of interval calculation. Because no approximation is used, RIPF can acquire narrower intervals of voltage phase results than polar interval power flow (PIPF), which is an excellent method raised before. However, RIPF is no better than PIPF in estimating intervals of voltage amplitude, since it could not satisfy the voltage equations of interval power flow. Therefore, a mixed interval power flow (MIPF) is established to incorporate merits of RIPF in computing intervals of voltage phase and PIPF's advantages in evaluating intervals of voltage magnitude. The IEEE118 system is analyzed to conclude that MIPF can acquire more accurate power flow intervals than PIPF.}
}
@article{zou-pen-cai-wuh-hul-16-aa-forens,
author = {Tiefang Zou and Haitao Peng and Ming Cai and Hequan Wu and Lin Hu}
title = {A {Taylor}-Affine Arithmetic for Analyzing the Calculation Result Uncertainty in Accident Reconstruction}
fullhash = {8878670bfcc23cac2a692f76cef292de}
journal = {Forensic Science International}
volume = {266}
year = {2016}
pages = {502-510}
numpages = {9}
doi = {10.1016/j.forsciint.2016.07.011}
abstract = {In order to analyze the uncertainty of a reconstructed result, the Interval Algorithm (IA), the Affine Arithmetic (AA) and the Modified Affine Arithmetic (MAA) were introduced firstly, and then a Taylor-Affine Arithmetic (TAA) was proposed based on the MAA and Taylor series. Steps of the TAA, especially in analyzing uncertainty of a simulation result were given. Through the preceding five numerical cases, its application was demonstrated and its feasibility was validated. Results showed that no matter other methods (The IA, AA, the Upper and Lower bound Method, the Finite Difference Method) work well or bad, the TAA work well, even under the condition that the MAA cannot work in some cases because of the division/root operation in these models. Furthermore, in order to make sure that the result obtained from the TAA can be very close to the accurate interval, a simple algorithm was proposed based on the sub-interval technique, its feasibility was validated by two other numerical cases. Finally, a vehicle--pedestrian test was given to demonstrate the application of the TAA in practice. In the vehicle--pedestrian test, the interval [35.5, 39.1]km/h of the impact velocity can be calculated according to steps of the TAA, such interval information will be more useful in accident responsibility identification than a single number. This study will provide a new alternative method for uncertainty analysis in accident reconstruction.}
}
@phdthesis{rad-16-aa-simul,
author = {Carna Radojicic}
title = {Symbolic Simulation of Mixed-Signal Systems with Extended Affine Arithmetic}
institution = {Technische Universit{\"{a}}t Kaiserslautern}
fullhash = {f73fcbd491592cbfc292b48f58b66ce3}
note = {Doctoral thesis; advisor: Christoph Grimm}
year = {2016}
pages = {115}
numpages = {1}
urlraw = {http://nbn-resolving.de/urn:nbn:de:hbz:386-kluedo-44837}
url = {http://nbn-resolving.de/urn:nbn:de:hbz:386-kluedo-44837}
abstract = {Mixed-signal systems combine analog circuits with digital hardware and software systems. A particular challenge is the sensitivity of analog parts to even small deviations in parameters, or inputs. Parameters of circuits and systems such as process, voltage, and temperature are never accurate; we hence model them as uncertain values (`ncertainties'). Uncertain parameters and inputs can modify the dynamic behavior and lead to properties of the system that are not in specified ranges. For verification of mixed- signal systems, the analysis of the impact of uncertainties on the dynamical behavior plays a central role. Verification of mixed-signal systems is usually done by numerical simulation. A single numerical simulation run allows designers to verify single parameter values out of often ranges of uncertain values. Multi-run simulation techniques such as Monte Carlo Simulation, Corner Case simulation, and enhanced techniques such as Importance Sampling or Design-of-Experiments allow to verify ranges -- at the cost of a high number of simulation runs, and with the risk of not finding potential errors. Formal and symbolic approaches are an interesting alternative. Such methods allow a comprehensive verification. However, formal methods do not scale well with heterogeneity and complexity. Also, formal methods do not support existing and established modeling languages. This fact complicates its integration in industrial design flows. In previous work on verification of Mixed-Signal systems, Affine Arithmetic is used for symbolic simulation. This allows combining the high coverage of formal methods with the ease-of use and applicability of simulation. Affine Arithmetic computes the propagation of uncertainties through mostly linear analog circuits and DSP methods in an accurate way. However, Affine Arithmetic is currently only able to compute with contiguous regions, but does not permit the representation of and computation with discrete behavior, e.g. introduced by software. This is a serious limitation: in mixed-signal systems, uncertainties in the analog part are often compensated by embedded software; hence, verification of system properties must consider both analog circuits and embedded software. The objective of this work is to provide an extension to Affine Arithmetic that allows symbolic computation also for digital hardware and software systems, and to demonstrate its applicability and scalability. Compared with related work and state of the art, this thesis provides the following achievements: 1. The thesis introduces extended Affine Arithmetic Forms (XAAF) for the representation of branch and merge operations. 2. The thesis describes arithmetic and relational operations on XAAF, and reduces over-approximation by using an LP solver. 3. The thesis shows and discusses ways to integrate this XAAF into existing modeling languages, in particular SystemC. This way, breaks in the design flow can be avoided. The applicability and scalability of the approach is demonstrated by symbolic simulation of a Delta-Sigma Modulator and a PLL circuit of an IEEE 802.15.4 transceiver system.}
}
@inproceedings{rad-gri-16-aa-formal,
author = {Carna Radojicic and Christoph Grimm}
title = {Formal Verification of Mixed-Signal Designs Using Extended Affine Arithmetic}
fullhash = {2b9bb83dc73cec5c268fbe5cf51f8f9d}
booktitle = {2016 12th Conference on Ph.D. Research in Microelectronics and Electronics (PRIME)}
month = {6}
year = {2016}
pages = {1-4}
numpages = {4}
doi = {10.1109/PRIME.2016.7519482}
abstract = {The complexity and heterogeneity of today's mixed-signal systems makes verification a challenge. A particular challenge is the sensitivity of analog parts to variations in parameters, inputs, or initial conditions. We present a methodology for formal verification of mixed-signal systems that verifies the impact of variations of parameters, inputs, or initial conditions on specified properties. Compared with state of the art, the proposed methodology can be integrated easily in existing design flows, handles analog and digital parts, and offers improved scalability. The method is applied on a third order $\Sigma \Delta$ Modulator for verifying the stability property. The results show that our approach is using one simulation run able to find the input sequence that could lead to the undesired system behavior. These values are often not trivial and most likely would never be detected by traditional simulation-based techniques.}
}
@article{wan-qin-16-aa-mechopt,
author = {Shaobo Wang and Xiangyun Qing}
title = {A Mixed Interval Arithmetic/Affine Arithmetic Approach for Robust Design Optimization With Interval Uncertainty}
fullhash = {7d7d99a702f3bfa885e43b55521da26a}
journal = {Journal of Mechanical Design}
month = {02}
number = {4}
volume = {138}
year = {2016}
pages = {article 041403}
numpages = {-1}
doi = {10.1115/1.4032630}
abstract = {Uncertainty is ubiquitous throughout engineering design processes. Robust optimization (RO) aims to find optimal solutions that are relatively insensitive to input uncertainty. In this paper, a new approach is presented for single-objective RO problems with an objective function and constraints that are continuous and differentiable. Both the design variables and parameters with interval uncertainties are represented as affine forms. A mixed interval arithmetic (IA)/affine arithmetic (AA) model is subsequently utilized in order to obtain affine approximations for the objective and feasibility robustness constraint functions. Consequently, the RO problem is converted to a deterministic problem, by bounding all constraints. Finally, nonlinear optimization solvers are applied to obtain a robust optimal solution for the deterministic optimization problem. Some numerical and engineering examples are presented in order to demonstrate the advantages and disadvantages of the proposed approach. The main advantage of the proposed approach lies in the simplicity of the conversion from a nonlinear RO problem with interval uncertainty to a deterministic single-looped optimization problem. Although this approach cannot be applied to problems with black-box models, it requires a minimal use of IA/AA computation and applies some widely used advanced solvers to single-looped optimization problems, making it more suitable for applications in engineering fields.}
}
@article{xuc-guw-gao-son-men-fan-16-aa-improv,
author = {Chao Xu and Wei Gu and Fei Gao and Xiaohui Song and Xiaoli Meng and Miao Fan}
title = {Improved Affine Arithmetic Based Optimisation Model for Interval Power Flow Analysis}
fullhash = {e7c23571990b86799bbd548213c7656d}
journal = {IET Generation, Transmission {\&} Distribution}
number = {15}
volume = {10}
year = {2016}
pages = {3910-3918}
numpages = {9}
doi = {10.1049/iet-gtd.2016.0601}
abstract = {Power flow (PF) problem need to be further studied when confronted with uncertainties brought in by the increasing use of renewable energy. This study proposes a new solution method based on linear approximation of the affine arithmetic (AA) based PF model and optimal solution technique incorporated with boundary load flow framework under generation and load data uncertainties. In each iteration solution step, non-linear interval PF problem is modelled by the approximation technique with AA. Boundaries of state variables are explored by solving linear programming models with constraints reformed at given operating points. After optimisation process, new operating point is obtained and updated for further iteration solution step. Application of the proposed methodology is implemented in several IEEE benchmark test systems and results are demonstrated in details. Comparisons between the previous interval method and Monte Carlo simulations verify the effectiveness and better performance of the proposed method.}
}
@inproceedings{avi-can-sae-fel-15-aa-powdem,
author = {Fernanda {\'{A}}vila and Claudio Ca{\~{n}}izares and Doris S{\'{a}}ez and Felipe Valencia}
title = {Load Modelling Using Affine Arithmetic for Demand Side Management}
fullhash = {4de449dd7c4ff00b4ab7cc4b40f4f3f0}
booktitle = {Proc. IEEE PES Innovative Smart Grid Technologies Latin America (ISGT LATAM 2015)}
month = {10}
year = {2015}
pages = {456-460}
numpages = {5}
doi = {10.1109/ISGT-LA.2015.7381198}
abstract = {Demand side management (DSM) programs aim at reducing energy consumption on the demand side, which benefits both consumers and utilities. These programs could also help maintain the critical balance between generation and demand in isolated microgrids. In this case, demand is a treated as a significant uncertainty in the context of dispatching the power resources of a microgrid. Thus, in this paper the uncertainty of consumer response to DSM signals are treated as intervals to represent the corresponding range of variation based on affine arithmetic. The total range of total grid consumption is taken as the sum of intervals in each house, using data obtained through surveys regarding usage of appliances to obtain the individual consumer intervals. The proposed affine arithmetic model is then applied to the prediction of possible range variation in a one-day ahead load forecast. Finally, the presented technique is demonstrated using data obtained in an actual microgrid deployed in Huatacondo, Chile.}
}
@inbook{bos-pel-sol-15-aa-road,
author = {Gaetano Bosurgi and Orazio Pellegrino and Giuseppe Sollazzo}
title = {Road Design Problems Solved by Affine Arithmetic}
publisher = {CRC}
fullhash = {e8b2fc9cbd776ffc90ff143304114fd3}
isbn = {9781315690490}
year = {2015}
pages = {431-434}
numpages = {4}
abstract = {In this paper we propose the use of techniques based on range numbers for solving problems of road designing. We applied the Affine Arithmetic to verify the driver's visibility in a horizontal curve, providing very interesting results and confirming its usefulness even in the most complex analytical expressions. The preparation of charts such as those proposed in this paper also makes this method directly accessible to those not so familiar with this technique.}
}
@article{sch-olb-bar-15-aa-simutools,
author = {Oliver Scharf and Markus Olbrich and Erich Barke}
title = {Split and Merge Strategies for Solving Uncertain Equations Using Affine Arithmetic}
fullhash = {5c34757c95560d2b540f8642657dff18}
journal = {EAI Endorsed Transactions on Industrial Networks and Intelligent Systems (INIS)}
month = {8}
number = {9}
volume = {3}
year = {2015}
doi = {10.4108/eai.24-8-2015.2260594}
abstract = {The behaviour of systems is determined by various parameters. Due to several reasons like e. g. manufacturing tolerances these parameters can have some uncertainties. Corner Case and Monte Carlo simulations are well known approaches to handle uncertain systems. They sample the corners and random points of the parameter space, respectively. Both require many runs and do not guarantee the inclusion of the worst case. As alternatives, range based approaches can be used. They model parameter uncertainties as ranges. The simulation outputs are ranges which include all possible results created by the parameter uncertainties. One type of range arithmetic is the affine arithmetic, which allows to maintain linear correlations to avoid over-approximation. An equation solver based on affine arithmetic has been proposed earlier. Unlike many other range based approaches it can solve implicit non-linear equations. This is necessary for analog circuit simulation. For large uncertainties the solver suffers from convergence problems. To overcome these problems it is possible to split the parameter ranges, calculate the solutions separately and merge them again. For higher dimensional systems this leads to excessive runtimes as each parameter is split. To minimize the additional runtime several split and merge strategies are proposed and compared using two analog circuit examples.}
}
@inproceedings{gat-pas-per-hen-15-aa-voc,
author = {Paula O. La Gatta and Jo{\~{a}}o A. Passos Filho and Jos{\'{e}} L. R. Pereira and Ricardo M. Henriques}
title = {An Affine Arithmetic Method to Identify Voltage Control Areas for Secondary Voltage Control}
fullhash = {b20841f956bb5935b7d5639e938e082c}
booktitle = {Proc. 2015 IEEE Eindhoven PowerTech}
month = {6}
year = {2015}
pages = {1-6}
numpages = {6}
doi = {10.1109/PTC.2015.7232612}
abstract = {This paper aims to present a methodology to identify Voltage Control Areas (VCA) for Secondary Voltage Control (SVC). The proposed methodology consists in the representation of data uncertainty using Affine Arithmetic (AA) applied to the Power Flow Analysis. The uncertainty used in this paper is associated only with reactive power demand, mainly due to forecast errors. The IEEE 14 bus test system and the IEEE 118 bus system are used to evaluate the proposed methodology and the results presented validate and indicate the effectiveness of using an AA-based method to identify VCA.}
}
@inproceedings{abe-pas-mud-16-aa-powflow,
author = {Yoseph Mekonnen Abebe and Mallikarjuna Rao Pasumarthi and Gopichand Naik Mudavath}
title = {Load Flow Analysis of Uncertain Power System Through Affine Arithmetic}
publisher = {Springer}
fullhash = {2fbb938c80f9bea36a23d835358694f2}
booktitle = {Microelectronics, Electromagnetics and Telecommunications - Proceedings of ICMEET 2015}
location = {New Delhi}
isbn = {978-81-322-2728-1}
number = {372}
series = {LNEE}
year = {2016}
pages = {217-231}
numpages = {15}
abstract = {On this paper a novel load flow analysis using complex affine arithmetic (AA) based on Gauss-Seidel method for uncertain system is proposed. The Gauss-Seidel algorithm is used to find the uncertainty in each bus, which is the partial deviation value of the buses. The proposed algorithm is applied on an IEEE-14, 30 and 57 bus test systems. For comparison purpose a probabilistic load flow analysis based on Monte Carlo method is used. The proposed method is tested for different uncertainty level and in all the test cases AA based method is faster in convergence and gives slightly conservative bound than the probabilistic Monte Carlo approach.}
}
@phdthesis{vac-15-aa-thesis,
author = {Alfredo Vaccaro}
title = {Affine Arithmetic for Power and Optimal Power Flow Analyses in the Presence of Uncertainties}
institution = {University of Waterloo}
fullhash = {40325c9a3d98c5800e14f986b3783b17}
year = {2015}
urlraw = {http://hdl.handle.net/10012/9307}
url = {http://hdl.handle.net/10012/9307}
abstract = {Optimal power system operation requires intensive numerical analyses to study and improve system security and reliability. To address this issue, Power Flow (PF) and Optimal Power Flow (OPF) analyses are important tools, since they are the foundation of many power engineering applications. For the most common formalization of these problems, the input data are specified using deterministic variables resulting either from a snapshot of the system or defined by the analyst based on several assumptions about the system under study. This approach provides problem solutions for a single system state, which is deemed representative of the limited set of system conditions corresponding to the data assumptions. Thus, when the input conditions are uncertain, numerous scenarios need to be evaluated.}
}
@inproceedings{xue-xie-xie-pan-ren-nie-wan-che-15-aa-3phflow,
author = {Feng Xue and Weilun Xie and Peicheng Xie and Zhengning Pan and Jiayi Ren and Yinghui Nie and Xuchong Wang and Ming Chen}
title = {Unbalanced Three-Phase Distribution System Power Flow with Distributed Generation using Affine Arithmetic}
fullhash = {aa91b0c4d683d3073f7beb6cd2999bca}
booktitle = {5th International Conference on Electric Utility Deregulation and Restructuring and Power Technologies (DRPT 2015)}
month = {11}
year = {2015}
pages = {822-829}
numpages = {8}
doi = {10.1109/DRPT.2015.7432341}
abstract = {Due to the high penetration of the uncertain distributed generations (DGs), including wind power generators (WTG), photovoltaic (PV) power generation and plug-in hybrid electric vehicle (PHEV), the power flow of the distribution system has become asymmetry and uncertain. Based on the combination of interval and affine arithmetic, this paper establishes the model of three-phase load with uncertain DGs, and proposes a three-phase forward-backward sweep power flow algorithm for the distribution system. Compared with the deterministic power flow, the proposed algorithm can not only analyze the impact of the uncertain load and DGs on the power flow, but also reduce the overestimation of bounds yielded by correlation of variables in interval arithmetic (IA). IEEE 33-bus test systems are used to analyze the influence of the penetration of distributed generations on the distribution system.}
}
@inproceedings{abe-rao-15-aa-powline,
author = {Yoseph Mekonnen Abebe and P. Mallikarjuna Rao}
title = {Overhead Transmission Line Sag, Tension and Length Calculation using Affine Arithmetic}
fullhash = {2be93f3f8b4321af9130aa6d2d38a379}
booktitle = {2015 IEEE Power, Communication and Information Technology Conference (PCITC)}
month = {10}
year = {2015}
pages = {211-216}
numpages = {6}
doi = {10.1109/PCITC.2015.7438162}
abstract = {Transmission line sag, tension and conductor length varies with the variation of temperature due to thermal expansion and elastic elongation. Beside thermal effect, wind pressure and ice accumulation creates a horizontal and vertical loading on the conductor respectively. Such changes make the calculated data uncertain and require an uncertainty model. A novel affine arithmetic (AA) based transmission line sag, tension and conductor length calculation for parabolic curve is proposed and the proposed method is tested for different test cases. The results are compared with Monte Carlo (MC) and interval arithmetic (IA) methods. The AA based result gives a more conservative bound than MC and IA method in all the cases.}
}
@article{son-yuk-lix-han-15-aa-trosc,
author = {Haiyang Song and Kaiping Yu and Xiangyang Li and Jingyong Han}
title = {Affine Arithmetic Applied to Transient Statistical Energy Analysis of a Two-Oscillator System}
fullhash = {eed0752717f0d1989c7614cf9b9409b1}
journal = {Mechanics Research Communications}
volume = {70}
year = {2015}
pages = {12-16}
numpages = {5}
doi = {10.1016/j.mechrescom.2015.08.009}
abstract = {This paper applies affine arithmetic to transient statistical energy analysis (SEA) of a two-oscillator system, and the influence of the measurement errors of parameters on predicted transient energy is revealed. By considering the damping loss factors and coupling loss factors with measurement errors as interval variables, the mathematical expressions of the total energy interval and the peak energy interval can finally be derived. Then two flat plates which are perpendicular to each other and joined together through bolts are exploited as numerical example to demonstrate the feasibility and effectiveness of the presented approach. Meanwhile, the structural transient energy calculated from the presented approach is compared with that obtained from the traditional method which did not consider the measurement errors of damping loss factors and coupling loss factors.}
}
@inproceedings{mos-mun-smi-15-aa-proving,
author = {Mariano M. Moscato and C{\'{e}}sar A. Mu{\~{n}}oz and Andrew P. Smith}
title = {Affine Arithmetic and Applications to Real-Number Proving}
publisher = {Springer}
fullhash = {be8e0b67ab3e0036e9c567ffa42f6b4e}
booktitle = {Interactive Theorem Proving}
series = {LNCS}
volume = {9236}
year = {2015}
pages = {294-309}
numpages = {16}
doi = {10.1007/978-3-319-22102-1_20}
abstract = {Accuracy and correctness are central issues in numerical analysis. To address these issues, several self-validated computation methods have been proposed in the last fifty years. Their common goal is to provide rigorously correct enclosures for calculated values, sacrificing a measure of precision for correctness. Perhaps the most widely adopted enclosure method is interval arithmetic. Interval arithmetic performs well in a wide range of cases, but often produces excessively large overestimations, unless the domain is reduced in size, e.g., by subdivision. Many extensions of interval arithmetic have been developed in order to cope with this problem. Among them, affine arithmetic provides tighter estimations by taking into account linear correlations between operands. This paper presents a formalization of affine arithmetic, written in the Prototype Verification System (PVS), along with a formally verified branch-and-bound procedure implementing that model. This procedure and its correctness property enables the implementation of a PVS strategy for automatically computing upper and lower bounds of real-valued expressions that are provably correct up to a user-specified precision.}
}
@article{vac-pis-zob-15-aa-enerhub,
author = {Alfredo Vaccaro and Cosimo Pisani and Ahmed Faheem Zobaa}
title = {Affine Arithmetic-Based Methodology for Energy Hub Operation-Scheduling in the Presence of Data Uncertainty}
fullhash = {e30cd336389c167c8ce7f6b9b38c045c}
journal = {IET Generation, Transmission {\&} Distribution}
number = {13}
volume = {9}
year = {2015}
pages = {1544-1552}
numpages = {9}
doi = {10.1049/iet-gtd.2015.0197}
abstract = {In this study, the role of self-validated computing for solving the energy hub-scheduling problem in the presence of multiple and heterogeneous sources of data uncertainties is explored and a new solution paradigm based on affine arithmetic is conceptualised. The benefits deriving from the application of this methodology are analysed in details, and several numerical results are presented and discussed.}
}
@article{loi-ter-vac-wal-15-aa-smartgr,
author = {Vincenzo Loia and Vladimir Terzija and Alfredo Vaccaro and Peter Wall}
title = {An Affine-Arithmetic-Based Consensus Protocol for Smart-Grid Computing in the Presence of Data Uncertainties}
fullhash = {da77cfa6af1eee281a3c497e50f2f7f3}
journal = {IEEE Transactions on Industrial Electronics}
number = {5}
volume = {62}
year = {2015}
pages = {2973-2982}
numpages = {10}
doi = {10.1109/TIE.2014.2363046}
abstract = {The recent advances in cooperative and bio-inspired computing based on self-organizing sensor networks have been recognized as a very promising enabling technology for future smart grids (SGs). Although many papers outline the important role of these paradigms in decentralized SG monitoring and control, they do not explicitly consider the effect of the uncertainties that characterize the actual power system environment. The effects of these uncertainties could compromise the convergence of consensus-based algorithms in the form of unwanted changes in the convergence time, increasing the convergence errors and compromising the repeatability of the convergence. To address this problem, this paper proposes a new consensus protocol based on the use of affine arithmetic. The application of this reliable consensus protocol to decentralized SG computing is explained in detail. Furthermore, several numerical results are presented and discussed to demonstrate the effectiveness of the proposed methodology.}
}
@article{din-tri-man-sti-can-15-aa-unceletr,
author = {Tongyu Ding and Riccardo Trinchero and Paolo Manfredi and Igor S. Stievano and Flavio G. Canavero}
title = {How Affine Arithmetic Helps Beat Uncertainties in Electrical Systems}
fullhash = {e36e6e0ed8f4fd8af31306bde565ab7e}
journal = {IEEE Circuits and Systems Magazine}
number = {4}
volume = {15}
year = {2015}
pages = {70-79}
numpages = {10}
doi = {10.1109/MCAS.2015.2484198}
abstract = {The ever-increasing impact of uncertainties in electronic circuits and systems is requiring the development of robust design tools capable of taking this inherent variability into account. Due to the computational inefficiency of repeated design trials, there has been a growing demand for smart simulation tools that can inherently and effectively capture the results of parameter variations on the system responses. To improve product performance, improve yield and reduce design cost, it is particularly relevant for the designer to be able to estimate worst-case responses. Within this framework, the article addresses the worst-case simulation of lumped and distributed electrical circuits. The application of interval-based methods, like interval analysis, Taylor models and affine arithmetic, is discussed and compared. The article reviews in particular the application of the affine arithmetic to complex algebra and fundamental matrix operations for the numerical frequency-domain simulation. A comprehensive and unambiguous discussion appears in fact to be missing in the available literature. The affine arithmetic turns out to be accurate and more efficient than traditional solutions based on Monte Carlo analysis. A selection of relevant examples, ranging from linear lumped circuits to distributed transmission-line structures, is used to illustrate this technique.}
}
@article{rum-kas-15-aa-impl,
author = {Siegfried M. Rump and Masahide Kashiwagi}
title = {Implementation and Improvements of Affine Arithmetic}
fullhash = {de6d4113027d836b8bc18e4a45c706b7}
journal = {Nonlinear Theory and Its Applications, IEICE}
number = {3}
volume = {6}
year = {2015}
pages = {341-359}
numpages = {19}
doi = {10.1587/nolta.6.341}
abstract = {Affine arithmetic is a well-known tool to reduce the wrapping effect of ordinary interval arithmetic. We discuss several improvements both in theory and in terms of practical implementation. In particular details of INTLAB's affine arithmetic toolbox are presented. Computational examples demonstrate advantages and weaknesses of the approach.}
}
@article{wan-han-wul-15-aa-uncert,
author = {Shouxiang Wang and Liang Han and Lei Wu}
title = {Uncertainty Tracing of Distributed Generations via Complex Affine Arithmetic Based Unbalanced Three-Phase Power Flow}
fullhash = {f89084bd0a9864e11b8c6068170ac83a}
journal = {IEEE Transactions on Power Systems}
number = {6}
volume = {30}
year = {2015}
pages = {3053-3062}
numpages = {10}
doi = {10.1109/TPWRS.2014.2377042}
abstract = {Variations of load demands and generations bring multiple uncertainties to power system operation. Under this situation, power flows become increasingly uncertain, especially when significant distributed generations (DGs), such as wind and solar, are integrated into power systems. In this paper, a Complex Affine arithmetic based unbalanced Three-phase Forward-Backward Sweep power flow model (CATFBS) is proposed to study the impacts of uncertainties in unbalanced three-phase distribution systems. An index of Relative Influence of Uncertain Variables on Outcomes (RIUVO) is proposed for quantifying the impacts of individual uncertain factors on power flows and bus voltages. The CATFBS method is tested on the modified IEEE 13-bus system and a modified 292-bus system. Numerical results show that the proposed method outperforms the Monte Carlo method for exploring the impacts of uncertainties on the operation of distribution systems. The proposed CATFBS method can be used by power system operators and planners to effectively monitor and control unbalanced distribution systems under various uncertainties.}
}
@inproceedings{nin-rev-14-aa-fastnodir,
author = {Jordan Ninin and Nathalie Revol}
title = {Accurate and Effcient Implementation of Affine Arithmetic Using Floating-Point Arithmetic}
publisher = {University of W{\"{u}}rzburg}
fullhash = {1ce6aec94129689f572fefe038e40fe2}
booktitle = {Abstracts of 16th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2014)}
location = {W{\"{u}}rzburg, Germany}
month = {9}
year = {2014}
pages = {125-126}
numpages = {2}
abstract = {Affine arithmetic is one of the extensions of interval arithmetic that aim at counteracting the variable dependency problem. With affine arithmetic, defined in [5] by Stolfi and Figueiredo, variables are represented as ane combination of symbolic noises. It differs from the generalized interval arithmetic, defined by Hansen in [1], where variables are represented as affine combination of intervals. Non-affne operations are realized through the introduction of a new noise, that accounts for nonlinear terms. Variants of ane arithmetic have been proposed, they aim at limiting the number of noise symbols. Let us mention [4] by Messine and [6] by Vu, Sam-Haroud and Faltings to quote only a few. The focus here is on the implementation of affine arithmetic using floating-point arithmetic, specified in [2]. With floating-point arithmetic, an issue is to handle roundoff errors and to incorporate them in the final result, so as to satisfy the inclusion property, which is the fundamental property of interval arithmetic. In [4], [5] and [6], roundoff errors are accounted for in a manner that implies frequent switches of the rounding mode; this incurs a severe time penalty. Implementations of these variants are available in YalAA, developed by Kiel [2]. We propose an implementation that uses one dedicated noise symbol for accumulated roundoff errors. For accuracy purposes, the roundoff error $\epsilon$ of each arithmetic operation is computed exactly via EFT (Error Free Transforms). For efficiency purposes, the rounding mode is never switched. Instead, a brute-force bound on the roundoff error $\varepsilon$ incurred by the accumulation of the $\epsilon$s mentioned above is used. Experimental results are presented. The proposed implementation is one of the most accurate and its execution time is significantly reduced; it can be up to 50{\%} faster than other implementations. Furthermore, the use of a FMA (Fused Multiply-and-Add) reduces the cost of the EFT and the overall performance is even better}
}
@mastersthesis{gba-14-aa-fatig-thesis,
author = {Audrey J-M Gbaguidi}
title = {Fatigue Damage Prognosis of Internal Delamination in Composite Plates Under Cyclic Compression Loadings Using Affine Arithmetic as Uncertainty Propagation Tool}
institution = {Embry-Riddle Aeronautical University}
fullhash = {5c1a3415bef50ccbacc3e76a33cad53f}
month = {12}
year = {2014}
urlraw = {https://commons.erau.edu/edt/267}
url = {https://commons.erau.edu/edt/267}
abstract = {Structural health monitoring (SHM) has become indispensable for reducing maintenance costs and increasing the in-service capacity of a structure. The increased use of lightweight composite materials in aircraft structures drastically increased the effects of fatigue induced damage on their critical structural components and thus the necessity to predict the remaining life of those components. Damage prognosis, one of the least investigated fields in SHM, uses the current damage state of the system to forecast its future performance by estimating the expected loading environments. A successful damage prediction model requires the integration of technologies in areas like measurements, materials science, mechanics of materials, and probability theories, but most importantly the quantification of uncertainty in all these areas. In this study, Affine Arithmetic is used as a method for incorporating the uncertainties due to the material properties into the fatigue life prognosis of composite plants subjected to cyclic compressive loadings. When loadings are compressive in nature, the composite plates undergo repeated buckling-unloading of the delaminated layer which induces mixed modes I and II states of stress at the tip of the delamination in the plates. The Kardomateas model-based prediction law is used to predict the growth of the delamination, while the integration of the effects of the uncertainties for modes I and II coefficients in the fatigue life prediction model is handled using Affine Arithmetic. The Mode I and Mode II interlaminar fracture toughness and fatigue characterization of the composite plates are first experimentally studied to obtain the material coefficients and fracture toughness, respectively. Next, these obtained coefficients are used in the Kardomateas law to predict the delamination lengths in the composite plates while using Affine Arithmetic to handle their uncertainties. At last, the fatigue characterization of the composite plates during compressive-buckling loadings in experimentally studied, and the delamination lengths are compared with the predicted values to check the performance of Affine Arithmetic as an uncertainty propagation tool.}
}
@article{gba-kim-14-aa-fatig,
author = {Audrey Gbaguidi and Daewon Kim}
title = {Fatigue Damage Prognosis Using Affine Arithmetic}
fullhash = {15ddd7dd762dc1fae69610e6b14a494b}
journal = {AIP Conference Proceedings}
number = {1}
volume = {1581}
year = {2014}
pages = {719-726}
numpages = {8}
doi = {10.1063/1.4864891}
abstract = {Among the essential steps to be taken in structural health monitoring systems, damage prognosis would be the field that is least investigated due to the complexity of the uncertainties. This paper presents the possibility of using Affine Arithmetic for uncertainty propagation of crack damage in damage prognosis. The structures examined are thin rectangular plates made of titanium alloys with central mode I cracks and a composite plate with an internal delamination caused by mixed mode I and II fracture modes, under a harmonic uniaxial loading condition. The model-based method for crack growth rates are considered using the Paris Erdogan law model for the isotropic plates and the delamination growth law model proposed by Kardomateas for the composite plate. The parameters for both models are randomly taken and their uncertainties are considered as defined by an interval instead of a probability distribution. A Monte Carlo method is also applied to check whether Affine Arithmetic (AA) leads to tight bounds on the lifetime of the structure.}
}
@inproceedings{pir-can-bha-vac-14-aa-microgr,
author = {Mehrdad Pirnia and Claudio Ca{\~{n}}izares and Kankar Bhattacharya and Alfredo Vaccaro}
title = {An Affine Arithmetic Approach for Microgrid Dispatch with Variable Generation and Load}
fullhash = {5fd87eb032fb4a0ee0d987f6f6f44604}
booktitle = {Proc. 2014 Power Systems Computation Conference}
month = {8}
year = {2014}
pages = {1-8}
numpages = {8}
doi = {10.1109/PSCC.2014.7038382}
abstract = {A self-validated computing (SVC) method, based on affine arithmetic (AA) is proposed in this paper to solve the optimal power flow (OPF) problem for microgrids with renewable sources of variable generation. In the AA-based OPF formulation, all the state and control variables are presented in affine form, to represent the variable load, and wind and solar generation. Hence, the OPF model becomes an interval-based model with upper and lower bounds to represent the uncertain variables. To check the accuracy of the AA-based method, the resulted intervals are compared against those obtained from Monte-Carlo Simulation (MCS), in a 13-bus microgrid test system. The obtained real power generation intervals for thermal generators are used to determine the reserves required in dispatchable generators in the short-term to properly supply for the variability of load and intermittent renewable generation sources.}
}
@phdthesis{mun-14-aa-thesis,
author = {Juan Carlos Munoz Guerrero}
title = {Affine Arithmetic Based Methods for Power Systems Analysis Considering Intermittent Sources of Power}
institution = {University of Waterloo}
fullhash = {d476b26b4ce8aa79344a046c1d4f2aa8}
month = {1}
year = {2014}
urlraw = {http://hdl.handle.net/10012/8161}
url = {http://hdl.handle.net/10012/8161}
abstract = {Intermittent power sources such as wind and solar are increasingly penetrating electrical grids, mainly motivated by global warming concerns and government policies. These intermittent and non-dispatchable sources of power affect the operation and control of the power system because of the uncertainties associated with their output power. Depending on the penetration level of intermittent sources of power, the electric grid may experience considerable changes in power flows and synchronizing torques associated with system stability, because of the variability of the power injections, among several other factors. Thus, adequate and efficient techniques are required to properly analyze the system stability under such uncertainties. A variety of methods are available in the literature to perform power flow, transient, and voltage stability analyses considering uncertainties associated with electrical parameters.}
}
@article{lin-qin-lia-xio-14-aa-surfint,
author = {Hongwei Lin and Yang Qin and Hongwei Liao and Yunyang Xiong}
title = {Affine Arithmetic-Based {B}-Spline Surface Intersection with {GPU} Acceleration}
fullhash = {67ea3b96a5375de9955bbe67aef7821d}
journal = {IEEE Transactions on Visualization and Computer Graphics}
month = {2}
number = {2}
volume = {20}
year = {2014}
pages = {172-181}
numpages = {10}
doi = {10.1109/TVCG.2013.237}
abstract = {Because the B-spline surface intersection is a fundamental operation in geometric design software, it is important to make the surface intersection operation robust and efficient. As is well known, affine arithmetic is robust for calculating the surface intersection because it is able to not only find every branch of the intersection, but also deal with some singular cases, such as surface tangency. However, the classical affine arithmetic is defined only for the globally supported polynomials, and its computation is very time consuming, thus hampering its usefulness in practical applications, especially in geometric design. In this paper, we extend affine arithmetic to calculate the range of recursively and locally defined B-spline basis functions, and we accelerate the affine arithmetic-based surface intersection algorithm by using a GPU. Moreover, we develop efficient methods to thin the strip-shaped intersection regions produced by the affine arithmetic-based intersection algorithm, calculate the intersection points, and further improve their accuracy. The many examples presented in this paper demonstrate the robustness and efficiency of this method.}
}
@article{guw-luo-din-men-she-14-aa-powflow,
author = {Wei Gu and Lizi Luo and Tao Ding and Xiaoli Meng and Wanxing Sheng}
title = {An Affine Arithmetic-Based Algorithm for Radial Distribution System Power Flow with Uncertainties}
fullhash = {4018bc72fd0aac2e61f0b0c2680ab8eb}
journal = {International Journal of Electrical Power {\&} Energy Systems}
volume = {58}
year = {2014}
pages = {242-245}
numpages = {4}
doi = {10.1016/j.ijepes.2014.01.025}
abstract = {This letter presents an algorithm for radial distribution system power flow in the presence of uncertainties. To reduce the overestimation of bounds yielded by correlation of variables in interval arithmetic (IA), affine arithmetic (AA) is applied in this study to carry out tests of distribution system power flow. Compared with the algorithm based on IA, the proposed algorithm narrows the gap between the upper and lower bounds of the power flow solution. IEEE 33-bus and 69-bus test systems are used to demonstrate the effectiveness of the proposed algorithm.}
}
@article{pir-can-bha-vac-14-aa-powflow,
author = {Mehrdad Pirnia and Claudio A. Ca{\~{n}}izares and Kankar Bhattacharya and Alfredo Vaccaro}
title = {A Novel Affine Arithmetic Method to Solve Optimal Power Flow Problems With Uncertainties}
fullhash = {7d480b49249ac8b9e54e5a72b71185ba}
journal = {IEEE Transactions on Power Systems}
number = {6}
volume = {29}
year = {2014}
pages = {2775-2783}
numpages = {9}
doi = {10.1109/TPWRS.2014.2316114}
abstract = {An affine arithmetic (AA) method is proposed in this paper to solve the optimal power flow (OPF) problem with uncertain generation sources. In the AA-based OPF problem, all the state and control variables are treated in affine form, comprising a center value and the corresponding noise magnitudes, to represent forecast, model error, and other sources of uncertainty without the need to assume a probability density function (pdf). The proposed AA-based OPF problem is used to determine the operating margins of the thermal generators in systems with uncertain wind and solar generation dispatch. The AA-based approach is benchmarked against Monte Carlo simulation (MCS) intervals in order to determine its effectiveness. The proposed technique is tested and demonstrated on the IEEE 30-bus system and also a real 1211-bus European system.}
}
@inproceedings{uew-sur-13-aa-theorems,
author = {Prapeepat Uewichitrapochana and Athasit Surarerks}
title = {Signed-symmetric function approximation in affine arithmetic}
fullhash = {2bfd20492962cca45922a625f098a097}
booktitle = {10th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology}
month = {5}
year = {2013}
pages = {1-6}
numpages = {6}
doi = {10.1109/ECTICon.2013.6559630}
abstract = {One major cause of the error explosion is the overestimation of a non-affine function introducing a new noise symbol term with non-minimum coefficient. This paper proposes theorems and its proofs to construct the best univariate affine approximation to a non-affine function in the exception case, Signed-symmetric function, that the existing theorem is not sufficient to determine the optimum one. And, as the result, it shows the use by evaluating the power function and approximating sine function.}
}
@article{vac-can-bha-13-aa-optpowflow,
author = {Alfredo Vaccaro and Claudio A. Ca\~{n}izares and Kankar Bhattacharya}
title = {A Range Arithmetic-Based Optimization Model for Power Flow Analysis Under Interval Uncertainty}
fullhash = {c81eaa923b5743c2fcf994f651e7f7ab}
journal = {IEEE Transactions on Power Systems}
month = {5}
number = {2}
volume = {28}
year = {2013}
pages = {1179-1186}
numpages = {8}
doi = {10.1109/TPWRS.2012.2214405}
abstract = {This paper presents a novel framework based on range arithmetic for solving power flow problems whose input data are specified within real compact intervals. Reliable interval bounds are computed for the power flow problem, which is represented as an optimization model with complementary constraints to properly represent generator bus voltage controls, including reactive power limits and voltage recovery processes. It is demonstrated that the lower and upper bounds of the power flow solutions can be obtained by solving two determinate optimization problems. Several numerical results are presented and discussed, demonstrating the effectiveness of the proposed methodology and comparing it to a previously proposed affine arithmetic based solution approach.}
}
@article{elo-ela-13-aa-complex,
author = {Hassan B. M. El-Owny and A. E. A. Elaraby}
title = {New Extension of Affine Arithmetic: Complex Form}
fullhash = {ee2092ff50b9f836227107fadd0faac6}
journal = {Journal of Communications and Computer Engineering}
note = {The DOI seems to be fake}
number = {1}
volume = {3}
year = {2013}
pages = {11-14}
numpages = {4}
doi = {10.20454/jcce.2013.457}
abstract = {In this paper we present the new complex form for affine arithmetic (AA) which is a self-verifying computational approach that keeps track of first-order correlation between uncertainties in the data and intermediate and final results. In this paper we propose a higher-order extension satisfying the requirements of genericity, arbitrary-order and self-verification, comparing the resulting method with other well-known high-order extensions of AA.}
}
@article{roc-13-aa-markov,
author = {Claudio M Rocco S}
title = {Affine Arithmetic for Assessing the Uncertainty Propagation on Steady-State Probabilities of {Markov} Models Owing to Uncertainties in Transition Rates}
fullhash = {39271fa277b731eb40bcec0f4492645f}
journal = {Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability}
number = {5}
volume = {227}
year = {2013}
pages = {523-533}
numpages = {11}
doi = {10.1177/1748006X13485189}
abstract = {This article proposes the use of affine arithmetic as an alternative approach for assessing the effects of uncertainties of transition rates on the steady-state probabilities for each possible state of a system represented by a Markov model. Affine arithmetic is an extension of interval arithmetic, able to track ``the dependency between variables throughout calculations'' and to provide strict bounds. Several examples illustrate the proposed approach. Results are compared with other approaches, such as interval arithmetic, Monte Carlo simulation or solving linear systems of simultaneous equations.}
}
@article{din-bor-guo-sun-wuw-zha-13-aa-powflow,
author = {Tao Ding and Rui Bo and Qinglai Guo and Hongbin Sun and Wenchuan Wu and Boming Zhang}
title = {A Non-Iterative Affine Arithmetic Methodology for Interval Power Flow Analysis of Transmission Network}
fullhash = {aa1029f6ee1346e7994d607b52239277}
journal = {Zhongguo Dianji Gongcheng Xuebao / Proceedings of the Chinese Society of Electrical Engineering}
month = {7}
note = {In Chinese}
number = {19}
volume = {33}
year = {2013}
pages = {76-83}
numpages = {8}
abstract = {Power flow exhibits uncertainty when power injections randomly fluctuate. The variables of power flow problem are correlated through the constraints set forth by power flow equations. Affine arithmetic can effectively overcome the correlation and therefore is adopted in this paper. In order to improve the computational efficiency, non-iterative method was proposed, which can transform the interval power flow problem into optimization problems. Linear programming, nonlinear programming and quadratic programming optimization models were established to obtain intervals of nodal voltage magnitude, phase angle, branch active power and reactive power, respectively. Simulation results of 9-bus and 57-bus test system show that the power flow interval obtained by the proposed algorithm is nearly the same as that by iterative algorithms, whereas the branch power flow results are better. The interval power flow problem can be solved quickly and efficiently with time complexity approximately $O(m^3)$. As well, parallel computing can be implemented to achieve promising improvement in computation time.}
}
@article{wan-han-zha-13-aa-powflow,
author = {Shouxiang Wang and Liang Han and Peng Zhang}
title = {Affine Arithmetic-based DC Power Flow for Automatic Contingency Selection with Consideration of Load and Generation Uncertainties}
fullhash = {a26356fe340f92a82a24a8c897a3a1b7}
journal = {Electric Power Components and Systems}
publisher = {Taylor {\&} Francis}
number = {8}
volume = {42}
year = {2014}
pages = {852-860}
numpages = {9}
doi = {10.1080/15325008.2014.896433}
abstract = {Power systems operate under uncertainties from variations of loads and generation with time. Power flow in a power system becomes increasingly uncertain, especially when renewable energy, such as wind and photovoltaic generation, is integrated. In this article, a DC power flow approach based on affine arithmetic is proposed to quantify risks due to uncertainties. The proposed method has been applied to the automatic contingency selection under uncertainties. Numerical studies on IEEE test systems (30, 57, and 118 buses) have proved that the DC power flow approach based on affine arithmetic is a fast and reliable method for power system planning and operation when considering uncertainty. Test cases have shown that the proposed method is as fast as the conventional DC power flow while it is much more efficient than the Monte Carlo method. Thus, the DC power flow approach based on affine arithmetic serves as a new and effective decision support tool for planners and operators to cope with high levels of renewable energy penetration, electric vehicle load integration, and other uncertain scenarios.}
}
@article{rad-gri-sch-rat-13-aa-mixsig,
author = {Carna Radojicic and Christoph Grimm and Florian Schupfer and Michael Rathmair}
title = {Verification of Mixed-Signal Systems with Affine Arithmetic Assertions}
fullhash = {edbaf52fcd85c9882dbce8bb85ab475e}
journal = {VLSI Design}
number = {3}
volume = {2013}
year = {2013}
pages = {article 239064}
numpages = {-1}
doi = {10.1155/2013/239064}
abstract = {Embedded systems include an increasing share of analog/mixed-signal components that are tightly interwoven with functionality of digital HW/SW systems. A challenge for verification is that even small deviations in analog components can lead to significant changes in system properties. In this paper we propose the combination of range-based, semisymbolic simulation with assertion checking. We show that this approach combines advantages, but as well some limitations, of multirun simulations with formal techniques. The efficiency of the proposed method is demonstrated by several examples.}
}
@article{soa-13-aa-allsols,
author = {R. de P. Soares}
title = {Finding all real solutions of nonlinear systems of equations with discontinuities by a modified affine arithmetic}
fullhash = {af9097c2d1803bc436b3ebde6f23c09b}
journal = {Computers {\&} Chemical Engineering}
volume = {48}
year = {2013}
pages = {48-57}
numpages = {10}
doi = {h10.1016/j.compchemeng.2012.08.002}
abstract = {Chemical engineering is a rich area when comes to nonlinear systems of equations, possibly with multiple solutions, (unbounded) discontinuities, or functions which become undefined in terms of real values. In this work, a new approach is proposed for finding all real solutions of such systems within prescribed bounds. A modified affine arithmetic is used in an interval Newton method plus generalized bisection. A special constraint propagation is used to automatically remove regions where the functions are undefined for real numbers. Results for test problems have shown that the proposed implementation requires less computation effort than similar methods available in the literature for small continuous systems. Further, the method is able to find all real solutions of nonlinear systems of equations even when there are unbounded discontinuities or when functions become undefined within the given variable bounds.}
}
@article{sta-car-soa-13-aa-phaseq,
author = {P.B. Staudt and N.S.M. Cardozo and R. de P. Soares}
title = {Phase Stability Analysis using a Modified Affine Arithmetic}
fullhash = {4a0238b824f29b1d7b840df652735962}
journal = {Computers {\&} Chemical Engineering}
volume = {53}
year = {2013}
pages = {190-200}
numpages = {11}
doi = {10.1016/j.compchemeng.2013.03.011}
abstract = {Phase stability analysis is a crucial step in the determination of multiphase equilibrium. This analysis by the tangent plane distance (TPD) minimization is a well-known technique, as well as the difficulties in providing guarantees that the global minimum has been found. On this regard, interval methods are powerful tools since they provide such guarantees. In this work, an interval Newton method plus generalized bisection, based on a modified affine arithmetic, is used to reliably find all possible stationary points of the TPD function. Additionally, an improved convergence test is suggested as well as a special treatment for mole fraction weighted averages. Several mixtures with up to 5 components, including LLE island type ternary systems, were studied. Both activity coefficient models and cubic equations of state were considered. For all the cases tested, the proposed modified affine arithmetic method was superior to other interval-based methods.}
}
@inproceedings{vak-lan-boi-13-aa-finerr,
author = {Shervin Vakili and J. M. Pierre Langlois and Guy Bois}
title = {Finite-Precision Error Modeling using Affine Arithmetic}
fullhash = {9afb8b2c8b1cbf9e2f98882b7665df2b}
booktitle = {2013 IEEE International Conference on Acoustics, Speech and Signal Processing}
month = {5}
year = {2013}
pages = {2591-2595}
numpages = {5}
doi = {10.1109/ICASSP.2013.6638124}
abstract = {This paper introduces a new approach for finite-precision error modeling based on affine arithmetic. The paper demonstrates that there is a common hazard in affine arithmetic-based error modeling methods described in the literature. The hazard is linked to early substitution of the signal terms that emerge in operations such as multiplication and division. The paper proposes postponed substitution combined with function maximization to address this problem. The paper also proposes a modification in the error propagation process to enhance the error modeling accuracy. An existing word length optimization method is reproduced to evaluate the efficiency of this modification. The results demonstrate that the proposed modification can improve the hardware area results by up to 7.0{\%} at the expense of negligible complexity overhead.}
}
@article{vak-lan-boi-13-aa-bitwd,
author = {Shervin Vakili and J. M. Pierre Langlois and Guy Bois}
title = {Enhanced Precision Analysis for Accuracy-Aware Bit-Width Optimization Using Affine Arithmetic}
fullhash = {9afb8b2c8b1cbf9e2f98882b7665df2b}
journal = {IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems}
number = {12}
volume = {32}
year = {2013}
pages = {1853-1865}
numpages = {13}
doi = {10.1109/TCAD.2013.2277978}
abstract = {Bit-width allocation has a crucial impact on hardware efficiency and accuracy of fixed-point arithmetic circuits. This paper introduces a new accuracy-guaranteed word-length optimization approach for feed-forward fixed-point designs. This method uses affine arithmetic, which is a well-known analytical technique, for both range and precision analyses. This paper introduces an acceleration technique and two new semianalytical algorithms for precision analysis. While the first algorithm follows a progressive search strategy, the second one uses a tree-shaped search method for fractional width optimization. The algorithms offer two different time-complexity/cost efficiency tradeoffs. The first algorithm has polynomial complexity and achieves comparable results with existing heuristic approaches. The second algorithm has exponential complexity, but it achieves near-optimal results compared to the exhaustive search method. A commonly used set of case studies is used to evaluate the efficiency of the proposed techniques and algorithms in terms of optimization time and hardware cost. The first and second algorithms achieve 10.9{\%} and 13.1{\%} improvements in area, respectively, over uniform fractional width allocation. The proposed acceleration technique reduces the complexity of the fractional width selection problem by an average of 20.3{\%}.}
}
@inproceedings{mun-can-bja-vac-13-aa-transtab,
author = {Juan Mu{\~{n}}oz and Claudio Ca{\~{n}}izares and Kankar Bhattacharya and Alfredo Vaccaro}
title = {Affine Arithmetic Based Methods for Voltage and Transient Stability Assessment of Power Systems with Intermittent Generation Sources}
fullhash = {07e0478a7e770b388332254342b05801}
booktitle = {2013 IREP Symposium Bulk Power System Dynamics and Control - IX Optimization, Security and Control of the Emerging Power Grid}
month = {8}
year = {2013}
pages = {1-12}
numpages = {12}
doi = {10.1109/IREP.2013.6629363}
abstract = {This paper presents novel methods based on Affine Arithmetic (AA) for voltage and transient stability assessment of power systems, considering uncertainties of power injections from intermittent generation sources such as wind and solar. These methods, which are an alternative to the well-known sampling and probabilistic based approaches, are able to improve the computational efficiency as compared to simple Monte Carlo (MC) simulations, with reasonably accuracy. The presented AA-based voltage stability method computes the hull of PV curves associated with the assumed uncertainties, and results are discussed in detail for a 5-bus test system. On the other hand, the proposed AA-based transient stability method solves the set of differential-algebraic equations in affine form, based on a trapezoidal integration approach, that yields the hull of the dynamic response of the system, for a set of assumed contingencies and uncertainties; this approach is demonstrated on a generator-infinite bus test system. In both presented stability studies, MC simulations are used for comparison and validation purposes.}
}
@article{mun-can-bha-vac-13-aa-voltstab,
author = {Juan Mu{\~{n}}oz and Claudio Ca{\~{n}}izares and Kankar Bhattacharya and Alfredo Vaccaro}
title = {An Affine Arithmetic-Based Method for Voltage Stability Assessment of Power Systems With Intermittent Generation Sources}
fullhash = {07e0478a7e770b388332254342b05801}
journal = {IEEE Transactions on Power Systems}
number = {4}
volume = {28}
year = {2013}
pages = {4475-4487}
numpages = {13}
doi = {10.1109/TPWRS.2013.2276424}
abstract = {This paper presents a novel method based on affine arithmetic (AA) for voltage stability assessment of power systems considering uncertainties associated with operating conditions, which may be attributed to intermittent generation sources, such as wind and solar. The proposed AA-based method reduces the computational burden as compared to Monte Carlo (MC) simulations, and also improves the accuracy as compared to some analytical approaches. The proposed method is tested using two study cases: first, a 5-bus test system is used to illustrate the proposed technique in detail, and thereafter a 2383-bus test system to demonstrate its practical application. The results are compared with those obtained using MC simulations to verify the accuracy and computational burden of the proposed AA-based method, and also with respect to a previously proposed technique to estimate parameter sensitivities in voltage stability assessment.}
}
@article{ska-12-aa-bestmul,
author = {Iwona Skalna}
title = {Algorithm for min-range multiplication of affine forms}
fullhash = {79a5c772124ee18ed172632c5bcc2850}
journal = {Numerical Algorithms}
volume = {63}
year = {2013}
pages = {601-614}
numpages = {14}
doi = {10.1007/s11075-012-9644-0}
abstract = {Affine arithmetic produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as round-off errors. Elementary operations on affine forms are redefined so they result in affine forms. Affine-linear operations result straightforwardly in affine forms. Non-linear operators, such as multiplication, must be approximated by affine forms. Choosing the appropriate approximation is a big challenge. The reason is that different approximations may be more accurate for specific purposes. This paper presents an efficient method for computing the minimum range (min-range) affine approximation of the product of arbitrary affine forms that do not contain zero properly. Numerical experiments are carried out to demonstrate the essential features of the proposed approach, especially its usefulness for bounding ranges of functions for global optimisation and for finding roots of functions.}
}
@inproceedings{ska-12-aa-linsys,
author = {Iwona Skalna}
title = {Enclosure for the Solution Set of Parametric Linear Systems with Non-affine Dependencies}
publisher = {Springer}
fullhash = {79a5c772124ee18ed172632c5bcc2850}
booktitle = {Proc. International Conference on Parallel Processing and Applied Mathematics (PPAM 2011)}
isbn = {978-3-642-31500-8}
year = {2012}
pages = {513-522}
numpages = {10}
abstract = {The problem of solving linear systems whose coefficients are nonlinear functions of parameters varying within prescribed intervals is investigated. A new method for outer interval solution of such system is proposed. In order to reduce memory usage, nonlinear dependencies between parameters are handled using revised affine arithmetic. Some numerical experiments which aim to show the properties of the proposed method are reported.}
}
@article{gou-gal-put-12-aa-zonjoin,
author = {Eric Goubault and Tristan {Le Gall} and Sylvie Putot}
title = {An Accurate Join for Zonotopes, Preserving Affine Input/Output Relations}
fullhash = {da80323aaef8cbbbd94d1fa57e3628f7}
journal = {Electronic Notes in Theoretical Computer Science}
note = {Proceedings of the Fourth International Workshop on Numerical and Symbolic Abstract Domains, NSAD 2012}
volume = {287}
year = {2012}
pages = {65-76}
numpages = {12}
doi = {10.1016/j.entcs.2012.09.007}
urlraw = {https://www.sciencedirect.com/science/article/pii/S157106611200059X}
url = {https://www.sciencedirect.com/science/article/pii/S157106611200059X}
abstract = {Zonotopes are a convenient abstract domain for the precise analysis of programs with numerical variables. Compared to the domain of convex polyhedra, it is less expensive and may easily handle non-linear assignments. However, the classical join operator of this abstract domain does not always preserve linear invariants, unlike the convex hull. We present a global join operator that preserves some affine relations. We end up by showing some experiments conducted on the constrained Taylor1+ domain of Apron.}
}
@inproceedings{iha-kan-12-aa-allsol,
author = {Kosuke Ihara and Yuchi Kanzawa}
title = {All Solution Algorithm for Parameter-Dependent Nonlinear Equations Using Affine Arithmetic}
fullhash = {f70ad700217f4ddfa13133f6acb7731c}
booktitle = {Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications}
year = {2012}
pages = {419-422}
numpages = {4}
doi = {10.15248/proc.1.419}
urlraw = {https://www.ieice.org/publications/proceedings/bin/pdf_link.php?fname=419.pdf&iconf=NOLTA&year=2012&vol=1&number=419&lang=E}
url = {https://www.ieice.org/publications/proceedings/bin/pdf_link.php?fname=419.pdf&iconf=NOLTA&year=2012&vol=1&number=419&lang=E}
abstract = {A new all solution algorithm is proposed for parameter-dependent nonlinear equations. In this algorithm, affine arithmetic[1], which is more accurate than interval arithmetic[2], is used for an existence test and two non-existence tests of a solution. The efficiency of the proposed algorithm is verified by some numerical examples.}
}
@inproceedings{mia-12-aa-varilog,
author = {R{\v{a}}zvan-C{\v{a}}t{\v{a}}lin Mial{\c{t}}u}
title = {A {SystemVerilog} Approach in System Validation with Affine Arithmetic}
fullhash = {0177d4e8f0605b325023d7216d19362d}
booktitle = {International Semiconductor Conference (CAS 2012)}
month = {10}
volume = {2}
year = {2012}
pages = {407-410}
numpages = {4}
doi = {10.1109/SMICND.2012.6400746}
abstract = {This paper introduces an original approach to system modeling for performance analysis and optimization. The method presented herein theoretical background is the mathematical field of affine arithmetic chosen for its intrinsic data representation optimal to analysis of the mitigation of variations and refinement of deviations and error analysis. The chosen language of SystemVerilog is beneficial for it is allowing the integration of the validation process and of the verification process for the specific class of mixed signal electrical circuits and systems.}
}
@inproceedings{kra-olb-bar-12-aa-neural,
author = {Anna Krause and Markus Olbrich and Erich Barke}
title = {Enclosing the Modeling Error in Analog Behavioral Models Using Neural Networks and Affine Arithmetic}
fullhash = {226bd7e70657c0a2075b41e9e45a3b73}
booktitle = {2012 International Conference on Synthesis, Modeling, Analysis and Simulation Methods and Applications to Circuit Design (SMACD)}
month = {9}
year = {2012}
pages = {5-8}
numpages = {4}
doi = {10.1109/SMACD.2012.6339403}
abstract = {One all-time challenge in behavioral modeling is to minimize the modeling error while still profiting from a simplified representation of an analog circuit. In many cases the modeling error is known, but up to now it was only an indicator for the quality of the model. Its influence on errors during simulation could not be evaluated. We present a flow for the generation of behavioral models based on neural networks which uses affine arithmetic to guarantee enclosing the modeling error. We also demonstrate that the approach can also be applied to modeling the effects of parameter deviations.}
}
@inproceedings{rad-sch-rat-gri-12-aa-assert,
author = {Carna Radojicic and Florian Schupfer and Michael Rathmair and Christoph Grimm}
title = {Assertion-Based Verification of Signal Processing Systems with Affine Arithmetic}
fullhash = {ee01b4c154992cbf535e683e487f07e9}
booktitle = {Proceeding of the 2012 Forum on Specification and Design Languages}
location = {Vienna, Austria}
month = {9}
year = {2012}
pages = {20-26}
numpages = {7}
abstract = {This paper describes a novel method to verify typical properties of systems with parameter deviations. These parameter deviations have an impact on a system behavior causing it to deviate from its ideal behavior. The concept of Affine Arithmetic is used to model typical errors and uncertainties in today's analog-mixed signal systems. The method which is proposed is a combination of assertions and Affine Arithmetic where Affine Arithmetic is also used to model typical properties of signal processing specifications. The target of the proposed method is to speed up the detection of errors in system behavior, which cause the violation of system properties. The applicability of the proposed method is demonstrated through two small examples.}
}
@inproceedings{rat-sch-rad-gri-12-aa-simul,
author = {Michael Rathmair and Florian Schupfer and Carna Radojicic and Christoph Grimm}
title = {Extended Framework for System Simulation with Affine Arithmetic}
fullhash = {10fc768c1c483a32c13a3c2a35e917a7}
booktitle = {Proceeding of the 2012 Forum on Specification and Design Languages}
month = {9}
year = {2012}
pages = {168-175}
numpages = {8}
abstract = {This paper describes a calculation framework for the simulation of systems being affected by parameter deviations. These unpredictable factors are described with the help of affine arithmetic noise symbols in the modeling domain. Methods for a semi-symbolic output signal calculation containing linear and nonlinear operators must be provided. For a nonlinear composition of two affine arithmetic forms an appropriate approximation algorithm is required. This can be implemented and selected dependent on the objective of the simulation. Additional functionalities for debugging and tracing of noise symbols can improve analysis properties of simulation results. In the following work an extended framework structure for affine arithmetic system simulations containing an application specific selection of necessary approximated forms and advanced noise symbol management methods is introduced.}
}
@misc{kas-12-aa-reduce,
author = {Masahide Kashiwagi}
title = {An Algorithm to Reduce the Number of Dummy Variables in Affine Arithmetic}
fullhash = {043579a540ab675a207a5254db9be693}
howpublished = {Slides presented on 2012-09-28 at SCAN'2012, Nobosibirsk, Russia. Available online}
month = {9}
year = {2012}
urlraw = {http://conf.nsc.ru/files/conferences/scan2012/139995/Kashiwagi-scan2012.pdf}
url = {http://conf.nsc.ru/files/conferences/scan2012/139995/Kashiwagi-scan2012.pdf}
}
@inproceedings{pir-can-bha-vac-12-aa-powflow,
author = {Mehrdad Pirnia and Claudio A. Ca{\~{n}}izares and Kankar Bhattacharya and Alfredo Vaccaro}
title = {An Affine Arithmetic Method to Solve the Stochastic Power Flow Problem Based on a Mixed Complementarity Formulation}
fullhash = {7d480b49249ac8b9e54e5a72b71185ba}
booktitle = {2012 IEEE Power and Energy Society General Meeting}
year = {2012}
pages = {1-7}
numpages = {7}
doi = {10.1109/PESGM.2012.6345100}
abstract = {An affine-based stochastic power flow problem is proposed in this paper. First, a novel optimization-based model of the power flow problem using complementarity conditions to properly represent generator bus voltage controls, including reactive power limits and voltage recovery is presented. This model is then used to solve the stochastic power flow problem to obtain operational intervals for power flow variables based on an Affine Arithmetic (AA) method to consider active and reactive power demand uncertainties. The proposed AA algorithm is tested on a 14-bus test system and the results are then compared with the Monte-Carlo Simulation (MCS) results. The AA method shows slightly more conservative bounds; however, it is faster and does not need any information regarding statistical distributions of random variables.}
}
@article{kie-12-aa-yalaa,
author = {Stefan Kiel}
title = {YalAA: Yet Another Library for Affine Arithmetic}
fullhash = {6a0e0f87d055c9a205aba69fbd40b3a7}
journal = {Reliable Computing}
volume = {16}
year = {2012}
pages = {114-129}
numpages = {16}
urlraw = {http://interval.louisiana.edu/reliable-computing-journal/volume-16/reliable-computing-16-pp-114-129.pdf}
url = {http://interval.louisiana.edu/reliable-computing-journal/volume-16/reliable-computing-16-pp-114-129.pdf}
abstract = {In this paper, we present YalAA, a new library for affine arithmetic. Recently, affine arithmetic has been given increased attention even from outside the traditional verified computing community, for example, in the areas of circuit design, GPU based rendering of implicit objects and global optimization. Furthermore, several improvements to the original affine model were proposed. However, a fully verified, object-oriented implementation supporting at least some of the extensions is currently not available. The goal of YalAA is to provide a wide range of elementary functions and to allow the user to incorporate improvements for the original affine model easily. In contrast to other available libraries, YalAA also provides verified implementations of non-convex or non-concave elementary functions. Our library has a policy based design. That is, the user can alter different aspects in the library's behavior to reflect variations of the original model while relying on the same code base. Because affine arithmetic is often used in combination with interval arithmetic, we followed the principles of the upcoming P1788 interval standard during the design process. Therefore, YalAA can be integrated seamlessly into an existing interval arithmetic environment.}
}
@article{bou-gou-gou-put-12-aa-pboxes,
author = {O. Bouissou and E. Goubault and J. Goubault-Larrecq and Sylvie Putot}
title = {A Generalization of {p-Boxes} to Affine Arithmetic}
fullhash = {7baa2bda32227fdbedbde6c2025f88a3}
journal = {Computing}
volume = {94}
year = {2012}
pages = {189-201}
numpages = {13}
doi = {10.1007/s00607-011-0182-8}
abstract = {We often need to deal with information that contains both interval and probabilistic uncertainties. P-boxes and Dempster--Shafer structures are models that unify both kind of information, but they suffer from the main defect of intervals, the wrapping effect. We present here a new arithmetic that mixes, in a guaranteed manner, interval uncertainty with probabilities, while using some information about variable dependencies, hence limiting the loss from not accounting for correlations. This increases the precision of the result and decreases the computation time compared to standard p-box arithmetic.}
}
@article{lic-che-yix-tao-11-aa-reliab,
author = {Chun-yang Li and Xun Chen and Xiao-shan Yi and Jun-yong Tao}
title = {Interval-Valued Reliability Analysis of Multi-State Systems}
fullhash = {29aee0c8d9b9aa123c40bf0a0171ef32}
journal = {IEEE Transactions on Reliability}
month = {3}
number = {1}
volume = {60}
year = {2011}
pages = {323-330}
numpages = {8}
doi = {10.1109/TR.2010.2103670}
abstract = {In previous studies that analyzed the reliability of multi-state systems, the precise values of the state performance levels and state probabilities of multi-state components were required. In many cases, however, there are insufficient data to obtain the state probabilities of components precisely. A method is proposed in this paper to analyze the reliability of multi-state systems when the available data of components are insufficient. Based on the Bayesian approach and the imprecise Dirichlet model, the interval-valued state probabilities of components are obtained instead of precise values. The interval universal generating function is developed, and the corresponding operators are defined to estimate the interval-valued reliability of multi-state systems. Affine arithmetic is used to improve the interval-valued reliability. A numerical example illustrates the proposed method. The results show that the proposed method is efficient when state performance levels and/or state probabilities of components are uncertain and/or imprecise.}
}
@inproceedings{hua-lac-rob-11-aa-imprec,
author = {Jiawei Huang and John Lach and Gabriel Robins}
title = {Analytic Error Modeling for Imprecise Arithmetic Circuits}
fullhash = {8760ebed35d510bd1450bf91ea6d2661}
booktitle = {Proc. 7th IEEE Workshop on Silicon Errors in Logic - System Effects (SELSE 2011)}
location = {Urbana-Champaigh, IL, USA}
month = {3}
note = {Proceedings not available.}
year = {2011}
urlraw = {https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.298.381&rep=rep1&type=pdf}
url = {https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.298.381&rep=rep1&type=pdf}
abstract = {Imprecise hardware challenges the traditional notion that correctness is an immutable priority, by systematically trading off efficacy (precision) against efficiency (power, area, performance, and cost). Evaluating the impact of such tradeoffs on output quality using, e.g., Monte Carlo simulations is a time-consuming and non-deterministic process. This paper presents two analytic modeling techniques for evaluating error properties and output quality in imprecise arithmetic circuits, based on Interval Arithmetic and Affine Arithmetic. Experiments show that these techniques offer significant speedups over previous methods, as well as promising estimation accuracy}
}
@inproceedings{huy-muc-11-aa-dotpr,
author = {Thang Viet Huynh and Manfred M\"{u}cke}
title = {Error Analysis and Precision Estimation for Floating-Point Dot-Products Using Affine Arithmetic}
fullhash = {59aeef75f98845283cdb9429fe8456fc}
booktitle = {The 2011 International Conference on Advanced Technologies for Communications (ATC 2011)}
month = {8}
year = {2011}
pages = {319-322}
numpages = {4}
doi = {10.1109/ATC.2011.6027495}
abstract = {One challenging task for VLSI and reconfigurable system design is the identification of the smallest number format possible to implement a given numerical algorithm guaranteeing some final accuracy while minimising area used, execution time and power. We apply affine arithmetic, an extension to interval arithmetic, to estimate the rounding error of different floating-point dot-product variants. The validity of the estimated error bounds is demonstrated using extensive simulations. We derive the analytical models for rounding errors over a wide range of parameters and show that affine arithmetic with a probabilistic bounding operator is able to provide a tighter bound compared to conventional forward error analysis. Due to the tight bounds, minimum mantissa bit width for hardware implementation can be determined and comparison of different dot-product variants is possible. Our presented models allow for an efficient design space exploration and are key to specialised code generators.}
}
@inbook{pen-loo-vas-11-aa-symbreg,
author = {Cassio Pennachin and Moshe Looks and J. A. de Vasconcelos}
title = {Improved Time Series Prediction and Symbolic Regression with Affine Arithmetic}
publisher = {Springer}
fullhash = {5bae54cd58e3a3f44a4e718b351fe367}
booktitle = {Genetic Programming Theory and Practice IX}
isbn = {978-1-4614-1770-5}
year = {2011}
pages = {97-112}
numpages = {16}
doi = {10.1007/978-1-4614-1770-5_6}
abstract = {We show how affine arithmetic can be used to improve both the performance and the robustness of genetic programming for problems such as symbolic regression and time series prediction. Affine arithmetic is used to estimate conservative bounds on the output range of expressions during evolution, which allows us to discard trees with potentially infinite bounds, as well as those whose output range lies outside the desired range implied by the training dataset. Benchmark experiments are performed on 15 symbolic regression problems as well as 2 wellknown time series problems. Comparison with a baseline genetic programming system shows a reduced number of fitness evaluations during training and improved generalization on test data, completely eliminating extreme errors. We also apply this technique to the problem of forecasting wind speed on a real world dataset, and the use of affine arithmetic compares favorably with baseline genetic programming, feedforward neural networks and support vector machines.}
}
@inproceedings{nin-mes-10-aa-globopt,
author = {Jordan Ninin and Fr\'{e}d\'{e}ric Messine}
title = {A Mixed Affine Reformulation Method for Global Optimization}
fullhash = {e0f0c216180fbf5aa77575639a1903e0}
booktitle = {Proc. Tolouse Global Optimzation Workshop (TOGO 2010)}
year = {2010}
pages = {101-104}
numpages = {4}
urlraw = {http://www.lix.polytechnique.fr/togo10/TOGO10-proceedings.pdf}
url = {http://www.lix.polytechnique.fr/togo10/TOGO10-proceedings.pdf}
abstract = {An automatic method for constructing mixed integer linear relaxations of mixed integer non-convex optimization problems is proposed. This method is an extension of the affine reformulation technique [4] in order to consider mixed integer problems. The so-generated mixed integer linear program has exactly the same number of variables and of inequality constraints as the given problems. Thus, the resolution wastes less time. This technique is including in an interval branch and bound algorithm to improve the computation of lower bounds.}
}
@inproceedings{ska-10-aa-paramint,
author = {Iwona Skalna}
title = {Direct Method for Solving Parametric Interval Linear Systems with Non-affine Dependencies}
publisher = {Springer}
fullhash = {79a5c772124ee18ed172632c5bcc2850}
booktitle = {Parallel Processing and Applied Mathematics}
isbn = {978-3-642-14403-5}
year = {2010}
pages = {485-494}
numpages = {10}
abstract = {Many real-life problems can be modelled by systems of linear equations or safely transformed to the linear case. When uncertain model parameters are introduced by intervals, then a parametric interval linear system must properly be solved to meet all possible scenarios and yield useful results. In general case, system coefficients are nonlinear functions of parameters. The Direct Method for solving such systems is proposed. Affine arithmetic is used to handle nonlinear dependencies. Some illustrative examples are solved and the results are compared to the literature data produced by other methods.}
}
@inproceedings{gor-gou-put-10-aa-zonint,
author = {Khalil Ghorbal and Eric Goubault and Sylvie Putot}
title = {A Logical Product Approach to Zonotope Intersection}
fullhash = {7acaf97c3ed94143f8767d6a61ebc5b9}
booktitle = {Proceedings of the 22nd International Conference on Computer Aided Verification (CAV 2010)}
location = {Edinburgh, UK}
isbn = {364214294X}
year = {2010}
pages = {212-226}
numpages = {15}
doi = {10.1007/978-3-642-14295-6_22}
urlraw = {https://doi.org/10.1007/978-3-642-14295-6_22}
url = {https://doi.org/10.1007/978-3-642-14295-6_22}
abstract = {We define and study a new abstract domain which is a fine-grained combination of zonotopes with (sub-)polyhedric domains such as the interval, octagon, linear template or polyhedron domains While abstract transfer functions are still rather inexpensive and accurate even for interpreting non-linear computations, we are able to also interpret tests (i.e intersections) efficiently This fixes a known drawback of zonotopic methods, as used for reachability analysis for hybrid systems as well as for invariant generation in abstract interpretation: intersection of zonotopes are not always zonotopes, and there is not even a best zonotopic over-approximation of the intersection We describe some examples and an implementation of our method in the APRON library, and discuss some further interesting combinations of zonotopes with non-linear or non-convex domains such as quadratic templates and maxplus polyhedra.}
}
@inproceedings{tha-pos-ham-10-aa-symdemo,
author = {Balavelan Thanigaivelan and Adam Postula and Tara Julia Hamilton}
title = {Live Demo: {Affine} Arithmetic Concept Based Symbolic Circuit Analyser}
fullhash = {5157b12897b12acc20d1d9542fe3e3c7}
booktitle = {Proceedings of 2010 IEEE International Symposium on Circuits and Systems}
month = {5}
year = {2010}
pages = {2776-2776}
numpages = {1}
doi = {10.1109/ISCAS.2010.5537017}
abstract = {We implemented a Symbolic Circuit Analysis Software toolbox based on Affine Arithmetic. The toolbox is defined with small signal Affine MOSFET models along with conventional MOSFET models. As a case study the demonstration compares the use of affine models with conventional models in various circuit architectures. The symbolic analysis results are also compared with SPICE analysis results. This demonstration is associated with CAD and other tools for Analog Design track (Track 1.12).}
}
@article{zhu-che-son-lin-10-aa-truss,
author = {Zeng-qing Zhu and Jian-jun Chen and Zong-feng Song and Li-guang Lin}
title = {Non-Probabilistic Reliability Index of Bar Structures with Interval Parameters Based on Modified Affine Arithmetic}
fullhash = {fa2717cea400e4616f32dfefa14d8071}
journal = {Engineering Mechanics}
number = {2}
volume = {27}
year = {2010}
pages = {49-53}
numpages = {5}
urlraw = {http://engineeringmechanics.cn/en/article/id/909}
url = {http://engineeringmechanics.cn/en/article/id/909}
abstract = {Uncertain parameters are described as interval numbers, then, the reliability index equations of bar structures are obtained. A modified matrix affine arithmetic polynomial evaluation method, together with recursive derivative information, is proposed in this paper. Recognizing that affine forms and intervals variables can transform each other, affine forms of bounded uncertain variables and modified affine arithmetic including derivative information for univariate interval polynomial evaluation are introduced into the modeling and calculating of non-probabilistic reliability index. A hanger beam and a ten-bar truss structure are analyzed to illustrate the validity and feasibility of the presented procedures.}
}
@inproceedings{liu-che-zhu-10-aa-truss,
author = {G.L. Liu and J.J. Chen and Zeng Qing Zhu}
title = {Reliability Analysis of Interval Parameters Bar Structures with Affine Arithmetic}
publisher = {Trans Tech}
fullhash = {cb650a77e86d3e88a9fc5d13d94c44eb}
booktitle = {Proc. Materials and Product Technologies II}
month = {6}
series = {Advanced Materials Research}
volume = {118}
year = {2010}
pages = {201-205}
numpages = {5}
doi = {10.4028/www.scientific.net/AMR.118-120.201}
abstract = {By representing the uncertain parameters as interval numbers, the reliability index equations of bars structures were obtained. A modified matrix affine arithmetic polynomial evaluation method plus recursive derivative information was proposed in this paper, which keeps all powers of noise symbols without approximation. Based on the nature that affine forms and intervals variables could transform each other, affine forms of bounded uncertain variables and modified affine arithmetic including derivative information for interval univariate polynomial evaluation were introduced into modeling and calculating non-probabilistic reliability index. An extended beam example and a ten-bar truss structure example were provided to illustrate the validity and feasibility of the presented procedures.}
}
@inproceedings{fry-pas-com-10-aa-raytr,
author = {Oleg Fryazinov and Alexander Pasko and Peter Comninos}
title = {Extending Revised Affine Arithmetic for Fast Reliable Ray-Tracing of Procedurally Defined Implicit Surfaces}
fullhash = {3f0f061bcf2bb3eb98acc072f7dff75b}
booktitle = {Proc. International Conference on Computer Graphics Theory and Applications}
year = {2010}
pages = {199-207}
numpages = {9}
doi = {10.5220/0002847501990207}
abstract = {Fast and reliable rendering of implicit surfaces is an important research area in the field of implicit modelling. Direct rendering, namely ray-tracing, is shown to be a suitable technique for obtaining good-quality visualisations of implicit surfaces. We present a technique for reliable ray-tracing of arbitrary procedurally defined implicit surfaces by using a modification of Affine Arithmetic called Revised Affine Arithmetic. A wide range of procedurally defined implicit objects can be rendered using this technique including polynomial surfaces, constructive solids, pseudo-random objects, procedurally defined microstructures, and others. We compare our technique with other reliable techniques based on Interval and Affine Arithmetic to show that our technique provides the fastest, while still reliable, ray-surface intersections and ray-tracing. We also suggest possible modifications for the GPU implementation of this technique for real-time rendering of relatively simple implicit models and for near real-time rendering of complex implicit models.}
}
@misc{bou-gou-gou-put-10-aa-pbox-slid,
author = {O. Bouissou and E. Goubault and J. Goubault-Larrecq and S. Putot}
title = {A Generalization of {P}-boxes to Affine Arithmetic, and Applications to Static Analysis of Programs}
fullhash = {1c1aa46386402d66c2bf1470c3d66579}
howpublished = {Slides of presentation at SCAN 2010, Lyon.}
year = {2010}
urlraw = {http://scan2010.ens-lyon.fr/SCAN2010_Bouissou_Goubault_Goubault-Larrecq_Putot.pdf}
url = {http://scan2010.ens-lyon.fr/SCAN2010_Bouissou_Goubault_Goubault-Larrecq_Putot.pdf}
}
@article{fry-pas-com-10-aa-procsurf,
author = {Oleg Fryazinov and Alexander Pasko and Peter Comninos}
title = {Fast Reliable Interrogation of Procedurally Defined Implicit Surfaces using Extended Revised Affine Arithmetic}
fullhash = {3f0f061bcf2bb3eb98acc072f7dff75b}
journal = {Computers {\&} Graphics}
note = {Graphics for Serious Games Computer Graphics in Spain: a Selection of Papers from CEIG 2009 Selected Papers from the SIGGRAPH Asia Education Program}
number = {6}
volume = {34}
year = {2010}
pages = {708-718}
numpages = {11}
doi = {10.1016/j.cag.2010.07.003}
abstract = {Techniques based on interval and affine arithmetic and their modifications are shown to provide reliable function range evaluation for the purposes of surface interrogation. In this paper we present a technique for the reliable interrogation of implicit surfaces using a modification of affine arithmetic called revised affine arithmetic. We extend the range of functions presented in revised affine arithmetic by introducing affine operations for arbitrary functions such as set-theoretic operations with R-functions, blending and conditional operators. The obtained affine forms of arbitrary functions provide faster and tighter function range evaluation. Several case studies for operations using affine forms are presented. The proposed techniques for surface interrogation are tested using ray-surface intersection for ray-tracing and spatial cell enumeration for polygonisation. These applications with our extensions provide fast and reliable rendering of a wide range of arbitrary procedurally defined implicit surfaces (including polynomial surfaces, constructive solids, pseudo-random objects, procedurally defined microstructures, and others). We compare the function range evaluation technique based on extended revised affine arithmetic with other reliable techniques based on interval and affine arithmetic to show that our technique provides the fastest and tightest function range evaluation for fast and reliable interrogation of procedurally defined implicit surfaces.}
}
@article{zha-zha-zho-10-aa-accur,
author = {L. Zhang and Y. Zhang and W. Zhou}
title = {Tradeoff between Approximation Accuracy and Complexity for Range Analysis using Affine Arithmetic}
fullhash = {ebc06912d93c01271defcb6ce11393a7}
journal = {Journal of Signal Processing Systems}
volume = {61}
year = {2010}
pages = {279-291}
numpages = {13}
doi = {10.1007/s11265-010-0452-2}
abstract = {Digital signal processing algorithms are usually developed in floating-point arithmetic. After that floating-point to fixed-point transformation is performed to implement them on fixed-point devices, for higher speed, smaller area and lower power. During this transformation, range analysis is to find the minimum integer bit-widths for signals to prevent overflow. Existing state-of-the-art analytical methods for range analysis are generally based on Affine Arithmetic, which presents two approximation methods for non-affine operations. The Chebyshev approximation provides the best approximation with prohibitive computation expense. The trivial range estimation, which is very efficient for computation, over-estimates the range four times at the worst case. This paper presents a novel approach to let user decide tradeoff between approximation accuracy and complexity of Affine Arithmetic. Case studies and experiments are carried out to demonstrate its efficiency.}
}
@inproceedings{pen-loo-vas-10-aa-symbreg,
author = {Cassio L. Pennachin and Moshe Looks and Jo{\~{a}}o A. de Vasconcelos}
title = {Robust Symbolic Regression with Affine Arithmetic}
publisher = {Association for Computing Machinery}
fullhash = {d2ec377712d7ecf696896193d303794c}
booktitle = {Proceedings of the 12th Annual Conference on Genetic and Evolutionary Computation (GECCO '10)}
location = {Portland, Oregon, USA}
isbn = {9781450300728}
year = {2010}
pages = {917-924}
numpages = {8}
doi = {10.1145/1830483.1830648}
abstract = {We use affine arithmetic to improve both the performance and the robustness of genetic programming for symbolic regression. During evolution, we use affine arithmetic to analyze expressions generated by the genetic operators, estimating their output range given the ranges of their inputs over the training data. These estimated output ranges allow us to discard trees that contain asymptotes as well as those whose output is too far from the desired output range determined by the training instances. We also perform linear scaling of outputs before fitness evaluation. Experiments are performed on 15 problems, comparing the proposed system with a baseline genetic programming system with protected operators, and with a similar system based on interval arithmetic. Results show that integrating affine arithmetic with an implementation of standard genetic programming reduces the number of fitness evaluations during training and improves generalization performance, minimizes overfitting, and completely avoids extreme errors of unseen test data.}
}
@article{vac-can-vil-10-aa-power,
author = {Alfredo Vaccaro and Claudio A. Canizares and Domenico Villacci}
title = {An Affine Arithmetic-Based Methodology for Reliable Power Flow Analysis in the Presence of Data Uncertainty}
fullhash = {d39047bec6267456c5299f4e3f15cf71}
journal = {IEEE Transactions on Power Systems}
number = {2}
volume = {25}
year = {2010}
pages = {624-632}
numpages = {9}
doi = {10.1109/TPWRS.2009.2032774}
abstract = {Power flow studies are typically used to determine the steady state or operating conditions of power systems for specified sets of load and generation values, and is one of the most intensely used tools in power engineering. When the input conditions are uncertain, numerous scenarios need to be analyzed to cover the required range of uncertainty. Under such conditions, reliable solution algorithms that incorporate the effect of data uncertainty into the power flow analysis are required. To address this problem, this paper proposes a new solution methodology based on the use of affine arithmetic, which is an enhanced model for self-validated numerical analysis in which the quantities of interest are represented as affine combinations of certain primitive variables representing the sources of uncertainty in the data or approximations made during the computation. The application of this technique to the power flow problem is explained in detail, and several numerical results are presented and discussed, demonstrating the effectiveness of the proposed methodology, especially in comparison to previously proposed interval arithmetic's techniques.}
}
@article{deg-lom-roe-10-aa-finelm,
author = {D. Degrauwe and G. Lombaert and G. {De Roeck}}
title = {Improving interval analysis in finite element calculations by means of affine arithmetic}
fullhash = {d0856857a402631397ae03ac2d5e7be5}
journal = {Computers {\&} Structures}
number = {3}
volume = {88}
year = {2010}
pages = {247-254}
numpages = {8}
doi = {10.1016/j.compstruc.2009.11.003}
abstract = {Interval-based uncertainty models have proven to be well-suited for structural safety engineering with few data at hand. The practical use of interval analysis is hindered by the dependency problem, which leads to an overestimation of the uncertainty on the results. Affine arithmetic is a generalization of interval arithmetic that accounts for the relation between variables. By circumventing the dependency problem, it yields more accurate results. This paper presents a novel method to solve affine systems of linear equations, which allows for the application of affine arithmetic in finite element analysis. The proposed procedure is illustrated with three applications.}
}
@inproceedings{con-gur-liu-liu-zha-zho-zou-09-aa-static,
author = {Jason Cong and Karthik Gururaj and Bin Liu and Chunyue Liu and Zhiru Zhang and Sheng Zhou and Yi Zou}
title = {Evaluation of Static Analysis Techniques for Fixed-Point Precision Optimization}
fullhash = {c9ade27a85c44c77b69b52dde771a78f}
booktitle = {Proc. 17th IEEE Symposium on Field Programmable Custom Computing Machines}
month = {4}
year = {2009}
pages = {231-234}
numpages = {4}
doi = {10.1109/FCCM.2009.35}
abstract = {Precision analysis and optimization is very important when transforming a floating-point algorithm into fixed-point hardware implementations. The core analysis techniques are either based on dynamic analysis or static analysis. We believe in static error analysis, as it is the only technique that can guarantee the desired worst-case accuracy. In this paper we study various underlying arithmetic candidates that can be used in static error analysis and compare their computed sensitivities. The approaches studied include Affine Arithmetic(AA), General Interval Arithmetic (GIA) and Automatic Differentiation (Symbolic Arithmetic). Our study shows that symbolic method is preferred for expressions with higher order cancellation. For programs without strong cancellation, any method works fairly well and GIA slightly outperforms others. We also study the impact of program transformations on these arithmetics.}
}
@techreport{fry-pas-com-09-aa-procsurf-tr,
author = {O. Fryazinov and A. Pasko and P. Comninos}
title = {Fast Reliable Ray-tracing of Procedurally Defined Implicit Surfaces Using Revised Affine Arithmetic}
institution = {Bournemouth University}
fullhash = {b0fe593363e9ba88d1f73d94db97a645}
month = {10}
number = {TR-NCCA-2009-04}
year = {2009}
urlraw = {http://eprints.bournemouth.ac.uk/11708/}
url = {http://eprints.bournemouth.ac.uk/11708/}
abstract = {Fast and reliable rendering of implicit surfaces is an important area in the field of implicit modelling. Direct rendering, namely ray-tracing, is shown to be a suitable technique for obtaining good-quality visualisations of implicit surfaces. We present a technique for reliable ray-tracing of arbitrary procedurally defined implicit surfaces by using a modification of Affine Arithmetic called Revised Affine Arithmetic. A wide range of procedurally defined implicit objects can be rendered using this technique including polynomial surfaces, constructive solids, pseudo-random objects, procedurally defined microstructures, and others. We compare our technique with other reliable techniques based on Interval and Affine Arithmetic to show that our technique provides the fastest, while still reliable, ray-surface intersections and ray-tracing. We also suggest possible modifications for the GPU implementation of this technique for real-time rendering of relatively simple implicit models and for near real-time for complex implicit models.}
}
@article{zhu-che-09-aa-eigen,
author = {Zeng-Qing Zhu and Jian-Jun Chen}
title = {Dynamic Eigenvalue Analysis of Structures with Interval Parameters Based on Affine Arithmetic}
fullhash = {c53373ff65b70b76bd4472ddceb9bba4}
journal = {tructural Engineering and Mechanics}
number = {4}
volume = {33}
year = {2009}
pages = {539-542}
numpages = {4}
}
@inproceedings{min-zhe-qin-09-aa-fuzzmark,
author = {Yu Min and He Zhengyou and Qian Qingquan}
title = {Fuzzy Availability Based on Affine Arithmetic and {Markov} Process}
fullhash = {556c25ca90b480ff0e22b4eb3ea41f54}
booktitle = {Proc. Second International Conference on Intelligent Computation Technology and Automation}
month = {10}
volume = {2}
year = {2009}
pages = {704-707}
numpages = {4}
doi = {10.1109/ICICTA.2009.405}
abstract = {This paper proposes a novel approach of fuzzy availability modeling based on probabilistic approach, employs affine arithmetic together with Markov process. Availability models based on Markov process allow failure, maintenance to be specified in a concise manner. Under a Markov model from which transient availability and steady-state availability can be easily obtained. In practice, sometimes we can't have the exact values of failure rate, repairing rate or other system parameters but with some uncertainty about these values. This is quite common for systems in modern large-scale monitor system and the control system, where many system parameters are not normalized. For the deviation explosion of fuzzy dependability analysis based on fuzzy set or interval arithmetic, affine forms of bounded uncertain quantities and affine arithmetic are introduced into modeling and calculating the dependability index. Using Markov process and affine arithmetic, the probability of each state at moment t can be calculated and then the fuzzy dependability curve of system will be built.}
}
@inproceedings{bah-rev-09-aa-solver,
author = {Ali Baharev and Endre R{\'{e}}v}
title = {A Complete Nonlinear System Solver Using Affine Arithmetic}
fullhash = {e8a905f33d5eb3200e31663f08e60e47}
booktitle = {Proc. Interval Analysis and Constraint Propagation for Applications (IntCP 2009)}
location = {Lisbon, Portugal}
month = {9}
year = {2009}
pages = {17-33}
numpages = {17}
urlraw = {https://www.researchgate.net/profile/Ali-Baharev/publication/237677929_A_complete_nonlinear_system_solver_using_affine/links/0a85e5382708b1dfab000000/A-complete-nonlinear-system-solver-using-affine.pdf#page=24}
url = {https://www.researchgate.net/profile/Ali-Baharev/publication/237677929_A_complete_nonlinear_system_solver_using_affine/links/0a85e5382708b1dfab000000/A-complete-nonlinear-system-solver-using-affine.pdf#page=24}
abstract = {A general framework is presented for finding all solutions to systems of nonlinear equations, or proving that there is no solution to the problem. Components of this framework are interval arithmetic, affine arithmetic, constraint propagation based on directed acyclic graph (DAG) representation of the problem, a pruning technique based on linear programming and a local approximate optimizer. The proposed method is implemented in C++ and will be released as an open source standalone solver. An important feature of the solver is that it is interfaced with the AMPL\textdegree{}R modeling system. Numerical results on difficult benchmarks and comparisons to other solvers are given. The favorable effect of exploiting common subexpressions is also discussed. Real-world problems from chemical engineering, having multiple solutions, are successfully solved.}
}
@inproceedings{tha-pos-ham-09-aa-mosfet,
author = {Balavelan Thanigaivelan and Adam Postula and Tara Julia Hamilton}
title = {A Modified {MOSFET} Small-Signal Model Based on Affine Arithmetic Concepts}
fullhash = {5157b12897b12acc20d1d9542fe3e3c7}
booktitle = {2009 Asia Pacific Conference on Postgraduate Research in Microelectronics Electronics (PrimeAsia)}
month = {1}
year = {2009}
pages = {73-76}
numpages = {4}
doi = {10.1109/PRIMEASIA.2009.5397443}
abstract = {We modified small-signal MOSFET models with affine arithmetic concepts. The model elements were redistributed so they are meaningfully grouped and better correlated. The modified models are more compact, easier to handle (even at the highest accuracy level the model leads to simpler circuit expressions) and provide better insight into the analysis of large analog circuits. The results presented in this paper show that the modified models generate compact circuit expressions with a significant reduction in the number of terms. This leads to reduced computational complexity and simulation time when compared with analysis using conventional models.}
}
@article{bah-ach-rev-09-aa-distill,
author = {Ali Baharev and Tobias Achterberg and Endre R{\'{e}}v}
title = {Computation of an extractive distillation column with affine arithmetic}
fullhash = {3ed342cb8a90811e5620de8bf2b4d320}
journal = {Journal of the American Institute of Chemical Engineers}
number = {7}
volume = {55}
year = {2009}
pages = {1695-1704}
numpages = {10}
doi = {10.1002/aic.11777}
abstract = {The need of reliably solving systems of nonlinear equations often arises in the everyday practice of chemical engineering. In general, standard methods cannot provide theoretical guarantee for convergence to a solution, cannot reliably find multiple solutions, and cannot prove nonexistence of solutions. Interval methods provide tools to overcome these problems, thus achieving reliability. To the authors' best knowledge, computation of distillation columns with interval methods have not yet been considered in the literature. This article presents significant enhancements compared with a previously published interval method of the authors. The proposed branch-and-prune algorithm is guaranteed to converge, and is fairly general at the same time. If no solution exists then this information is provided by the method as a result. Power of the suggested method is demonstrated by solving, with guaranteed convergence, even the MESH equations of a 22 stage extractive distillation column with a ternary mixture.}
}
@article{kno-hij-ken-sch-han-hag-09-aa-raytr,
author = {A. Knoll and Y. Hijazi and A. Kensler and M. Schott and C. Hansen and H. Hagen}
title = {Fast Ray Tracing of Arbitrary Implicit Surfaces with Interval and Affine Arithmetic}
fullhash = {9d202617048c8ff8275c3b82456eecf7}
journal = {Computer Graphics Forum}
number = {1}
volume = {28}
year = {2009}
pages = {26-40}
numpages = {15}
doi = {10.1111/j.1467-8659.2008.01189.x}
abstract = {Abstract Existing techniques for rendering arbitrary-form implicit surfaces are limited, either in performance, correctness or flexibility. Ray tracing algorithms employing interval arithmetic (IA) or affine arithmetic (AA) for root-funding are robust and general in the class of surfaces they support, but traditionally slow. Nonetheless, implemented efficiently using a stack-driven iterative algorithm and SIMD vector instructions, these methods can achieve interactive performance for common algebraic surfaces on the CPU. A similar algorithm can also be implemented stacklessly, allowing for efficient ray tracing on the GPU. This paper presents these algorithms, as well as an inclusion-preserving reduced affine arithmetic (RAA) for faster ray-surface intersection. Shader metaprogramming allows for immediate and automatic generation of symbolic expressions and their interval or affine extensions. Moreover, we are able to render even complex forms robustly, in real-time at high resolution.}
}
@inproceedings{hij-hag-han-joy-08-aa-iauseful,
author = {Y. Hijazi and H. Hagen and C. D. Hansen and K. I. Joy}
title = {Why Interval Arithmetic is So Useful}
publisher = {Gesellschaft f\"{u}r Informatik}
fullhash = {93b5df6931743cd0e8e0241faaef8df2}
booktitle = {Visualization of Large and Unstructured Data Sets}
isbn = {978-3-88579-441-7}
year = {2008}
pages = {148-163}
numpages = {16}
urlraw = {https://dl.gi.de/handle/20.500.12116/4619}
url = {https://dl.gi.de/handle/20.500.12116/4619}
abstract = {Interval arithmetic was introduced by Ramon Moore [Moo66] in the 1960s as an approach to bound rounding errors in mathematical computation. The theory of interval analysis emerged considering the computation of both the exact solution and the error term as a single entity, i.e. the interval. Though a simple idea, it is a very powerful technique with numerous applications in mathematics, computer science, and engineering. In this survey we discuss the basic concepts of interval arithmetic and some of its extensions, and review successful applications of this theory in particular in computer science.}
}
@inproceedings{zhu-che-08-aa-dynstruc,
author = {Zeng Qing Zhu and J.J. Chen}
title = {Dynamic Response Analysis for Structures with Interval Parameters Based on Affine Arithmetic}
publisher = {Trans Tech Publications Ltd}
fullhash = {479d6351bc1584833a2796c466acc76a}
booktitle = {Advanced Materials Research}
month = {6}
volume = {44-46}
year = {2008}
pages = {157-164}
numpages = {8}
doi = {10.4028/www.scientific.net/AMR.44-46.157}
abstract = {This paper aims to study the uncertainty of the MDOF structural dynamic response, taking not only the interval characteristics of structural physical parameters and geometric dimension, but also the interval characteristics of applied load simultaneously . By means of the description of the interval parameters of uncertain structure with affine forms, the interval structural dynamic equation is studied, and an improved affine arithmetic based on interval division is presented, where correlations between the interval elements in eigenvalue and responses equations are considered, independent uncertain parameters are transformed to affine forms, and the solution of eigenvalue and response equations are transformed into the corresponding certain ones. With general affine arithmetic, the eigenvalue of each order and response bounds are determined by searching for the maximum and minimum in the solutions. Finally, some mathematical examples and a further engineering application confirm the feasibility and validity of this approach.}
}
@mastersthesis{cha-08-aa-stan-thesis,
author = {Bryan Chan}
title = {Static Analysis for Efficient Affine Arithmetic on {GPUs}}
institution = {University of Waterloo}
fullhash = {b05477d3fb03cbac9c42f3781e68acb6}
month = {1}
year = {2008}
urlraw = {http://hdl.handle.net/10012/3571}
url = {http://hdl.handle.net/10012/3571}
abstract = {Range arithmetic is a way of calculating with variables that hold ranges of real values. This ability to manage uncertainty during computation has many applications. Examples in graphics include rendering and surface modeling, and there are more general applications like global optimization and solving systems of nonlinear equations. This thesis focuses on affine arithmetic, one kind of range arithmetic. The main drawbacks of affine arithmetic are that it taxes processors with heavy use of floating point arithmetic and uses expensive sparse vectors to represent noise symbols. Stream processors like graphics processing units (GPUs) excel at intense computation, since they were originally designed for high throughput media applications. Heavy control flow and irregular data structures pose problems though, so the conventional implementation of affine arithmetic with dynamically managed sparse vectors runs slowly at best. The goal of this thesis is to map affine arithmetic efficiently onto GPUs by turning sparse vectors into shorter dense vectors at compile time using static analysis. In addition, we look at how to improve efficiency further during the static analysis using unique symbol condensation. We demonstrate our implementation and performance of the condensation on several graphics applications.}
}
@misc{bil-08-aa-hiorder-errata,
author = {Giuseppe Bilotta}
title = {Errata for ``Self-Verified Extension of Affine Arithmetic to Arbitrary Order''}
fullhash = {5140f52cecdf2021026ae0961f0227cf}
howpublished = {In the journal \emph{Le Matematiche}}
note = {Errata (missing references) for~\cite{}.}
urlraw = {https://lematematiche.dmi.unict.it/index.php/lematematiche/article/download/67/64}
url = {https://lematematiche.dmi.unict.it/index.php/lematematiche/article/download/67/64}
}
@article{bil-08-aa-hiorder,
author = {Giuseppe Bilotta}
title = {Self-Verified Extension of Affine Arithmetic to Arbitrary Order}
fullhash = {5140f52cecdf2021026ae0961f0227cf}
journal = {Le Matematiche}
note = {See errata (missing references)~\cite{bil-08-aa-hiorder-errata}}
number = {I}
volume = {LXIII}
year = {2008}
pages = {15-30}
numpages = {16}
urlraw = {https://lematematiche.dmi.unict.it/index.php/lematematiche/article/download/47/46}
url = {https://lematematiche.dmi.unict.it/index.php/lematematiche/article/download/47/46}
abstract = {Affine Arithmetic (AA) is a self-verifying computational approach that keeps track of first-order correlation between uncertainties in the data and intermediate and final results. In this paper we propose a higher-order extension satisfying the requirements of genericity, arbitrary-order and self-verification, comparing the resulting method with other well-known high-order extensions of AA.}
}
@inproceedings{fre-olb-bar-08-aa-circ,
author = {M. Freisfeld and M. Olbrich and E. Barke}
title = {Circuit Simulations with Uncertainties Using Affine Arithmetic and Piecewise Affine Statemodels}
fullhash = {475d037670067f739f5c9915713ca399}
booktitle = {9th International Conference on Solid-State and Integrated-Circuit Technology (2008)}
month = {10}
year = {2008}
pages = {496-499}
numpages = {4}
doi = {10.1109/ICSICT.2008.4734572}
abstract = {Especially in the field of safety critical applications the impact of process variations on the behavior of integrated circuits can not be neglected. Usually, simulation is costly due to the need for multiple simulation runs with different technology parameter settings. In the past, semi-symbolic-simulation applying affine arithmetic (AA), as a suitable range arithmetic, has produced very motivating results on netlistlevel. Using AA reduces the complexity without loosing generality of the simulation results. This paper covers the formulation of effective highlevel circuit models. We use piecewise affine functions to model the circuit and to provide fast simulations while simultaneously achieving good convergence. The equations are separated into matrices for linear mapping and matrices for the representation of the system's state. We formulate the uncertainties within the system with affine terms entered in the system matrices. Furthermore, we present the relevant solution algorithms for the semi symbolic simulation and their implementation.}
}
@article{sun-lij-mad-wan-08-aa-chyield,
author = {Jin Sun and Jun Li and Dongsheng Ma and Janet M. Wang}
title = {{Chebyshev} Affine-Arithmetic-Based Parametric Yield Prediction Under Limited Descriptions of Uncertainty}
fullhash = {309c1c6bff603ff4a25e8cb9d96303d4}
journal = {IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems}
month = {10}
number = {10}
volume = {27}
year = {2008}
pages = {1852-1865}
numpages = {14}
doi = {10.1109/TCAD.2008.2003300}
abstract = {Due to the hard-to-measure distributions of real process data, it is difficult to provide accurate parametric yield prediction for modern circuit design. Most existing approaches are not able to handle the uncertain distribution properties coming from the process data. Other approaches are inadequate in considering correlations among the distributions of variations. This paper suggests a new approach that not only takes care of correlations among distributions but also provides a low-cost and efficient computation scheme. The proposed method approximates the parameter variations with Chebyshev affine arithmetic (CAA) to capture both the uncertainty and nonlinearity in a cumulative distribution function. The CAA-based probabilistic range presentation describes, both fully and partially, specified process and environmental parameters. Thus, we are able to predict the probability bounds for leakage consumption with unknown dependences among variations. The end result is the chip-level parametric yield estimation based on leakage prediction. Experimental results demonstrate that the new approach provides a reliable bound estimation, which leads to a 20{\%} yield improvement compared with only using the intervals of partially specified uncertainties.}
}
@article{bah-rev-08-aa-equilib,
author = {Ali Baharev and Endre R{\'{e}}v}
title = {Reliable Computation of Equilibrium Cascades with Affine Arithmetic}
fullhash = {e8a905f33d5eb3200e31663f08e60e47}
journal = {Journal of the American Institute of Chemical Engineers}
number = {7}
volume = {54}
year = {2008}
pages = {1782-1797}
numpages = {16}
doi = {10.1002/aic.11490}
abstract = {Computing the steady state of multistage counter-current processes like distillation, extraction, or absorption is the equivalent to finding solutions for large-scale nonlinear equation systems. The conventional solution techniques are fast and efficient if a good estimation is available, but are prone to fail, and do not provide information about the reason for the failure. This is the main motive to apply reliable methods in solving them. Reliable computations are usually realized with interval methods. A reliable root finding method is presented, based on affine arithmetic (AA), a recently developed linearization technique and interval method. AA is compared here to another linearization technique, the widely known Interval Newton method. The studied examples seem to indicate superiority of the novel method over the traditional one. The comparison is made with a pruning technique not state-of-the-art, but analogous in the two compared methods. AA can be combined with constraint propagation (CP), or linear programming (LP) aided CP, as pruning techniques. The combined techniques, AA/CP and AA/LP are studied and compared. AA/LP proves to be preferable because of its robustness. Short distillation columns are successfully computed with the proposed AA/LP method.}
}
@misc{gou-put-08-aa-perturb,
author = {Eric Goubault and Sylvie Putot}
title = {Perturbed Affine Arithmetic for Invariant Computation in Numerical Program Analysis}
fullhash = {76fa0e927f81bc74745420b7d50a13de}
howpublished = {Preprint online at the arXiv repository, article 0807.2961}
year = {2008}
urlraw = {https://arxiv.org/abs/0807.2961}
url = {https://arxiv.org/abs/0807.2961}
abstract = {We completely describe a new domain for abstract interpretation of numerical programs. Fixpoint iteration in this domain is proved to converge to finite precise invariants for (at least) the class of stable linear recursive filters of any order. Good evidence shows it behaves well also for some non-linear schemes. The result, and the structure of the domain, rely on an interesting interplay between order and topology.}
}
@inproceedings{zho-bia-07-aa-wdlen,
author = {S. Zhou and J. Bian}
title = {{OWL}: {Word}-Length Optimization via Extreme Value Theory and Affine Arithmetic/Interval Arithmetic Model}
publisher = {Acta}
fullhash = {d8b3f800136414c632d4e87990057e68}
booktitle = {Proc. of Circuits, Signals, and Systems 2007}
year = {2007}
pages = {article 573-046}
numpages = {-1}
abstract = {In the high level synthesis from languages such as C, word-length of variables is one of the key issues on VLSI design optimization. This paper proposes a new automated approach -- OWL, for optimizing word-lengths of fixed-point designs. OWL is based on static analysis via extreme value theory and affine/interval arithmetic. It describes methods to minimize both the integer and fraction parts of fixed-point signals. For range analysis, it employs a semi-analytical approach with extreme value distribution model to identify the number of integer bits, while for precision analysis, it uses AAIA error model to find the optimum number of fraction bits. The reduction percentage of word-length is over 60{\%} in integer part and over 45{\%} in fraction part.}
}
@inproceedings{liu-luk-son-tan-zen-07-aa-ellips,
author = {Xuexin Liu and Wai-Shing Luk and Yu Song and Pushan Tang and Xuan Zeng}
title = {Robust Analog Circuit Sizing Using Ellipsoid Method and Affine Arithmetic}
fullhash = {e8740e9526458eae21f619fa8bfddff5}
booktitle = {2007 Asia and South Pacific Design Automation Conference}
month = {1}
year = {2007}
pages = {203-208}
numpages = {6}
doi = {10.1109/ASPDAC.2007.357986}
abstract = {Analog circuit sizing under process/parameter variations is formulated as a mini-max geometric programming problem. To tackle such problem, we present a new method that combines the ellipsoid method and affine arithmetic. Affine arithmetic is not only used for keeping tracks of variations and correlations, but also helps to determine the sub-gradient at each iteration of the ellipsoid method. An example of designing a CMOS operational amplifier is given to demonstrate the effectiveness of the proposed method. Finally numerical results are verified by SPICE simulation.}
}
@article{ceb-kre-cho-lon-ngu-lud-bar-07-aa-expert,
author = {Martine Ceberio and Vladik Kreinovich and Sanjeev Chopra and Luc Longpr\'{e} and Hung T. Nguyen and Bertram Lud\"{a}scher and Chitta Baral}
title = {Interval-Type and Affine Arithmetic-Type Techniques for Handling Uncertainty in Expert Systems}
fullhash = {a8b7253d64abc9d9485279e58e600314}
journal = {Journal of Computational and Applied Mathematics}
note = {Special Issue on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2004)}
number = {2}
volume = {199}
year = {2007}
pages = {403-410}
numpages = {8}
doi = {10.1016/j.cam.2005.08.030}
abstract = {Expert knowledge consists of statements $S_j$ (facts and rules). The facts and rules are often only true with some probability. For example, if we are interested in oil, we should look at seismic data. If in 90{\%} of the cases, the seismic data were indeed helpful in locating oil, then we can say that if we are interested in oil, then with probability 90{\%} it is helpful to look at the seismic data. In more formal terms, we can say that the implication ``if oil then seismic'' holds with probability 90{\%}. Another example: a bank $A$ trusts a client $B$, so if we trust the bank $A$, we should trust $B$ too; if statistically this trust was justified in 99{\%} of the cases, we can conclude that the corresponding implication holds with probability 99{\%}. If a query $Q$ is deducible from facts and rules, what is the resulting probability $p(Q)$ in $Q$? We can describe the truth of $Q$ as a propositional formula $F$ in terms of $S_j$, i.e., as a combination of statements $S_j$ linked by operators like $\wedge$, $\vee$, and $\neg$; computing $p(Q)$ exactly is NP-hard, so heuristics are needed. Traditionally, expert systems use technique similar to straightforward interval computations: we parse $F$ and replace each computation step with corresponding probability operation. Problem: at each step, we ignore the dependence between the intermediate results $F_j$; hence intervals are too wide. Example: the estimate for $P(A\vee \neg A)$ is not 1. Solution: similar to affine arithmetic, besides $P(F_j)$, we also compute $P(F_j\wedge F_i)$ (or $P(F_{j1}\wedge \cdots \wedge F_{jd})$), and on each step, use all combinations of $\ell$ such probabilities to get new estimates. Results: e.g., $P(A\vee\neg A)$ is estimated as 1.}
}
@article{miy-kas-07-aa-exist,
author = {Shinya Miyajima and Masahide Kashiwagi}
title = {Existence Test for Solution of Nonlinear Systems Applying Affine Arithmetic}
fullhash = {23f720934d733e451fa0ea8c7f7f6e88}
journal = {Journal of Computational and Applied Mathematics}
note = {Special Issue on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2004)}
number = {2}
volume = {199}
year = {2007}
pages = {304-309}
numpages = {6}
doi = {10.1016/j.cam.2005.08.051}
abstract = {A computational test is proposed for existence of solution in nonlinear systems. In this test, an interval inclusion of Newton mapping is estimated applying affine arithmetic. Numerical examples are presented to show the efficiency of this test.}
}
@article{gam-mad-07-aa-raytr,
author = {M. Gamito and S. Maddock}
title = {Ray Casting Implicit Fractal Surfaces with Reduced Affine Arithmetic}
fullhash = {3c64fe009e686c67f41b01b5174ce3b8}
journal = {Visual Computer}
volume = {23}
year = {2007}
pages = {155-165}
numpages = {11}
doi = {10.1007/s00371-006-0090-7}
abstract = {A method is presented for ray casting implicit surfaces defined by fractal combinations of procedural noise functions. The method is robust and uses affine arithmetic to bound the variation of the implicit function along a ray. The method is also efficient due to a modification in the affine arithmetic representation that introduces a condensation step at the end of every non-affine operation. We show that our method is able to retain the tight estimation capabilities of affine arithmetic for ray casting implicit surfaces made from procedural noise functions while being faster to compute and more efficient to store.}
}
@inproceedings{gou-put-07-aa-under,
author = {Eric Goubault and Sylvie Putot}
title = {Under-Approximations of Computations in Real Numbers Based on Generalized Affine Arithmetic}
fullhash = {76fa0e927f81bc74745420b7d50a13de}
booktitle = {Static Analysis - Proc. International Static Analysis Symposium SAS 2007}
isbn = {978-3-540-74061-2}
series = {LNCS}
volume = {4634}
year = {2007}
pages = {137-152}
numpages = {16}
doi = {10.1007/978-3-540-74061-2_9}
abstract = {We build a new, implicitly relational abstract domain which gives accurate under-approximations of the set of real values that program variables can take. This statement is demonstrated both on a theoretical basis and on non-trivial numerical examples. It is, we believe, the first non-trivial under-approximating numerical domain in the static analysis literature.}
}
@inproceedings{sho-son-she-mar-wan-06-aa-raytr,
author = {Huahao Shou and Wenhao Song and Jie Shen and Ralph Robert Martin and Guo-jin Wang}
title = {A Recursive {Taylor} Method for Ray Casting Algebraic Surfaces}
fullhash = {d5cc8a0f348950a4a06cbfa4deaf633a}
booktitle = {Proc. International Conference on Computer Graphics {\&} Virtual Reality (CGVR 2006)}
location = {Las Vegas}
month = {6}
year = {2006}
pages = {196-202}
numpages = {7}
urlraw = {https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.65.2267&rep=rep1&type=pdf}
url = {https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.65.2267&rep=rep1&type=pdf}
abstract = {In this paper, we propose a recursive Taylor method for ray-casting algebraic surfaces. The performance of this approach is compared with four other candidate approaches to raycasting: using interval arithmetic on the power form, using interval arithmetic on centered forms, using affine arithmetic, and using modified affine arithmetic. Test results show that the recursive Taylor method compares favorably to the other methods.}
}
@misc{gay-coe-hur-06-aa-lib-cpp,
author = {Olivier Gay and David Coeurjolly and Nathan J. Hurst}
title = {{Libaffa} - {C}$++$ Affine Arithmetic Library for {GNU}/{Linux}}
institution = {{\'{E}}cole Polithecnique F{\'{e}}d{\'{e}}rale de Lausanne}
fullhash = {ca71bea5229a04d3743830638fbe5a1b}
howpublished = {Project website}
month = {6}
year = {2006}
urlraw = {https://www.nongnu.org/libaffa/}
url = {https://www.nongnu.org/libaffa/}
}
@inbook{heu-gri-wal-06-aa-analog,
author = {Wilhelm Heupke and Christoph Grimm and Klaus Waldschmidt}
title = {Modeling Uncertainty in Nonlinear Analog Systems with Affine Arithmetic}
publisher = {Springer Netherlands}
fullhash = {9d095d81ee1e0337164a9abde75e7806}
booktitle = {Applications of Specification and Design Languages for SoCs: Selected papers from Forum on Specification {\&} Design Languages (FDL 2005)}
location = {Dordrecht}
chapter = {9}
isbn = {978-1-4020-4998-9}
year = {2006}
pages = {155-169}
numpages = {15}
doi = {10.1007/978-1-4020-4998-9_9}
abstract = {This chapter describes a semisymbolic method for the analysis of mixed signal systems. Aimed at control and signal-processing applications, it delivers a superset of all reachable values. The method that relies on affine arithmetic is precise for linear systems, but in the case of nonlinear systems approximations are needed. As a new term is added for each approximation, the number of approximation terms increases during simulation and therefore slows down the simulation. This leads to a quadratic time complexity in the number of time steps. A method to avoid this and an example implementation based on SystemC analog and mixed signal (AMS) are presented. Efficiency and time complexity of the improved semisymbolic simulation are analyzed and discussed.}
}
@article{sho-lin-mar-wan-06-aa-implsurf,
author = {Huahao Shou and Hongwei Lin and Ralph R. Martin and Guojin Wang}
title = {Modified Affine Arithmetic in Tensor Form for Trivariate Polynomial Evaluation and Algebraic Surface Plotting}
fullhash = {53d4cc0eca29f34531552ce2d4f3190f}
journal = {Journal of Computational and Applied Mathematics}
note = {Special Issue: The International Symposium on Computing and Information (ISCI2004)}
number = {1}
volume = {195}
year = {2006}
pages = {155-171}
numpages = {17}
doi = {10.1016/j.cam.2005.08.003}
\keyw{Interval arithmetic,Affine arithmetic,Algebraic surfaces}
abstract = {This paper extends the modified affine arithmetic in matrix form method for bivariate polynomial evaluation and algebraic curve plotting in 2D to modified affine arithmetic in tensor form for trivariate polynomial evaluation and algebraic surface plotting in 3D. Experimental comparison shows that modified affine arithmetic in tensor form is not only more accurate but also much faster than standard affine arithmetic when evaluating trivariate polynomials.}
}
@misc{sho-lin-mar-wan-06-aa-impldraft,
author = {Huahao Shou and Hongwei Lin and Ralph Martin and Guojin Wang}
title = {Modified Affine Arithmetic in Tensor Form}
fullhash = {4b7fb1642f497f76efa1cc20c2fc8eb2}
howpublished = {Preprint available.}
urlraw = {http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.353.4398&rep=rep1&type=pdf}
url = {http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.353.4398&rep=rep1&type=pdf}
abstract = {This paper extends the modified affine arithmetic in matrix form method for bivariate polynomial evaluation and algebraic curve plotting in 2D to modified affine arithmetic in tensor form for trivariate polynomial evaluation and algebraic surface plotting in 3D. Experimental comparison shows that modified affine arithmetic in tensor form is not only more accurate but also much faster than affine arithmetic when evaluating trivariate polynomials.}
}
@article{mes-tou-06-aa-quadr,
author = {F. Messine and A. Touhami}
title = {A General Reliable Quadratic Form: {An} Extension of Affine Arithmetic}
fullhash = {ea981f897875e2d7ef8b8f86806eaa77}
journal = {Reliable Computing}
year = {2006}
pages = {171-192}
numpages = {22}
doi = {10.1007/s11155-006-7217-4}
abstract = {In this article, a new extension of affine arithmetic is introduced. This technique is based on a quadratic form named general quadratic form. We focus here on the computation of reliable bounds of a function over a hypercube by using this new tool. Some properties of first quadratic functions and then polynomial ones are reported. In order to show the efficiency of such a method, ten polynomial global optimization problems are presented and solved by using an interval branch-and-bound based algorithm.}
}
@inproceedings{yup-hay-06-aa-bitwd,
author = {Yu Pu and Yajun Ha}
title = {An Automated, Efficient and Static Bit-Width Optimization Methodology towards Maximum Bit-Width-to-Error Tradeoff with Affine Arithmetic Model}
publisher = {IEEE Press}
fullhash = {fb849821e57d3666bc0ce27c8696d41e}
booktitle = {Proc. of the 2006 Asia and South Pacific Design Automation Conference (ASP-DAC '06)}
location = {Yokohama, Japan}
year = {2006}
pages = {886-891}
numpages = {6}
doi = {10.1145/1118299.1118500}
abstract = {Ideally, bit-width analysis methods should be able to find the most appropriate bit-widths to achieve the optimum bit-width-to-error tradeoff for variables and constants in high level DSP algorithms when they are implemented into hardware. The tradeoff enables the fixed-point hardware implementation to be area efficient but still within the allowed error tolerance. Unfortunately, almost all the existing static bit-width analysis methods are Interval Arithmetic (IA) based that may overestimate bit-widths and enable fairly pessimistic bit-width-to-error tradeoff. We have developed an automated and efficient bit-width optimization methodology that is Affine Arithmetic (AA) based. Experiments have proven that, compared to the previous static analysis methods, our methodology not only dramatically reduces the fractional bit-width by more than 35{\%} but also slightly reduces the integer bit-width. In addition, our probabilistic error analysis method further enlarges the bit-width-to-error tradeoff.}
}
@misc{mic-fou-lam-men-05-aa-berns,
author = {Dominique Michelucci and Sebti Foufou and Lo{\"{\i{}}}c Lamarque and David M{\'{e}}negaux}
title = {{Bernstein} Based Arithmetic Featuring {de} {Casteljau}}
fullhash = {30b720e422aa7f525871fce027feadd8}
howpublished = {Online document.}
year = {2005}
urlraw = {https://www.researchgate.net/publication/220991064_Bernstein_based_arithmetic_featuring_de_Casteljau}
url = {https://www.researchgate.net/publication/220991064_Bernstein_based_arithmetic_featuring_de_Casteljau}
abstract = {Bernstein based interval analysis permits to trace algebraic curves and surfaces. In this paper, we propose touse the classical de Casteljau algorithm to improve the efficiency of the Bernstein based method. The proposed tracing method gives significant results with functions of high degree. These results are illustrated and compared with other interval analysis approaches.}
}
@article{akh-05-aa-gauss,
author = {Ramil R. Akhmerov}
title = {Interval-Affine {Gaussian} Algorithm for Constrained Systems}
fullhash = {4945ff2df1c938cb9ab94fc6d5159017}
journal = {Reliable Computing}
volume = {11}
year = {2005}
pages = {323-341}
numpages = {19}
doi = {10.1007/s11155-005-0040-5}
abstract = {The paper presents interval-affine Gaussian algorithm for the interval linear systems $Ax = b$ subject to some constraints on real matrices $A$ from the interval matrix $A$. The interval-affine method is based on the so-called interval-affine arithmetic that allows to take the constraints into account during the computation of interval enclosures of the united solution set of the system $Ax = b$, and to make the estimates more accurate.}
}
@inproceedings{kan-ois-05-aa-exodesol,
author = {{Shin'ichi} Oishi Yuchi Kanzawa}
title = {Numerical Method of Proving Existence of Periodic Solution for Nonlinear {ODE} using Affine Arithmetic and Green's Function Expression}
fullhash = {86507c29feadc0e0040435f86e2144d3}
booktitle = {Proc. International Symposium on Nonlinear Theory and its Applications}
month = {10}
year = {2005}
pages = {753-756}
numpages = {4}
doi = {10.34385/proc.40.2-1-4-6}
abstract = {In this paper, a method is proposed to prove the existence of solutions for nonlinear ordinary equations on an essentially bounded functional space. For that, Affine Arithmetic extended to the functional space. Green's Function Expression is also used to prevent from overestimating integral arithmetic with Affine Arithmetic.}
}
@inproceedings{zou-zha-cai-zho-hon-05-aa-timing,
author = {Yi Zou and Mengsheng Zhang and Yici Cai and Qiang Zhou and Xianlong Hong}
title = {Minimum Error Based Affine Arithmetic for Variational Timing Analysis}
fullhash = {224f87efd0121a9c7b0c97a86255bec7}
booktitle = {6th International Conference on ASIC}
month = {10}
volume = {2}
year = {2005}
pages = {978-981}
numpages = {4}
doi = {10.1109/ICASIC.2005.1611491}
abstract = {In this paper, we describes a novel improvement for affine arithmetic which is widely used in statistical timing analysis and circuit simulation. The representation and calculation method for affine arithmetic is both simple and accurate comparing with previous works. The rule for calculating interval multiplication is simple and effective. It can both give a conservative bound and a minimum error affine approximation at the same time. An application for variational timing analysis demonstrated the usefulness of affine arithmetic}
}
@inproceedings{miy-kas-05-aa-maxmin,
author = {Shinya Miyajima and Masahide Kashiwagi}
title = {A Method Which Finds the Maxima and Minima of a Multivariable Function Applying Affine Arithmetic}
publisher = {Springer}
fullhash = {23f720934d733e451fa0ea8c7f7f6e88}
booktitle = {Proc. International Conference on Numerical Analysis and Its Applications (NAA 2004)}
isbn = {978-3-540-31852-1}
year = {2005}
pages = {424-431}
numpages = {8}
abstract = {A new method is proposed for finding all maxima and minima of a multivariable function $f$ in a box $X^0$. In this method, the maxima and the minima are calculated by dividing $X^0$ into subregions recursively and bounding the ranges of $f$ in the each subregion applying affine arithmetic and discarding the subregions which don't possess the possibility of including the point that the maximum (minimum) value occurs. Moreover, to discard more subregions in initial stage, i.e. to speed the new method, two algorithms are introduced. And to show the efficiency of the new method, some numerical examples are implemented.}
}
@techreport{gam-mad-05-aa-rayproc,
author = {Manuel N. Gamito and Steve C. Maddock}
title = {Ray Casting Implicit Procedural Noises with Reduced Affine Arithmetic}
institution = {University of Sheffield}
fullhash = {ef5ff785261b38719f5fe4337d8efc2f}
month = {4}
number = {CS--05--04}
year = {2005}
abstract = {A method for ray casting implicit surfaces, defined with procedural noise models, is presented. The method is robust in that it is able to guarantee correct intersections at all image pixels and for all types of implicit surfaces. This robustness comes from the use of an affine arithmetic representation for the quantity that expresses the variation of the implicit function along a ray. Affine arithmetic provides a bounding interval estimate which is tighter than the interval estimates returned by conventional interval arithmetic. Our ray casting method is also efficient due to a proposed modification in the data structure used to hold affine arithmetic quantities. This modified data structure ultimately leads to a reduced affine arithmetic model. We show that such a reduced affine arithmetic model is able to retain all the tight estimation capabilities of standard affine arithmetic, in the context of ray casting implicit procedural noises, while being faster to compute and more efficient to store. We also show that, without this reduced model, affine arithmetic would not have any advantage over the more conventional interval arithmetic for ray casting the class of implicit procedural surfaces that we are interested in visualizing.}
}
@inproceedings{jam-acu-05-aa-neural,
author = {Marcela Jamett and Gonzalo Acu{\~{n}}a}
title = {Comparative Assessment of Interval and Affine Arithmetic in Neural Network State Prediction}
fullhash = {50c6d6030a145ca1d3bda275191fca6f}
booktitle = {Advances in Neural Networks: Proc. International Symposium on Neural Networks}
series = {Lecture Notes in Computer Science}
volume = {3497}
year = {2005}
pages = {448-453}
numpages = {6}
doi = {10.1007/11427445_73}
abstract = {Two set theory methods, Interval and Affine Arithmetic, are used together with feedforward neural networks (FNN) in order to study their ability to perform state prediction in non-linear systems. Some fundamental theory showing the basic interval and affine arithmetic operations necessary to forward propagate through a FNN is presented and an application to a generic biotechnological process is performed confirming that due to the way the perturbations of the input data are considered, affine FNN perform better than interval ones.}
}
@mastersthesis{cho-05-aa-thesis,
author = {Sanjeev Chopra}
title = {Affine Arithmetic-Type Techniques for Handling Uncertainty in Expert Systems}
institution = {University of Texas at El Paso}
fullhash = {e1a3cf2e9b0392db1b70a4318053b974}
month = {12}
year = {2005}
abstract = {Expert knowledge consists of statements $S_j$: facts and rules. The expert's degree of confidence in each statement $S_j$ can be described as a (subjective) probability. For example, if we are interested in oil; we should look at seismic data (confidence 90{\%}); a bank $A$ trusts a client $B$, so if we trust $A$, we should trust $B$ too (confidence 99{\%}). If a query $Q$ is deducible from facts and rules, what is our confidence $P(Q)$ in $Q$? We can describe $Q$ as a propositional formula $F$ in terms of $S_j$; computing $P(Q)$ exactly is NP-hard, so heuristics are needed. Traditionally, expert systems use technique similar to straightforward interval computations: we parse $F$ and replace each computation step with corresponding probability operation. The problem with this approach is that at each step, we ignore the dependence between the intermediate results $F_j$; hence intervals are too wide. For example, the estimate for $P(A\vee \neg A)$ is not 1. In this thesis, we propose a new solution to this problem; similarly to affine arithmetic, besides $P(F_j)$, we also compute $P(F_j \wedge F_i)$ (or $P(F_{j1} \wedge\cdots\wedge F_{jk})$), and on each step, use all combinations of $\ell$ such probabilities to get new estimates. As a result, for the above stated e.g., $P(A\vee \neg A)$ is estimated as 1.}
}
@article{gri-heu-wal-05-aa-control,
author = {C. Grimm and W. Heupke and K. Waldschmidt}
title = {Analysis of Mixed-Signal Systems with Affine Arithmetic}
fullhash = {afd7325dd3294308ea703ce2f2794907}
journal = {IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems}
number = {1}
volume = {24}
year = {2005}
pages = {118-123}
numpages = {6}
doi = {10.1109/TCAD.2004.839469}
abstract = {This paper describes methods and a framework for the refinement and analysis of control and signal processing systems. The design starts with an executable specification and its allowed deviations. Refinement steps introduce models of analog or digital implementations, and augment the "ideal" behavior with different sources of uncertainty, such as noise or production tolerances. The framework verifies and analyzes the influence of these uncertainties on system properties using affine arithmetic.}
}
@inproceedings{lee-gaf-men-luk-05-aa-bitwidth,
author = {Dong-U Lee and Altaf Abdul Gaffar and Oskar Mencer and Wayne Luk}
title = {{MiniBit}: {Bit}-Width Optimization via Affine Arithmetic}
publisher = {Association for Computing Machinery}
fullhash = {de29784b176d82108a112d6413c0ad8b}
booktitle = {Proceedings of the 42nd Annual Design Automation Conference (DAC'05)}
location = {Anaheim, California, USA}
isbn = {1595930582}
year = {2005}
pages = {837-840}
numpages = {4}
doi = {10.1145/1065579.1065799}
\keyw{FPGA,simulated,bit-width,annealing,affine arithmetic,fixed-point}
abstract = {MiniBit, our automated approach for optimizing bit-widths of fixed-point designs is based on static analysis via affine arithmetic. We describe methods to minimize both the integer and fraction parts of fixed-point signals with the aim of minimizing circuit area. Our range analysis technique identifies the number of integer bits required. For precision analysis, we employ a semi-analytical approach with analytical error models in conjunction with adaptive simulated annealing to find the optimum number of fraction bits. Improvements for a given design reduce area and latency by up to 20{\%} and 12{\%} respectively, over optimum uniform fraction bit-widths on a Xilinx Virtex-4 FPGA.}
}
@inproceedings{gri-heu-wal-04-aa-semisym,
author = {Ch. Grimm and W. Heupke and K. Waldschmidt}
title = {Semi-Symbolic Modeling and Analysis of Noise in Heterogeneous Systems}
publisher = {ECSI}
fullhash = {81681970c0862125da40d2c14de8ffdd}
booktitle = {Proc. Forum on Specification and Design Languages (FDL 2004)}
location = {Lille, France}
month = {9}
year = {2004}
pages = {56-68}
numpages = {13}
abstract = {The article describes semi-symbolic methods for the analysis of control and signal processing systems, including static and dynamic uncertainties. This above mentioned semi-symbolic description of uncertainties is based upon affine arithmetic. A short introduction to affine arithmetic is given. As affine arithmetic is only able to describe static uncertainties, an extension for effects of dynamic uncertainties is described and its feasibility is demonstrated by an example that delivers the frequency dependent noise of an output stage of a delta-sigma converter.}
}
@phdthesis{miz-04-aa-nonex-thesis,
author = {Yoshio Mizuguchi}
title = {Evaluation of Removal Performance of Non-existence Region of Solution using Affine Arithmetic}
institution = {Waseda University}
fullhash = {a58beb2718d7c7e617274cc3ff317687}
note = {In Japanese}
year = {2004}
abstract = {[Google transtalion:] One of the methods for finding the solution of an equation such as a curve or a curved surface is to determine a region where a solution does not exist and remove the region. At this time, the interval arithmetic is used, but the interval arithmetic has a drawback that the obtained interval width is often extremely wide, so that the determination performance is not always good. Affine Arithmetic proposed by Stolfi is a method of interval arithmetic, and by converting a given interval into a format called Affine format, it is possible to perform calculations considering the correlation between variables. Therefore, it is possible to suppress an increase in the section width as in the conventional case. This time, in order to evaluate the removal performance of the non-existent region of the solution, we decided to verify it by using a concrete example.}
}
@techreport{cha-vor-pue-hon-04-aa-fixpt,
author = {Lawrence J. Chang and Yevgen Voronenko and Markus Pueschel and Inpyo Hong}
title = {Adaptive Mapping of Linear {DSP} Algorithms to Fixed-Point Arithmetic}
institution = {Carnegie-Mellon UNiversity}
fullhash = {063a9d4a9cade385888a16681589b786}
month = {9}
year = {2004}
pages = {25}
numpages = {1}
abstract = {Embedded DSP digital signal processing applications are typically implemented using fixed point arithmetic -- in hardware to reduce area requirements and increase throughput, but also in software since most embedded processors do not provide floating point arithmetic. Consequently, the developer is confronted with the difficult task of deciding on the fixed point format, i.e., the number of integer and fractional bits to avoid overflow and ensure sufficient accuracy. For software implementations, the entire bitwidth is fixed, typically at 32, which means that increasing the representable range number of integer bits reduces the available accuracy number of fractional bits and vice-versa. In this paper we present a compiler that translates a floating point C implementation of a linear DSP kernel, such as a discrete Fourier or wavelet transform, into a high accuracy fixed point C implementation. The inputs to the compiler are a floating point arithmetic C program and the range of the input vector elements. First, the compiler statically analyzes the program in a single pass using a recently developed tool that uses affine arithmetic modeling. Then, in the global mode, the compiler determines the global fixed point format with the least number of integer bits and thus the highest accuracy that guarantees to avoid overflow and outputs the corresponding code. More interesting is the local mode, in which the compiler determines the best format independently for each variable, thus further pushing the possible accuracy. The compiler is currently limited to straightline code an extension to loop code is in development.}
}
@techreport{ceb-krei-lon-saa-lud-bar-ngu-04-aa-compsec,
author = {Martine C. Ceberio and V. Kreinovich and L. Longpr{\'{e}} and E. Saad and B. Ludaescher and C. Baral and H. T. Nguyen}
title = {Affine Arithmetic-Type Techniques for Handling Uncertainty in Expert Systems, with Applications to Geoinformatics and Computer Security}
institution = {University of Texas at El Paso}
fullhash = {987adb65bae351688b9c77c6e7585bde}
note = {Withdrawn}
year = {2004}
abstract = {Accepted to SCAN'2004.}
}
@article{ipp-vac-vil-20-aa-transf,
author = {Lucio Ippolito and Alfredo Vaccaro and Domenico Villacci}
title = {The Use of Affine Arithmetic for Thermal State Estimation of Substation Distribution Transformers}
fullhash = {e17a9202f1871e35e7cfa5a87735f4a9}
journal = {COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering}
publisher = {Emerald}
number = {1}
volume = {23}
year = {2004}
abstract = {Thermal protection of mineral-oil-filled substation distribution transformers is of critical importance in power systems. The failure of such a transformer is a matter of significant concern for electrical utilities, not only for the consequent severe economic losses, but also because the utility response to a customer during outage condition is one of the major factors in determining the overall customer attitude towards the utility. Therefore, it is essential to estimate the thermal state of transformers during load cycling and, in presence of overload conditions, to evaluate the need to reduce the load current or to install another transformer bay. A method of solving the transformer's thermal model, considering explicitly the source of uncertainty affecting its parameters, is required. In this paper, such an activity is developed by an interval-based approach, which provides the calculation of the inner and outer solution in the hot-spot temperature or top-oil temperature estimation process, keeping track of correlation between uncertain quantities.}
}
@inproceedings{lop-caf-car-nie-04-aa-cycles,
author = {J. A. L{\'{o}}pez and G. Caffarena and C. Carreras and O. Nieto-Taladriz}
title = {Analysis of limit cycles by means of affine arithmetic computer-aided tests}
fullhash = {e974125cd24c6659835573591d416da6}
booktitle = {2004 12th European Signal Processing Conference}
location = {Vienna, Austria}
year = {2004}
pages = {991-994}
numpages = {4}
abstract = {Modern analysis of the finite word-length properties of DSP algorithms is typically carried out using CAD tools. The objectives of this paper are threefold: (i) to show that affine arithmetic is specially well-suited for quantization analysis of digital filters; (ii) to present a new tool based on multiple-precision affine arithmetic elements that allows fast characterization of the signal ranges in linear systems; and (iii) to develop an exhaustive-search algorithm that detects limit cycles and requires less computation time than existing approaches. Final comparative results confirm the validity of the limit cycles detection algorithm on second-order shift and delta-operator realizations.}
}
@article{miy-kas-04-aa-impdiv,
author = {Shinya Miyajima and Masahide Kashiwagi}
title = {A Dividing Method Utilizing the Best Multiplication in Affine Arithmetic}
fullhash = {23f720934d733e451fa0ea8c7f7f6e88}
journal = {IEICE Electronics Express}
number = {7}
volume = {1}
year = {2004}
pages = {176-181}
numpages = {6}
doi = {10.1587/elex.1.176}
abstract = {Affine arithmetic (AA) is a variant of interval arithmetic. In AA, it is difficult to realize the efficient nonlinear binomial operations. The purpose of this paper is to propose a new dividing method which consists of a reciprocal and the best multiplication. And the performances of the new method and the conventional method are compared through some numerical examples.}
}
@inproceedings{gri-heu-wal-04-aa-refsig,
author = {Ch. Grimm and W. Heupke and K. Waldschmidt}
title = {Refinement of mixed-signals systems with affine arithmetic}
fullhash = {81681970c0862125da40d2c14de8ffdd}
booktitle = {Proceedings Design, Automation and Test in Europe Conference and Exhibition}
volume = {1}
year = {2004}
pages = {372-377 Vol.1}
numpages = {-1}
doi = {10.1109/DATE.2004.1268875}
abstract = {This paper describes a framework for the refinement of control and signal processing functions. The design starts with an executable specification, and allowed deviations thereof. Refinement steps introduce models of analog or digital implementations, and augment the 'ideal' behavior with different sources of uncertainty. The framework verifies and analyzes the influence of these uncertainties on system properties using affine arithmetic.}
}
@article{pic-vac-vil-04-aa-thermpow,
author = {Antonio Piccolo and Alfredo Vaccaro and Domenico Villacci}
title = {Thermal Rating Assessment of Overhead Lines by Affine Arithmetic}
fullhash = {8aefaa4ea53cd56b0715d6b391c51ac8}
journal = {Electric Power Systems Research}
number = {3}
volume = {71}
year = {2004}
pages = {275-283}
numpages = {9}
doi = {10.1016/j.epsr.2004.01.018}
abstract = {The increasing complexity of transmission networks can raise significantly the load flows during and following serious system disturbances. In this context accurate thermal rating assessment of overhead lines represents an essential requirement in order to obtain a sensible increase of the infrastructure exploitation assuring at the same time a reliable functioning of the power networks. This necessitates reliable calculation models which should be able to predict the thermal behaviour on short and long time horizons and in the presence of data uncertainties deriving from several sources such as non stationary load and fluctuating operating conditions. To address this problem, in the paper the use of Affine Arithmetic, an enhanced model for numerical computation, is proposed. Using AA, the thermal rating solution is computed taking into account the model parameter uncertainty interdependencies as well as the diversity of uncertainty sources. Simulation studies are presented and discussed in order to prove the effectiveness of the proposed methodology in addressing the problem of uncertainty analysis in both static and dynamic thermal rating assessment.}
}
@article{ned-kre-sta-04-aa-others,
author = {Nedialko S. Nedialkov and Vladik Kreinovich and Scott A. Starks}
title = {Interval Arithmetic, Affine Arithmetic, {Taylor} Series Methods: {Why}, What Next?}
fullhash = {32addac2c289d1de0b5028b934134a0d}
journal = {Numerical Algorithms}
volume = {37}
year = {2004}
pages = {325-336}
numpages = {12}
doi = {10.1023/B:NUMA.0000049478.42605.cf}
abstract = {In interval computations, the range of each intermediate result $r$ is described by an interval $\mathbf{r}$. To decrease excess interval width, we can keep some information on how $r$ depends on the input $x=(x_1,...,x_n)$. There are several successful methods of approximating this dependence; in these methods, the dependence is approximated by linear functions (affine arithmetic) or by general polynomials (Taylor series methods). Why linear functions and polynomials? What other classes can we try? These questions are answered in this paper.}
}
@inproceedings{fan-rut-che-03-aa-bitwd,
author = {C.F. Fang and R.A. Rutenbar and Tsuhan Chen}
title = {Fast, Accurate Static Analysis for Fixed-Point Finite-Precision Effects in {DSP} Designs}
fullhash = {609bddbda0a40969a6997febe7482575}
booktitle = {Proc. International Conference on Computer Aided Design (ICCAD-2003)}
month = {11}
year = {2003}
pages = {275-282}
numpages = {8}
doi = {10.1109/ICCAD.2003.159701}
abstract = {Translating digital signal processing (DSP) software into its finite-precision hardware implementation is often a time-consuming task. We describe a new static analysis technique that can accurately analyze finite-precision effects arising from fixed-point implementations of DSP algorithms. The technique is based on recent interval representation methods from affine arithmetic, and the use of new probabilistic bounds. The resulting numerical error estimates are comparable to detailed statistical simulation, but achieve speedups of four to five orders of magnitude by avoiding actual bit-true simulation. We show error analysis results on both feed forward and feedback DSP kernels.}
}
@article{miy-03-aa-muldiv,
author = {S. Miyajima}
title = {On the Multiplicaton and the Division of the Affine Arithmetic}
fullhash = {5dc0e2775e8a17e8e49a9e3d9082971c}
journal = {IEICE Transactions on Fundamentals}
number = {3}
volume = {J86-A}
year = {2003}
pages = {232-240}
numpages = {9}
urlraw = {https://ci.nii.ac.jp/naid/10011629913/en/}
url = {https://ci.nii.ac.jp/naid/10011629913/en/}
}
@inproceedings{heu-gri-wal-03-aa-mixsig,
author = {Wilhelm Heupke and Christoph Grimm and Klaus Waldschmidt}
title = {A New Method for Modeling and Analysis of Accuracy and Tolerances in Mixed-Signal Systems}
publisher = {ECSI}
fullhash = {9d095d81ee1e0337164a9abde75e7806}
booktitle = {Proc. Forum on specification and Design Languages (FDL 2003)}
location = {Frankfurt, Germany}
month = {9}
year = {2003}
pages = {82-90}
numpages = {9}
urlraw = {http://www.systemc-ams.org/documents/fdl2003_heupke.pdf}
url = {http://www.systemc-ams.org/documents/fdl2003_heupke.pdf}
abstract = {Tolerances are a very important property of a design. This paper presents a method for simulating tolerances in signal processing and control systems on the system level using affine arithmetic. The method is compard to simulation based on interval arithmetic or Monte-Carlo method and goes into details of the shortcomings of interval arithmetic. An introduction to affine arithmetic is given and the corresponding model of tolerances is described. The concept of the implementation is given and a case study of a control loop modeled on the system level is presented. This small system is simulated using affine arithmetic and is based on an early prototype of SystemC-AMS.}
}
@inproceedings{pic-vac-vil-03-aa-thermpow-conf,
author = {A. Piccolo and A. Vaccaro and D. Villacci}
title = {An Affine Arithmetic Based Methodology for the Thermal Rating Assessment of Overhead Lines in the Presence of Data Uncertainty}
fullhash = {f745e478df8debb28b4847a642f87351}
booktitle = {2003 IEEE Bologna Power Tech Conference Proceedings}
month = {6}
volume = {4}
year = {2003}
pages = {7}
numpages = {1}
doi = {10.1109/PTC.2003.1304743}
abstract = {The increasing complexity of transmission networks can raise significantly the load flows during and following serious system disturbances. In this context accurate thermal rating assessment of overhead lines represents an essential requirement in order to obtain a sensible increase of the infrastructure exploitation assuring at the same time a reliable functioning of the power networks. This necessitates reliable calculation models which should be able to predict the thermal behaviour on short and long time horizons and in the presence of data uncertainties deriving from several sources such as nonstationary load and fluctuating operating conditions. To address this problem, in the paper the use of Affine Arithmetic, an enhanced model for numerical computation, is proposed. Using AA, the thermal rating solution is computed taking into account the model parameter uncertainty interdependencies as well as the diversity of uncertainty sources. Simulation studies are presented and discussed in order to prove the effectiveness of the proposed methodology in addressing the problem of uncertainty analysis in both static and dynamic thermal rating assessment.}
}
@article{miy-miy-kas-03-aa-newdiv,
author = {Shinya Miyajima and Takatomi Miyata and Masahide Kashiwagi}
title = {A New Dividing Method in Affine Arithmetic}
fullhash = {b533a2949ac7620f6b228c29c9244b38}
journal = {IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences}
month = {9}
number = {9}
volume = {E86-A}
year = {2003}
pages = {2192-2196}
numpages = {5}
abstract = {Affine arithmetic is a kind of interval arithmetic defined by Stolfi et al. In affine arithmetic, it is difficult to realize the efficient nonlinear binomial operations. The purpose of this letter is to propose a new dividing method which is able to supply more suitable evaluation than the old dividing method. And this letter also shows the efficiency of the new dividing method by numerical examples.}
}
@inproceedings{sho-lin-mar-wan-03-aa-modif,
author = {Huaohao Shou and Hongwei Lin and Ralph Martin and Guojin Wang}
title = {Modified Affine Arithmetic Is More Accurate than Centered Interval Arithmetic or Affine Arithmetic}
publisher = {Springer}
fullhash = {52714653327b46e579299aba11c9b283}
booktitle = {Mathematics of Surfaces: Proc. 10th IMA International Conference}
location = {Leeds, UK}
series = {LNCS}
volume = {2768}
year = {2003}
pages = {355-365}
numpages = {11}
doi = {10.1007/978-3-540-39422-8_23}
abstract = {In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a box-shaped region: (i) using interval arithmetic in centered form is always more accurate than standard affine arithmetic, and (ii) modified affine arithmetic is always more accurate than interval arithmetic in centered form. Test results show that modified affine arithmetic is not only more accurate but also much faster than standard affine arithmetic. We thus suggest that modified affine arithmetic is the method of choice for evaluating algebraic functions, such as implicit surfaces, over a box.}
}
@misc{gay-03-aa-lib-cpp-fr,
author = {Olivier Gay}
title = {{Libaa} - Une Librairie {C}$++$ d' \guillemotleft{} Affine Arithmetic \guillemotright{}}
institution = {{\'{E}}cole Polithecnique F{\'{e}}d{\'{e}}rale de Lausanne}
fullhash = {256c570126c65f1b6f4035ad62dd3357}
howpublished = {Online document}
month = {2}
year = {2003}
urlraw = {https://www.nongnu.org/libaffa/libaffa.pdf}
url = {https://www.nongnu.org/libaffa/libaffa.pdf}
}
@inproceedings{fan-che-rut-03-aa-fperr,
author = {C. F. Fang and T. Chen and R. Rutenbar}
title = {Floating-point Error Analysis Based on Affine Arithmetic}
fullhash = {47f1c904455024c0a6abd0b54c4b4f98}
booktitle = {Proc. 2003 International Conf. on Acoustic, Speech and Signal Processing}
year = {2003}
doi = {10.1109/ICASSP.2003.1202428}
abstract = {During the development of floating-point signal processing systems, an efficient error analysis method is needed to guarantee the output quality. We present a novel approach to floating-point error bound analysis based on affine arithmetic. The proposed method not only provides a tighter bound than the conventional approach, but also is applicable to any arithmetic operation. The error estimation accuracy is evaluated across several different applications which cover linear operations, nonlinear operations, and feedback systems. The accuracy decreases with the depth of computation path and also is affected by the linearity of the floating-point operations.}
}
@misc{fan-rut-che-03-aa-dspround-draft,
author = {Claire Fang Fang and Rob A. Rutenbar and Tsuhan Chen}
title = {Efficient Static Analysis of Fixed-Point Error in {DSP}: {Applications} via Affine Arithmetic Modeling}
fullhash = {da080ea37d7bc13ec2f27320e9e571b6}
howpublished = {Draft of article, online}
urlraw = {http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.6.4363&rep=rep1&type=pdf}
url = {http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.6.4363&rep=rep1&type=pdf}
abstract = {Translating digital signal processing (DSP) software into its finite-precision hardware implementation is often a timeconsuming task. We describe a new static analysis technique that can accurately analyze finite-precision effects arising from fixed-point implementations of DSP algorithms. The technique is based on recent interval representation methods from affine arithmetic, and the use of new probabilistic bounds. The resulting numerical error estimates are comparable to detailed statistical simulation, but achieve speedups of four to five orders of magnitude by avoiding actual bittrue simulation. We show error analysis results on several common DSP kernels.}
}
@inproceedings{fan-rut-pus-che-03-aa-dspround,
author = {Claire Fang Fang and Rrob A. Rutenbar and Markus P\"{u}schel and Tsuhan Chen}
title = {Toward efficient static analysis of finite-precision effects in DSP applications via affine arithmetic modeling}
fullhash = {ed3d02bf6ece28b6b5ff3b7417fbaf70}
booktitle = {Proc. 40th annual Design Automation Conference (DAC'03)}
year = {2003}
doi = {10.1145/775832.775960}
abstract = {We introduce a static error analysis technique, based on smart interval methods from affine arithmetic, to help designers translate DSP codes from full-precision floating-point to smaller finite-precision formats. The technique gives results for numerical error estimation comparable to detailed simulation, but achieves speedups of three orders of magnitude by avoiding actual bit-level simulation. We show results for experiments mapping common DSP transform algorithms to implementations using small custom floating point formats.}
}
@article{mar-sho-voi-bow-wan-02-aa-compar,
author = {Ralph Martin and Huahao Shou and Irina Voiculescu and Adrian Bowyer and Guojin Wang}
title = {Comparison of Interval Methods for Plotting Algebraic Curves}
fullhash = {ebb757e388004a733327c07d6a10789d}
journal = {Computer Aided Geometric Design}
number = {7}
volume = {19}
year = {2002}
pages = {553-587}
numpages = {35}
doi = {10.1016/S0167-8396(02)00146-2}
abstract = {This paper compares the performance and efficiency of different function range interval methods for plotting f(x,y)=0 on a rectangular region based on a subdivision scheme, where f(x,y) is a polynomial. The solution of this problem has many applications in CAGD. The methods considered are interval arithmetic methods (using the power basis, Bernstein basis, Horner form and centred form), an affine arithmetic method, a Bernstein coefficient method, Taubin's method, Rivlin's method, Gopalsamy's method, and related methods which also take into account derivative information. Our experimental results show that the affine arithmetic method, interval arithmetic using the centred form, the Bernstein coefficient method, Taubin's method, Rivlin's method, and their related derivative methods have similar performance, and generally they are more accurate and efficient than Gopalsamy's method and interval arithmetic using the power basis, the Bernstein basis, and Horner form methods.}
}
@inbook{bow-mar-sho-voi-02-aa-csg,
author = {Adrian Bowyer and Ralph Martin and Huahao Shou and Irina Voiculescu}
title = {Affine Intervals in a CSG Geometric Modeller}
publisher = {Springer}
fullhash = {97459e67aaac0c778e2fb7fc11b53456}
booktitle = {Uncertainty in Geometric Computations}
chapter = {1}
isbn = {978-1-4615-0813-7}
year = {2002}
pages = {1-14}
numpages = {14}
doi = {10.1007/978-1-4615-0813-7_1}
urlraw = {https://doi.org/10.1007/978-1-4615-0813-7_1}
url = {https://doi.org/10.1007/978-1-4615-0813-7_1}
abstract = {Our CSG modeller, svLis, uses interval arithmetic to categorize implicit functions representing primitive shapes against boxes; this allows an efficient implementation of recursive spatial division to localize the primitives for a variety of purposes, such as rendering or the computation of integral properties.}
}
@inbook{sho-mar-wan-voi-bow-02-aa-curve,
author = {Huahao Shou and Ralph Martin and Guojin Wang and Irina Voiculescu and Adrian Bowyer}
title = {Affine Arithmetic and {Bernstein} Hull Methods for Algebraic Curve Drawing}
publisher = {Springer}
fullhash = {38f2642d57cf38f0baa05209d6135777}
booktitle = {Uncertainty in Geometric Computations}
chapter = {12}
isbn = {978-1-4615-0813-7}
year = {2002}
pages = {143-154}
numpages = {12}
doi = {10.1007/978-1-4615-0813-7_12}
abstract = {We compare approaches to the location of the algebraic curve $f(x,y) = 0$ in a rectangular region of the plane, based on recursive use of conservative estimates of the range of the function over a rectangle. Previous work showed that performing interval arithmetic in the Bernstein basis is more accurate than using the power basis, and that affine arithmetic in the power basis is better than using interval arithmetic in the Bernstein basis. This paper shows that using affine arithmetic with the Bernstein basis gives no advantage over affine arithmetic with the power basis. It also considers the Bernstein coefficient method based on the convex hull property, which has similar performance to affine arithmetic.}
}
@misc{kre-ber-02-aa-selfval,
author = {V. Kreinovich and D. J. Berleant}
title = {Interval Computations}
fullhash = {4d64560d86b948b4989ed20dd02c4376}
howpublished = {WWW document}
year = {2002}
urlraw = {http://www.cs.utep.edu/interval-comp/main.html}
url = {http://www.cs.utep.edu/interval-comp/main.html}
}
@techreport{gol-vog-met-02-aa-cue-tr,
author = {Siome Goldenstein and Christian Vogler and Dimitris Metaxas}
title = {Cue Integration using Affine Arithmetic and Gaussians}
institution = {University of Pennsylvania}
fullhash = {7471d32d2fe43e7473b8f5126a2f2cf2}
number = {MS-CIS-02-06}
year = {2002}
abstract = {In this paper we describe how the connections between affine forms, zonotopes, and Gaussian distributions help us devise an automated cue integration technique for tracking deformable models. This integration technique is based on the confidence estimates of each cue. We use affine forms to bound these confidences. Affine forms represent bounded intervals, with a well-defined set of arithmetic operations. They are constructed from the sum of several independent components. An n-dimensional affine form describes a complex convex polytope, called a zonotope. Because these components lie in bounded intervals, Lindeberg's theorem, a modified version of the central limit theorem,can be used to justify a Gaussian approximation of the affine form. We present a new expectation-based algorithm to find the best Gaussian approximation of an affine form. Both the new and the previous algorithm run in $O(n 2m)$ time, where $n$ is the dimension of the affine form, and $m$ is the number of independent components. The constants in the running time of new algorithm, however, are much smaller, and as a result it runs 40 times faster than the previous one for equal inputs. We show that using the Berry-Esseen theorem it is possible to calculate an upper bound for the error in the Gaussian approximation. Using affine forms and the conversion algorithm, we create a method for automatically integrating cues in the tracking process of a deformable model. The tracking process is described as a dynamical system, in which we model the force contribution of each cue as an affine form. We integrate their Gaussian approximations using a Kalman filter as a maximum likelihood estimator. This method not only provides an integrated result that is dependent on the quality of each on of the cues, but also provides a measure of confidence in the final result. We evaluate our new estimation algorithm in experiments, and we demonstrate our deformable model-based face tracking system as an application of this algorithm.}
}
@inproceedings{lem-hed-bar-02-aa-circ,
author = {Andreas Lemke and Lars Hedrich and Erich Barke}
title = {Analog Circuit Sizing Based on Formal Methods using Affine Arithmetic}
fullhash = {b2f756d93f43606fb308b567801bf85c}
booktitle = {Proc. ICCAD-2002 - International Conference on Computer Aided Design}
location = {San Jose, California}
month = {11}
year = {2002}
pages = {486-489}
numpages = {4}
doi = {10.1145/774572.774643}
abstract = {We present a novel approach to optimization-based variation-tolerant analog circuit sizing. Using formal methods based on affine arithmetic, we calculate guaranteed bounds on the worst-case behavior and deterministically find the global optimum of the sizing problem by means of branch-and-bound optimization. To solve the nonlinear circuit equations with parameter variations, we define a novel affine-arithmetic Newton operator that gives a significant improvement in computational efficiency over an implementation using interval arithmetic. The calculation of guaranteed worst-case bounds and the global optimization are demonstrated by a prototype implementation.}
}
@inproceedings{kan-ois-02-aa-ode,
author = {Yuchi Kanazawa and {Shin'ichi} Oishi}
title = {A Numerical Method of Proving the Existence of Solutions for Nonlinear {ODEs} using Affine Arithmetic}
fullhash = {532f9862e44a63ac736e6f0291707584}
booktitle = {Proc. SCAN'02 -- 10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics}
month = {9}
year = {2002}
}
@article{sho-mar-voi-02-aa-matr,
author = {Huahao Shou and Ralph Martin and Irina Voiculescu and Adrian Bowyer and Guojin Wang}
title = {Affine Arithmetic in Matrix Form for Polynomial Evaluation and Algebraic Curve Drawing}
fullhash = {9fc60cbeeac12e27eeb8b043c741644e}
journal = {Progress in Natural Science}
month = {1}
note = {Some sources give only I. V. as author, others omit H. S. and G.W.}
number = {1}
volume = {12}
year = {2002}
pages = {77-81}
numpages = {5}
abstract = {This paper shows how tight bounds for the range of a bivariate polynomial can be found using a matrix method based on affine arithmetic. Then, this method is applied to drawing an algebraic curve with a hierarchical algorithm, which demonstrates that more accurate answers can be obtained more rapidly than using conventional interval arithmetic.}
}
@article{mes-02-aa-jucs,
author = {F. Messine}
title = {Extentions of Affine Arithmetic: {Application} to Unconstrained Global Optimization}
fullhash = {68058f090a609e3bf008855ed01087bd}
journal = {Journal of Universal Computer Science}
month = {11}
number = {11}
volume = {8}
year = {2002}
pages = {992-1015}
numpages = {24}
urlraw = {http://www.jucs.org/jucs_8_11/extentions_of_affine_arithmetic}
url = {http://www.jucs.org/jucs_8_11/extentions_of_affine_arithmetic}
}
@inproceedings{bue-02-aa-estim,
author = {Katja B{\"{u}}hler}
title = {Implicit Linear Interval Estimations}
publisher = {ACM Press}
fullhash = {7f583fc9561299cbcddbda1ab8aceaca}
booktitle = {Proc. 18th Spring Conference on Computer Graphics}
location = {Budmerice, Slovakia}
isbn = {1-58113-608-0}
year = {2002}
pages = {123-132}
numpages = {10}
doi = {http://doi.acm.org/10.1145/584458.584479}
}
@techreport{mes-01-aa-incl-tr,
author = {F. Messine}
title = {Theorem on Affine or Quadratic Inclusion Functions}
institution = {Universit{\'{e}} de Pau, D{\'{e}}partement d'Informatique}
fullhash = {68058f090a609e3bf008855ed01087bd}
month = {10}
number = {R2I 01-03}
year = {2001}
pages = {2}
numpages = {1}
}
@inproceedings{gol-vog-met-02-aa-cue-cv,
author = {Siome Goldenstein and Christian Vogler and Dimitris Metaxas}
title = {Affine Arithmetic Based Estimation of Cue Distributions in Deformable Model Tracking}
fullhash = {7471d32d2fe43e7473b8f5126a2f2cf2}
booktitle = {Proc. 2001 Conference on Computer Vision and Pattern Recognition (CVPR 2001)}
isbn = {0-7695-1272-0}
month = {12}
note = {Not in the Proceedings' table of contents?}
year = {2001}
pages = {???-???}
numpages = {-1}
abstract = {In this paper we describe a statistical method for the integration of an unlimited number of cues within a deformable model framework. We treat each cue as a random variable, each of which is the sum of a large number of local contributions with unknown probability distribution functions. Under the assumption that these distributions are independent, the overall distributions of the generalized cue forces can be approximated with multidimensional Gaussians, as per the central limit theorem. Estimating the covariance matrix of these Gaussian distributions, however, is difficult, because the probability distributions of the local contributions are unknown. We use affine arithmetic as a novel approach toward overcoming these difficulties. It lets us track and integrate the support of bounded distributions without having to know their actual probability distributions, and without having to make assumptions about their properties. We present a method for converting the resulting affine forms into the estimated Gaussian distributions of the generalized cue forces. This method scales well with the number of cues. We apply a Kalman filter as a maximum likelihood estimator to merge all Gaussian estimates of the cues into a single best fit Gaussian. Its mean is the deterministic result of the algorithm, and its covariance matrix provides a measure of the confidence in the result. We demonstrate in experiments how to apply this framework to improve the results of a face tracking system.}
}
@inproceedings{kik-kas-01-aa-ode,
author = {Tomoyuki Kikuchi and Masahide Kashiwagi}
title = {Elimination of Non-existence Regions of the Solution of Nonlinear Equations using Affine Arithmetic}
fullhash = {d6cacfc6e8d52ce2f85b858e29b7ab51}
booktitle = {Proc. International Symposium on Nonlinear Theory and its Applications (NOLTA 2001)}
month = {10}
year = {2001}
pages = {223-226}
numpages = {4}
}
@inproceedings{miy-kas-01-aa-poly,
author = {Takatomi Miyata and Masahide Kashiwagi}
title = {On Range Evaluation of Polynomials of Affine Arithmetic}
fullhash = {876558803ef4b00cad4fd35a145a0ea0}
booktitle = {Proc. NOLTA'01 - 2001 International Symposium on Nonlinear Theory and its Applications}
month = {10}
year = {2001}
}
@inproceedings{mar-sho-voi-wan-01-aa-comp,
author = {Ralph Martin and Huahao Shou and Irina Voiculescu and Guojin Wang}
title = {A Comparison of {Bernstein} Hull and Affine Arithmetic Methods for Algebraic Curve Drawing}
publisher = {Kluwer Academic Publishers}
fullhash = {981eefdfa51b38f05d662755ba336e78}
booktitle = {Proc. Uncertainty in Geometric Computations}
isbn = {0-7923-7309-X}
month = {7}
year = {2001}
pages = {143-154}
numpages = {12}
}
@inproceedings{bue-bar-00-aa-inter,
author = {Katja {B{\"{u}}hler} and Wilhelm Barth}
title = {A New Intersection Algorithm for Parametric Surfaces Based on Linear Interval Estimations}
fullhash = {e2762b5e25c947e4add01918d9b5fe27}
booktitle = {Proc. SCAN 2000 / Interval 2000 - 9th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics}
year = {2000}
pages = {???-???}
numpages = {-1}
}
@inproceedings{mic-00-aa-dyna,
author = {Dominique Michelucci}
title = {Reliable Computations for Dynamic Systems}
fullhash = {fed8d2213dce15125265914f669cdbdd}
booktitle = {Proc. SCAN 2000 / Interval 2000 - 9th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics}
year = {2000}
pages = {???-???}
numpages = {-1}
}
@misc{kur-00-aa-range,
author = {Toshiyuki Kurosaki}
title = {Range Evaluation of the Polynomial using Interval Arithmetic}
fullhash = {28ef6848673bfba76a429322dcf0ecc9}
howpublished = {WWW document}
year = {2000}
urlraw = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
url = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
}
@misc{kik-00-aa-ode,
author = {Tomoyuki Kikuchi}
title = {Elimination of Non-existence Region of the Solution of Nonlinear Equations using Affine Arithmetic}
fullhash = {9d1823afee87068ef4daf71dcbe1a93f}
howpublished = {WWW document}
year = {2000}
urlraw = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
url = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
}
@misc{shi-00-aa-round,
author = {Kouichi Shiota}
title = {On the Rounding Error of the Affine Arithmetic}
fullhash = {64ce9e3274f3e26fda9a028f71a03dbc}
howpublished = {WWW document}
year = {2000}
urlraw = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
url = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
}
@misc{miy-00-aa-div,
author = {Shinya Miyajima}
title = {On the Improvement of the Division of the Affine Arithmetic}
fullhash = {f6ec6a3eaf614995958c07c39c9761f9}
howpublished = {WWW document}
year = {2000}
urlraw = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
url = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
}
@article{fem-spa-00-aa-eletr,
author = {N. Femia and G. Spagnuolo}
title = {True Worst-Case Circuit Tolerance Analysis using Genetic Algorithm and Affine Arithmetic - {Part} {I}}
fullhash = {0020a21fa9bbe33df85791ec6d71bab3}
journal = {IEEE Transactions on Circuits and Systems}
month = {9}
number = {9}
volume = {47}
year = {2000}
pages = {1285-1296}
numpages = {12}
doi = {10.1109/81.883323}
abstract = {In this paper, a new approach to the calculation of the true worst case in circuit tolerance analysis (TWC-CTA) with parameters characterized by large uncertainties is presented. It is based on the joint use of genetic algorithms (GAs) and affine arithmetic (AA). The GAs are used to minimize the underestimation error which affects stochastic methods while the AA is applied to minimize the overestimation error which affects interval arithmetic methods in TWC-CTA problems. The joint GA-AA approach presented in this paper ensures a reliable TWC evaluation.}
}
@inproceedings{zha-mar-00-aa-polycurv,
author = {Q. Zhang and R. R. Martin}
title = {Polynomial Evaluation using Affine Arithmetic for Curve Drawing}
fullhash = {21f84fed7cc67ccf3f95bbc7cefc135e}
booktitle = {Proc. of Eurographics UK 2000 Conference}
isbn = {0-9521097-9-4}
year = {2000}
pages = {49-56}
numpages = {8}
}
@inproceedings{voi-ber-bow-mar-zha-00-aa-surf,
author = {Irina Voiculescu and Jakob Berchtold and Adrian Bowyer and Ralph R. Martin and Qijiang Zhang}
title = {Interval and Affine Arithmetic for Surface Location of Power- and {Bernstein}-form Polynomials}
publisher = {Springer}
fullhash = {2ddaa8be739dd95b0cba7436a2fbadec}
booktitle = {Proc. Mathematics of Surfaces IX}
isbn = {1-85233-358-8}
month = {9}
year = {2000}
pages = {410-423}
numpages = {14}
abstract = {This paper describes a problem of interest in CSG modelling, namely the location of implicit polynomial surfaces in space. It is common for surfaces defined by implicits to be located using interval arithmetic. However, the method only gives conservative bounds for the values of the function inside a region of interest. This paper gives two possible ways of producing tighter bounds. One involves using a Bernstein--form representation of the implicit polynomials used as input to the method. The other fine--tunes the method itself by employing careful use of affine arithmetic --- a more sophisticated version of interval arithmetic. As both methods contribute significant improvements, we speculate about combining the two into a fast and accurate method for surface location.}
}
@article{kan-ois-99-aa-ode-tr,
author = {Yuchi Kanazawa and {Shin'ichi} Oishi}
title = {A Numerical Method to Prove the Existence of Solutions for Nonlinear {ODEs} Using Affine Arithmetic}
\list{publisher}{2}{%
{Institute of Electronics, Information}%
{Communication Engineers}%
}
fullhash = {532f9862e44a63ac736e6f0291707584}
journal = {IEICE Technical Report}
month = {5}
note = {See also~\cite{kan-ois-02-aa-ode}}
number = {42}
volume = {99}
year = {1999}
pages = {39-44}
numpages = {6}
abstract = {In this paper, we shall propose a method to prove the existence of the solutions for nonlinear ordinary differential equa tions using affine arithmetic instead of using interval arithmetic. In the case using interval arithmetic, we must use Krawczyk-like operator, which is the mean value form for Newton operator,because of the disadvantage of interval arithmetic. Otherwise, we use affine arithmetic to Newton operator directly since affine arithmetic is effective to variables-dependent calculations. In other words, we need not Mean value theorem any more.}
}
@techreport{mes-99-aa-new-tr,
author = {F. Messine}
title = {New Affine Forms in Interval Branch and Bound Algorithms}
institution = {Universit{\'{e}} de Pau, D{\'{e}}partement d'Informatique}
fullhash = {68058f090a609e3bf008855ed01087bd}
month = {10}
note = {Submitted in Journal of Universal Computer Science}
number = {R2I-99-02}
year = {1999}
}
@inproceedings{kas-99-aa-div,
author = {Masahide Kashiwagi}
title = {On Division of the Affine Arithmetic}
fullhash = {043579a540ab675a207a5254db9be693}
booktitle = {Proc. NOLTA'99 - 1999 International Symposium on Nonlinear Theory and its Applications}
month = {11}
year = {1999}
}
@article{kas-99-aa-div-tr,
author = {Masahide Kashiwagi}
title = {On the Division of the Affine Arithmetic}
\list{publisher}{2}{%
{The Institute of Electronics, Information}%
{Communication Engineers}%
}
fullhash = {043579a540ab675a207a5254db9be693}
journal = {IEICE Technical Report}
month = {3}
note = {In Japanese. See also~\cite{kas-99-aa-div}}
number = {662}
volume = {98}
year = {1999}
pages = {23-29}
numpages = {7}
urlraw = {https://ci.nii.ac.jp/naid/110003292159/en/}
url = {https://ci.nii.ac.jp/naid/110003292159/en/}
abstract = {In this report we propose a new method for the division of the affine arithmetic. Affine arithmetic is an extension of the interval arithmetic and it can give narrower result than conventional interval arithmetic. In the affine arithmetic, operations for addition, subtraction, multiplication and 1-variable functions are known, but, operation for division is not known and so division is generally done by decomposing $x/y$ into $x\times (1/y)$. In this report we propose a direct operation for division of affine arithmetic. We also show that our method gives better result than the decomposition method.}
}
@mastersthesis{cus-99-aa-thesis,
author = {Affonso de {Cusatis Jr.}}
title = {Tra{\c{c}}ado de Raios de Superf{\'{\i{}}}cies Impl{\'{\i{}}}citas com Aritm{\'{e}}tica Afim}
institution = {Departamento de Inform{\'{a}}tica, Pontif{\'{\i{}}}cia Universidade Cat{\'{o}}lica do Rio de Janeiro}
fullhash = {eb9cb875b0828bf12b40361586f3c839}
month = {4}
note = {(in Portuguese)}
year = {1999}
}
@inproceedings{cus-fig-gat-99-aa-rt,
author = {Affonso de {Cusatis Jr.} and Luiz H. Figueiredo and Marcelo Gattass}
title = {Interval Methods for Ray Casting Surfaces with Affine Arithmetic}
fullhash = {1a714aa34c43bb764eee54d813f19e20}
booktitle = {Proceedings of SIBGRAPI'99 - the 12th Brazilian Symposium on Computer Graphics and Image Processing}
year = {1999}
pages = {65-71}
numpages = {7}
doi = {10.1109/SIBGRA.1999.805711}
abstract = {We study the performance of affine arithmetic as a replacement for interval arithmetic in interval methods for ray casting implicit surfaces. Affine arithmetic is a variant of interval arithmetic designed to handle the dependency problem, and which has improved several interval algorithms in computer graphics.}
}
@misc{kaw-99-aa-sol,
author = {Osamu Kawakami}
title = {On All Solution Algorithms using Affine Arithmetic}
fullhash = {79efc67a40d7ada222fa56da8fe64bad}
howpublished = {WWW document}
year = {1999}
urlraw = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
url = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
}
@article{sak-kas-98-aa-allsol-tr,
author = {Takayu Sakiyama and Masahide Kashiwagi}
title = {An Algorithm for Finding All Solutions Using Affine Arithmetic}
fullhash = {22924dce4795dde37535bc8a5433521a}
journal = {IEICE Technical Report}
month = {2}
note = {In Japanese.}
number = {532}
volume = {97}
year = {1998}
pages = {49-55}
numpages = {7}
abstract = {This paper proposes a method for finding all solutions of nonlinear equations with guaranteed accuracy, which is improved by using affine arithmetic. The feature of affine arithmetic is that it keeps the correlation between variables(input and intermediate variables), unlike interval arithmetic. Affine arithmetic is superior to interval arithmetic in point of providing tighter intervals. Therefore the algorithm using affine arithmetic can remove the nonexistent regions of the solution more efficiently than using interval arithmetic. Some numerical examples show the efficiency of the proposed algorithm.}
}
@inproceedings{kas-98-aa-allsol,
author = {M. Kashiwagi}
title = {An All Solution Algorithm using Affine Arithmetic}
fullhash = {e3a3b4ae6e94552118f33c291c90150b}
booktitle = {NOLTA'98 - 1998 International Symposium on Nonlinear Theory and its Applications}
location = {Crans-Montana, Switzerland}
month = {9}
note = {(withdrawn?)}
year = {1998}
pages = {14-17}
numpages = {4}
}
@inproceedings{egi-fem-spa-98-aa-cir,
author = {L. Egiziano and N. Femia and G. Spagnuolo}
title = {New Approaches to the True Worst-Case Evaluation in Circuit Tolerance and Sensitivity Analysis -- {Part} {II}: {Calculation} of the Outer Solution using Affine Arithmetic}
fullhash = {121e8ebf0c34246a722f7272d200e9a6}
booktitle = {Proc. COMPEL'98 - 6th Workshop on Computer in Power Electronics}
location = {Villa Erba, Italy}
month = {7}
year = {1998}
pages = {19-22}
numpages = {4}
abstract = {For pt. I see ibid., p.133-9 (1998). In this paper a new approach to the calculation of the true worst-case in circuit tolerance analysis (TWC-CTA) based on affine arithmetic is presented. The method proposed solves the problem of maximizing the intrinsic overestimation error which affects interval arithmetic methods in the calculation of the TWC in CTA problems with parameters characterized by large tolerances. The joint application of the technique presented in this paper with that one presented in part I for the calculation of an underestimated solution ensures a reliable and efficient TWC evaluation.}
}
@inproceedings{mes-mah-98-aa-mopt,
author = {F. Messine and A. Mahfoudi}
title = {Use of Affine Arithmetic in Interval Optimization Algorithms to Solve Multidimensional Scaling Problems}
fullhash = {88a9cbdf8feb7141effb90ddeddfe40a}
booktitle = {Proc. SCAN'98 - IMACS/GAMM International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics}
location = {Budapest, Hungary}
month = {9}
note = {To appear in {\em Reliable Computing}.}
year = {1998}
pages = {22-25}
numpages = {4}
}
@article{hei-slu-sei-98-aa-shader,
author = {W. Heidrich and Ph. Slusallek and Hans-Peter Seidel}
title = {Sampling Procedural Shaders using Affine Arithmetic}
fullhash = {5ff3deac8efb4e5b425b76ffff43f620}
journal = {ACM Transactions on Graphics (TOG)}
month = {7}
number = {3}
volume = {17}
year = {1998}
pages = {158-176}
numpages = {19}
}
@misc{nag-98-aa-sol,
author = {Yuuichirou Nagashima}
title = {An Effective Method of Division of Affine Arithmetic}
fullhash = {20a48cbe5ef49f5ff918a14d245bf242}
howpublished = {WWW document}
year = {1998}
urlraw = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
url = {http://www.kashi.info.waseda.ac.jp/Non-linear/thesis-e.html}
}
@article{oku-yas-tok-98-aa-segint-tr,
author = {Hisanobu Okuda and Seigo Yasutome and Nobuki Tokura}
title = {Decision of Segments Intersection under Input Data Expressed in Affine Arithmetic}
\list{publisher}{2}{%
{IEICE - Institute of Electronics, Information}%
{Communication Engineers}%
}
fullhash = {e530c3d411fa09a288cb0f45d0aa725d}
journal = {IEICE Technical Report}
month = {6}
note = {See also~\cite{oku-yas-tok-98-aa-segint}}
number = {137}
volume = {98}
year = {1998}
pages = {23-30}
numpages = {8}
urlraw = {https://ci.nii.ac.jp/naid/110003191662/en/}
url = {https://ci.nii.ac.jp/naid/110003191662/en/}
abstract = {The computational geometry algorithms which give consideration to imprecise input don't assume a correlations between the input errors. If a predicate which decide a geometrical property keep track of the errors, it can give more precise information. In this paper, we propose a method which gives consideration of a correlation of input errors in 1st degree, using Affine Arithmetic and show how a result of a decision of segments intersection is affected by the consideration of correlation of input errors and the range's overestimation caused by Affine Arithmetic, comparing with a current method.}
}
@inproceedings{oku-yas-tok-98-aa-segint,
author = {Hisanobu Okuda and Seigo Yasutome and Nobuki Tokura}
title = {Decision of Segments Intersection under Input Data Expressed in Affine Arithmetic}
\list{publisher}{2}{%
{IEICE - Institute for Electronics, Information}%
{Communication Engineers of Japan}%
}
fullhash = {e530c3d411fa09a288cb0f45d0aa725d}
booktitle = {Proc. of TGCOMP Meeting - Technical Group on foundation of COMPuting}
note = {See also~\cite{oku-yas-tok-98-aa-segint-tr}}
year = {1998}
}
@techreport{hei-97-aa-libs-tr,
author = {W. Heidrich}
title = {A Compilation of Affine Arithmetic Versions of Common Math Library Functions}
institution = {Universit{\"{a}}t Erlangen-N{\"{u}}rnberg}
fullhash = {70bd1b144bff2ea027bcdaa06d3a5fb4}
number = {3}
year = {1997}
}
@techreport{hei-slu-sei-96-aa-shtr,
author = {W. Heidrich and Ph. Slusallek and Hans-Peter Seidel}
title = {Sampling Procedural Shaders using Affine Arithmetic}
institution = {Lehrstuhl f{\"{u}}r Informatik IX, Friedrich-Alexander-Universit{\"{a}}t, Erlangen-N{\"{u}}rnberg}
fullhash = {5ff3deac8efb4e5b425b76ffff43f620}
month = {12}
number = {11/1996}
year = {1996}
}
@inproceedings{fig-96-aa-surfint,
author = {L. H. {de Figueiredo}}
title = {Surface Intersection using Affine Arithmetic}
fullhash = {f0c91cb3bc748d828da19ddfb4a63f6c}
booktitle = {Proc. Graphics Interface '96}
month = {5}
year = {1996}
pages = {168-175}
numpages = {8}
}
@inproceedings{pin-pra-gan-95-aa-symb,
author = {Axel Pinz and Manfred Prantl and Harald Ganster}
title = {Affine Matching of Intermediate Symbolic Representation}
fullhash = {3724b92ae4a1cb0a6606e007949a8de9}
booktitle = {Proc. CAIP'95 - 6th International Conference on Computer Analysis of Images and Patterns}
series = {Lecture Notes in Computer Science}
volume = {870}
year = {1995}
pages = {359-367}
numpages = {9}
}
@techreport{tsa-92-aa-line-tr,
author = {Frank Chee-Da Tsai}
title = {Statistical Approach to Affine Invariant Matching with Line Features}
institution = {Robotics Research Laboratory - Courant Institute of Mathematical Sciences - New York University}
fullhash = {b147bea9a3f50d1c9003bf3d57256665}
month = {11}
number = {TR1992-621}
year = {1992}
pages = {25}
numpages = {1}
}