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- 2-Stage approach for the optimal power flow of photovoltaic penetrated distribution systems
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- Bunaiyan, Amin (
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- 2017
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- The thesis presents a 2â€“stage model for achieving feasible optimal solution along with numerical validation for the exactness of the available second order cone relaxation in the distribution optimal power flow problem. To this end, the IEEE 4â€“bus, 13â€“bus, 34â€“bus and 123â€“bus distribution power systems are adapted by balancing the various feeder segments and by populating selected nodes with photovoltaic inverters. The different test feeders are studied under typical daily power profiles. Results generated by Matlab Software for Disciplined Convex Programming (CVX) and the nonlinear solver (fmincon) suggest that second order cone relaxation for the branch flow model is exact only for small radial systems but not necessary for large radial systems.
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- Thesis (M.S.)--University of Colorado Denver, 2017.
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- Includes bibliographical references .
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- by Amin Bunaiyan.
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2-STAGE APPROACH FOR THE OPTIMAL POWER FLOW OF
PHOTOVOLTAIC-PENETRATED DISTRIBUTION SYSTEMS by AMIN BUNAIYAN B.S., University of Colorado Denver, 2016 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering Program 2017 This thesis for the Master of Science degree by Amin Bunaiyan has been approved for the Electrical Engineering Program by Fernando Mancilla-David, Chair Jae Do Park Dan Connors Date: May 13, 2017 Bunaiyan, Amin (M.S., Electrical Engineering) 2-Stage Approach for the Optimal Power Flow of Photovoltaic-Penetrated Distribution Systems Thesis directed by Associate Professor Fernando Mancilla-David ABSTRACT The thesis presents a 2-stage model for achieving feasible optimal solution along with numerical validation for the exactness of the available second order cone relaxation in the distribution optimal power flow problem. To this end, the IEEE 4-bus, 13-bus, 34-bus and 123-bus distribution power systems are adapted by balancing the various feeder segments and by populating selected nodes with photovoltaic inverters. The different test feeders are studied under typical daily power profiles. Results generated by Matlab Software for Disciplined Convex Programming (CVX) and the nonlinear solver (fmincon) suggest that second order cone relaxation for the branch flow model is exact only for small radial systems but not necessary for large radial systems. The form and content of this abstract are approved. I recommend its publication. Approved: Fernando Mancilla-David iii DEDICATION This thesis is dedicated to my family who have supported me since the beginning of studies. They provide me with a great source of motivation and inspiration. Finally, this thesis is dedicated to all those who believe that knowledge is power. ACKNOWLEDGMENTS Greatest thanks to my parents who have supported me completely through good times and bad. Also, I would like to express my deepest gratitude to my advisor, Dr. Fernando Mancilla-David, for his guidance, valuable advice and support during this research. v TABLE OF CONTENTS CHAPTER 1. INTRODUCTION..................................................... 1 2. DISTRIBUTION OPTIMAL POWER FLOW BALANCED FEEDER . 5 2.1 Original Branch Flow Model.................................... 5 2.2 Convex Relaxation of the BFM ................................. 7 3. BRANCH ANALYSIS.................................................. 10 3.1 Unbalanced Series Impedance Matrices......................... 10 3.1.1 Overhead lines........................................ 10 3.1.2 Underground Lines..................................... 11 3.2 Unbalanced Shunt Admittance Matrices......................... 13 3.2.1 Overhead Lines........................................ 13 3.2.2 Underground Lines..................................... 14 3.3 Approximating The Line Matrix................................ 15 3.3.1 Approximating the Series Impedance Matrix............. 15 3.3.2 Approximating the Shunt Admittance Matrix ............ 17 4. CASE STUDIES....................................................... 21 4.1 Introduction................................................. 21 4.2 Error Computation............................................ 22 4.3 Preparing Case Studies....................................... 23 4.3.1 Approximating a Balanced Feeder....................... 23 4.3.2 Synthesizing Load Profiles............................ 24 4.3.3 Synthesizing PV Generation Profiles................... 26 4.4 Case Studies................................................. 27 4.4.1 Case Study IEEE 4-bus feeder.......................... 27 4.4.2 Case Study IEEE 13-bus feeder......................... 30 4.4.3 Case Study IEEE 34-bus feeder......................... 33 vi 4.4.4 Case Study IEEE 123-bus feeder .................... 36 5. CONCLUSIONS.............................................. 39 REFERENCES........................................................... 40 APPENDIX A. MATHEMATICAL BACKGROUND......................................... 42 A.l Convexity................................................. 42 A. 1.1 Convex Set........................................ 42 A. 1.2 Convex Function .................................. 42 A. 1.3 Convexity conditions.............................. 43 A. 1.4 Relaxation........................................ 43 A. 2 Lagrangian Optimization................................... 44 A. 3 Duality................................................... 44 A.4 Karush-Kuhn-Tucker (KKT) conditions....................... 46 A.5 Semidefinite Programming SDP.............................. 47 A.6 Second Order Cone Programming SOCP......................... 47 vii LIST OF TABLES TABLE 3.1 Nomenclature and Definitions for Chapter 3................................. 20 4.1 IEEE 4-bus feeder: Line Configuration Data................................ 25 4.2 IEEE 123-bus feeder: Line Configuration Data.............................. 25 4.3 IEEE 4-bus feeder: tightness results...................................... 28 4.4 IEEE 4-bus feeder: duality gap and relative error of decision variables ... 29 4.5 IEEE 13-bus feeder: tightness results........................................ 31 4.6 IEEE 13-bus feeder: duality gap and relative error of decision variables ... 32 4.7 IEEE 34-bus feeder: tightness results........................................ 34 4.8 IEEE 34-bus feeder: duality gap and relative error of decision variables ... 35 4.9 IEEE 123-bus feeder: tightness results.................................... 37 4.10 IEEE 123-bus feeder: duality gap and relative error of decision variables . . 38 viii LIST OF FIGURES FIGURE 2.1 Balanced model of distribution feeder line connecting two buses.................... 5 3.1 unbalanced line feeder............................................................ 10 4.1 2-Stage approach.................................................................. 21 4.2 The IEEE 4-bus test feeder network................................................ 27 4.3 IEEE 4-bus feeder power profiles: total demanded power and total available power from PV generation............................................................. 27 4.4 The IEEE 13-bus test feeder network............................................... 30 4.5 IEEE 13-bus feeder power profiles: (a) total demanded power and total available power from PV generation, (b) individual PV panel profiles..................... 30 4.6 The IEEE 34-bus test feeder network............................................... 33 4.7 IEEE 34-bus feeder power profiles: (a) total demanded power and total available power from PV generation, (b) individual PV panel profiles..................... 33 4.8 The IEEE 123-bus test feeder network.............................................. 36 4.9 IEEE 123-bus feeder power profiles: (a) total demanded power and total available power from PV generation, (b) individual PV panel profiles..................... 36 A.l Examples of convex and nonconvex sets ............................................ 42 A.2 Definition of convex function .................................................... 42 A.3 1st order condition of convex function............................................ 43 A.4 (a) relaxed set, (b) relaxed function............................................. 44 A.5 Epigraph geometric interpretation for an optimization problem..................... 46 IX CHAPTER I INTRODUCTION The Optimal Power Flow (OPF) problem integrates the power flow (PF) problem and the economic dispatch (ED) problem that, known as a nonlinear problem [1], The ED finds the lowest generation cost that satisfies demand, while the PF counts for losses and the topology of the network. It can also identify the power transfer distribution function (PTDF). The OPF problem, which was introduced by J. Carpentier in French in 1962 [2] is NP-hard because of its nonlinearity. The problem is NP-hard because of its nonlinearity. However, it is a highly desirable problem that can be solved in many ways for different applications. More specifically, there are many techniques for solving this problem by linearizing or relaxing the nonlinearity. A decoupled (DC) power flow approximation was proposed in 1974 [3]. The DC OPF is solved through linear programming (LP), which counts for the topology of the system but not for losses of the system. From an economic point of view, the DC OPF works fairly well for transmission networks but not for distribution networks. The reason is that DC approximation works better when the X/R ratio is being sufficiently high [4], Distribution systems were designed to be unidirectional [5] and their X/R ratio is not high enough to use DC approximation. The problem has become more interesting on a distribution level as renewable resources have become more available to individual users. The AC OPF was solved using a nonlinear solver, but there is no guarantee its solution will converge to the global optimum; instead, it converges to a local minimum that is highly dependent on the initial points. Therefore, studies have taken the path of linearization and convex relaxation. The semidefinite programming (SDP) relaxation technique is one way to obtain a solution to a convex problem. In [6], the AC OPF was solved using SDP in 2008. SDP relaxation for the OPF problem was proposed using primal-dual interior point 1 algorithm that is a convex relaxation. It states that SDP works out as linear programming but in its matrices form. This method takes advantage of the strong duality of the LP. It was proven that hrst-order KKT optimum conditions are the same for primal and dual OPF problems. The disadvantage is that, due to limitations in computer capacity, problems in extremely large systems cant be solved [6]. Further, [7], [5] and [8] describe the use of SDP, whose globally optimal solution may be found with available numerical algorithms in polynomial time. In [7], SDP is used to obtain solutions to convex problems solvable in polynomial-time complexity. As a result, the method demonstrates the ability to reach the global optimal solution of the original nonconvex optimal power flow problem. In [5], SDP is used to obtain a computationally feasible convex reformulation. This is shown using real world photovoltaic (PV) generation and load profile data for an illustrative low-voltage residential distribution system. In [8], the OPF problem is solved by leveraging SDP through the use of a centralized computational device with reduced computational burden. Second order cone programming (SOCP) is another way to solve convex problems. In [9], it was proven that SOCP is a special case of SDP. This means that SDP is a generalization of SOCP. However, SOCP is more tractable optimization class than SDP. Therefore, SOCP should not be solved as SDP. Angle reparameterization along with second order cone (SOC) relaxation are additional techniques used to solve a convex OPF problem. In [10], the relaxation in the bus injection model (BIM) and the branch flow model (BFM) were presented along with a proven relationship between the two models. In [11], three types of sufficient conditions of exactness were investigated. The first condition involves power injections that require some of the active and reactive powers injected at the two ends of each line to not be bounded. In other words, upper and lower constraints of both ends of each line cannot be active at the two ends unless the objective function is strictly convex. The 2 second condition involves voltage magnitudes that require all the upper bounds of voltages to be inactive. The third condition refers to voltage angles, which requires small enough differences across each line. One main advantage of solving the OPF problem using SOC relaxation finding a globally optimal solution to the problem [11], If the solution found using the relaxation is exact, then the optimal solution to the original nonconvex OPF problem can be computed. Through the study in [11], it has been found that the topology of the network is an important factor when solving the OPF problem. The angle relaxation is exact for radial systems but not for mesh systems. The conditions are insufficient for mesh systems; however, they could be sufficient if the phase shifters are controlled in a strategic way that makes the system behave like a radial system [11], Another investigation in [4] shows that an additional sufficient condition is necessary for ensuring the exactness of the SOC relaxation for distribution systems. The condition is related to the thermal limits and capacity of the branches. This condition states that there can only be active reversed power, only reactive reversed power, or no reverse power flows through a line. This condition is important due to the presence of renewable energy devices capable of injecting power in the lines. The branch flow model, which involves relaxation and convexification, is one such design. The design is made to carry out optimization of mesh topologies in addition to radial networks. This phase includes reviewing the convex relationship, controlling load flow, checking on optimal power flow, and managing the entire power system. The power flow model focuses on different variables associated with the distributed power, which includes the voltages and injections of current together with power injections as to regulate power flow in the network [12]. The relaxed branch flow approach presented in [12]. Solving the OPF problem is carried out using two stages of relaxation. The initial step involves the elimination of the current and voltage angles while the purpose of the second stage is to approximate 3 the outcoming problem via a conic program. The relaxation steps have been proven equal through the radial systems [12]. The main contributions of this thesis are 1.) to numerically validate the exactness of the SOC relaxation in the BFM of the optimal power flow problem at the distribution level (DOPF) for balanced feeders, 2.) to investigate the strength of the solution found by the relaxed model, and 3.) to seek a true solution for the DOPF for different test feeders. This thesis is organized into the following chapters: Chapter II describes the original BFM and the two-stage process of SOC relaxation for balanced networks. Chapter III discusses the process of approximating balanced feeders of the available IEEE test feeders that are unbalanced by nature. Chapter IV shows the numerical results of the tightness, duality gap, and decision variable comparison for different case studies. Chapter V provides an analysis and conclusion to the case studies. 4 CHAPTER II DISTRIBUTION OPTIMAL POWER FLOW BALANCED FEEDER The nature of the OPF problem is that it is nonlinear and nonconvex. Therefore, there are different approaches to solving this optimization problem. Also, there are different setups of the model; the model used in this research is the branch flow model. In Fig. 2.1, Ijk represents the complex current flow through the line, Vj represents the complex voltages at bus j, the series line impedance is Zjk = Rjk + jXjk, and Sjk is the complex power flowing through the line. The branch flow model is designed based on the computation of the power flow through the line. The optimal power flow optimization is used to minimize power losses. Figure 2.1: Balanced model of distribution feeder line connecting two buses. 2.1 Original Branch Flow Model 5 The following model along with the relaxation are explained in [12]: min f(x) = Y Pi jeA/- subject to (2.1) Vk Vj Zjkljk, (2.2) Sjk = Vjl*k, (2.3) sj = 4 Vi2l!f) k:jk - Zyliul2 + j|Vj|2t|i), kiS\j (2.4) |Vmin| < |Vj| < | V max 11 (2.5) | Ijk | | Imax 11 (2.6) Pf,min < Pf < Pjjmax (2.7) dj.min *4j dj.max' (2.8) In the above optimization problem, the objective function is described in (2.1), where it minimizes the produced active power that leads to minimizing the active power losses. Equation (2.2) represents Ohms law. Further, (2.3) represents the computation of apparent power flow through a line where denotes a conjugate. In (2.4), Sj is the net power injected at bus j, which is the bus of interest, where the power flows from i to j and then from j to k. The net injected power is computed as follows: >i U>j P-i ./b|j q-) (2.9) where p^pj1 are the generated and demanded active powers, and q^qj1 are the generated and demanded reactive powers. Furthermore, equations (2.5), (2.6), (2.7), (2.8) are the upper and lower bounds of the decision variables. This model is a nonconvex and nonlinear model. Its decision variables are complex 6 quantities, which are Ijk, Vj, Sjk, and Sj. Equation (2.3), and the left inequality of (2.5) are the main sources of nonlinearity. 2.2 Convex Relaxation of the BFM The first step of relaxation is performed by multiplying each side of (2.2) and (2.3) by their corresponding conjugates that attain (2.10) and (2.11), respectively. Also, it is essential to modify (2.4) by separating the real and imaginary parts, shown in (2.9), that attain (2.12) and (2.13). After applying the proposed step, the model of the optimization problem becomes: min f(x) = J^pf (2.1) jeAf subject to |Vkp = |V,|2 (SjtZjt* + ZJt V) + |ZJt|2|IJt|2, (2.10) Pjk + Q?k = |Vj|2|ijkl2, (2.11) p-p?= Erv-IhPii-RiiN2)- (2.12) k:js-k i:is-j 4 d = E (Q* IvjI2w) k:j>k -E(ci'i-xlI'il2 + lvil2T)' i:i>j (2.13) |Vmin|2 < |Vj|2 < |Vmax|2, (2.14) IT 12 ^ IT 12 I pjk | Pmaxl i (2.15) P? < p? < p? J- j^min l j J- j^max^ (2.7) q? < q? < q? (2.8) In the new formulation, the unknown variables become |Vj|2, |Ijk|2, Pjk, Qjk, Pj and q_j. The relaxation only considers the magnitudes of the complex quantities, called the angle-relaxation model. This is more of reparametrization than an actual mathematical relaxation. This relaxation is always exact for radial systems [10]. This 7 model is implemented using fmincon solver, which is a nonlinear solver built in Matlab software. However, the formulation remains nonconvex and nonlinear due to (2.11). Therefore, the second relaxation applied is the SOC, which relaxes the equality of (2.11) to an inequality (2.16). The model will become as follows: min f(x) = J^pf (2.1) jeAf subject to |Vkp = |V,|2 (SjtZjt* + ZJt V) + |ZJt|2|IJt|2, (2.10) Pjk + Qjl < |Vil2|iik|2. (2.16) p-p?= Xp-X
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PAGE 1 2{STAGEAPPROACHFORTHEOPTIMALPOWERFLOWOF PHOTOVOLTAIC{PENETRATEDDISTRIBUTIONSYSTEMS by AMINBUNAIYAN B.S.,UniversityofColoradoDenver,2016 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofScience ElectricalEngineeringProgram 2017 PAGE 2 ThisthesisfortheMasterofSciencedegreeby AminBunaiyan hasbeenapprovedforthe ElectricalEngineeringProgram by FernandoMancilla{David,Chair JaeDoPark DanConnors Date:May13,2017 ii PAGE 3 Bunaiyan,Amin(M.S.,ElectricalEngineering)2{StageApproachfortheOptimalPowerFlowofPhotovoltaic{Pen etratedDistribution SystemsThesisdirectedbyAssociateProfessorFernandoMancilla{David ABSTRACT Thethesispresentsa2{stagemodelforachievingfeasibleoptimal solutionalongwith numericalvalidationfortheexactnessoftheavailablesecondorde rconerelaxationin thedistributionoptimalpowerrowproblem.Tothisend,theIEEE4{ bus,13{bus, 34{busand123{busdistributionpowersystemsareadaptedbyba lancingthevarious feedersegmentsandbypopulatingselectednodeswithphotovolta icinverters.The dierenttestfeedersarestudiedundertypicaldailypowerprole s.Resultsgenerated byMatlabSoftwareforDisciplinedConvexProgramming(CVX)andth enonlinear solver(fmincon)suggestthatsecondorderconerelaxationfort hebranchrowmodelis exactonlyforsmallradialsystemsbutnotnecessaryforlargera dialsystems. Theformandcontentofthisabstractareapproved.Irecommen ditspublication. Approved:FernandoMancilla{David iii PAGE 4 DEDICATION Thisthesisisdedicatedtomyfamilywhohavesupportedmesincetheb eginningofmy studies.Theyprovidemewithagreatsourceofmotivationandinspir ation.Finally, thisthesisisdedicatedtoallthosewhobelievethatknowledgeispowe r. iv PAGE 5 ACKNOWLEDGMENTS Greatestthankstomyparentswhohavesupportedmecompletely throughgoodtimes andbad.Also,Iwouldliketoexpressmydeepestgratitudetomyadv isor,Dr. FernandoMancilla{David,forhisguidance,valuableadviceandsuppo rtduringthis research. v PAGE 6 TABLEOFCONTENTS CHAPTER 1.INTRODUCTION................................12.DISTRIBUTIONOPTIMALPOWERFLOW{BALANCEDFEEDER..5 2.1OriginalBranchFlowModel.......................52.2ConvexRelaxationoftheBFM.....................7 3.BRANCHANALYSIS..............................10 3.1UnbalancedSeriesImpedanceMatrices.................10 3.1.1Overheadlines..........................103.1.2UndergroundLines.......................11 3.2UnbalancedShuntAdmittanceMatrices.................13 3.2.1OverheadLines.........................133.2.2UndergroundLines.......................14 3.3ApproximatingTheLineMatrix.....................15 3.3.1ApproximatingtheSeriesImpedanceMatrix.........153.3.2ApproximatingtheShuntAdmittanceMatrix........17 4.CASESTUDIES.................................21 4.1Introduction................................214.2ErrorComputation............................224.3PreparingCaseStudies..........................23 4.3.1ApproximatingaBalancedFeeder...............234.3.2SynthesizingLoadProles...................244.3.3SynthesizingPVGenerationProles..............26 4.4CaseStudies................................27 4.4.1CaseStudyIEEE4{busfeeder.................274.4.2CaseStudyIEEE13{busfeeder................304.4.3CaseStudyIEEE34{busfeeder................33 vi PAGE 7 4.4.4CaseStudyIEEE123{busfeeder...............36 5.CONCLUSIONS.................................39 REFERENCES.....................................40APPENDIX A.MATHEMATICALBACKGROUND......................42 A.1Convexity.................................42 A.1.1ConvexSet............................42A.1.2ConvexFunction........................42A.1.3Convexityconditions......................43A.1.4Relaxation............................43 A.2LagrangianOptimization.........................44A.3Duality...................................44A.4Karush-Kuhn-Tucker(KKT)conditions.................4 6 A.5SemideniteProgrammingSDP.....................47A.6SecondOrderConeProgrammingSOCP................47 vii PAGE 8 LISTOFTABLES TABLE3.1NomenclatureandDenitionsforChapter3................ ...20 4.1IEEE4{busfeeder:LineCongurationData............... ...25 4.2IEEE123{busfeeder:LineCongurationData............. ...25 4.3IEEE4{busfeeder:tightnessresults................... ...28 4.4IEEE4{busfeeder:dualitygapandrelativeerrorofdecisionvar iables...29 4.5IEEE13{busfeeder:tightnessresults.................. ....31 4.6IEEE13{busfeeder:dualitygapandrelativeerrorofdecisionva riables...32 4.7IEEE34{busfeeder:tightnessresults.................. ....34 4.8IEEE34{busfeeder:dualitygapandrelativeerrorofdecisionva riables...35 4.9IEEE123{busfeeder:tightnessresults................. ....37 4.10IEEE123{busfeeder:dualitygapandrelativeerrorofdecision variables..38 viii PAGE 9 LISTOFFIGURES FIGURE2.1Balancedmodelofdistributionfeederlineconnectingtwobuses. .......5 3.1 unbalancedlinefeeder. ..............................10 4.12{Stageapproach.................................2 1 4.2 TheIEEE4{bustestfeedernetwork. .......................27 4.3 IEEE4{busfeederpowerproles:totaldemandedpowerandto talavailablepower fromPVgeneration ................................27 4.4 TheIEEE13-bustestfeedernetwork. .......................30 4.5 IEEE13{busfeederpowerproles:(a)totaldemandedpowera ndtotalavailable powerfromPVgeneration,(b)individualPVpanelproles. ...........30 4.6 TheIEEE34-bustestfeedernetwork. .......................33 4.7 IEEE34{busfeederpowerproles:(a)totaldemandedpowera ndtotalavailable powerfromPVgeneration,(b)individualPVpanelproles. ...........33 4.8 TheIEEE123-bustestfeedernetwork. ......................36 4.9 IEEE123{busfeederpowerproles:(a)totaldemandedpower andtotalavailable powerfromPVgeneration,(b)individualPVpanelproles. ...........36 A.1Examplesofconvexandnonconvexsets.................. ..42 A.2Denitionofconvexfunction..........................4 2 A.31storderconditionofconvexfunction................... ...43 A.4(a)relaxedset,(b)relaxedfunction.................... ...44 A.5Epigraphgeometricinterpretationforanoptimizationproblem.. ......46 ix PAGE 10 CHAPTERI INTRODUCTION TheOptimalPowerFlow(OPF)problemintegratesthepowerrow(PF )problemand theeconomicdispatch(ED)problemthat,knownasanonlinearprob lem[1].TheED ndsthelowestgenerationcostthatsatisesdemand,whilethePF countsforlosses andthetopologyofthenetwork.Itcanalsoidentifythepowertra nsferdistribution function(PTDF).TheOPFproblem,whichwasintroducedbyJ.Carp entierinFrench in1962[2]isNP-hardbecauseofitsnonlinearity.TheproblemisNP{ha rdbecauseof itsnonlinearity.However,itisahighlydesirableproblemthatcanbeso lvedinmany waysfordierentapplications.Morespecically,therearemanyte chniquesforsolving thisproblembylinearizingorrelaxingthenonlinearity.Adecoupled(DC)powerrowapproximationwasproposedin1974[3].T heDCOPFis solvedthroughlinearprogramming(LP),whichcountsforthetopo logyofthesystem butnotforlossesofthesystem.Fromaneconomicpointofview,th eDCOPFworks fairlywellfortransmissionnetworksbutnotfordistributionnetwo rks.Thereasonis thatDCapproximationworksbetterwhentheX/Rratioisbeingsuc ientlyhigh[4]. Distributionsystemsweredesignedtobeunidirectional[5]andtheir X/Rratioisnot highenoughtouseDCapproximation.Theproblemhasbecomemoreinterestingonadistributionlevelasre newableresources havebecomemoreavailabletoindividualusers.TheACOPFwassolved usinga nonlinearsolver,butthereisnoguaranteeitssolutionwillconverge totheglobal optimum;instead,itconvergestoalocalminimumthatishighlydepend entonthe initialpoints.Therefore,studieshavetakenthepathoflinearizat ionandconvex relaxation.Thesemideniteprogramming(SDP)relaxationtechniqueisonewayt oobtaina solutiontoaconvexproblem.In[6],theACOPFwassolvedusingSDPin2 008.SDP relaxationfortheOPFproblemwasproposedusingprimal{dualinter iorpoint 1 PAGE 11 algorithmthatisaconvexrelaxation.ItstatesthatSDPworksout aslinear programmingbutinitsmatricesform.Thismethodtakesadvantage ofthestrong dualityoftheLP.Itwasproventhatrst{orderKKToptimumcond itionsarethesame forprimalanddualOPFproblems.Thedisadvantageisthat,dueto limitationsin computercapacity,problemsinextremelylargesystemscan'tbeso lved[6]. Further,[7],[5]and[8]describetheuseofSDP,whosegloballyoptima lsolutionmaybe foundwithavailablenumericalalgorithmsinpolynomialtime.In[7],SDPis usedto obtainsolutionstoconvexproblemssolvableinpolynomial{timecomplex ity.Asa result,themethoddemonstratestheabilitytoreachtheglobalop timalsolutionofthe originalnonconvexoptimalpowerrowproblem.In[5],SDPisusedtoo btaina computationallyfeasibleconvexreformulation.Thisisshownusingre alworld photovoltaic(PV)generationandloadproledataforanillustrative low{voltage residentialdistributionsystem.In[8],theOPFproblemissolvedbylev eragingSDP throughtheuseofacentralizedcomputationaldevicewithreduce dcomputational burden.Secondorderconeprogramming(SOCP)isanotherwaytosolvecon vexproblems.In [9],itwasproventhatSOCPisaspecialcaseofSDP.ThismeansthatS DPisa generalizationofSOCP.However,SOCPismoretractableoptimizatio nclassthanSDP. Therefore,SOCPshouldnotbesolvedasSDP.Anglereparameterizationalongwithsecondordercone(SOC)relax ationareadditional techniquesusedtosolveaconvexOPFproblem.In[10],therelaxatio ninthebus injectionmodel(BIM)andthebranchrowmodel(BFM)wereprese ntedalongwitha provenrelationshipbetweenthetwomodels.In[11],threetypesof sucientconditions ofexactnesswereinvestigated.Therstconditioninvolvespower injectionsthat requiresomeoftheactiveandreactivepowersinjectedatthetwo endsofeachlineto notbebounded.Inotherwords,upperandlowerconstraintsofb othendsofeachline cannotbeactiveatthetwoendsunlesstheobjectivefunctionisst rictlyconvex.The 2 PAGE 12 secondconditioninvolvesvoltagemagnitudesthatrequirealltheup perboundsof voltagestobeinactive.Thethirdconditionreferstovoltageangles ,whichrequires smallenoughdierencesacrosseachline.OnemainadvantageofsolvingtheOPFproblemusingSOCrelaxationn dinga globallyoptimalsolutiontotheproblem[11].Ifthesolutionfoundusing therelaxation isexact,thentheoptimalsolutiontotheoriginalnonconvexOPFpr oblemcanbe computed.Throughthestudyin[11],ithasbeenfoundthatthetop ologyofthe networkisanimportantfactorwhensolvingtheOPFproblem.Thean glerelaxationis exactforradialsystemsbutnotformeshsystems.Thecondition sareinsucientfor meshsystems;however,theycouldbesucientifthephaseshifte rsarecontrolledina strategicwaythatmakesthesystembehavelikearadialsystem[11 ]. Anotherinvestigationin[4]showsthatanadditionalsucientcondit ionisnecessary forensuringtheexactnessoftheSOCrelaxationfordistributions ystems.The conditionisrelatedtothethermallimitsandcapacityofthebranche s.Thiscondition statesthattherecanonlybeactivereversedpower,onlyreactiv ereversedpower,orno reversepowerrowsthroughaline.Thisconditionisimportantdueto thepresenceof renewableenergydevicescapableofinjectingpowerinthelines.Thebranchrowmodel,whichinvolvesrelaxationandconvexication, isonesuch design.Thedesignismadetocarryoutoptimizationofmeshtopologie sinadditionto radialnetworks.Thisphaseincludesreviewingtheconvexrelations hip,controllingload row,checkingonoptimalpowerrow,andmanagingtheentirepower system.The powerrowmodelfocusesondierentvariablesassociatedwiththe distributedpower, whichincludesthevoltagesandinjectionsofcurrenttogetherwith powerinjectionsas toregulatepowerrowinthenetwork[12].Therelaxedbranchrowapproachpresentedin[12].SolvingtheOPFp roblemis carriedoutusingtwostagesofrelaxation.Theinitialstepinvolvest heeliminationof thecurrentandvoltageangleswhilethepurposeofthesecondsta geistoapproximate 3 PAGE 13 theoutcomingproblemviaaconicprogram.Therelaxationstepshav ebeenproven equalthroughtheradialsystems[12].Themaincontributionsofthisthesisare1.)tonumericallyvalidatethe exactnessof theSOCrelaxationintheBFMoftheoptimalpowerrowproblematthe distribution level(DOPF)forbalancedfeeders,2.)toinvestigatethestrengt hofthesolutionfound bytherelaxedmodel,and3.)toseekatruesolutionfortheDOPFfor dierenttest feeders.Thisthesisisorganizedintothefollowingchapters:Chapt erIIdescribesthe originalBFMandthetwo{stageprocessofSOCrelaxationforbalan cednetworks. ChapterIIIdiscussestheprocessofapproximatingbalancedfee dersoftheavailable IEEEtestfeedersthatareunbalancedbynature.ChapterIVsh owsthenumerical resultsofthetightness,dualitygap,anddecisionvariablecomparis onfordierentcase studies.ChapterVprovidesananalysisandconclusiontothecases tudies. 4 PAGE 14 CHAPTERII DISTRIBUTIONOPTIMALPOWERFLOW{BALANCEDFEEDER ThenatureoftheOPFproblemisthatitisnonlinearandnonconvex.T herefore,there aredierentapproachestosolvingthisoptimizationproblem.Also,t herearedierent setupsofthemodel;themodelusedinthisresearchisthebranchr owmodel. Z jk =R jk + j X jk I jk Busj Y jk 2 = j B jk 2 Busk V j V k Figure2.1:Balancedmodelofdistributionfeederlineconnectingtwo buses. 2.1OriginalBranchFlowModel InFig.2.1,I jk representsthecomplexcurrentrowthroughtheline,V j representsthe complexvoltagesatbusj,theserieslineimpedanceisZ jk =R jk + j X jk ,andS jk isthe complexpowerrowingthroughtheline.Thebranchrowmodelisdesignedbasedonthecomputationofthep owerrowthrough theline.Theoptimalpowerrowoptimizationisusedtominimizepowerlos ses. 5 PAGE 15 Thefollowingmodelalongwiththerelaxationareexplainedin[12]: min f ( x )= X j 2N p gj (2.1) subjectto V k =V j Z jk I jk ; (2.2) S jk =V j I jk ; (2.3) s j = X k:j k (S jk j j V j j 2 B jk 2 ) X i:i j (S ij Z ij j I ij j 2 + j j V j j 2 B ij 2 ) ; (2.4) j V min jj V j jj V max j ; (2.5) j I jk jj I max j ; (2.6) p gj ; min p gj p gj ; max ; (2.7) q gj ; min q gj q gj ; max : (2.8) Intheaboveoptimizationproblem,theobjectivefunctionisdescrib edin(2.1),whereit minimizestheproducedactivepowerthatleadstominimizingtheactive powerlosses. Equation(2.2)representsOhm'slaw.Further,(2.3)representst hecomputationof apparentpowerrowthroughalinewhere denotesaconjugate. In(2.4),s j isthenetpowerinjectedatbusj,whichisthebusofinterest,wher ethe powerrowsfromitojandthenfromjtok.Thenetinjectedpower iscomputedas follows: s j =(p gj p dj )+ j (q gj q dj ) : (2.9) wherep gj ; p dj arethegeneratedanddemandedactivepowers,andq gj ; q dj arethe generatedanddemandedreactivepowers.Furthermore,equations(2.5),(2.6),(2.7),(2.8)aretheuppera ndlowerboundsofthe decisionvariables.Thismodelisanonconvexandnonlinearmodel.Itsdecisionvariablesa recomplex 6 PAGE 16 quantities,whichareI jk ,V j ,S jk ,ands j .Equation(2.3),andtheleftinequalityof(2.5) arethemainsourcesofnonlinearity. 2.2ConvexRelaxationoftheBFM Therststepofrelaxationisperformedbymultiplyingeachsideof(2 .2)and(2.3)by theircorrespondingconjugatesthatattain(2.10)and(2.11),re spectively. Also,itisessentialtomodify(2.4)byseparatingtherealandimagina ryparts,shown in(2.9),thatattain(2.12)and(2.13).Afterapplyingtheproposedstep,themodeloftheoptimizationpr oblembecomes: min f ( x )= X j 2N p gj (2.1) subjectto j V k j 2 = j V j j 2 (S jk Z jk +Z jk S jk )+ j Z jk j 2 j I jk j 2 ; (2.10) P 2jk +Q 2jk = j V j j 2 j I jk j 2 ; (2.11) p gj p dj = X k:j k P jk X i:i j (P ij R ij j I ij j 2 ) ; (2.12) q gj q dj = X k:j k (Q jk j V j j 2 B jk 2 ) X i:i j (Q ij X ij j I ij j 2 + j V j j 2 B ij 2 ) ; (2.13) j V min j 2 j V j j 2 j V max j 2 ; (2.14) j I jk j 2 j I max j 2 ; (2.15) p gj ; min p gj p gj ; max ; (2.7) q gj ; min q gj q gj ; max : (2.8) Inthenewformulation,theunknownvariablesbecome j V j j 2 j I jk j 2 ,P jk ,Q jk ,p j andq j Therelaxationonlyconsidersthemagnitudesofthecomplexquantit ies,calledthe \angle{relaxationmodel."Thisismoreofreparametrizationthanana ctual mathematicalrelaxation.Thisrelaxationisalwaysexactforradials ystems[10].This 7 PAGE 17 modelisimplementedusingfminconsolver,whichisanonlinearsolverbu iltinMatlab software.However,theformulationremainsnonconvexandnonlineardueto( 2.11).Therefore, thesecondrelaxationappliedistheSOC,whichrelaxestheequalityof (2.11)toan inequality(2.16).Themodelwillbecomeasfollows: min f ( x )= X j 2N p gj (2.1) subjectto j V k j 2 = j V j j 2 (S jk Z jk +Z jk S jk )+ j Z jk j 2 j I jk j 2 ; (2.10) P 2jk +Q 2jk j V j j 2 j I jk j 2 ; (2.16) p gj p dj = X k:j k P jk X i:i j (P ij R ij j I ij j 2 ) ; (2.12) q gj q dj = X k:j k (Q jk j V j j 2 B jk 2 ) X i:i j (Q ij X ij j I ij j 2 + j V j j 2 B ij 2 ) ; (2.13) j V min j 2 j V j j 2 j V max j 2 ; (2.14) j I jk j 2 j I max j 2 ; (2.15) p gj ; min p gj p gj ; max ; (2.7) q gj ; min q gj q gj ; max : (2.8) Thismodeliscalledthe\SOC{relaxationmodel",whichisaconvexmode l,butits exactnessisrelatedtotheexplainedsucientcondition.Themodel isimplemented usingCVX,whichisapackageforspecifyingandsolvingconvexprogr ams[13]. Animportantstepistorecovertheanglevaluesaftersolvingeither the angle{relaxationmodelortheSOC{relaxationmodel.Thedistribute dalgorithmof 8 PAGE 18 anglerecoverywaspresentedin[12],anditisasfollows: \ I jk = \ V j \ S jk (2.17) \ V j = \ ( j V j j 2 \ Z jk S jk )(2.18) 9 PAGE 19 CHAPTERIII BRANCHANALYSIS Linesegmentsarecongureddierentlybetweenoverheadandun dergroundlines.For example,theIEEE123{bustesthaselevenvaryingconnectiontyp esforoverheadlines andonecongurationfortheundergroundline.Impedancematric esareprovidedin [14]foreachcongurationandareunbalanced.Alltestfeedersin[14]areunbalancedfeeders,whichisnumericallyr ecognizedby lookingatthelinesegmentdata.Becausetheprovidedmatricesare notreadytobe approximated,abackwardstepismadebyrecalculatingthematrice susingCarson's equationandtheKronreduction. 3.1UnbalancedSeriesImpedanceMatrices 3.1.1Overheadlines Z aa =R a +jX a I a I b I c Busj (sendingend) Z bb Z cc Z ac Z ab Z bc a b c Busk (receivingend) Figure3.1: unbalancedlinefeeder. Usingthecongurationdataprovidedin[14]andapplyingittoCarson 'sequations (3.1),(3.2),theprimitiveimpedancematrixisbuilt[15]. z ii =r i +r e +j0 : 12134 ln( D e GMR )(3.1) z ij =r e +j0 : 12134 ln( D e D ij )(3.2) 10 PAGE 20 Theprimitiveimpedancematrixiscomputedasfollows: [Z prim ]=[Z abcn ]= 266666664 z aa z ab z ac z an z ba z bb z bc z bn z ca z cb z cc z cn z na z nb z nc z nn 377777775 Fromthismatrix,theKronreduction(3.3)isusedtocomputetheph aseimpedance matrix[Z abc ],whichisanequivalentmatrixthatincludestheeectoftheneutral phase. Z ij =z ij z in z nj z nn (3.3) Thephaseimpedancematrixisasfollows: [Z abc ]= 266664 Z aa Z ab Z ac Z ba Z bb Z bc Z ca Z cb Z cc 377775 Thismatrixisunsymmetricalbecauseitso{diagonalelementsaren otequivalent.The computedmatricesdomatchalltheprovidedmatrices[14]fortheo verheadline congurationsofalldierenttestfeeders.3.1.2UndergroundLinesInsometestfeedersin[14],thereareundergroundlinecongurat ionsthatare \ConcentricNeutralCable,"whichhas3phases/3wires.Therefor e,computing[Z abc ] isnotthesameasoverheadlineconguration[16].Sinceeachcableha sitsown conductors,[Z prim ]iscalculatedusingthemodiedCarson'sequations(3.4){(3.11)as 11 PAGE 21 follows: [Z prim ]= 264 [Z ij ][Z in ] [Z nj ][Z nn ] 375 6 6 where: [Z ij ]= 266664 z aa z ab z ac z ba z bb z bc z ca z cb z cc 377775 ; suchthat: z ii =r+r e +j0 : 12134 ln( D e GMR )(3.4) z ij =r e +j0 : 12134 ln( D e D ij )(3.5) [Z in ]= 266664 z aa n z ab n z ac n z ba n z bb n z bc n z ca n z cb n z cc n 377775 ; suchthat: z ii n =r+r e +j0 : 12134 ln( D e R )(3.6) z ij n =r e +j0 : 12134 ln( D e D i n j )(3.7) [Z nj ]= 266664 z a n a z a n b z a n c z b n a z b n b z b n c z c n a z c n b z c n c 377775 ; suchthat: z i n i =r+r e +j0 : 12134 ln( D e R )(3.8) z i n j =r e +j0 : 12134 ln( D e D i n j )(3.9) 12 PAGE 22 and [Z nn ]= 266664 z a n a n z a n b n z a n c n z b n a n z b n b n z b n c n z c n a n z c n b n z c n c n 377775 suchthat: z i n i n =r+r e +j0 : 12134 ln( D e R )(3.10) z i n j n =r e +j0 : 12134 ln( D e D i n j n )(3.11) Similarapproachisusedfortheoverheadlines,fromprimitivematrix[Z prim ],theKron reduction(3.12)isusedtocomputethephaseimpedancematrix[Z abc ][16]. [Z abc ]=[Z ij ] [Z in ][Z nn ] 1 [Z nj ](3.12) 3.2UnbalancedShuntAdmittanceMatrices 3.2.1OverheadLinesUsingthesamemethodologyofthecomputationoftheseriesimpeda ncematrices,a stepbackisneededtocomputetheshuntcapacitivematricesbefo reapplyingany approximation.Accordingto[16],theprimitivepotentialcoecientm atrix[P prim ]is neededintheprocessofcomputingthelineadmittancematrixY abc iscomputed. [P prim ]hasthesizeofthenumberofconductorsbythenumberofcondu ctors.Forthe overheadcongurations,thelinesarethreephase{fourwires.T heprimitivepotential coecientmatrixisasfollows: [P prim ]=[P abcn ]= 266666664 p aa p ab p ac p an p ba p bb p bc p bn p ca p cb p cc p cn p na p nb p nc p nn 377777775 13 PAGE 23 Theelementsoftheprimitivepotentialcoecientmatrixarecomput edasfollows: p ii = 1 2 0 ln( S ii RD i )(3.13) p ij = 1 2 0 ln( S ij D ij )(3.14) UsingKronreduction3.15,thephasepotentialcoecientmatrix[P abc ]isasfollows: P ij =p ij p in p nj p nn (3.15) [P abc ]= 266664 P aa P ab P ac P ba P bb P bc P ca P cb P cc 377775 Now,itispossibletocomputethephasecapacitivematrixthatis[C abc ]=[P abc ] 1 Then,theshuntadmittancematrix[Y abc ]iscomputedasfollows: [Y abc ]=j2 [C abc ]= 266664 Y aa Y ab Y ac Y ba Y bb Y bc Y ca Y cb Y cc 377775 Theresultsofthismatrixdomatchtheprovideddataofdierentfe edersin[14].The matrixhasunequalo{diagonalelements,whichmakethematrices asymmetrical. 3.2.2UndergroundLinesAgain,someofthefeedershavecongurationsthatareundergr ound.Thecomputation oftheY abc matrixisdierentthantheoverheadlines.Sinceallphasesoftheco ncentric neutralstrandaregrounded,thereisonlythecapacitancebetw eeneachphaseof groundY pg tobecomputed. Y pg =j2 f 2 0 r ln( R b RD c ) 1 k ln( kRD s R b ) (3.16) 14 PAGE 24 Thisimpliesthatallphaseshavethesamecharge[16],whichresultsint hefollowing matrix: [Y abc ]= 266664 Y pg 00 0Y pg 0 00Y pg 377775 Finally,thisisthelastsanitycheckofthecomputedmatriceswiththe provided matricesin[14]. 3.3ApproximatingTheLineMatrix 3.3.1ApproximatingtheSeriesImpedanceMatrixThegoalistoapproximatethephaseimpedancematrixwithabalance dsymmetric matrixthathasalldiagonalelements(self{impedanceelementsZ s )equaltoeachother andallodiagonalelements(mutualimpedanceelementsZ s )equaltoeachother[15]. Thisapproachhasbeenusedtosolvepowerrowandfaultproblems.Therefore,recalculatinganapproximatedbalancedversionofthe primitiveimpedance matrixZ prim : b requiresconsideringageometricmeandistance(GMD)foroverhea dand undergroundlinecongurations,whichwillmaketheCarson'sequat ions(3.2),(3.5), (3.7),(3.9),and(3.11)changeto: z ij =r e +j0 : 12134 ln( D e GMD )(3.17) whereGMD= 3 p D ab D bc D ca forphase{phasemutualimpedancesand GMD= 3 p D an D bn D cn forphase{neutralmutualimpedances. AfterapplyingtheKronreductiontoZ prim : b ,thebalancedimpedancematrixZ abc : b will be: [Z abc : b ]= 266664 Z aa Z ab Z ac Z ba Z bb Z bc Z ca Z cb Z cc 377775 = 266664 Z s Z m Z m Z m Z s Z m Z m Z m Z s 377775 15 PAGE 25 Thismatrixissymmetrical;allo{diagonalelementsareequaltoeac hother,andall diagonalelementsarealsoequaltoeachother.Thismatrixcansub stituteZ abc inorder tocomputeanapproximatedsolution.OutofZ abc : b ,thesequenceimpedancematrixZ 012 iscalculatedasthefollowing: [Z 012 ]= 266664 Z 0 00 0Z 1 0 00Z 2 377775 where: Z 0 =Z s +2Z m (3.18) Z 1 =Z 2 =Z s Z m (3.19) Fromthismatrix,threedierentmodelsofapproximatedmatricesc ouldbebuilt[15]: 1.TheSequencePhaseImpedanceModelZ SEQ [Z SEQ ]= 1 3 266664 (2Z 1 +Z 0 )(Z 0 +Z 1 )(Z 0 +Z 1 ) (Z 0 +Z 1 )(2Z 1 +Z 0 )(Z 0 +Z 1 ) (Z 0 +Z 1 )(Z 0 +Z 1 )(2Z 1 +Z 0 ) 377775 ThismatrixhappenstohaveexactentriesofZ abc : b ,whichisthematrixcalculated usingtheGMD.Assumingabalancedpowerloadandgeneration,thism atrix couldbeusedtosolveonephaserepresentation. 2.TheModiedSequencePhaseImpedanceModelZ MD [Z MD ]= 1 3 266664 (2Z 1 +Z 0 )00 0(2Z 1 +Z 0 )0 00(2Z 1 +Z 0 ) 377775 Theo{diagonalelementsofthismatrix,consideringthemutualph asecoupling betweenphases,canbeeliminated.Also,itisonephaserepresenta tionthathas beenusedinthecasestudyoftheapproximatedmodelinthecoming chapters. 16 PAGE 26 3.ThePositiveSequencePhaseImpedanceModelZ POS [Z POS ]= 266664 Z 1 00 0Z 1 0 00Z 1 377775 Thismatrixonlyconsidersthepositivesequencephase.Numerically, thevalues areslightlyo.Therefore,itmightnotbethebestchoicefortheca sestudy. 3.3.2ApproximatingtheShuntAdmittanceMatrixAccordingto[17],itispossibletoapproximateP abc whereP abc : b hasthefollowing structure: [P abc : b ]= 266664 P s P m P m P m P s P m P m P m P s 377775 usingtheresultantmatrixtocomputethefollowing: [Y abc : b ]=j2 [C abc ]=j2 [P abc ] 1 = 266664 Y s Y m Y m Y m Y s Y m Y m Y m Y s 377775 ThisisdoneusingthesametechniqueforZ abc bycomputingGMDandconsideringline transpose,whichleadstothefollowingmatrix,called(sequencepot entialcoecient 17 PAGE 27 matrixP 012 ): [P 012 ]=[A] 1 [P abc : b ][A](3.20) where: A= 266664 1111 2 1 2 377775 (3.21) and: =1 e j120 (3.22) Equation3.20couldbecomputedfollowingthesameprocessofseries impedances: [P 012 ]= 266664 P 0 00 0P 1 0 00P 2 377775 where: P 0 =P s +2P m (3.23) P 1 =P 2 =P s P m (3.24) usingtheresultantP 012 tocomputeC 012 andY 012 asthefollowing: [C 012 ]=[P 012 ] 1 = 266664 C 0 00 0C 1 0 00C 2 377775 and: [Y 012 ]=j2 [C 012 ]= 266664 Y 0 00 0Y 1 0 00Y 2 377775 18 PAGE 28 Fortheundergroundline,itisnotnecessarytoapproximatenorto havethesequence matrixsincethematrixcomputedoriginallyisdiagonalwiththeequivale ntdiagonal element.Approximatingtheshuntadmittancematrixfortheoverh eadlinewillfollow thesameprocessusedwhenapproximatingtheserieslineimpedance Inordertohave thesamefeaturesattheapproximatedmatrices. 1.TheSequencePhaseAdmittanceModelY SEQ [Y SEQ ]= 1 3 266664 (2Y 1 +Y 0 )(Y 0 +Y 1 )(Y 0 +Y 1 ) (Y 0 +Y 1 )(2Y 1 +Y 0 )(Y 0 +Y 1 ) (Y 0 +Y 1 )(Y 0 +Y 1 )(2Y 1 +Y 0 ) 377775 Similartotheseriesimpedancework,thismatrixhappenstohavethe same entriesasY abc : b .Assumingabalancedpowerloadandgeneration,thismatrix couldbeusedtosolveonephaserepresentation. 2.TheModiedSequencePhaseAdmittanceModelY MD [Y MD ]= 1 3 266664 (2Y 1 +Y 0 )00 0(2Y 1 +Y 0 )0 00(2Z 1 +Y 0 ) 377775 FollowingthesameassumptionmadeforZ MD ,thismatrixconsidersthemutual phasecouplingbetweenphaseswheretheo{diagonalelementsca nbeeliminated. Y MD isusedinthecasestudyoftheapproximatedmodelinfuturechapt ers. 3.ThePositiveSequencePhaseAdmittanceModelY POS [Y POS ]= 266664 Y 1 00 0Y 1 0 00Y 1 377775 Similarly,thismatrixconsidersonlythepositivesequencephase.Num erically,the valuesareslightlyo.Therefore,itmightnotbethebestchoicefor thiscase study. 19 PAGE 29 Table3.1:NomenclatureandDenitionsforChapter3. zii(n = mi)diagonalimpedances zij(n = mi)o{diagonalimpedances ri(n = mi)conductorresistance re=1 : 588 10 3f(n = mi)resistanceofCarson'sequivalentofearthreturnconductor De=2160 ( f)1 2 (ft)functionofearthresistivityandfrequency Dij(ft)thedistancebetweenconductoriandj GMD(ft)thegeometricmeandistancebetweenphasesGMR(ft)conductorgeometricmeanradiusf=60(Hz)frequency =100(n = m)averageearthresistivity [Zij](n = mi)phase{phaseconductormatrix [Zin](n = mi)phase{neutralofconductormatrix [Znj](n = mi)neutral{phaseconductormatrix [Znn](n = mi)neutral{neutralofconductormatrix k{numberofconcentricstrandsdc(inch)diameterofthephaseconductor ds(inch)diameterofthephasestrand dod(inch)outsidediameter R=dod ds 24(ft)distancebetweenthecenterofthephase conductortoitsownstrand r(n = mi)phaseconductorresistance rs(n = mi)neutralstrandresistance rcn=rs k(n = mi)equivalentresistanceoftheconcentric neutral GMRs(ft)strandgeometricmeanradius GMRcn= k p GMRskRk 1(ft)equivalentgeometricmeanradiusofthe concentricneutral Sii(ft)distancebetweenthephaseanditsimage Sij(ft)distancebetweenphaseiandimageofphasej RDi(inch)radiusoftheconductor P(mi F)mutualpotentialcoecient 0=1 : 424 10 2(mi F)permittivityoftheair Rb=dod ds 2(ft)distancebetweenthecenterofthe phaseconductortoitsownstrand RDc(inch)radiusofthephaseconductor RDs(inch)radiusoftheneutralstrand r=2 : 3074(mi F)permittivityofthemedium (Cross{LinkedPolyethylene) 20 PAGE 30 CHAPTERIV CASESTUDIES 4.1Introduction ConditionsunderwhichSOC-basedrelaxationsareexactfortheOP Fproblemare currentlybeinginvestigatedin[4]and[11].Incontrast,therstpa rtofthiscasestudy investigatesthenumericalresultsoftherelaxedmodel.Thegoalis totestthe exactnessofthisSOCrelaxationbycheckingiftheinequalityof(2.16 )istightat dierentpowerlevels.Thesecondpartofthecasestudyinvestiga testhedistance betweentheresultsandthetrueresultsbyusingthe2{stagemod elshowinginFig4.1. Thisapproachleadstondingafeasiblepointusingthenonlinearsolve rbecausethe relaxedsolutionmaybeinfeasible. Model 2 BFM SOC relaxed convex CVX Model 1 angle relaxed exact nonlinear fmincon Recover Angle Figure4.1:2{Stageapproach. RecallingModel1:\SOC{relaxationmodel"andModel2:\angle{relax ationmodel," whichareexplainedindetailinchapter2.Model1isanexactmodelbu tnonlinear. ThismodelisimplementedusingthefminconsolverinMatlab.Model2isa relaxed andconvexmodel.ThismodelisimplementedusingCVXinMatlab.Thede cision variablessolutionofModel2willbefedtothenonlinearsolverasinitial guessvalues forsolvingModel2. 21 PAGE 31 TheCVXalgorithmwasconguredwiththefollowingsettings:semide nite programmingSDPT3solverandprecisioncontrolledtoasettoleranc ecalled`best'[13]. Forthefminconsolver,thenonlineartolerancewasassignedas 10 13 toinsure tightness. 4.2ErrorComputation Foreachcasestudy,threedierenttypesoferrorswerecompu tedatdierenttimes. 1.Tightness Recallthefollowingequationsfromchapter2. P 2jk +Q 2jk = j V j j 2 j I jk j 2 ; (2.11) P 2jk +Q 2jk j V j j 2 j I jk j 2 ; (2.16) Theaboveequationsaretheonlydierencebetweenthetwomodels .The tightnesscomparestheright{handsidewiththeleft{handsideoft heinequalityin (2.16),wheretheleft{handsideisLHS=P 2jk +Q 2jk ,andtheright{handsideis RHS= j V j j 2 j I jk j 2 Thetightnessrelativeerroriscomputedasfollows: rel = j LHS RHS j j RHS j 100(4.1) Thisrelativeerroriscomputedperlineforeachsystemateachtime. Andto attainexactness,theerrorshouldbesmallenoughtoimplythatLH S=RHS. 2.DualityGap: Thedualitygapcomparestheobjectivevaluescomputedinthetwom odels. Therewillbeoneerrorcalculationpertime.Thesolutioncomputedbytherelaxedmodelisthelowerbound(L),w herethe solutioncomputedbytheexactmodelistheupperbound(U).Ther elativeerror iscomputedasfollows: DG= U L U 100 : (4.2) 22 PAGE 32 3.DecisionVariables: Thedecisionvariablesxofbothmodelsare j V j j 2 j I jk j 2 ,P jk ,Q jk ,p j andq j .The computederrorwillcompareboththesolutionfoundbytherelaxed model(x 0 ) andthesolutionfoundbytheexactmodel(x).Therelativeerroris computedas follows: DX= x x 0 max(x ; x 0 ) 100(4.3) 4.3PreparingCaseStudies Inordertoconductthecasestudy,therearesomeassumptions andapproximations thathavebeenmade.Thisisnecessarybecausethetestfeeders providedby[14]are unbalanceddistributionsystemsandthealgorithmthatisusedonlyc onsidersbalanced systems.Therefore,preparingthebenchmarkisnecessaryfor therelaxedBFM. 4.3.1ApproximatingaBalancedFeederDatain[14]showsthatsomeofthepositionsareconsistentlyclosed .Anassumptionis madethat,forallclosedswitches,theimpedanceisnegligible,creat ingashortcircuit [18].Oneimportantapproximationisthatofthebalancebranchimpedanc esandadmittance matrices.Approximationdoneforthebranchimpedancematricesw asobtainedby following[15]and[16].Followingthesamemethodology,theapproximat ionofthe branchadmittancematrixfollows[17].Theseapproximationswereex plainedpreviously inchapter3.Therststepoftheproposedapproximationisconsideringthegeo metricmeandistance toCarson'sequationasrepresentedin[15]toobtainanapproximat edprimitivematrix. AfterapplyingtheKronreduction,theresultantmatrixisabalance dmatrixasfollows: 23 PAGE 33 [ Z abc ]= 266664 Z s Z m Z m Z m Z s Z m Z m Z m Z s 377775 Inthismatrix, Z s istheselfimpedanceelementand Z m isthemutualimpedance element.Matchingtheassumptionmadein[18],assumingabalancedpo werloadand generation,thismatrixcouldbeusedtosolveonephaserepresent ationbynot consideringthemutualphasecouplingofphases.Thisresultsinthe eliminationofthe o{diagonalelements,whichgeneratesthematrixcalledthemodie dsequencephase impedancematrix[Z MD ][15]: [ Z MD ]= 266664 Z s 00 0 Z s 0 00 Z s 377775 Consequently,thematrixcalledthemodiedsequencephaseadmitt ancematrix[Y MD ] [17]is: [ Y MD ]= 266664 Y s 00 0 Y s 0 00 Y s 377775 Forexample,Tables4.1,4.2showtheapproximatedresultsforeach congurationat thedierenttestfeederswhere Z s = r + jx and Y s = jb 4.3.2SynthesizingLoadProlesThetestfeederdatain[14]providesspotloaddatathatcontainsd ierenttypesofload connections.However,allloadsarerepresentedaskWorkVARat eachphase. Thesecasestudiesfollowsomeapproximationsthataremadein[18].O neassumption isthatthethreephaseloadsareuniformlydistributed.Duetothist ypeofdistribution, 24 PAGE 34 Table4.1:IEEE4{busfeeder:LineCongurationData congurationrxbnumber(n/mi)(n/mi)(S/mi) 10.48370.99325.763 Table4.2:IEEE123{busfeeder:LineCongurationData congurationrxbnumber(n/mi)(n/mi)(S/mi) 10.48370.99325.76320.48370.99325.76330.48370.99325.76340.48370.99325.76350.48370.99325.76360.48370.99325.76370.47371.0254.851880.47371.0254.851891.32921.34744.5193101.32921.34744.5193111.32921.34744.5193121.52410.740667.2242 thethreephasesareidentical,whichallowsthemtobedecoupledand presentedasa singlephase.TheIEEE4{busfeederisassumedtohavethesameloadatalltimes, whichisnot realisticbutservesthepurposeofatestforsuchasmallsystem. However,forallother testfeeders,theloadvalueprovidedisusedasthepeakvaluesoth atthelineswon'tbe overloaded.Coincidingwiththecasestudy,thereactivepowerwas computedwitha 25 PAGE 35 powerfactorof0.9asproposedin[5].Thesynthesizedtemporaldis tributionload prole,demonstratedingures4.3,4.5,4.7,and4.9,foreachindivid ualcaseis rerectedbyatypicaldailypowerdemandsimilarto[19].Thenodalload was distributedacrossthenodesproportionallytotheloadleveldescr ibedin[14]. 4.3.3SynthesizingPVGenerationProlesIntheoriginaldata,devicesincludingvoltageregulatorsandshunt capacitorshave beenusedinordertomaintainvoltagelevels.Inthesecasestudies, thevoltage regulatorsandshuntcapacitorswillnotbeconsidered.However, addingthePVpanels willsubstituteforthefunctionalityofallthesystem'svoltageregu latorsandshunt capacitors.ThelocationandactivepowerratingofthePVcomponentswerecho senatrandom, andtheirindividualproleswillbeshownforeverysinglecase.Theco mputationofthe reactivepowerisasfollows[20]: S rated =1 : 1 P max (4.4) Q max = q S 2rated P 2max (4.5) 26 PAGE 36 4.4CaseStudies 4.4.1CaseStudyIEEE4{busfeeder InniteBus [I 12 ] 2000 ft. 123 4 2500 ft. Load PV [I 34 ] Figure4.2: TheIEEE4{bustestfeedernetwork. 05101520 time(hours) 0 500 1000 1500 2000ActivePower(kW) TotalDemand TotalPV Figure4.3: IEEE4{busfeederpowerproles:totaldemandedpowerandto talavailablepower fromPVgenerationTheIEEE4{busfeederisthesmallestfeederprovidedin[14].Thesys temiscreatedto beatestfeederwiththepossibilityofchoosingdierentspecicatio ns.Inthiscase study,abalancedcloseconnectionloadhasbeenchosen.Theloadis designedtobe1.8 MWatapowerfactorof0.9lagging.Thevoltagelevelis12.47kVatthe substation busthatislocatedatbus1asshowninFig.4.2.Astep{downtransfor merhasbeen chosen,wherethesecondaryvoltageis4.16kV.A4-wirelinecongu rationhasbeen chosen,andtheapproximatedlineimpedancesareshowninTable4.1.Aspointedout,abalancedcloseconnectionloadhasbeenchosen.T heloadisdesigned tobe1.8MWatapowerfactorof0.9lagging.Thisloadhasbeenkeptas aconstantat alltimesforsimplicity.Specically,thefeederhasbeenmappedclose toDenver InternationalAirport(DIA),acommerciallocationthatisopenmo stofthetime. 27 PAGE 37 Table4.3:IEEE4{busfeeder:tightnessresults DemandTotalPVCVXfmincon TimeP(kW)P(kW)max( rel )max( rel ) 05a.m.18000.004 : 24 10 10 3 : 05 10 13 06a.m.18000.004 : 24 10 10 3 : 05 10 13 07a.m.18000.004 : 24 10 10 3 : 05 10 13 08a.m.180012.701 : 01 10 10 1 : 07 10 12 09a.m.1800113.601 : 86 10 09 6 : 91 10 13 10a.m.1800370.332 : 78 10 09 5 : 92 10 14 11a.m.1800967.311 : 37 10 11 6 : 05 10 14 12p.m.18001259.591 : 45 10 11 1 : 45 10 11 01p.m.18001315.374 : 55 10 12 4 : 55 10 12 02p.m.18001217.851 : 16 10 11 1 : 16 10 11 03p.m.18001054.132 : 55 10 11 2 : 02 10 14 04p.m.1800715.899 : 35 10 09 2 : 51 10 12 05p.m.1800323.392 : 32 10 09 2 : 02 10 14 06p.m.1800187.931 : 36 10 09 2 : 73 10 12 07p.m.180043.911 : 34 10 09 1 : 46 10 13 AccordingtothewebsiteoftheU.S.DepartmentofEnergy,thePV farmnexttoDIA hasaratingof1.9MW.ThelatitudeandlongitudeofthesePVgenerat ionswere providedtotherenewablesninjawebsite(https://www.renewables .ninja/)toshowdata onavailablepoweronApril10,2014.ThebluelineinFig.4.3displaystheloadproleofactivepowerduringthed ay,while theredlinerepresentstheavailableactivepowerfromthePVgener ator.Table4.3 showstheoptimizationresultsoffteencasestudiesatdierentt imesthroughoutthe dayincludingtheactivepowerdemanded,availableactivepowerfrom thePVpanels, 28 PAGE 38 Table4.4:IEEE4{busfeeder:dualitygapandrelativeerrorofdecis ionvariables DemandTotalPVCVXfminconDGmax(DX) TimeP(kW)P(kW)L(kW)U(kW)(%)(%) 05a.m.1800.000.001864.951864.950.0000.02406a.m.1800.000.001864.951864.950.0000.02407a.m.1800.000.001864.951864.950.0000.02408a.m.1800.0012.701864.891864.890.0000.00509a.m.1800.00113.601864.491864.490.0000.00510a.m.1800.00370.331863.561863.560.0000.00011a.m.1800.00967.311861.951861.950.0000.03212p.m.1800.001259.591861.651861.650.0000.00001p.m.1800.001315.371861.701861.700.0000.00002p.m.1800.001217.851861.651861.650.0000.00003p.m.1800.001054.131861.801861.800.0000.02504p.m.1800.00715.891862.531862.530.0000.00505p.m.1800.00323.391863.721863.720.0000.00106p.m.1800.00187.931864.201864.200.0000.00007p.m.1800.0043.911864.771864.770.0000.001 andthemaximumrelativeerrorofthetightnessforbothmodels.Ta ble4.4showsthe theoptimalvaluesoftherelaxedmodelandtheexactmodel,thedu alitygapofthose optimalvalues,andtherelativeerroroftheworstdecisionvariable s. 29 PAGE 39 4.4.2CaseStudyIEEE13{busfeeder 650 646645 632633 634 611 684 671692675 680 652 PV P V PV P V Figure4.4: TheIEEE13-bustestfeedernetwork. 05101520 0 500 1000 1500ActivePower(kW) (a) TotalDemand TotalPV 05101520 time(hours) 0 50 100ActivePower(kW) (b) Bus680 Bus632 Bus684 Bus633 Figure4.5: IEEE13{busfeederpowerproles:(a)totaldemandedpowera ndtotalavailable powerfromPVgeneration,(b)individualPVpanelproles.TheIEEE13{bustestfeederpresentedinFig.4.4isdesignedtoope rateatanominal voltageof4.16kV[14].Theloadproleusedinthiscasestudyisobtaine dfrom[14], 30 PAGE 40 Table4.5:IEEE13{busfeeder:tightnessresults DemandTotalPVCVXfmincon TimeP(kW)P(kW)max( rel )max( rel ) 05a.m.356.000.002 : 06 10 08 2 : 78 10 11 06a.m.302.6090.789 : 40 10 06 4 : 11 10 11 07a.m.356.00181.567 : 99 10 06 1 : 69 10 10 08a.m.445.00242.081 : 51 10 05 9 : 79 10 13 09a.m.534.00302.601 : 71 10 06 5 : 49 10 12 10a.m.551.80347.995 : 10 10 08 3 : 37 10 13 11a.m.516.20363.125 : 83 10 07 4 : 53 10 12 12p.m.498.40363.125 : 70 10 08 2 : 22 10 13 01p.m.551.80363.127 : 21 10 08 2 : 19 10 13 02p.m.569.60332.866 : 98 10 08 2 : 28 10 11 03p.m.587.40272.344 : 33 10 07 6 : 19 10 12 04p.m.587.40211.825 : 96 10 08 1 : 20 10 11 05p.m.747.60151.302 : 74 10 07 1 : 59 10 11 06p.m.1068.0030.261 : 11 10 08 6 : 41 10 13 07p.m.640.800.003 : 15 10 07 3 : 79 10 12 whichshowsthepowerrowresultas1.1MW.Thisvalueisusedasthepe akvalueso thatthelineswillnotbeoverloaded.ThebluelineinFig.4.5(a)displayst heload proleofactivepowerduringtheday.Thesynthesizedtemporald istributionload prole,demonstratedinFig.4.5(a),isrerectedbyatypicaldailypo werdemandsimilar to[19].Thenodalloadwasdistributedacrossthenodesproportion allytotheloadlevel describedin[14].InFig.4.5(a),theredlinerepresentstheaggrega tedavailableactive powerfromthePVgenerators.TheindividualprolesareshowninF ig.4.5(b).Table 4.5showstheoptimizationresultsoffteencasestudiesatdieren ttimesthroughout 31 PAGE 41 Table4.6:IEEE13{busfeeder:dualitygapandrelativeerrorofdec isionvariables DemandTotalPVCVXfminconDGmax(DX) TimeP(kW)P(kW)L(kW)U(kW)(%)(%) 05a.m.356.000.00357.59357.590.0000.00006a.m.302.6090.78303.27303.270.0000.05607a.m.356.00181.56356.59356.590.0000.70608a.m.445.00242.08445.86445.860.0007.68609a.m.534.00302.60535.20535.200.0000.00710a.m.551.80347.99552.98552.980.0000.00011a.m.516.20363.12517.12517.120.0000.20812p.m.498.40363.12499.20499.200.0000.00201p.m.551.80363.12552.97552.970.0000.00202p.m.569.60332.86570.96570.960.0000.27903p.m.587.40272.34589.21589.210.0000.01604p.m.587.40211.82589.64589.640.00012.36905p.m.747.60151.30752.61752.610.0000.00406p.m.1068.0030.261082.341082.340.0000.00007p.m.640.800.00646.01646.010.0000.000 thedayincludingtheactivepowerdemanded,availableactivepowerf romthePV panels,andthemaximumrelativeerrorofthetightnessforbothmo dels.Table4.6 showstheoptimalvaluesoftherelaxedmodelandtheexactmodel, thedualitygapof thoseoptimalvalues,andtherelativeerroroftheworstdecisionv ariables. 32 PAGE 42 4.4.3CaseStudyIEEE34{busfeeder 800 802806 808810 812814850 816 818 820 822 824 826 828 864858 832 834 842 844 846 848 860836840 862838 890 888 852 830854 856 P V PV PV PV Figure4.6: TheIEEE34-bustestfeedernetwork. 05101520 0 100 200 300ActivePower(kW) (a) TotalDemand TotalPV 05101520 time(hours) 0 20 40ActivePower(kW) (b) Bus856 Bus806 Bus802 Bus820 Figure4.7: IEEE34{busfeederpowerproles:(a)totaldemandedpowera ndtotalavailable powerfromPVgeneration,(b)individualPVpanelproles.TheIEEE34{bustestfeederpresentedinFig.4.6isdesignedtoope rateatanominal voltageof24.9kV[14].Theloadproleusedinthiscasestudywasobta inedfrom[14], whichshowsthepowerrowresultas240kW.Thisvalueisusedasthep eakvalueso thatthelineswillnotbeoverloaded.ThebluelineinFig.4.7(a)displayst heload proleofactivepowerduringtheday.Thesynthesizedtemporald istributionload 33 PAGE 43 Table4.7:IEEE34{busfeeder:tightnessresults DemandTotalPVCVXfmincon TimeP(kW)P(kW)max( rel )max( rel ) 05a.m.80.000.001 : 15 10 03 6 : 32 10 10 06a.m.68.0027.605 : 16 10 04 1 : 55 10 10 07a.m.80.0055.201 : 70 10 04 2 : 40 10 10 08a.m.100.0073.601 : 27 10 04 9 : 60 10 10 09a.m.120.0092.001 : 27 10 04 5 : 24 10 10 10a.m.124.00105.801 : 33 10 04 8 : 30 10 10 11a.m.116.00110.401 : 50 10 04 3 : 93 10 09 12p.m.112.00110.401 : 43 10 04 2 : 01 10 10 01p.m.124.00110.401 : 39 10 04 3 : 27 10 10 02p.m.128.00101.205 : 01 10 05 3 : 05 10 10 03p.m.132.0082.809 : 19 10 05 1 : 27 10 10 04p.m.132.0064.401 : 02 10 04 1 : 56 10 10 05p.m.168.0046.001 : 43 10 04 3 : 72 10 10 06p.m.240.009.201 : 95 10 04 2 : 53 10 10 07p.m.144.000.007 : 01 10 03 1 : 88 10 10 prole,demonstratedinFig.4.7(a),isrerectedbyatypicaldailypo werdemandsimilar to[19].Thenodalloadwasdistributedacrossthenodesproportion allytotheloadlevel describedin[14].InFig.4.7(a),theredlinerepresentstheaggrega tedavailableactive powerfromthePVgenerators.TheindividualprolesareshowninF ig.4.7(b).Table 4.7showstheoptimizationresultsoffteencasestudiesatdieren ttimesthroughout thedayincludingtheactivepowerdemand,availableactivepowerfro mthePVpanels, andthemaximumrelativeerrorofthetightnessforbothmodels.Ta ble4.8showsthe optimalvaluesoftherelaxedmodelandtheexactmodel,theduality gapofthose 34 PAGE 44 Table4.8:IEEE34{busfeeder:dualitygapandrelativeerrorofdec isionvariables DemandTotalPVCVXfminconDGmax(DX) TimeP(kW)P(kW)L(kW)U(kW)(%)(%) 05a.m.80.000.0080.8880.880.0000.00006a.m.68.0027.6068.6168.610.0000.47107a.m.80.0055.2080.7280.720.0000.00008a.m.100.0073.60101.03101.030.0000.00009a.m.120.0092.00121.43121.430.0000.00010a.m.124.00105.80125.52125.520.0000.23811a.m.116.00110.40117.35117.350.0007.59412p.m.112.00110.40113.27113.270.0001.77001p.m.124.00110.40125.52125.520.0000.00002p.m.128.00101.20129.62129.620.0000.00003p.m.132.0082.80133.76133.760.0000.00104p.m.132.0064.40133.84133.840.0000.54405p.m.168.0046.00171.21171.210.00010.05106p.m.240.009.20247.39247.390.0000.03607p.m.144.000.00146.63146.630.0000.000 optimalvalues,andtherelativeerroroftheworstdecisionvariable s. 35 PAGE 45 4.4.4CaseStudyIEEE123{busfeeder PV33 PV250 PV18 PV66 PV13 PV52 PV95 PV300 PV114 PV450 PV75 PV85 24 35 37 36 38 39 29 30 31 32 28 25 23 20 22 40 43 19 49 47 48 44 42 151 51 50 46 45 41 135 27 26 21 6 5 150 149 1 7 12 8 9 2 10 14 11 56 55 87 89 91 93 94 96 53 152 16 15 34 17 4 3 112 110 111 610 61 160 67 72 76 80 81 82 62 63 64 65 86 59 58 60 57 88 90 92 97 103 102 107 105 106 108 109 113 84 77 78 79 73 74 71 101 70 68 69 100 99 98 197 104 83 PV Figure4.8: TheIEEE123-bustestfeedernetwork. 0 5 10 15 20 0 500 1000 1500 (a)ActivePower(kW) TotalDemand TotalPV 0 5 10 15 20 0 20 40 60 (b) time(hours)ActivePower(kW) Bus450 Bus33,52,66 Bus85,151 Bus75,114,300 Bus13,18,95 Bus250 Figure4.9: IEEE123{busfeederpowerproles:(a)totaldemandedpower andtotalavailable powerfromPVgeneration,(b)individualPVpanelproles.TheIEEE123{bustestfeederpresentedinFig.4.8isdesignedtoop erateatanominal voltageof4.16kV[14].Theapproximatedlinecongurationsareshow ninTable4.2. 36 PAGE 46 Table4.9:IEEE123{busfeeder:tightnessresults DemandTotalPVCVXfmincon TimeP(kW)P(kW)max( rel )max( rel ) 05a.m.4000.001 : 34 10 05 1 : 37 10 09 06a.m.340130.802 : 84 10 03 1 : 30 10 10 07a.m.400261.605 : 31 10 03 7 : 57 10 10 08a.m.500348.808 : 54 10 03 2 : 01 10 10 09a.m.600436.004 : 38 10 04 2 : 65 10 10 10a.m.620501.404 : 28 10 03 7 : 29 10 10 11a.m.580523.208 : 30 10 03 2 : 21 10 09 12p.m.560523.206 : 67 10 03 4 : 02 10 10 01p.m.620523.202 : 63 10 03 1 : 54 10 10 02p.m.640479.603 : 83 10 05 5 : 49 10 10 03p.m.660392.401 : 91 10 03 7 : 23 10 10 04p.m.660305.202 : 31 10 04 3 : 88 10 10 05p.m.840218.004 : 06 10 05 7 : 48 10 10 06p.m.120043.601 : 21 10 05 8 : 03 10 11 07p.m.7200.006 : 79 10 07 2 : 49 10 10 Theloadproleusedinthiscasestudyisobtainedfrom[14],whichshow sthepower rowresultas1.2MW.Anadditionalloadwasaddedtobus610because anoldversion of[14]showsthatthisbusisconnectedtoaninductionmachine.This valueisusedas thepeakvaluesothatthelineswillnotbeoverloaded.ThebluelineinFig .4.9(a) displaystheloadproleofactivepowerduringtheday.Thesynthes izedtemporal distributionloadprole,demonstratedinFig.4.9(a),isrerectedby atypicaldaily powerdemandsimilarto[19].Thenodalloadwasdistributedacrossth enodes proportionallytotheloadleveldescribedin[14].InFig.4.9(a),there dlinerepresents 37 PAGE 47 Table4.10:IEEE123{busfeeder:dualitygapandrelativeerrorofd ecisionvariables DemandTotalPVCVXfminconDGmax(DX) TimeP(kW)P(kW)L(kW)U(kW)(%)(%) 05a.m.4000.00401.52401.520.0000.00006a.m.340130.80340.42341.580.340198.73207a.m.400261.60400.22401.500.317196.20908a.m.500348.80500.29501.480.236189.28409a.m.600436.00600.38601.460.179192.77610a.m.620501.40620.32620.950.100197.10111a.m.580523.20580.25580.460.037199.16012p.m.560523.20560.22561.110.159197.28701p.m.620523.20620.32622.990.429199.26602p.m.640479.60640.41641.420.159197.31603p.m.660392.40660.76661.540.118199.71504p.m.660305.20661.23661.880.098196.00205p.m.840218.00843.67843.670.0007.98106p.m.120043.601212.921212.920.0000.32107p.m.7200.00724.95724.950.0000.000 theaggregatedavailableactivepowerfromthePVgenerators.Th eindividualproles areshowninFig.4.9(b).Table4.9showstheoptimizationresultsoff teencasestudies atdierenttimesthroughoutthedayincludingtheactivepowerdem and,available activepowerfromthePVpanels,andthemaximumrelativeerroroft hetightnessfor bothmodels.Table4.10showstheoptimalvaluesoftherelaxmodela ndtheexact model,thedualitygapofthoseoptimalvalues,andtherelativeerro roftheworst decisionvariables. 38 PAGE 48 CHAPTERV CONCLUSIONS BasedontheresultsoftheapproximatedbalancedIEEE4{bus,13 {bus,34{bus,and 123{bustestfeederspresentedinchapter4,theSOCrelaxation providesastrong solutionwhencomparedtothetightnessresults.Thisindicatestha tthenumerical resultsofthetightnessforalltestfeedersatdierentloadandg enerationlevelsare closeto0%fortheradialmodiedsystems.However,resultsshow thatthesmallerthe testfeeder,thebetteristhetightness.Thedualitygapresultscomputedforthedierentfeedersindicate thattheobjective valueswereexactforallcasesexceptfortheIEEE123{bustest feederthathasthe highestgaptobe0.43%.Thisindicatesthattherelaxedmodelndsa nobjective functionthatiscloseenoughtoafeasiblelocaloptimumwhichismostlik elytobethe globaloptimum.However,thepercenterrorofdecisionvariablesdemonstratest hatfeasiblesolutionsare notnecessarilythesameasthosecomputedbytherelaxedmodel.W iththeexception oftheIEEE123{bustestfeeder,theerrorvaluesforalltestsw erebelow15%,which relativelysmall.However,theerrorcomputedfortheIEEE123{bu stestfeederwere largeenoughthatboththetotaldispatchandthepowerrowwere dierent. Withthehelpofthe2{stagemodel,resultsindicatethat,forsmalls ystems,the relaxationisstrongenoughandthesecondstageisabletodetermin etheglobal optimum.However,the2{stagemodelwasabletodetermineafeas iblelocalminimum thatmayormaynotbetheglobaloptimumfortheIEEE123{bustes tfeederevenif theobjectivevaluesaresimilarinthetwomodels.ItworthpointingoutthatSOCrelaxationisapowerfultoolthatfa cilitatedthe developmentofthe2{stagemodelbyprovidingtheinitialvaluestot henonlinearsolver. Otherwise,thesolverwouldstartfromaratpointwithnoguarante eofndingglobal optimality.Therefore,furtherinvestigationisneededtoapproac hglobalsolution. 39 PAGE 49 REFERENCES [1]A.J.Wood,B.F.Wollenberg,andG.B.Shebl, PowerGeneration,Operation,and Control,3rdEdition .Hoboken,NewJersey,USA:Wiley,2014. [2]J.Carpentier,\Contributiontotheeconomicdispatchproblem," Bulletindela SocieteFrancoisedesElectriciens ,vol.3,no.8,pp.431{447,1962. 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[21]L.GanandS.H.Low,\Convexrelaxationsandlinearapproximat ionforoptimal powerrowinmultiphaseradialnetworks,"in PowerSystemsComputationConference ,Aug.2014,pp.1{9. [22]S.BoydandL.Vandenberghe, ConvexOptimization .UK:CambridgeUniversity Press,2009. 41 PAGE 51 APPENDIXA MATHEMATICALBACKGROUND Thisappendixhighlightsthemathematicalbackgroundneededfort hestudyperformedinthisthesis.Foracomprehensiveunderstandingofthema terialsinthisappendix,reading[22]ishighlyrecommended. A.1Convexity A.1.1ConvexSet S R n isconvexif x;y 2 S = ) x + y 2 S forall ; 0with + =1 convex set not a convex set FigureA.1:Examplesofconvexandnonconvexsets A.1.2ConvexFunction f : R n R isconvexifthedomainisaconvexsetand f ( x +(1 ) y ) f ( x )+(1 ) f ( y ) forall x;y 2 dom f; 0 1. (x, f(x)) (y, f(y)) FigureA.2:Denitionofconvexfunction A.1.3Convexityconditions 42 PAGE 52 1stordercondition:If f ( x )isdierentiable,anditsgradientis 5 f ( x ),then f ( x )isconvexif: f ( y ) f ( x )+ 5 f ( x ) T ( y x )forall x;y 2 dom f 2ndordercondition:If f ( x )istwicedierentiable,anditsHessianis 5 2 f ( x ),then f ( x )isconvexif: 5 2 f ( x ) 0PositiveSemidenite(PSD) f (y) (x, f(x)) f (x) + f (x) (y y) T FigureA.3:1storderconditionofconvexfunction. Tosumup,anequalitysethastobeanane(linear)functiontobeco nvex,whichisis knownashyperplane.Foraconvexinequalityset,thiscanbeanan ecalledhalfspaceor anyfunctionthatsatisesoneofthetwoconditions.Forexample, aquadraticfunction f ( x )= x T Px +2 q T x + r isconvexonlyif P 0. A.1.4Relaxation OriginalProblem: U = min f 0 ( x )(A.1) subjectto x 2 X (A.2) RelaxedProblem: L = min f 0 ( x )(A.3) subjectto x 2 X r (A.4) Relaxedif X X r then L U 43 PAGE 53 f(x) U L (a) (b) FigureA.4:(a)relaxedset,(b)relaxedfunction A.2LagrangianOptimization Considerthefollowingoptimizationfunction: min f 0 ( x )(A.5) subjectto f i ( x ) 0(A.6) h j ( x )=0(A.7) Thelagrangianfunctionoftheproblemisasfollows: L ( x;; )= f 0 ( x )+ N i X i =1 ( i f i ( x ))+ N j X j =1 ( j h j ( x ))(A.8) thescalar i and j arecalledlagrangianmultipliers.Thevector and arecalleddual variables. A.3Duality Inordertoexplainduality,wewillbeusingLinearProgramming(LP): ConsideraPrimalfunctioninthestandardofLP p = min C T x (A.9) subjectto A x =b(A.10) x 0(A.11) 44 PAGE 54 ThelagrangianfunctionofthePrimalproblemisasfollows: L ( x;; )= b T +(A T +C) T x (A.12) ThedualfunctioninthestandardofLPisasfollows: d = max b T (A.13) subjectto A T +C=0(A.14) 0(A.15) Thelagrangianfunctionofthedualproblemisasfollows: g( ; )= b T +inf x ((A T +C) T x )(A.16) Lowerboundpropertystatesthatif 0 theng( ; ) p where and arethedualvariables, and p istheoptimalvalueoftheprimalproblem. Whensolvingtheconcavedualproblem,wearendingthemax ( ). 1.Dualisinfeasible2.Primalisinfeasible3.Bothdualandprimalareinfeasible4.Botharefeasible(typically) Solutionofthedualproblemis d Weakduality d p .Thisholdforconvexandnonconvexproblems. 45 PAGE 55 Strongduality d = p .Thisdoesnotgenerallyholdbutusuallyholdsforconvex problems. UsingSlater'sConstraintsQualication,whichareexemptfromthelin earinequalityconstraintsbutmustsatisfynonlinearinequalityconstraints,s trongdualityfor SDPissolvable. Theepigraphgeometricinterpretationing.A.5. t u p d duality gap u + t } set of all possible pairs of x values feasible FigureA.5:Epigraphgeometricinterpretationforanoptimizationpr oblem. A.4Karush-Kuhn-Tucker(KKT)conditions Suppose f i ( x )and h j ( x )aredierentiableand x ; and arealocaloptimum solution: 1. f i ( x ) 0, h j ( x )=0(PrimalFeasibility) 2. i 0(DualFeasibility) 3. @L ( x ; ; ) @x i =0(LagrangianOptimality) 4. i f i ( x )=0(ComplementarySlackness) 46 PAGE 56 A.5SemideniteProgrammingSDP TheSDPcanbesimilartolinearprogramming,butitisinitsmatrixform.T he followingexamplehasA 0 andA i assymmetricconstantmatrices.Thetracefunction sumsupthediagonalelementsofamatrix. mintr (A 0 x)(A.17) subjectto tr (A i x)=b i ;i =1 ;:::;m (A.18) x 0(A.19) TheDualSDP: max b T y (A.20) subjectto A 0 m X i =1 ( y i A i ) 0(A.21) A.6SecondOrderConeProgrammingSOCP TheSOCPisproventobeaspecialtypeofSDP.However,theSOCPis more tractablethanSDP,whichisanadvantage.Thefollowingexamplesho wsatypicalway toexpresstheSOCP. min f T x (A.22) subjectto k A i x +b i k 2 c Ti x +d i i =1 ;:::;m (A.23) 47 |