Affine Arithmetic-Based Methods for Uncertain Power System Analysis

158 Pages·2022·12.047 MB·English

by  Alfredo Vaccaro, Antonio Pepiciello

AFFINE
ARITHMETIC-
BASED METHODS
FOR UNCERTAIN
POWER SYSTEM
ANALYSIS

AFFINE
ARITHMETIC-
BASED METHODS
FOR UNCERTAIN
POWER SYSTEM
ANALYSIS
ALFREDOVACCARO
ANTONIOPEPICIELLO
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Contents
Preface xi
Acknowledgments xiii
1. Uncertaintymanagementinpowersystems 1
1.1. Samplingmethods 2
1.2. Analyticalmethods 3
1.3. Approximatemethods 3
1.4. Non-probabilisticmethods 4
References 6
2. Elementsofreliablecomputing 9
2.1. Intervalarithmetic 9
2.2. Affinearithmetic 10
2.3. SolvinguncertainequationsbyAA 14
2.4. Reliablesolutionofnon-linearequations 16
2.5. Reliablesolutionsofconstrainedoptimizationproblems 18
References 22
3. Uncertainpowerflowanalysis 23
3.1. Problemformulation 24
3.2. Affinearithmeticbasedsolutionofthepowerflowequations 25
3.3. Numericalresults 29
3.4. Robustformulationofthepowerflowequations 39
3.5. Casestudy 42
References 47
4. Uncertainoptimalpowerflowanalysis 49
4.1. Mathematicalbackground 49
4.2. Numericalresults 55
4.3. Robustformulationofoptimalpowerflowproblems 58
4.4. Casestudy 62
4.5. Remarks 63
References 63
vii
viii Contents
5. UnifiedAA-basedsolutionofuncertainPFandOPFproblems 65
5.1. Theoreticalframework 65
5.2. Applications 70
5.3. Numericalresults 72
5.4. Computationalrequirements 79
5.5. Remarks 79
References 79
6. Uncertainpowersystemreliabilityanalysis 81
6.1. MarkovChains 82
6.2. UncertainMarkovChainsanalysisbyAA 85
6.3. Casestudies 88
References 91
7. Uncertainanalysisofmulti-energysystems 93
7.1. Optimalschedulingofanenergyhub 94
7.2. Casestudy 98
References 104
8. Enablingmethodologiesforreducingthecomputationalburden
inAA-basedcomputing 105
8.1. PFanalysis 105
8.2. OPFanalysis 106
8.3. AA-basedcomputing 106
8.4. Numericalresults 108
8.5. Remarks 122
References 122
9. Uncertainvoltagestabilityanalysisbyaffinearithmetic 123
9.1. AA-basedcalculationofPVcurves 128
9.2. Numericalresults 129
9.3. Remarks 133
References 133
10.Reliablemicrogridsschedulinginthepresenceofdata
uncertainties 135
10.1. Deterministicoptimization 136
10.2. Robustoptimization 138
Contents ix
10.3. Affinearithmetic-basedoptimization 138
10.4. Numericalresultsanddiscussion 140
10.5. Remarks 143
References 143
Index 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 mathematicaltools can support powersystem operators in ad-
dressing 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,
whichshouldbedefinedbytheanalysteitherfromasnapshotofthepower
systemorinferredonthebasisofsomeassumptionsontheanalyzedsystem.
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 uncer-
tain 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
combinationsofprimitivenoisesymbols,whicharesymbolicvariablesrep-
resenting 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 con-
ventional and widely used formulation adopted in interval-based Newton
methods. Thanks to this feature a reliable estimation of the solutions hull
canbereliablyestimatedbyconsideringtheparameteruncertaintycorrela-
tions and the cumulative effect of all the uncertainty sources.
These results can be generalized in the task of solving a generic mathe-
maticalprogrammingproblemunderuncertainty,whichcanbeformalized
by an equivalent deterministic problem, defining a coherent set of min-
imization, equality, and inequality affine arithmetic-based operators. This
xi