Table Of ContentMathematical Models and Algorithms for
Power System Optimization
Mathematical Models and
Algorithms for Power System
Optimization
Modeling Technology for Practical Engineering
Problems
Mingtian Fan
Zuping Zhang
Chengmin Wang
AcademicPressisanimprintofElsevier
125LondonWall,LondonEC2Y5AS,UnitedKingdom
525BStreet,Suite1650,SanDiego,CA92101,UnitedStates
50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates
TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom
©2019ChinaElectricPowerPress.PublishedbyElsevierInc.Allrightsreserved.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical,including
photocopying,recording,oranyinformationstorageandretrievalsystem,withoutpermissioninwritingfromthepublisher.
Detailsonhowtoseekpermission,furtherinformationaboutthePublisher’spermissionspoliciesandourarrangementswith
organizationssuchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite:www.
elsevier.com/permissions.
ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher(otherthanasmaybe
notedherein).
Notices
Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperiencebroadenourunderstanding,
changesinresearchmethods,professionalpractices,ormedicaltreatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingandusinganyinformation,
methods,compounds,orexperimentsdescribedherein.Inusingsuchinformationormethodstheyshouldbemindfuloftheir
ownsafetyandthesafetyofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeanyliabilityforanyinjury
and/ordamagetopersonsorpropertyasamatterofproductsliability,negligenceorotherwise,orfromanyuseoroperationofany
methods,products,instructions,orideascontainedinthematerialherein.
LibraryofCongressCataloging-in-PublicationData
AcatalogrecordforthisbookisavailablefromtheLibraryofCongress
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ISBN978-0-12-813231-9
ForinformationonallAcademicPresspublications
visitourwebsiteathttps://www.elsevier.com/books-and-journals
Publisher:GlynJones
AcquisitionEditor:GlynJones
EditorialProjectManager:NaomiRobertson
ProductionProjectManager:PremKumarKaliamoorthi
CoverDesigner:VictoriaPearson
TypesetbySPiGlobal,India
Abstract
A number of mathematical models and algorithms are presented in this book for solving the
practicalproblemsinplanning,operation,control,andmarketingdecisionsforpowersystems.
Itfocusesoneconomicdispatching,generatormaintenancescheduling,loadflow,optimalload
flow, load optimization,reactive optimization,load frequency control, transient stability, and
electricity marketing where mathematical models are transformed into relatively standard
optimization models to make optimization applications possible. The optimization models
discussedincludelinear(0–1,integer,mixed-integer),nonlinear,mixedinteger,andnonlinear
mixedintegermodels.Bothnumericalandnon-numericaloptimizationalgorithmsareusedin
this book, the former (mathematical programming approachs) includes linear programming,
nonlinear programming, mixed integer programming and dynamic programming, the latter
(rules based approaches) includes Genetic Algorithm (GA), Simulated Annealing (SA), and
ExpertSystem(ES).Basedontheauthors’extensiveresearchexperienceindevelopingmodels
andalgorithmsforpowersystemoptimization,thisbookalsoprovidesanin-depthanalysisof
some practical modeling techniques which are seldom explained comprehensively in the
existingtextbooks,bothfromtheoreticalandpracticalstandpoints,forexample,validitytesting
ofdata,typesettingofvariables,specialsettingoflimitvaluesofvariables,specialsettingof
constraints, and preprocessing of parameter and data. These techniques can be effectively
applied to the modeling of power system optimization problems. Therefore, the readers of
Mathematical Models and Algorithms for Power System Optimization will gain important
insights into: how to transform the practical problems into mathematical models, how to
develop the standard optimal mathematical models and utilize commercially available and
reliableprogrammingsoftware,howtodealwithvariousissuesthataffecttheperformanceofa
model, and how to evaluate the effectiveness of the models.
The authors hope that the ideas and practices of the modeling techniques presented in this
bookwillbeinformativeandhelpfulforthefuturemodelingresearchonpowersystems.This
book will be a useful reference for those in universities and research institutes who are
actively engaged in power system optimization.
xiii
Preface
Thepracticalmodelsforpowersystemplanning,operation,control,andelectricitymarketsare
provided in this book based on the authors’ research achievements in the development of
mathematical models and algorithms. The models include optimization models (linear,
nonlinear, mixed integer, nonlinear mixed integers), differential equations, difference
equations,andtimeseriesmodels.Thisbooknotonlyusesnumericalalgorithms(mathematical
programming methods), such as linear programming, nonlinear programming, mixed
integerprogramminganddynamicprogramming,butalsousessomenon-numericalalgorithms
such as Genetic Algorithm (GA), Simulated Annealing (SA), and Expert System (ES). The
mathematical models and calculation methods provided in this book have been proven by
typicalcalculationexamplesorappliedinengineeringpractices.Therefore,thisbookfollowsa
highly original and very practical approach.
The current research results on modeling technology for power systems can be found in
research papers and textbooks. However, research papers mainly focus on related theoretical
aspects,whereastextbooksemphasizegeneralknowledge,butneitherdescribesthemodeling
process in detail. Considering both theoretical and practical aspects, this book not only
introduces the methods and processes for the development of optimization models but also
providessomepracticaltechniques,suchasmutualtransformationofvariablesandfunctions,
transformation of equation types, and transformation of constraints. It also provides some
specialtechniquessuchassettingofvariabletypesandpreprocessingofdataandparameters.
The practical techniques mentioned above allow us to solve the modeling problems
encountered in new generation power systems more effectively.
The power system is a typical large-scale man-made system, though all the conventional
componentshaveacompletemodel,anynewcomponentneedstohaveanewmodelsoastobe
connected to the power system. To properly handle the new coming problems in power
system planning, operation, and control, the development of corresponding optimization
mathematical models and investigation of feasible algorithms should take many
relationships into consideration, such as the relationships between old and new components,
betweenoldandnewmodels,betweenthepowersystemandtheexternalenvironment,toname
a few.
xv
Preface
In recent years, there has been an evident tendency for a large number of distributed
resources, such as distributed generation, energy storage devices, and interactive loads, to
beconnectedtothepowergrid.Inaddition,informationandcommunicationtechnologieshave
beenwidelyappliedinmanyfieldsofpowersystems.Toadapttothisnewprogress,manynew
relationshipsneedtobedealtwithandmanynewmodelsneedtobedeveloped,andtraditional
mathematical models of power systems need to be further improved.
The modeling of power systems is extremely challenging due to the complexity of practical
problems,whichrequiresfairlygoodmathematicalknowledgeanddeepunderstandingofthe
physicalsystem.Althoughthereasonablereproducibilityofmathematicalmodelsallowsusto
simulate practical problems more effectively, the selecting of an optimization model nearly
always involves compromise among conflicting goals, such as discrete versus continuous,
accurate versus approximate, simple use versus comprehensive analysis, etc. The modeling
techniques for power system optimization deserve to be discussed in depth in this book.
Fourtypesofbasicvariablesareconsideredinthesteady-stateanalysisandcalculationofthe
powersysteminthisbook:activepower,reactivepower,voltage,andphaseangle(namelyP,
Q,U,andθ).Amongthem,activepowerandreactivepowercanbedividedintoactivepower
generationoutputandreactivepowergenerationoutput(P ,Q ),andactiveloadandreactive
G G
powerload(P ,Q ),respectively.Occasionally,the“P”and“Q”onthenodeareconsideredas
L L
thecorrespondingimpedancesratherthanthevariables.Besidesthebasicvariablesdescribed
previously,twomorevariablesareconsideredinthetransientcalculationofthepowersystem:
thepowerangleδandtheangularfrequencyorrotationalspeedofthegeneratorω¼2πf(where
f is the system frequency).
Chapter 1 introduces the fundamental issues of modeling techniques deduced from practical
engineering problems, including some general and special modeling techniques. It provides
some ideas for the setting of variables and functions, the selection of model types, and the
selection of algorithms, all of which provide main aspects for power system model
constructions and solutions.
Therestofthebookisdividedintofourparts:operation,planning,control,andmarketingfor
powersystems.Allfourpartsdescribethemathematicalmodelsandthecalculationmethodsto
optimizethevariablesP,Q,U,andθ,fromdifferentpointsofview.Thefirstpartcomprising
Chapters 2 and 3 focuses on the power generation operation plan, which optimizes the
generated output of the generator hourly, daily, and yearly. The second part of the book,
Chapters 4, 5 and 6, focuses on the investment and operation planning of the power network,
which optimizes the variables active power P (including P and P ), reactive power Q ,
G L G
voltage U, phase angle θ, transformer ratio T, capacitor bank C, and reactor bank R in hourly
and yearly cycles. The third part of this book, Chapters 7 and 8, describes the power system
control on small or large disturbances in a second and millisecond time cycle, which mainly
optimizesvariablessuchasthegeneratoroutputP ,thepowerangleδofthegenerator,andthe
G
xvi
Preface
systemfrequencyf.Thelastpart,Chapter9,integratestheprinciplesofmicroeconomicsinto
thepracticaloperationofthepowersystemandestablishesanoptimaldecisionmodelforallthe
market participants based on the Nash equilibrium and the Walrasian general equilibrium.
Chapter 2 studies the optimization model of daily economic dispatch of a pump storage plant
in a practical multiregional system in a province in China. This chapter describes how to
optimize the arrangement for the generator output P within a daily cycle based on hourly
G
intervals, of which the power output of each generating unit is treated as a continuous
variable and pump storage output as a discrete variable. It proposes a mixed-integer
programming (MIP)-based optimization model with both linear objective function and the
constraintsand twocategoriesofvariables(continuousand discrete).The MIP methodisthen
used to solve the problem. The proposed model effectively optimizes the operation of the
pump storage plant and meets all constraints, thus achieving the goal of shifting the peak
load and filling the valley of the load curve. Therefore, it has a high relevance for the
current smart operation of the power grid.
Chapter3focusesontheoptimizationmodeloftheannualgeneratormaintenancescheduling
(GMS). This chapter describes how to optimize the arrangement for the generator output
PG within an annual cycle based on hourly intervals. The GMS model based on fuzzy logic
dynamic programming is proposed. Because GMS constraints (such as maintenance window
interval, spare capacity, maintenance manpower, regional maintenance capability, and
generator maintenance time) cannotbe overlapped,the concept of a fuzzy set, which handles
the boundary of the objective function and constraints of GMS, is used to obtain a more
feasible solution for GMS. The objective function and constraint function in the GMS
model are both linear functions whose variables are continuous variables. Knowledge based
on expert systems is also used in the solution process. The method has been effectively
applied to GMS problems in an actual provincial power system.
Chapter 4 deals with two types of new power flow models, ill-conditioned power flow and
discreteoptimalpowerflows,bywayofconstructionofobjectivefunctionandconstraints.This
chapterfirstdescribehowtodevelopanewpowerflowmodelbasedonthecombinationofthe
simulated annealing (SA) method and the Newton-Raphson power flow method. Then, it
describeshowtodevelopadiscreteoptimalpowerflow(discreteOPF)modelbyconstructinga
linearobjectivefunctionwithP ,Q ,U,andθasconstraints.ThediscreteOPFmodelissolved
G G
by the successive linear programming (SLP) based algorithm and the approximate mixed-
integerlinearprogrammingalgorithms,inwhichamethodtochangetheincrementofvariables
in the iterative calculation of the linear programming is applied. Both models have been
successfully applied to practical power systems.
Chapter 5addressesthemodelsforminimizingload curtailmentandmaximizingload supply
capability based on the DC power flow algorithm to optimize the load P , where U and θ
L
aretreatedasconstants.Thischapterfirstdescribehowtodevelopthenodeloadminimization
xvii
Preface
modelofthenodeloadcurtailmentintheeventoffaults,wherenodeloadcurtailment(P )isa
C
variable (node load P is a limit), and the objective function is to minimize the sum of node
L
loadcurtailmentP .Then,thischapterpresentsthemaximizingloadsupplycapabilitymodel
C
ofthenodeunderthenormalconditionthatthenodeloadP isavariable,wheretheobjective
C
function is to maximize the sum of the node power supply and load P . Both models are
L
applicable to the actual situation of urban power grids.
Chapter 6 studies the discrete optimal reactive power (VAR) planning (a mixed-integer
nonlinear programming problem) models for some actual power systems. This chapter
describes how to develop a discrete VAR planning optimization model based on successive
linear programming (SLP), where the number “C” of the capacitor bank, “R” of the reactor
bank, and “T” of the transformer tap ratio are treated as discrete variables, and the other
variables (P, Q, U, and θ) are treated as continuous variables. First, a single state
discreteoptimalVARplanningmodelisgiven.Then,amultistatemodelwithashapeofablock
diagonalmatrixisproposed,inwhichthecorrespondingdecompositioncoordinationalgorithm
is also presented by decomposing, coordinating, and solving all states to minimize the total
investment in reactive power equipment. This chapter also combines expert rules, fuzzy
mathematical concepts, and GA algorithms with traditional optimization methods to
improve the possibility of obtaining discrete solutions. The results of practical test systems
showthattheproposedalgorithmcaneffectivelysolvethediscreteoptimizationVARproblems
of power systems.
Chapter7addressesthemodelofloadfrequencycontrolundersmalldisturbances.Basedonthe
Z-transformloadfrequencyfeedforwardcontrolmethod,thischapterdescribeshowtodevelop
a model and algorithm for controlling the power angular acceleration of the generator in the
given interval level of seconds to maintain the frequency of the generator. First, the power
system load disturbance model is established by the identification method. Then, the system
stateestimatorsareconstructedaccordingtothehierarchicaldecompositionprinciple.Finally,
theloadfrequencycontrolrulesarederivedaccordingtotheinvarianceprinciple.Furthermore,
thischapteralsoproposesthreepracticalmathematicalmodeltransformationmethods,suchas
the eigenvalue method, the logarithmic matrix expansion method and the successive
approximation method, to make the transformation of difference equations into differential
equations, and the mutual transformation of differential transfer functions. The results of
simulationshowedthatthecontrolmethodproposedcaneffectivelycontroldifferenttypesof
disturbances in power systems.
Chapter8studiesthelocalstabilitycontrolproblemofpowersystemsunderlargedisturbances.
Based on the decoupling control method, this chapter introduces a new state space that can
stably monitor the operation of the system based on local measurements without losing
synchronizationinthecaseoflargedisturbances,andprovidesrulestocontrolthestabilityof
the entire system in two stages with only locally applied stability control measurements.
xviii
Preface
In addition, this chapter has mathematically proven that the newly constructed state space
isobservable,decoupled,andtopologicallyequivalenttotheoriginalstatespaceofthesystem.
Basedonthetwostagecontrolcriteriagiveninthechapter,newmathematicalmodelsforstage
controlandintegratedcomputingprocesseshavebeendeveloped.Finally,thechapterexplores
therealisticfeasibilityofthedefinedcriteriaandmethodologiesviathecasestudyoftheoffline
calculation.
Chapter9focusesonthedecision-makingmodelinthepowermarket.Thischapterstudiesthe
single commodity market with transactions of only active power and the multicommodity
market with transactions of both active and reactive power, using the power pool mode. This
chapter establishes an optimal decision model, and illustrates that this model and the
competition equilibrium model are consistent in form. It indicates that the result of decision
optimizationhasreachedacompetitiveequilibrium.Basedonthecharacteristicsofthepower
systems, the accounting pricing method is used to distribute the loss of a power transmission
network and the cost of transmission congestion reasonably among market participants. This
eliminates market surpluses and avoids unfair posttrade distribution issues.
There are three appendices in this book. Appendix A describes the approximate algorithm
forMIP(whichhasbeenappliedinChapters4and6).AppendixBpresentsthederivationofthe
difference expressions for transformer T and shunt capacitor C in the optimization model
proposedinChapter6.AppendixCintroducesthederivationofthedecouplingbenchmarkδ
ei
proposed in Chapter 8 by using the DC power flow calculation method.
Finally, the authors gratefully appreciate the edification and inspiration of several respected
mentors, the contributions of collaborators, as well as the participation of several graduate
students, especially the assistance of Dr. Su Aoxue, who made the book more concise and
morereflectiveoftheauthors’mostinnovativework.Theauthorsarealsoparticularlygrateful
to Dr. Liu Yunren, a retired engineer from the California Independent System Operator
(CAISO) in the United States, who carefully read the manuscripts of the book and made
valuable comments. The authors also thank to Dr. QianXin, who provided support on the
English proof reading and promoted the publishing of the book.
The authorshopeto helpimprove theprofessionalskillsof power engineersaswell as senior
undergraduates and graduates from the relevant universities in their work on the modeling
technology of power system optimization.
Fan Mingtian
Zhang Zuping
Wang Chengmin
August 2017
xix
