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AFFINE ARITHMETICBASED METHODS FOR UNCERTAIN POWER SYSTEM ANALYSIS

AFFINE ARITHMETICBASED METHODS FOR UNCERTAIN POWER SYSTEM ANALYSIS

ALFREDO VACCARO ANTONIO PEPICIELLO

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2022 Elsevier Inc. All rights reserved.

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The MathWorks does not warrant the accuracy of the text or exercises in this book.

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No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions.

This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

ISBN: 978-0-323-90502-2 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Charlotte Cockle Acquisitions Editor: Graham Nisbet Editorial Project Manager: Sara Valentino Production Project Manager: Manju Thirumalaivasan Designer: Mark Rogers Typeset by VTeX

Contents Preface

xi

Acknowledgments

xiii

1. Uncertainty management in power systems

1

1.1. Sampling methods

2

1.2. Analytical methods

3

1.3. Approximate methods

3

1.4. Non-probabilistic methods

4

References

6

2. Elements of reliable computing 2.1. Interval arithmetic

9 9

2.2. Affine arithmetic

10

2.3. Solving uncertain equations by AA

14

2.4. Reliable solution of non-linear equations

16

2.5. Reliable solutions of constrained optimization problems

18

References

22

3. Uncertain power flow analysis

23

3.1. Problem formulation

24

3.2. Affine arithmetic based solution of the power flow equations

25

3.3. Numerical results

29

3.4. Robust formulation of the power flow equations

39

3.5. Case study

42

References

47

4. Uncertain optimal power flow analysis

49

4.1. Mathematical background

49

4.2. Numerical results

55

4.3. Robust formulation of optimal power flow problems

58

4.4. Case study

62

4.5. Remarks

63

References

63 vii

viii

Contents

5. Unified AA-based solution of uncertain PF and OPF problems 5.1. Theoretical framework 5.2. Applications 5.3. Numerical results 5.4. Computational requirements 5.5. Remarks References

65 65 70 72 79 79 79

6. Uncertain power system reliability analysis

81

6.1. Markov Chains 6.2. Uncertain Markov Chains analysis by AA 6.3. Case studies References

82 85 88 91

7. Uncertain analysis of multi-energy systems

93

7.1. Optimal scheduling of an energy hub 7.2. Case study References

8. Enabling methodologies for reducing the computational burden in AA-based computing 8.1. PF analysis 8.2. OPF analysis 8.3. AA-based computing 8.4. Numerical results 8.5. Remarks References

9. Uncertain voltage stability analysis by affine arithmetic 9.1. AA-based calculation of PV curves 9.2. Numerical results 9.3. Remarks References

10. Reliable microgrids scheduling in the presence of data uncertainties 10.1. Deterministic optimization 10.2. Robust optimization

94 98 104

105 105 106 106 108 122 122

123 128 129 133 133

135 136 138

Contents

10.3. Affine arithmetic-based optimization 10.4. Numerical results and discussion 10.5. Remarks References Index

ix

138 140 143 143 145

In memory of my Father (Alfredo Vaccaro) To my family (Antonio Pepiciello)

Preface Reliable power system operation requires complex numerical analyses aimed at studying and improving the security and resiliency of electrical grids. Many mathematical tools can support power system operators in addressing this challenging issue, such as power flow and optimal power flow analyses, static/dynamic security assessment, and power system reliability analysis. The conventional formalization of these problems is based on the assumption that the input data are specified by deterministic parameters, which should be defined by the analyst either from a snapshot of the power system or inferred on the basis of some assumptions on the analyzed system.

Consequently, the solutions computed by solving these deterministic problems are rigorously valid only for the considered power system state, which represents a limited set of system conditions. Thus, when the input data are uncertain, numerous scenarios need to be evaluated in the task of computing robust problem solutions.

To address this problem, this Book introduces the basic elements of Affine Arithmetic-based computing, outlining its important role in uncertain power system analysis.

Affine Arithmetic is an enhanced model for reliable computing in which the input data and the problem variables are represented by affine combinations of primitive noise symbols, which are symbolic variables representing the independent uncertainty sources affecting the input data (i.e. exogenous uncertainties), or approximation errors generated during the computations (i.e. endogenous uncertainties).

After introducing the mathematical foundations of Affine Arithmetic, its deployment in solving the most fundamental power system operation problems is presented and discussed. In these contexts the adoption of Affine Arithmetic-based computing allows formalizing the power system state equations in a more convenient formalism compared to the conventional and widely used formulation adopted in interval-based Newton methods. Thanks to this feature a reliable estimation of the solutions hull can be reliably estimated by considering the parameter uncertainty correlations and the cumulative effect of all the uncertainty sources.

These results can be generalized in the task of solving a generic mathematical programming problem under uncertainty, which can be formalized by an equivalent deterministic problem, defining a coherent set of minimization, equality, and inequality affine arithmetic-based operators. This xi

xii

Preface

Affine Arithmetic-based formulation allows computing feasible and robust solution enclosures of complex uncertain optimization problems, reducing the approximation errors, and the estimation over-conservatism.

The deployment of this computing paradigm is applied for solving a wide range of power system operation problems, as far as optimal power flow, voltage stability analysis, equipment reliability assessment, and microgrid scheduling are concerned.

Finally, to reduce the computational burden of Affine Arithmetic-based computing, this Book analyzes the role of formal methods for knowledge discovery from historical power system operation data, which could also play an important role in optimally identifying the parameters of the affine forms describing the uncertain input parameters.

To this aim, a data processing technique-based on Principal component Analysis is introduced for identifying complex features and hidden relationships potentially describing regularities in the problem solutions, hence reducing the problem cardinality and the algorithm complexity. Thanks to this technique it is also possible to define a formal connection between the principal components and the noise symbols of the uncertain variables, which furnish an effective method for the optimal identification of the affine forms.

Alfredo Vaccaro

Acknowledgments The Authors wish to thank Prof. Claudio A. Canizares who inspired our research work. Special thanks go to Prof. Kankar Bhattacharya and Dr. Juan Carlos Munoz for their valuable contributions in applying Affine Arithmetic-based computing in voltage stability analysis, and Prof. Alberto Berizzi for his useful insights about the relations between affine arithmetic and robust optimization.

xiii

CHAPTER 1

Uncertainty management in power systems State of the art and enabling methodologies Power system operators require sophisticated numerical tools aimed at analyzing the current and the expected operation states, guaranteeing secure and reliable grid operation. In this context, numerical programming represents the mathematical backbone supporting the most fundamental power engineering tools, such as Power Flow (PF) and Optimal Power Flow (OPF) analyses, state estimation, voltage and frequency regulation, and contingency analysis. These studies could also support the development of advanced operations tools for integrated systems management, such as reliability and resilience assessment of multi-carrier energy systems, or optimal energy flow management in distributed energy hubs, which are relevant applications in the light of the evolution of modern power grids towards more integrated, smarter, and sustainable systems.

Numerical programming problems in power system analysis are conventionally formalized by deterministic models, whose parameters are assumed to be exactly known on the basis of several assumptions on the current power system state. However, both the models and their parameters are intrinsically affected by complex and correlated uncertainty sources, which can be roughly classified in aleatory and epistemic. The former is irreducible, i.e. it has to be accepted and cannot be controlled, whereas the latter is related to the accuracy of measurements, models, and parameters characterizing the system.

Uncertainty sources are mostly related to the complex dynamics of active and reactive power supply and demand, which may vary due to different causes [1]: • Variable generation mix depending on the market clearing.

• The increasing penetration of non-dispatchable generators based on renewable energy sources.

• The unpredictable fluctuation of loads due to unexpected events.

• The increasing geographical dispersion of energy generation.

Affine Arithmetic-Based Methods for Uncertain Power System Analysis https://doi.org/10.1016/B978-0-32-390502-2.00008-9

Copyright © 2022 Elsevier Inc. All rights reserved.

1

2

Affine Arithmetic-Based Methods for Uncertain Power System Analysis

The presence of uncertainties hinders the effectiveness of deterministic methods. Indeed, they can affect the problem solutions to a considerable extent. Hence, reliable solution paradigms, which incorporate information about data uncertainty are required. These paradigms allow to characterize the uncertainty associated to input data, and to assess its propagation over iterative calculations, hence providing a certain level of confidence about the problem solutions, and assessing their sensitivity to changes in input data uncertainties.

The scientific literature proposes a wide range of methodologies to handle uncertainty in power system analysis [2,3], and the most promising are summarized in the next sections.

1.1 Sampling methods Sampling methods consist in generating a set of scenarios, given some random variables with a defined probability distribution function. Monte Carlo is the best example of such methods: although it can provide very accurate solutions, it involves a large computational burden, which is due to the large number of problems that have to be solved, one for each scenario [4]. This condition prevents its effective utilization for large-scale power system analysis, where the computational burden could be prohibitively expensive [5,6].

Due to this reason, much research effort has been put in proposing enhancements of Monte Carlo methods, which try to improve the sampling process at the cost of accepting an increasing level of risk. For example, [7] integrates Latin hypercube sampling and Cholesky decomposition in the Monte Carlo method. In [8], a hybrid solution algorithm based on deterministic annealing expectation maximization algorithm and Markov Chain Monte Carlo is proposed. Correlation between wind variables through an extended Latin hypercube sampling is presented in [9]. Finally, chance constrained programming problems can be employed for solving uncertain OPF analysis, in which Monte Carlo and deterministic optimization are combined to handle the stochastic features of the input variables [10].

As previously highlighted, these methods require the definition of a proper trade-off between accuracy and computational burden, hence they are useful only when a defined degree of risk can be accepted.

Uncertainty management in power systems

3

1.2 Analytical methods Analytical methods require a simplification of the real problem and a certain number of assumptions in order to make the problem tractable [11]. Some simplifications can be obtained by characterizing the output random variables through multi-linearization [12], convolution techniques or Fourier Transform [13].

An example of these methods is the cumulant method [14,15] or its combination with the Gram-Charlier expansion [15] or the integration of Von Mises functions, to handle discrete distributions [16]. Another formulation is presented in [6], through chance-constrained programming model.

However, many of these techniques require the linearization of PF equations, which are non-linear, and this process generates additional inaccuracy.

The shortcomings of analytical techniques are multiple, as discussed in [17–20]. One example could be the common assumption of statistical independence of the input data, or the approximation of the real probability distribution with an ideal random variable. In this sense, the lack of determinism, which is characteristic of power systems, especially as an increasing amount of dispersed generators is being installed, can be a problem in the correct formulation of the probabilistic models. Furthermore, when validation of these techniques is performed, the assumptions made are often not supported by empirical evidence.

1.3 Approximate methods Approximate methods, such as point estimate methods, have been proposed as possible alternatives to both analytical and sampling methods [21]. These methods aim at approximating the statistical properties of the variables resulting as outputs of PF and OPF analyses. For example, in case of first or second order method, the mean and the variance are propagated through the calculations, by using Taylor series expansion [22].

In case of the point estimation methods, a certain number of deterministic problems, in particular m, which is the number of uncertain variables, has to be solved in order to estimate the statistical moments of the problem solutions [23]. Extensions of this method are, for example, Hong’s point estimate method [24], which are much faster but their accuracy is guaranteed only for low uncertainty ranges, and the linearization is a good approximation only around the nominal operation point.

Other approaches are based on primal-dual interior point method [25], which allows computing the sensitivity of the uncertain parameters with

4

Affine Arithmetic-Based Methods for Uncertain Power System Analysis

respect to the problem solution. Robust design theory can also be used to approximate OPF solutions in the presence of multiple uncertainty sources [26].

Finally, approximate methods have their shortcomings as well. For example two-point estimate methods are not feasible for analyzing large scale systems, especially in the presence of a large number of random variables.

1.4 Non-probabilistic methods The spectrum of available methods for carrying out reliable analyses in power systems in the presence of uncertainty sources has enriched over the last decades. The theory of evidence [27] and the theory of possibility [28] are two examples.

These mathematical formulations are associated to the imprecise human knowledge about the system, which allows only imprecise estimates of values and relations between variables [29]. An example in power systems is related to wind energy production: indeed, wind can be measured only locally, but its spatial distribution over a large area is difficult using probabilities. Additionally, weather forecasts provide qualitative information, which is difficult to be represented in a probabilistic way. Therefore, non-probabilistic modeling and simulations can be useful to power system engineers.

The application of fuzzy-set theory to power system analysis has been proposed in several papers [30–32]. In this case, both input data and constraints are modeled through fuzzy numbers, which are particular instances of fuzzy sets [33].

Self-validated computing is another way of representing uncertainty. Its advantage is the possibility of keeping track of the uncertainty propagation through the calculations. This propagation does not require much information about the uncertainty sources affecting the problem variables [34]. The first example of self-validated models is Interval Arithmetic (IA), which represents each variable through interval of real numbers, without a probability structure [35]. A set of operations can be defined over these intervals, such that the final computed interval is guaranteed to contain the uncertain value of the output variable which it represents.

IA has been proposed as a tool for handling uncertainty in power systems by many authors [18,19,36,37]. However, the solution obtained in this case is affected by the dependency problem and the wrapping effect [34,38], which could reduce the effectiveness of the method in terms of results

Uncertainty management in power systems

5

interpretation. This is mainly due to the impossibility of IA to represent correlations between uncertain variables. These problems require special structures of the Jacobian matrices (e.g. M-matrices, H-matrices, diagonally dominant matrices, tri-diagonal matrices) [39] in order for IA to be effective in solving power system operation problems, which are commonly not found in real systems.

To overcome these limitations, Affine Arithmetic-based (AA) methods for uncertain power system analysis have been proposed. In this paradigm, each variable is defined by a first degree polynomial, represented by a central value and a number of partial deviations, each one representing a statistical independent uncertainty source. In this way, the equations describing the problem under study can be expressed more conveniently and the correlation between the uncertain input variables can be effectively taken into account.

The main advantage of this paradigm is that it is suitable for solving large-scale problems, both computationally and in terms of accuracy and robustness of the obtained solution. In [40] and in [41], AA methods have been adopted to solve uncertain PF and OPF problems, by representing all the state and control variables by affine forms, without assuming a probability density function for each of them. This represents a promising alternative to stochastic information management in smart grids [42].

Several paper explored this idea, started in [43], of the application of AA to power system analysis. In [44], the effect of network uncertainties was processed through AA for solving the state estimation problem in presence of both synchrophasors and conventional measurements.

In [45], the complementarity conditions were used to represent bus voltage control through an AA-based model, which returned operation intervals for PF variables in the presence of uncertainty. In [46], an uncertain PF problem is solved, by formalizing it through interval PF and solving a quadratic programming optimization.

Additional applications and benefits deriving from the application of AA in power systems have been proposed in [40,41,47–49]. In particular, some of them are planning and operation in the presence of a large penetration of renewable energy sources, electric vehicles integration or demand response. The main advantage of AA over IA derives by its capacity in narrowing the tolerance interval of the problem solutions, which IA-based computations tend to over-estimate, especially when integrated in iterative solution schemes.

6

Affine Arithmetic-Based Methods for Uncertain Power System Analysis

In this book, solution methodologies based on AA are introduced to solve the most fundamental power system operation problems. AA is an enhanced model of self-validated numerical analysis, which represents the variables of interest as affine combinations of primitive variables describing statistical independent uncertainty sources. These combinations allow describing the effects of complex and correlated sources of exogenous uncertainty, and approximation errors involved in the computations.

This methodology can be enhanced by combining it with advanced techniques for information discovery and data compression, such as principal component analysis. In this way, the parameters of the affine forms describing the input data uncertainty can be rigorously identified on the basis of historical information, and the cardinality of the considered problem can be sensibly reduced.

References [1] G. Verbic, C. Cañizares, Probabilistic optimal power flow in electricity markets based on a two-point estimate method, IEEE Transactions on Power Systems 21 (4) (2006) 1883–1893.

[2] P. Chen, Z. Chen, B. Bak-Jensen, Probabilistic load flow: A review, in: Proceedings of the 3rd International Conference on Deregulation and Restructuring and Power Technologies, DRPT 2008, 2008, pp. 1586–1591.

[3] B. Zou, Q. Xiao, Solving probabilistic optimal power flow problem using quasi Monte Carlo method and ninth-order polynomial normal transformation, IEEE Transactions on Power Systems 29 (1) (2014) 300–306.

[4] M. Hajian, W. Rosehart, H. Zareipour, Probabilistic power flow by Monte Carlo simulation with Latin supercube sampling, IEEE Transactions on Power Systems 28 (2) (2013) 1550–1559.

[5] C. Su, Probabilistic load-flow computation using point estimate method, IEEE Transactions on Power Systems 20 (4) (2005) 1843–1851.

[6] H. Zhang, P. Li, Probabilistic analysis for optimal power flow under uncertainty, IET Generation, Transmission & Distribution 4 (5) (2010) 553–561.

[7] H. Yu, C. Chung, K. Wong, H. Lee, J. Zhang, Probabilistic load flow evaluation with hybrid Latin hypercube sampling and Cholesky decomposition, IEEE Transactions on Power Systems 24 (2) (2009) 661–667.

[8] H. Mori, W. Jiang, A new probabilistic load flow method using MCMC in consideration of nodal load correlation, in: Proceedings of the 15th International Conference on Intelligent System Applications to Power Systems, 2009, pp. 1–6.

[9] H. Yu, W.D. Rosehart, Probabilistic power flow considering wind speed correlation of wind farms, in: Proceedings of the 17th Power Systems Computation Conference, 2011, pp. 1–7.

[10] M. Hajian, W.D. Rosehart, H. Zareipour, Probabilistic power flow by Monte Carlo simulation with Latin supercube sampling, IEEE Transactions on Power Systems 28 (2) (2012) 1550–1559.

[11] A. Schellenberg, W. Rosehart, J. Aguado, Cumulant-based probabilistic optimal power flow (P-OPF) with Gaussian and gamma distributions, IEEE Transactions on Power Systems 20 (2) (2005) 773–781.

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[12] A.S. Meliopoulos, G.J. Cokkinides, X.Y. Chao, A new probabilistic power flow analysis method, IEEE Transactions on Power Systems 5 (1) (1990) 182–190.

[13] R. Allan, A. da Silva, R. Burchett, Evaluation methods and accuracy in probabilistic load flow solutions, IEEE Transactions on Power Apparatus and Systems PAS-100 (5) (1981) 2539–2546.

[14] R. Allan, M. Al-Shakarchi, Probabilistic techniques in ac load-flow analysis, Proceedings of the Institution of Electrical Engineers 124 (2) (1977) 154–160.

[15] P. Zhang, S.T. Lee, Probabilistic load flow computation using the method of combined cumulants and Gram-Charlier expansion, IEEE Transactions on Power Systems 19 (1) (2004) 676–682.

[16] L. Sanabria, T. Dillon, Stochastic power flow using cumulants and Von Mises functions, International Journal of Electrical Power & Energy Systems 8 (1) (1986) 47–60.

[17] A. Dimitrovski, K. Tomsovic, Boundary load flow solutions, IEEE Transactions on Power Systems 19 (1) (2004) 348–355.

[18] F. Alvarado, Y. Hu, R. Adapa, Uncertainty in power system modeling and computation, in: Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, 1992, pp. 754–760.

[19] A. Vaccaro, D. Villacci, Radial power flow tolerance analysis by interval constraint propagation, IEEE Transactions on Power Systems 24 (1) (2009) 28–39.

[20] Z. Wang, F. Alvarado, Interval arithmetic in power flow analysis, IEEE Transactions on Power Systems 7 (3) (1992) 1341–1349.

[21] M. Madrigal, K. Ponnambalam, V. Quintana, Probabilistic optimal power flow, in: Proceedings of the IEEE Canadian Conference on Electrical and Computer Engineering, vol. 1, 1998, pp. 385–388.

[22] X. Li, Y. Li, S. Zhang, Analysis of probabilistic optimal power flow taking account of the variation of load power, IEEE Transactions on Power Systems 23 (3) (2008) 992–999.

[23] G. Verbic, C. Canizares, Probabilistic optimal power flow in electricity markets based on a two-point estimate method, IEEE Transactions on Power Systems 21 (4) (2006) 1883–1893.

[24] J.M. Morales, J. Perez-Ruiz, Point estimate schemes to solve the probabilistic power flow, IEEE Transactions on Power Systems 22 (4) (2007) 1594–1601.

[25] A. Mohapatra, P. Bijwe, B. Panigrahi, Optimal power flow with multiple data uncertainties, Electric Power Systems Research 95 (2013) 160–167.

[26] C. Boonchuay, K. Tomsovic, F. Li, W. Ongsakul, Robust optimization-based DC optimal power flow for managing wind generation uncertainty, AIP Conference Proceedings 1499 (1) (2012) 31–35.

[27] G. Shafer, A Mathematical Theory of Evidence, vol. 1, Princeton University Press, Princeton, 1976.

[28] L.A. Zadeh, Fuzzy sets, Information and Control 8 (3) (1965) 338–353.

[29] P. Smets, Imperfect information: Imprecision and uncertainty, in: Uncertainty Management in Information Systems, Springer, 1997, pp. 225–254.

[30] P. Bijwe, G.V. Raju, Fuzzy distribution power flow for weakly meshed systems, IEEE Transactions on Power Systems 21 (4) (2006) 1645–1652.

[31] V. Miranda, J. Saraiva, Fuzzy modelling of power system optimal load flow, in: Proceedings of the Power Industry Computer Application Conference, 1991, pp. 386–392.

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[33] X. Guan, W.E. Liu, A.D. Papalexopoulos, Application of a fuzzy set method in an optimal power flow, Electric Power Systems Research 34 (1) (1995) 11–18.

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Affine Arithmetic-Based Methods for Uncertain Power System Analysis

[34] J. Stolfi, L.H. De Figueiredo, Self-validated numerical methods and applications, in: Proceedings of the Monograph for 21st Brazilian Mathematics Colloquium, Citeseer, 1997.

[35] R. Moore, Methods and Applications of Interval Analysis, vol. 2, SIAM, 1979.

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CHAPTER 2

Elements of reliable computing Theoretical foundations 2.1 Interval arithmetic In IA, each variable x is assumed to be “unknown but bounded” in an interval of real numbers X = [xinf , xsup ], also known as the tolerance of x.

The following theorem is of foremost importance to define operations on intervals [1]: Theorem 2.1 (Fundamental invariant of range analysis for IA). ∀ : p → q , globally Lipschitz with bounded slope. There exists an interval extension I : p → q such that:

∀ x1 , .., xp ∈ X1 , .., Xp ⇒ x1 , .., xp ∈ I X1 , .., Xp

Extending elementary operations, such as sums or products, to intervals is trivial. For example, given two intervals X1 = [x1,inf , x1,sup ] and X2 = [x2,inf , x2,sup ], some basic operations are: • Sum:

X1 + X2 = x1,inf + x2,inf , x1,sup + x2,sup •

Difference: X1 − X2 = x1,inf − x2,sup , x1,sup + x2,inf

•

(2.1)

(2.2)

Product: X1 · X2 = [ min(x1,inf x2,inf , x1,inf x2,sup , x1,sup x2,inf , x1,sup x2,sup ), max(x1,inf x2,inf , x1,inf x2,sup , x1,sup x2,inf , x1,sup x2,sup )]

•

(2.3)

Division:

X1 /X2 = x1,inf , x1,sup ·

1

1 x2,sup , x

2,inf

0 ∈/ x2,inf , x2,sup

(2.4)

More complex functions can be obtained by composing these primitive operators [1,2]. Thanks to Theorem 2.1, the interval resulting from these operations is guaranteed to enclose the real range of function values.

Affine Arithmetic-Based Methods for Uncertain Power System Analysis https://doi.org/10.1016/B978-0-32-390502-2.00009-0

Copyright © 2022 Elsevier Inc. All rights reserved.

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10

Affine Arithmetic-Based Methods for Uncertain Power System Analysis

IA has been applied to solve systems of linear [3,4] and non-linear equations [5], and optimization problems [6,7]. The obtained solutions, however, can yield unexpected results [8,9], due to the lack of IA in representing the correlation between uncertain variables, which leads to the wrapping problem, as illustrated in Fig. 2.1 [9]. This problem can be demonstrated by considering an harmonic oscillator evolving through the state space ẋ1 = x2 , ẋ2 = −x1 . At t = 0, it is represented as a rectangle, whose length and height are parallel to cartesian axes. Since the evolution of the rectangle leads to another one which is not parallel to the axes, the evolution of the interval at t requires adding a series of spurious states, represented by the black region (rotated rectangle) A B C D . Due to this reason, the evolution of the interval, in a few iterations, diverges, covering the entire phase space.

Another example is given by this simple operation: X1 · (X2 + X3 ) ⊂ (X1 · X2 + X1 · X3 )

(2.5)

The resulting interval is much wider than the true range, and long computation chains may lead to aberrant solutions. This phenomenon requires adopting special computing techniques [10].

Finally, the “dependency problem” is another limiting aspect of interval computation, which is a consequence of the definition of the difference operator (2.2): X − X = [xinf , xsup ] − [xinf , xsup ] = [xinf − xsup , xsup + xinf ] = 0

(2.6)

This problem is a consequence of the inability of IA in discriminating relations between uncertainty sources, which are always assumed to be independent. If the interval variables are independent, then the results are correct, such as for X1 = [1, 2] and X2 = [1, 2]: in this case, their difference X1 − X2 = [−1, 3], leads to correct results. On the other hand, if the interval variables are the same, the overestimation error is extremely large.

To address these limitations, more advanced paradigms, based on AA, will be discussed in next section.

2.2 Affine arithmetic AA, which has been introduced in [2], is a self-validated computing method that allows modeling errors, imprecise data, and truncation errors. The

Elements of reliable computing

11

Figure 2.1 Dynamic behavior of an oscillator analyzed through IA.

approach originates from IA, but it keeps track of correlations between the uncertain variables. For this reason, the resulting bounds are much tighter and the divergence of the solution intervals observed in IA can be avoided or mitigated [11].

In AA, an uncertain variable x is represented by an affine form, which is a first degree polynomial: x̂ = x0 + x1 ε1 + x2 ε2 + ... + xp εp

(2.7)

where x0 and xk are known real coefficients representing the central value and the partial deviations of the affine form x̂, respectively.

The variables εk , called the noise symbols, are defined over the interval [−1, 1]. The noise symbols represent independent uncertainty sources affecting the problem variables, which can be induced by computation errors or exogenous uncertain sources.

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Affine Arithmetic-Based Methods for Uncertain Power System Analysis

Given an affine form, it can be converted to an interval by defining the radius rad(x̂), the upper bound x, the lower bound x, and the range |X |. rad(x̂) :=

p

|xk |

(2.8)

|xk |

(2.9)

|xk |

(2.10)

k=1

x := x0 +

p

k=1

x := x0 −

p

k=1

|X | := [x, x]

(2.11)

Operations on affine forms require the replacement of real operators with their associated AA-based version. An affine operator (x̂, ŷ) can be defined to map a generic function (x, y) to the affine domain. This allows computing an affine form for ζ = (x, y), which is coherent with the input affine forms (x̂, ŷ).

An useful feature of linear functions of affine forms, is that their affine representation can be obtained by rearranging all the original noise symbols into an affine form, as in the following equations: x̂ ± ŷ = (x0 ± y0 ) + (x1 ± y1 )ε1 + (x2 ± y2 )ε2 + ... + (xp ± yp )εp

(2.12)

λx̂ = (λx0 ) + (λx1 )ε1 + (λx2 )ε2 + ... + (λxp )εp

(2.13)

x̂ ± λ = (x0 ± λ) + x1 ε1 + x2 ε2 + ... + xp εp

(2.14)

Unfortunately, non-linear functions do not allow ζ to be expressed only in terms of the original noise symbols: ζ̂ = (x̂, ŷ) = (x0 + x1 ε1 + x2 ε2 + ... + xp εp , y0 + y1 ε1 + y2 ε2 + ... + yp εp )

(2.15) In this case, a proper approximating affine function should be defined: a (x̂, ŷ) = ζ0 + ζ1 ε1 + ζ2 ε2 + ... + εp ζp

(2.16)

Elements of reliable computing

13

this non-affine operator approximates the function (x̂, ŷ) reasonably well over its domain: ẑ = a (x̂, ŷ) + ζp+1 εp+1 = ζ0 + ζ1 ε1 + ζ2 ε2 + ... + ζp εp + ζp+1 εp+1

(2.17)

where the last term represents the residual or approximation error: e∗ (x̂, ŷ) = (x̂, ŷ) − a (x̂, ŷ)

(2.18)

In order to handle this approximation error, an additional noise symbol εp+1 is added in (2.17), in order to properly bound the corresponding approximation errors: ζp+1 >

max

(ε1 ,ε2 ,...,εp )

e∗ (x̂, ŷ)

(2.19)

Furthermore, affine approximations of different functions, such as a in (2.16), can be different depending on the required accuracy, and the maximum acceptable computational burden. The following approximation represents a decent trade-off between these requirements: a (x̂, ŷ) = α x̂ + β ŷ + ξ

(2.20)

where the unknown coefficients α , β , and ξ can be identified by the following theorem: Theorem 2.2 (Chebyshev approximation theorem for univariate functions). Given a bounded and twice differentiable function , defined in some interval xI = [xinf , xsup ], whose second derivative does not change sign inside x.

Let a (x̂) = α x̂ + ξ be its Chebyshev affine approximation in xI . Then: α=

(xsup ) − (xinf ) xsup − xinf

ξ=

(u) + r (u)

2

− αu

d (u) =α dx

r (u) = α u + (xsup ) − αχsup and the maximum absolute error is: ζp+1 = |

(u) − r (u)

2

|

In this case, the optimal coefficients α and ξ can be easily calculated.

Indeed, by considering the line interpolating the points (xinf , f (xinf )) and u) (xsup , f (xsup )), the slope d ( dx = α can be obtained.

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Affine Arithmetic-Based Methods for Uncertain Power System Analysis

Given all these properties, AA can be defined as an intermediate paradigm between Taylor forms and zonotopes, as described in [12], which is characterized by a set of advantages, lacking from other first order approximation paradigms.

2.3 Solving uncertain equations by AA A particular class of non-linear operation is the multiplication between two affine forms x̂ and ŷ: x̂ ∗ ŷ = x0 y0 +

n

(x0 yh + y0 xh )εh +

n n

(xh yj )εh εj

(2.21)

h=1 j=1

h=1

This non-affine operation is traditionally approximated by bounding the second order terms: n n

(xh yj )εh εj ≤

h=1 j=1

n

h=1

|xh |

n

|yj | = R(x̂)R(ŷ)

(2.22)

j=1

where the R(.) is the radius of the affine form. Starting from this result, a new noise symbol describing the approximation error can be defined as follows: x̂ ∗ ŷ = x0 y0 +

n

(x0 yh + y0 xh )εh + R(x̂)R(ŷ)εk

(2.23)

h=1

A set of uncertain non-linear equations, where the non-linearity derives from the multiplication of affine forms, can be solved by using the mathematical operators proposed in [13].

In particular, let’s consider the following non-linear equation: x̂2 = ẑF = 4 + 0.8ε1 + 0.4ε2

(2.24)

The solution of this equation requires computing the central value x0 and the partial deviations x1 and x2 of the affine form x̂, which shares the same primitive noise symbols of the known term ẑF , such that: x̂2 = x20 + 2x0 x1 ε1 + 2x0 x2 ε2 + R(x̂)2 ε3 = 4 + 0.8ε1 + 0.4ε2

(2.25)