Table Of ContentWei Wei · Jianhui Wang
Modeling and
Optimization of
Interdependent
Energy
Infrastructures
Modeling and Optimization of Interdependent
Energy Infrastructures
Wei Wei • Jianhui Wang
Modeling and Optimization
of Interdependent Energy
Infrastructures
123
WeiWei JianhuiWang
StateKeyLaboratoryofPowerSystems DepartmentofElectricalandComputer
DepartmentofElectricalEngineering Engineering
Beijing,China SouthernMethodistUniversity
Dallas,TX,USA
ISBN978-3-030-25957-0 ISBN978-3-030-25958-7 (eBook)
https://doi.org/10.1007/978-3-030-25958-7
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Preface
MotivationofThisBook
The everlasting consumption of fossil fuels with limited reserves amid climate
change and environmental pollution arises public awareness of sustainable devel-
opment, which calls for technology innovations as well as policy reformations to
encourage energy saving, enhance system efficiency, and harness green resources.
Nowadays, our energy supply infrastructures are experiencing rapid changes in
two aspects. First, renewable energies are utilized at various levels of our energy
systems. This initiative alleviates the dependence on traditional fossil fuels and
remarkably reduces carbon emissions. However, their volatility could impose
significant challenges on real-time power balancing of the electric power grid and
mayevenwrecksystemsecurity.Second,systemintegrationhasbecomeaprevalent
issue so as to unleash synergetic potentials across multiple energy carriers. The
scalablestorageofheatmakesitanattractivemediumforrenewableenergystorage;
the relatively low carbon impact of natural gas, its large reserve in shale rocks,
the fast-response capability of gas-fired power generation units, and the growing
maturity of power-to-gas technology will underpin a nexus of renewables, natural
gas,andelectricityinthecomingdecades.Moreover,thebidirectionalinformation
and energy flows allow electric vehicles to play an increasingly influential role
in future power grids. The complementary natures of different energy resources
precipitate the integration of traditionally independent energy systems, such as
the power system, natural gas system, heating system, and even the transportation
system,creatingenergyflowinterdependenceacrossinvolvedphysicalsystems.
Thetransitiontoacleanerandmoresustainableenergyindustryisaffectingthe
currentstructureofenergysystemsrangingfromthenational,state,andcityscales
down to the distribution, residential, and building levels. However, this ambitious
plannecessitatesbrand-newregulatoryframeworks,strategiceconomicincentives,
innovative business models, and active participation of consumers as responsive
demands.
vii
viii Preface
Withtheprogressofenergysystemintegration,decision-makingproblemsforall
involvedstakeholdersarebecomingmorechallenging,complicated,andinteractive.
Weperceivethenecessityofdisseminatingfundamentalmethodsinenergysystem
modeling as well as advanced theory and methodology in mathematical program-
ming which could address these problems faced by the broad energy research
community.
OrganizationofThis Book
Thisbookwillcoversomepromisingandactiveresearchtopicsandwillcomprehen-
sively address the modeling and optimization issues in the analysis, management,
and operation of interdependent energy systems from technical, economic, and
regulatory perspectives. Based on thorough mathematical methodologies, we can
makebestuseofpotentialsynergeticeffectsandachievedesiredgoals.Thisbookis
organizedasfollows:
Chapter 1 provides an overview on the functionality of several representative
physical systems which will be discussed in this book, including the electric
powergrid,naturalgasnetwork,districtheatingnetwork,andurbantransportation
network.Thecomplementarynaturalandsynergeticeffectsamongdifferentenergy
resources are revealed. The advantages, benefits, and challenges lying ahead,
brought by energy system integration, are introduced. The remaining chapters are
dedicatedtospecialformsofintegratedenergysystems.
Chapter 2 discusses the power system with renewable power integration. The
mainpurposeofthischapteristotacklethenon-convexityofpowerflowequations
as well as uncertainties brought by renewable integration. In view of the funda-
mental role of power flow in power system analysis, we give a thorough tutorial
on various power flow models in Sect.2.2, including AC power flow, DC power
flow, branch flow, and its linear approximation. Then, we proceed to the optimal
power flow problem and its variations, which actually come down to optimize a
specific objective function subject to power flow constraints. Convex relaxation
methods are employed to cope with the intrinsic non-convexity. We propose a
sequentialconvexoptimizationmethodtorecoverahigh-qualitysolutionwhenthe
convex relaxation is inexact, which usually happens in the maximum loadability
problem and the multi-objective optimal power flow problem. Next, we commit
effortstocopewithuncertaintiesresultingfromrenewableintegration.Depending
on the available information on the uncertainty, we apply two kinds of robust
optimization approaches to the hourly ahead energy and reserve dispatch problem
andtheday-aheadunitcommitmentproblem,aimingatimprovingsystemreliability
attheminimumcost.Weproposetheconceptofdispatchableregion,whichexactly
characterizes how much uncertainty can be dealt with in real-time operation with
a fixed energy and reserve dispatch strategy and propose an integer programming
approach to generate the boundaries of the dispatchable region. This concept
inspiresmanyinterestingapplications.Forexample,controlandoptimizationofthe
Preface ix
dispatchableregion,aswellasdata-drivenreliabilityassessmentwithoutprobability
distribution,areelaboratedinthischapter.
Chapter 3 is devoted to the integrated gas-power system. Such integration is
usuallyseenattheinter-areatransmissionlevel.InSect.3.2,weprovidethemathe-
maticalmodelofanaturalgaspipelinenetworkandsummarizetwoapproximations
forthegasflowpropagation equation,whichisapartialdifferentialequation.The
steady-state approximation is known as the Weymouth equation, and the quasi
steady-state approximation is usually referred to as the line pack model. Both of
them render algebraic equations. In the rest of this chapter, convex optimization
methodsforgas-powerflow,equilibriaofcoupledgas-powermarkets,andresilient
operationofgas-powersystemsarediscussed.
Chapter 4 is devoted to the integrated heat-power system. Such integration
is usually seen at the distribution level, as long-distance delivery of heat is not
economically efficient. In Sect.4.2, we introduce the mathematic model of a
districtheatingpipelinenetwork,calledthehydraulic-thermalmodel.Werevealthe
decompositionstructureoftheoptimalhydraulic-thermalflowproblemandpropose
amethodtodeterminethenearoptimalhydraulicconditionsinthesystem,suchthat
thethermalstateoptimizationgivesrisetoaconvexprogramwithlinearconstraints.
In the rest of this chapter, market-based decentralized operation schemes for the
heat-power system, equilibria of coupled heat-power markets, strategic behaviors
ofenergyhubsinelectricityandheatingmarkets,andcapacityplanningofenergy
hubsarepresented.
Chapter 5 is devoted to the city-sized transportation network and power distri-
bution network which are coupled by electric vehicles and on-road fast-charging
stations. In Sect.5.2, we present the mathematical descriptions of two traffic user
equilibrium models, which determine the steady-state vehicular flow distributions
inthetransportationnetwork.Differentfromotherenergyflowproblems,inwhich
the system is controlled by a central authority, the equilibrium in a transportation
networkisaspontaneousoutcomeofindividualrationalitiesoftravelers.Apractical
algorithm is introduced to solve large-scale instances by dynamically generating
active paths. In the rest of this chapter, important issues covering the topics in
network flow equilibria and optimization, system operation with uncertain traffic
demands,vulnerabilityanalysis,andcapacityexpansionplanningarestudied.
Given the interdependence across multiple energy systems, a central issue
encountered in energy markets is the transaction policy. Imitating the pool-based
powermarket,wegeneralizethemarginalcost-basedenergypricingschemetothe
companionenergysystemsinChaps.3–5andproposeabilateraltradingframework
for the integrated energy systems. Because the energy price should be retrieved
from the dual variables from the corresponding market clearing problems, the
analysisofstrategicbehaviorsofmarketparticipantsandcalculationofequilibrium
points can be challenging. We establish fundamental computation tools for such
interactivedecision-makingproblemsbasedonsophisticatedoptimizationtheories
andmethodologies.
Theformalcontentsofthisbookcenteronengineeringoptimizationproblemsof
the interdependent energy systems. Due to their complexity, it is usually difficult
x Preface
to clearly describe both the specific problem formulation and the mathematical
substance simultaneously, although we have tried our best to do so. In this book,
engineering problems in practice are set up in the main body, namely, Chaps.2–
5. Four advanced optimization techniques which are frequently used throughout
the main body are comprehensively tutored in the appendices, where optimization
problems are cast as compact matrix form in order to highlight the mathemat-
ical substance. The majority of materials in this part are concentrated from the
operationalresearchliteratureandpresentedinamoresystematicandapplication-
oriented manner after our filtering and refinement. Particular tricks discovered in
ourownresearchcanbequiteusefulforcertainproblems.Comprehensivesurveys
on the state-of-the-art development in respective fields are provided at the end of
each chapter. As such, we hope the readers can get clear landscape and concise
summariesofthesemethodswithoutbeingtrappedintedioussymbolsandnotations
forspecificapplicationsorabstrusemathematicaltheory.Theminimalbackground
knowledge required to understand these contents is tantamount to the level of an
introductorycourseonmathematicalprogrammingoroperationalresearch.
AppendixAreviewsbasicconceptsinconvexoptimization.Themostattractive
featuresofconvexoptimizationproblemsaretheirpolynomial-timesolvabilityand
zero duality gap under mild conditions. The linear programming and its duality
theoryareextendedtoconiclinearprogramminginthischapter.Thiscomparative
description helps readers who are not familiar with convex optimization develop
intuitiveunderstandingandnecessaryskillsforapplyingconiclinearprogramming
anddualitytechniquesintheirproblems,suchasthesemidefiniteprogrammingand
second-orderconeprogrammingencounteredintheoptimalpowerflowstudies.We
pay special attention to the most general quadratic programs which include non-
convex functions in constraints or the objective function or both. Such a problem
frequentlyarisesinpower flowoptimization,robustoptimization,andequilibrium
problems.Particularly,itisshownthatwhentheconstraintsarelinear,aquadratic
programwithanon-convexobjectivefunctioncanbeequivalentlyformulatedviaa
mixed-integerlinearprogramwithoutexploitingapproximation.
Appendix B summarizes linearization techniques in mixed-integer linear pro-
grams.Itiscommonknowledgethatreal-worldoptimizationproblemsareusually
non-convex and a non-convex program is generally NP-hard to solve globally.
Nevertheless, the progress in state-of-the-art solvers makes it possible for solving
a large-scale mixed-integer linear program in reasonable time, although it is also
NP-hard in the worst case. More importantly, many non-convex programs can be
approximatedviamixed-integerlinearprograms,whichcanbesolvedinreasonable
time. In this chapter, we summarize various formulation tricks which can convert
non-convex terms into linear expressions with integer variables, offering a viable
option to approximately solve particular classes of hard optimization problems
(globally) via mixed-integer linear programs. These techniques are particularly
useful for dealing with univariate nonlinear functions, bilinear terms, and logical
(disjunctive)constraints.
Appendix C illuminates the basics of robust optimization approaches, catego-
rizedbythedescriptionsofuncertainty(whetherdistributionalinformationistaken
Preface xi
into account) and the way of mitigating uncertainty (whether a recourse action
is allowed). It is demonstrated that for a fairly broad class of uncertain set, a
robustoptimizationproblemadmitsatractablecounterpart,whichcanbeefficiently
solved. In other cases, the problem can be solved in a decomposition fashion, and
thedifficultyrestsinrobustfeasibilitycheckorthedeterminationoftheworst-case
scenarioforagivenstrategy,whichcanbeformulatedasamax-minproblem.The
developmentofrobustoptimizationtheoryandmethodologyhasalreadyputitinto
applicationonrealengineeringproblems,whichisdesiredbytheelectricitysector
due to the volatility of ubiquitous renewable generation. This method is attracting
researchattentionatanunprecedentedlevel.
AppendixDshedslightsonequilibriumproblems.Forsingle-levelequilibrium
problems, we discuss the standard Nash game and the generalized Nash game.
We show how these problems that entail simultaneous decision-making can be
solved in a distributed manner via a best response iteration algorithm. As special
cases of the standard Nash game, matrix game and potential game are also
elucidated with more dedicated treatments. For bilevel equilibrium problems,
we elaborate bilevel program (single-leader-single-follower Stackelberg game),
mathematical program with equilibrium constraints (single-leader-multi-follower
Stackelberg game), and equilibrium program with equilibrium constraints (multi-
leader-multi-follower Stackelberg game). We set forth solution methodologies for
these challenging problems mainly based on mixed-integer linear programming
techniques. Specifically, the linear complementarity problem is discussed which
servesasthebasisfortheequivalentreformulationofbilevelgames.Inaddition,we
discuss several useful market models abstracted from our own research which are
generalenoughtoencompassawidespectrumofpracticalenergymarketproblems.
Webelievethesemodelshelppractitionersunderstandthemathematicalsubstance
behindcomplexequationsandapplyrelatedtheorymoreconveniently.
Althoughalotofmathematicaltheoremsareinvolvedintheappendicesofthis
book, we avoid mathematical formality and do not focus on proofs of theoretical
results. Instead, we give their implications on algorithm development and still
maintain sufficient mathematical rigor for readers to understand the elegance of
optimization.
Fromtheapplicationperspective,mathematicaltricksperformedinlinearization,
convexification,androbustification,amongotherreformulationtechniquesthathelp
solveacomplicateddecision-makingproblem,areratherscrappyandfragmentary.
The recent upsurge of research on robust and distributionally robust optimization
makesitdifficultforbeginnerstoquicklygrasptheessenceofrobustoptimization.
We feel the necessity of something like a dictionary which encapsulates all useful
knowledgealongeachdirectioninaunifiedandconcisearchitecture.Thismotivates
the appendix part, and we believe it will make the book widely accessible and
helpfulforpractitionersintheirresearch.
