Development of Improved dc Network Model for Contingency Analysis. Puneet Sood

Transcription

1 Development of Improved dc Network Model for Contngency Analyss by uneet Sood A Thess resented n artal Fulfllment of the Requrements for the Degree Master of Scence Approved November 04 by the Graduate Supervsory Commttee: Danel Tylavsky, Char ay ttal Raa Ayyanar ARIZONA STATE UNIERSITY December 04

2 ABSTRACT The development of new polces favorng ntegraton of renewable energy nto the grd has created a need to relook at our exstng nfrastructure resources and at the way the power system s currently operated. Also, the needs of electrc energy markets and transmsson/generaton expanson plannng has created a nche for development of new computatonally effcent and yet relable, smple and robust power flow tools for such studes. The so called dc power flow algorthm s an mportant power flow tool currently n use. However, the accuracy and performance of dc power flow results s hghly varable due to the varous formulatons whch are n use. Ths has thus ntensfed the nterest of researchers n comng up wth better equvalent dc models that can closely match the performance of ac power flow soluton. Ths thess nvolves the development of novel hot start dc model usng a power transfer dstrbuton factors (TDFs approach. Ths document also dscusses the problems of ll-condtonng / rank defcency encountered whle dervng ths model. Ths model s then compared to several dc power flow models usng the IEEE 8-bus system and ERCOT nterconnecton both as the base case ac soluton and durng sngle-lne outage contngency analyss. The proposed model matches the base case ac soluton better than contemporary dc power flow models used n the ndustry.

3 ACKNOWLEDGEMENTS Frst, I would lke to express my sncere thanks to Dr. Tylavsky, my advsor for beng a commendable source of knowledge and nspraton. Snce the ncepton of ths research, he has guded me through several challenges n research. Despte numerous ptfalls, he has stood by me and motvated me to explore the depths and breadths of research. I would lke to extend to my thanks to my graduate commttee members Dr. Ayyanar and Dr. ttal for ther valuable feedback and suggestons to my thess document. I am grateful to the all the faculty members of power engneerng for the wonderful learnng experence they provded for the last two years. I would also lke to extend my grattude to my colleagues n the research group who have been a source of knowledge and shared ther deas. Gratefully acknowledged s the support from Consortum for Energy Relablty Technology Solutons (CERTS and ower System Engneerng Research Center (SERC. I am also grateful to the School of Electrcal, Computer and Energy Engneerng for provdng me wth the resources to carry out ths research. I am hghly ndebted to my sprtual master and parents, as ths work would not have been possble wthout ther love and support I receved over the years.

4 TABLE OF CONTENTS age LIST OF FIGURES... v LIST OF TABLES... v NOMENCLATURE... x CHATER INTRODUCTION.... Background.... Lterature Revew...3 Research Obectve 6.4 Thess Outlne... 7 DC OWER FLOW MODEL FORMULATION AC ower Flow Model for A Transmsson Lne. 8. Classcal DC ower Flow Model Dervaton...3 Errors Due to DC ower Flow Assumptons Generalzed DC ower Flow Model Cold Start or State Independent DC Model Hot Start or State Dependent DC Model TDF BASED DC SERIES ELEMENT MODEL Introducton.

5 CHATER age 3. General DC Seres Element Model 3.3 ower Transfer Dstrbuton Factors (TDFs Classcal DC TDF Dervaton Lnearzed AC TDFs Dervaton TDF-Based Optmzaton Approaches Converson from AC TDFs TO DC TDFs Obtanng the DC Network Model Summary. 4 4 MODEL ALIDATION AND NUMERICAL ILL-CONDITIONING Introducton Model aldaton Checks on the Equvalent DC TDFs Toward aldatng the Susceptance Evaluaton Algorthm roblems Assoscated wth Rank Defcency Topologcal Dependency of the Rank of Emprcal Analyss of Network Topologes Identfcaton of the Sub-Networks Equvalent Network Soluton wth Sub-Networks Summary. 59 v

6 CHATER age 5 NUMERICAL EXAMLES Introducton Case Studes and Descrpton Case Study : 7-Bus Model Case Study : IEEE-8 Bus Model Case Study 3: ERCOT Interconnecton CONCLUSION AND FUTURE WORK Concluson Future Work 83 REFERENCES..85 v

7 LIST OF FIGURES Fgure age. Model of a Transmsson Lne Connectng Bus And Bus Model of a hase Shftng Transformer Connectng Bus and Typcal DC Model of a Transmsson Lne Connectng Bus and....4 A Generalzed DC Model of a Branch Connectng Bus and DC Network Representng TDF for Branch AC Network Representng AC TDF for Branch Loss Modeled as Inectons (ostve/negatve Flowchart for Entre Network Equvalencng rocess Radal Network of Four Buses Three Bus Sample System Four Bus Sample Radal Network Three Bus Meshed Network Eght Bus Meshed Network Generalzed Network Bus Network Flowchart for Sub-Network Reactance Evaluaton Bus Network Model Comparson of Classcal DC and Equvalent Derved DC TDFs Comparson of ower Flow Errors for Dfferent Models at Base Case v

8 Fgure age 5.4 Comparson of Reactance Between Dfferent Models Comparson of Maxmum Absolute MW Error for Contngences Comparson of RMS Error for Contngences IEEE-8 Bus Model Comparson of ower Flow Errors for Dfferent Models at Base Case Comparson of Maxmum Absolute MW Error for Contngences Error Duraton Curve for Maxmum Absolute MW Error for Contngences Comparson of RMS Error for Contngences ERCOT Interconnecton Comparson of ower Flow Errors for Dfferent Models at Base Case Comparson of Maxmum Absolute MW Error for Contngences Error Duraton Curve for Maxmum Absolute MW Error for Contngences Comparson of RMS Error for Contngences v

9 LIST OF TABLES Table age 4. Classcal DC TDFs for Network n Fgure Case Study Detals Network arameters for 7-Bus Model AC TDFs (at Sendng End AC TDFs (at Recevng End Equvalent DC TDFs Classcal DC TDFs Reactance and ower Flow Comparson at Base Operatng ont Summary of Results for 7-Bus Model Summary of Results for IEEE Summary of Results for ERCOT Interconnecton... 8 v

10 NOMENCLATURE ac Alternatng current Bbranch Branch susceptance matrx Bbus Bus susceptance matrx b Susceptance of transmsson lne C CRR Bus-Branch ncdence matrx Congeston Revenue Rghts CS Bus-branch ncdence matrx for sub-network dc ɛ EI ERCOT FACTS FTR Drect current Tolerance value Eastern Interconnecton Electrc Relablty Councl of Texas Flexble ac transmsson system Fnancal Transmsson Rghts g Conductance of transmsson lne H HDC ISF J LODF Susceptance of transmsson lne Hgh voltage dc Inecton shft factor Jacoban matrx Lne Outage Dstrbuton Factor x

11 NERC NR OF OTDF North Amercan Electrc Relablty Corporaton Newton Raphson Optmal ower Flow Outage Transfer Dstrbuton Factor flow Network branch power flow vector ower flow from bus to bus n Bus necton vector ower flow from bus to bus Q bus TDF bus Non-generator bus or bus at AR lmt ower Transfer Dstrbuton Factor Generator bus G ower generated at bus L Load at bus shft bus Compensatng bus power necton vector for phase shfter shft branch Compensatng branch power necton vector for phase shfter k ower flow over branch k r Resstance of transmsson lne RS SCED SCUC Renewable ortfolo Standard Securty Constraned Economc Dspatch Securty Constraned Unt Commtment x

12 t Transformer tap rato TLR TL Ʌ Transmsson Loadng Relef Transmsson lannng Lambda matrx for man network Ʌ Augmented lambda matrx oltage magntude at bus oltage magntude at bus ɅS Lambda matrx for sub-network WECC Western Electrcty Coordnatng Councl x Reactance of transmsson lne xs Reactance vector for sub-network Ybus Admttance matrx y Admttance of transmsson lne z Impedance of transmsson lne αk Sendng end loss compensaton for branch k αk Recevng end loss compensaton for branch k γ Sngle Multpler for loss compensaton δ Transformer phase shft angle θ Angle at bus θ Angle at bus x

13 ϕ ower transfer dstrbuton factor matrx ϕ ac ac power transfer dstrbuton factor matrx Φ dc dc power transfer dstrbuton factor matrx Ψ t Loss necton matrx Transacted power between two buses x

14 INTRODUCTION. BACKGROUND In 0 the electrcty sector was the largest source of greenhouse gas emssons n US, contrbutng 3% of the total greenhouse gas emssons, whch s sgnfcantly better than the 40% share of electrcty sector n 009 for greenhouse gas emssons. Albet the contrbuton of electrcty sector towards greenhouse emssons has declned sgnfcantly but a lot needs to be done to further mprove on clean energy []-[]. The declne s the result of development of new envronmental polces and federal tax ncentves favorng ntegraton of renewable resources nto grd. Now more than 35 states have renewable energy targets n place [3]-[4]. In Calforna, for example, as of Aprl 0, the renewable portfolo standards (RS requres Calforna s electrc utltes to derve 33% of ther retal sales from elgble renewable energy resources by 00 [5]. Because of governmental energy polces and ncentves, the power ndustry s rapdly changng gears to accommodate renewable resources such as photovoltacs, wnd n ther energy portfolos and thus changng the tradtonal perspectve of operaton and desgn of power systems. Besdes the changes n the generaton n pattern, the 0-year plannng summary prepared by Western Electrcty Coordnatng Councl (WECC ndcated that there s a 4% expected rse n loads from 009 to 00, whch s.% compound annual growth rate [3]. However, to keep pace wth the current trends n power systems, there s an urgent requrement to develop new tools to study the rapdly evolvng power system.

15 It s a well-known that an ac power flow soluton s more accurate than the correspondng dc soluton for studes lke power flow, contngency analyss. But the tradtonal Newton-Raphson (NR algorthm used for the ac soluton requres an teratve procedure whch s qute tme ntensve and less approprate for carryng out such studes where qualtatve soluton at an ntal plannng phase s of prme concern. Also, the nterest n development of effcent dc power flow models has further ntensfed due to market applcatons (such as securty constraned economc dspatch (SCED, securty-constraned unt commtment (SCUC where prces are a functon of network congestons [6]. Conventonally, these approxmate dc power flow models were used extensvely to tackle convergence ssues common to the full ac OF, contngency screenng, transmsson loadng relef and medum-to-long term transmsson plannng [6]. The obectve of the work reported on here s to determne a dc model whch takes nto account branch resstance n the flow equatons and whch produces a better estmate of power flows durng transmsson lne outages.. LITERATURE REIEW Network equvalencng, as the name suggests, s a procedure whch reduces the complexty of the orgnal model to create a smplfed model ether through reducton n system sze or ease n computatonal requrements. Often these network equvalents are the results of dc approxmatons of ac network wth an emphass on preservng the orgnal network propertes as much as possble. Rch lterature developed over the

16 years s descrptve of the myrad dc power flow models used for specfc purposes lke contngency analyss, market analyss. Wth the recent upsurge n the electrc energy market applcatons stated earler, nterest has been renewed n the development of mproved dc power flow models that better replcate the ac network model performance. The stmul for such research s the desre for robustness, smplcty and speed of these models when used n the applcatons of nterest. The word dc n dc power flow comes from the use of old dc network analyzers, used to represent the seres reactance as proportonal seres resstance and the current to represent the correspondng MW flow on the network [6]. The smplest verson of dc power flow wthout any loss compensaton, s a further smplfcaton of fast decoupled power flow by completely neglectng Q- equaton and assumng constant p.u. voltage magntude. Wth these smplfcatons the dc power flow problem s reduced to solvng a lnear system of equatons [7] and []. Ths classcal dc seres-reactance model s wdely known as the orgnal dc power flow method. arants of dc models are legon and ther performance s affected by the loss compensaton technques nvoked and branch susceptances selected. Furthermore, these dc models are nherently approxmate and ther accuracy s very system and case dependent yet they provde sgnfcant nsght nto system behavor under dfferent operatng scenaros []. Ths research work s focused on the development of dc network models from ac network models, therefore t s vtal to study the mpacts of assumptons (.e. voltage magntude p.u., resstances neglected. on the dc model s 3

17 accuracy. The theoretcal mplcatons of the assumptons made to derve a dc network model s well descrbed n [8] whle [9] [] and [] dscuss the effect of these assumptons on practcal and realstc bulk energy systems. Reference [4] dscusses the mpact of flow controllng devces on dc power flow. Reference [6] presents the modfcaton n the standard assumptons by ntroducng nterval-valued dc power flow equatons to overcome voltage and parametrc uncertantes. So-called dc power flow models have been segregated nto dfferent categores n lterature. However, the more popular classfcaton of these models nclude: ncremental and non-ncremental models. The non-ncremental models are further categorzed nto hot start or state-dependent and cold start or state-ndependent dc models. The hot start dc models are based on the solved ac soluton. In these type of models the seres elements represent the power flow over the network and shunt elements model the loss nectons. Therefore, these models match the ac soluton losses at the base case. These types of models are used n real tme SCED usng a state estmator soluton [] and short / medum term plannng studes. The other varant.e. cold start dc models lacks a solved ac soluton; thus the loss estmates are ether neglected or an estmate s used that represent the losses as a percent of the net load. These type of models fnd utlzaton n the market based applcatons lke fnancal transmsson rghts (FTR / congeston revenue rghts (CRR [3] and long term plannng studes. 4

18 Another category of models are known as the ncremental models, whch compute the ncremental changes from a known ac or dc state [6]. These ncremental models are sub-dvded nto sparse matrx models and senstvty factor models. Sparse matrx models are based on the avalable base case ac/dc soluton. The devatons from the base case (.e. topologcal change branch outage are modeled as ncremental changes n the problem formulaton; ths s performed by factor-updatng n orgnal formulaton [6]. Whle n the senstvty factor model, the senstvty factors are a functon of sparse dc network matrx and/or network soluton. Computaton of these senstvty factors, lke power transfer dstrbuton factors (TDFs, lne outage dstrbuton factor (LODF, outage transfer dstrbuton factor (OTDF s descrbed n [4]. These senstvty factors are wdely used by system operators n the congeston modelng n market applcatons e.g., transmsson loadng relef (TLR procedure, by provdng fast approxmatons of the actve power flow changes due to varous system operatons [0]. References [5] [7] dscuss the formaton of network equvalents usng the senstvty factors and extendng t further to obtan the reduced network models whch tend to preserve network propertes. References [8] [9] provde an nsght nto the varaton of senstvty factors.e., TDFs wth multple loadng scenaros across dfferent systems. References [6], [7] and [37] present network aggregaton technques appled to the classcal dc formulatons, but, for a complex network they suffer from the rank defcency/numercal ll-condtonng problems. Also, when ths technque s appled to a set of nconsstent TDFs nstead of consstent classcal dc TDFs the soluton 5

19 becomes constrant dependent and yelds dfferent results for dfferent constrants appled. Often n deregulated electrcty markets, the market partcpants, n the open access envronment, want to maxmze ther proft, so they compete to obtan electrcal energy from a cheap source, whch may lead to congeston n the transmsson network and affect system securty and relablty [5]. Ths has ncreased the need to conduct dynamc securty and relablty assessment on a real tme bass. Therefore, contngency analyss plays an ntegral part of such a study. Transmsson plannng (TL standards defne relable system performance followng a loss of a sngle bulk electrc element, two or more bulk electrc elements, or followng extreme events [7]. NERC [8] [30], requres analyss of the followng contngences [7]: Resultng n a loss of a sngle element (Category B Resultng n a loss of two or more elements (Category C Extreme events forcng two or more elements removed or cascadng out of servce (Category D dc power flow models are frequently used to provde a nsght when such an enormous number of cases must be taken under consderaton..3 RESEARCH OBJECTIE Ths research focuses on the development of an mproved dc model whch produces a better estmate of power flows durng transmsson lne outages or contngences. Ths report covers followng areas: 6

20 . Basc dc power flow formulaton. Modelng of loss compensaton n dc models 3. Introducton to TDFs (both ac and dc 4. Detaled development of a seres element n the proposed dc model 5. Numercal ssues n the development of ths model 6. Comparson of the results of ths model wth other wdely accepted dc models.4 THESIS OUTLINE Ths thess s dvded nto four addtonal chapters: Chapter ntroduces reader to general dc power flow model formulaton and the underlyng assumptons for development of such models. Chapter 3 presents the new optmzaton based approach for the development of a seres element of the proposed model. Chapter 4 dscusses the rank defcency or ll-condtoned matrx ssues encountered n the development of the proposed model. Chapter 5 conducts the numercal llustraton on 7-bus, IEEE-8 bus system and ERCOT nterconnecton to demonstrate the accuracy of ths model compared to other more prevalent dc models. Chapter 6 provdes the summary of ths research and future scope of work. 7

21 DC OWER FLOW MODEL FORMULATION The power flow problem nvolves the soluton of a non-lnear system of equatons usng the tradtonal mplct methods lke Newton-Raphson and Gauss-Sedel. In contrast, a dc power flow algorthm explctly solves a lnear system of equatons, an equaton set that s far more computatonally effcent,.e., non-teratve and low storage requrements. The advantages of speed and robustness offered by such models, makes the dc power flow an attractve opton to pursue for several of the applcatons stated earler. In ths chapter, the ac model for the power flow over a transmsson lne s frst ntroduced and then the dc model assumptons are overlad to derve the classcal dc model formulaton. A more generalzed dc power flow model s then presented to the reader. Furthermore, cold-start and hot-start dc models wth and wthout loss compensaton respectvely, are descrbed n detal.. AC OWER FLOW MODEL FOR A TRANSMISSION LINE A smplfed ac network model for a transmsson lne connectng bus and bus s shown n Fgure.. The power flows on the branch at the sendng end bus and recevng end bus are gven by and respectvely. The actve power flow from bus to bus under steady state condtons s descrbed by the followng power-flow equaton: Re ( g b (. * 8

22 z =r + x y =g + b Fgure. Model of a transmsson lne connectng bus and bus Upon smplfcaton, g cos( b sn( (. Smlarly, g cos( b sn( (.3 Where, = voltage at bus n per unt = voltage at bus n per unt θ = angle at bus θ = angle at bus r = resstance of transmsson lne - n per unt x = reactance of transmsson lne - n per unt z = mpedance of transmsson lne - n per unt y = admttance of transmsson lne - n per unt g = conductance of transmsson lne - n per unt b = susceptance of transmsson lne - n per unt =ower flow from bus to bus n per unt 9

23 = ower flow from bus to bus n per unt It can be seen that the dfference of the flows at the sendng and recevng ends of the branch represent losses that occur on the branch. Occasonally, phase shftng transformers are found n the large networks to regulate the real power flow over the transmsson lnes. Therefore, a generalzed model for phase shftng transformer nvolvng taps s derved. z =r + x y =g + b Fgure. Model of a phase shftng transformer connectng bus and Fgure. shows a phase shftng transformer connectng buses and. A fcttous bus s added for clarty to separate the transformer mpedance from transformaton rato. It s to be noted that the phase-shft transformer makes the bus admttance matrx Ybus asymmetrc. ower flow at the sendng end s obtaned as follows [3] [33]: Re ( g t t g t t cos( b b * t sn( (.4 Smlarly, the recevng end power flow s gven as: g cos( b sn( (.5 t t 0

24 where, t = tap rato of the transformer. δ = phase shft angle of the transformer. CLASSICAL DC OWER FLOW MODEL DERIATION To derve the classcal dc power-flow model, the classcal assumptons are appled to the ac branch flow model wthout the phase shftng transformer. The followng assumptons are made for the dc power-flow model: Assumpton : Losses are neglected on the branch.e. resstance s neglected. r 0 g = 0 and b = -/x; Assumpton : oltage at the buses are approxmate to p.u. for all bus ; Assumpton 3: The angle dfference across the branch end s small such that sn (θ θ Assumpton 4: Transformer taps are gnored as voltage s per unt at each bus. t The ntroducton of these assumptons n the ac power flow model ntroduces errors whch shall be dscussed later n ths chapter. These assumptons modfy the actve power flow equatons for bus and bus derved n (.4 - (.5 as follows: b ( (.6

25 x Fgure.3 Typcal dc model of a transmsson lne connectng bus and The node balance equaton for an N+ bus system (one bus s slack or reference bus and L branch network usng (.6 yelds the generalzed equaton for dc power flow model as: n B bus (.7 where, n N ( =N bus necton vector G L, N G = ower generated at bus L = Load at bus Bbus = C T. dag(/x. C = N N bus susceptance matrx Θ C = Bus voltage angles = L N bus-branch ncdence matrx (/x = [/x, /x, /x3../xl] = susceptance of the network model dag(/x = L L dagonal matrx contanng network susceptance Also, the generalzed equaton for the power flow over the network branches can be wrtten as:

26 flow B branch (.8 where flow = L network branch power flow vector Bbranch = dag(/x. C = L N branch susceptance matrx The soluton nvolves solvng the lnear system of equatons, (.7, from whch nodal phase angles are obtaned usng sparse LU factorzaton of B busand then performng forward/backward substtuton. ower flow over the network branch s obtaned usng (.8. As mentoned earler, phase shfters alter the real power flow over the transmsson lne. Therefore, ts mplcatons on the dc power flow equatons must be accounted for. The power flow over a branch wth phase shfter s gven by: b ( (.9 Usng node balance equatons, the dc bus necton and branch flow equatons are modfed as: n flow B bus B branch shft bus shft branch (.0 where, shft bus and shft branch represent the power necton vectors that compensate for the phase shfter on both the bus power necton and branch flow equaton. 3

27 .3 ERRORS DUE TO DC OWER FLOW ASSUMTIONS Although, the obectve of ths research s not to analyze n detal ether the mpacts of assumptons used n dc model creaton or the mpacts of loadng on such a model, t s desrable to have a comprehensve vew of these dc-modelng assumptons. Assumpton # above states that the losses are neglected. At frst glance, t mples that ths assumpton results n an error of few percent over the network branch. However, these small errors over all the lnes accumulate and appear on the slack or reference bus. Therefore, for a large power system the power flows over the transmsson lnes n the vcnty of ths reference bus results n large errors. Ths effect wll be observed n the ERCOT nterconnecton model dscussed later n ths report when the comparson of ac and dc power flows wll be made at the base case and under contngency. The second assumpton above states that the bus voltages are assumed to be per unt. Ths assumpton drectly mples the absence of transformer taps and the absence of lne voltage drop. It s typcal of the NERC-MMWG models to have voltages n the range of 0.75 to.4 per unt. Therefore the real power-flow error due to ths assumpton s n the range % to 96%. Ths assumpton mpacts the AR flow over the lne and hence the effectve value of current, therefore, large devatons from ths assumpton n conuncton wth the loss neglectng assumpton (.e. R=0 can lead to large errors n the dc power flows obtaned from such a model. 4

28 The thrd assumpton above states that the angle dfference across a transmsson lne s small whch approxmates sn (θ θ. Ths assumpton s typcally accurate for short transmsson lnes. However, for a longer transmsson lnes, the angle across the lne may be larger. Ths may result n ntroducng sgnfcant error. For example a 40 o angle across the transmsson lne ntroduces an error of 8.6% due to ths assumpton. It may seem from the above dscusson that these assumptons lead to naccurate power flow results by a large margn. However, n practce ths may not hold true. Consder the followng argument. Frst recognzed that the dc network model s ust a lnear drect current dvder crcut model. Therefore, t follows that MW flows are dvded accordng to Ohm s and Krchoff s law where real power flows are analogous to current and bus angles are analogous to voltages. For example, f all the bus voltages are dentcal but not equal to per unt, then the assumpton of per unt bus voltages wll have no mpact on the error of the dc power flow model. As another example, when there s a radal lne no errors are nduced apart from losses due to the assumpton numbers one and three stated above. It often becomes dffcult to predct f the assumptons made wll accumulate errors or dsplay self-cancellaton propertes or propagate the MW flow naccuraces throughout dc network when power flows are obtaned from such a model. For more detals, an nterested reader should look at [9]-[0] whch dscuss the mpact of such assumpton on realstc power systems. 5

29 .4 GENERALIZED DC OWER FLOW MODEL The dc model dscussed so far s a lossless model nvolvng only seres elements.e. reactance or susceptance, therefore there s no provson of any loss compensaton. In ths secton a more generalzed dc power flow model s formulated. H α k α k Fgure.4 A generalzed dc model of a branch connectng bus and The generalzed dc power flow model for a transmsson shown n Fgure.4 [6], ndcates a seres element.e. reactance or susceptance represented by H. The flow over the branch s gven by: k H (. where k = ower flow over the transmsson lne H = Susceptance of the branch = = Angle dfference across the branch k = Sendng end loss compensaton for branch k = Recevng end loss compensaton for branch Ths model ncorporates losses (based on the solved ac soluton that occur over the entre network by modelng the (negatve power nectons at the respectve buses. It s 6

30 worthwhle to note that sum of k k represent the total loss that appears over the branch due any resstve element n the ac model. arous dc power flow models are now ntroduced wth respect to ths generalzed dc power flow model..4. COLD START OR STATE INDEENDENT DC MODEL Cold start dc models a.k.a. state ndependent models are a common type of dc model used when there s an absence of relable solved ac base case soluton. Therefore, these type of dc models do not account for the losses that occur over the system,.e. there s no loss compensaton. Due to the absence of loss modelng, ths model often leads to a less accurate power flow soluton. However, ths type of dc model s qute prevalent n ndustry and extensvely used for SCUC, FTR and long term plannng purposes. Mathematcally, t s defned as: H x (. 0 (No loss compensaton k k The power flow soluton usng ths model s obtaned as explaned n secton.. Another suggested approach to cold start models, where there s lack of good voltage/ar soluton, s to use a fxed-voltage ac power-flow soluton. In ths model, bus voltages and transformer taps are set to per unt.e. all buses are specfed as 7

31 buses wth no Ar lmt. Although the Ar flows obtaned from such a flat voltage ac soluton are completely wrong, the MW flows, net losses or loss dstrbuton obtaned from such an approach s better than the no-loss or classcal dc model..4. HOT START OR STATE DEENDENT DC MODEL In ths type of dc model, seres and shunt elements are developed based on the solved ac network soluton and reman fxed thereafter. Therefore, for a gven network topology a pror knowledge of the losses s obtaned and ncorporated as nectons or wthdrawals at the buses; therefore, the load generaton balance n ths model s smlar to the full blown ac model,.e. losses match exactly those of the ac model. These types of models are used n SCED usng a state estmator soluton [6] and short/medum term operatons/plannng studes. In ths secton, two common hot start dc models shall be dscussed..4.. Sngle multpler or Net loss dspersal dc model Ths type of dc model s smlar to the classcal dc model descrbed above. However, the total losses are dstrbuted across the network by scalng all the loads usng a constant multpler. Ths constant multpler s defned as a rato of total generaton (load + losses to total load n the network. The sngle multpler s defned mathematcally as: 8

32 N N G L (.3 Other varatons for ths dc model nvolve the use of zonal multplers. Zonal multplers are obtaned n a smlar fashon to sngle multpler. In ths case, loads n respectve zones are scaled up by ther zonal multplers obtaned usng the load generaton balance n respectve zones. arameters correspondng to a generalzed model for the sngle multpler type loss compensaton s gven as: H x (.4 (,, L, N nodes.4.. Base pont matchng or Alpha-matchng dc model Ths model ntroduces localzed loss compensaton at the buses connected to the transmsson lne. Ths model matches the MW flows and nodal phase angles obtaned from the ac power flow soluton perfectly [6]. In ths model, the H parameter s specfed beforehand and the correspondng loss compensaton nectons (α s are then calculated based on solved ac power flow. Although, the detaled dervaton to obtan optmzed seres element H parameter shall be dealt wth n the next chapter, the mathematcal relatons for all the model parameters s gven as follows: 9

33 H k x derved o o H ( o (.5 k o o H ( o where o superscrpt denotes the values obtaned from the ac base case soluton. In comparson to all other dc models dscussed so far, ths model best fts tself to the solved ac soluton and can predct the small perturbaton around the operatng pont. The dscusson on dc power flow models s ncomplete wthout ntroducng reader to the nonlneartes assocated wth the contnuous actng control devces lke phase shfters, tap changers, HDC, FACTS devces, actve durng the outer loop of the ac power flow. For example, the phase-shfter angle wll vary wthn a prescrbed range to control the power flow whle t wll act as fxed-angle transformaton when the lower/upper lmt s ht. Ths leads to dscontnutes n the system model and may lead to naccurate MW flows over the lnes f modelng s not handled properly. If necessary such nonlneartes can be accounted for by usng teratve procedures but then some of the advantage of usng dc model s elmnated as t becomes more computatonally complex. The dscusson of the dc power model can be summarzed by statng that these models provde qualtatve nsght nto the system, however, the accuracy of these models vares over networks wth loadng condtons. Due to pervasve use of such 0

34 models n market applcatons, the accuracy of dc models s of great nterest especally when the crtcal paths are under consderaton.

35 3 TDF BASED DC SERIES ELEMENT MODEL 3. INTRODUCTION A dc model s made up of two dstnct element types, seres and shunt, and the methods used to get values for each element type may be, and often are, handled ndependently. The lossless seres element has unts of susceptance and s used to approxmate the power-flow angle-dfference relatonshp of a network branch. The dc seres elements act smlar to a current dvder network and these elements dvde the power across the network branches n accordance wth Ohm s law. The shunt element has unts of power and models the effect of losses n some fashon. The focus of ths chapter s the development of an optmzed dc seres element model for the base pont matchng or alpha-matchng dc model ntroduced n the prevous chapter. Frst, the mathematcal nter-relatonshp of power transfer dstrbuton factor (TDF and reactance shall be derved. Second, the problems assocated wth ac TDFs shall be consdered and ther converson to equvalent dc TDFs s dscussed. Fnally, an optmzaton problem s then formulated to derve the equvalent dc model from these derved dc TDFs. 3. GENERAL DC SERIES ELEMENT MODEL The dc power flow formulaton for an N+ buses (N non-reference bus and reference/slack bus and L branches model s represented by (.7 and (.8 matrx equatons repeated below:

36 n B bus (3. flow B branch (3. On substtutng (3. n (3., flow B branch B (3.3 bus n Therefore, flow n (3.4 where, B branch B bus (3.5 = L N TDF matrx or senstvty factor matrx. Ths gves the relatonshp of TDF matrx to the power flow over network branches and bus nectons at respectve buses. 3.3 OWER TRANSFER DISTRIBUTION FACTORS (TDFs A power transfer dstrbuton factor s defned as the lnear senstvty of lne flow to the necton at partcular bus and wthdrawal at snk bus. If the amount of power t s nected at bus k (necton bus and same amount of power s wthdrawn from bus N+ (snk bus [34], we defne the TDF for the lne connectng bus and bus as: N k t (3.6 where, 3

37 = change n power flow over branch N k t = TDF for branch due to necton at bus and wthdrawal at slack bus = ower transacted between bus and bus N+ (slack/reference bus Fgure 3. dc network representng TDF for branch In the smlar fashon, a matrx correspondng to the transacton between the each bus to reference bus s constructed, ths s known as the TDF ( matrx n (3.5 and the column correspondng to each necton bus s defned as necton shft factor (Ψ. The TDFs obtaned are functons of network branch parameters (.e. reactance and network topology. Any change to these parameters typcally leads to a change n these senstvty factors. However, the TDF matrx obtaned n (3.5 s based on the classcal dc model and ts assumptons. These classcal dc TDFs have unpredctable accuracy and can lead to 4

38 large errors n estmatng flow senstvtes when the network devates from the nomnal condtons, lke per unt bus voltage, or has branches wth low X/R ratos or large angle dfference across the lne; n contrast the ac TDFs do take nto account such senstvtes that arse due to network topology, branch parameters and network operatng pont. Therefore, ac TDFs can provde a better allocaton of MW s across the network than the classcal dc TDFs and can be used to obtan a better network dc equvalent model for market applcaton and plannng studes Classcal DC TDF dervaton In furtherance of the state goal of ths research, t s benefcal to recognze that the bus susceptance matrx ( B bus and branch susceptance matrx ( branch B can be wrtten n terms of the bus-branch ncdence matrx (C and network reactance (x as follows: B bus C T dag C x (3.7 B branch dag C x (3.8 where, C = L N Bus-branch ncdence matrx C T = Transpose of bus-branch ncdence matrx dag = x / x / x / x L LL and x x x x L 5

39 On substtutng (3.7 and (3.8 nto (3.5, dc TDFs as a functon of network parameter and topology are obtaned as: dag C C x T dag C x (3.9 Snce these TDFs are derved from an deal lossless model, s sad to be consstent n the followng three ways: The TDFs at the sendng and recevng end are exactly same TDFs are consstent along the necton shft factor.e. sum of nectons at the non-source buses s zero. TDFs are consstent across the necton shft factors.e. there exst a unque (can be unformly scaled values though set of reactance that can satsfy all the necton shft factors smultaneously Lnearzed AC TDFs dervaton The ncremental ac TDFs are defned by lnearzng the ac power flow equatons obtaned for branch ((. (.5 around the base operatng pont. Mathematcally, these ac TDFs are defned as: N k t, o o (3.0 where, = Incremental change n power flow over branch due to necton at bus k o = Angle from solved ac base case soluton 6

40 o = oltage magntude from solved ac base case soluton Fgure 3. ac network representng ac TDF for branch Alternatvely, these senstvtes can be obtaned from the fnal Jacoban formed n the Newton-Raphson method based ac power flow. Snce the approach nvolves evaluatng the Jacoban t makes sense to dscuss the basc ac power flow equatons [35] at ths pont The ac power flow equatons. For the typcal transmsson lne model shown n Fgure., the power balance equatons for bus are defned as: For Q bus (or for bus on Ar lmts 7

41 Q g cos( b sn( g cos( b sn( For bus (wthn Ar lmts (3. (3. sp g cos( b sn( (3.3 (3.4 where, = net real power necton to bus = G L Q = net reactve power necton to bus = Q G Q L G = net real power necton at bus due to generators Q G = net reactve power necton at bus due to generators L = net real power wthdrawn at bus due to loads Q L = net reactve power wthdrawn at bus due to loads sp = specfed voltage at bus f t s a bus The ncremental values of the unknown system varables,.e. and θ, are gven by: J Q J3 (3.5 J J 4 where, J J3 J J 4 (N m (N m = Jacoban matrx; m = number of buses 8

42 9 ( m N Q = represents real and reactve power msmatch ( m N = ncremental bus angle and voltage magntude where, calc calc b g Q b g cos( sn( sn( cos( calc calc calc calc B Q Q J G Q J G J B Q J b g Q J b g Q J b g J b g J 4 3 cos( sn( 4 sn( cos( 3 cos( sn( cos( sn( (3.6

43 The Jacoban matrx n (3.5 s a sparse matrx. Soluton of (3.5 s obtaned usng sparse LU factorzaton [35]-[36] of ths matrx and then sparse forward/backward substtuton Senstvty calculaton for bus voltage magntude and angle As stated earler, the Jacoban matrx provdes the senstvty of power nectons to both bus voltage magntudes and angles. Conversely, for an ncremental power necton at a partcular bus, (3.5 provdes the bus voltage magntude and angle senstvtes. Usng these bus voltage magntude and angle senstvtes, one can obtan the MW flow senstvtes; hence one can obtan the MW flow senstvty due to the ncremental necton at a gven bus whch s the defnton of ac TDF gven n (3.0. Defne the N+ th bus as the slack bus/snk bus and let the k th bus be the necton bus. Let transacted of dmenson N be the real power transacton vector between the source and the snk bus. Smlarly Qtransacted be of cardnalty N and represent the reactve power transacton vector. Snce the TDFs are defned for MW flow senstvty to ncremental MW necton at bus, the mathematcal formulaton [37] for these senstvtes s obtaned as: k transacted contans only 0 s and s. k transacted s at the k th poston correspondng to the necton bus k and the remanng elements of vector are 0 value correspondng to the non-necton bus. k Q transacted s 0 value for all ts element, as reactve power s not of nterest at ths stage. 30

44 Q k transacted k transacted [0,0,,0,,0,0] [0,0,,0,0,0,0] th k element (3.7 T T (3.8 Thus, (3.7 and (3.8 can now be substtuted n (3.5 to obtan the bus voltage magntude and angle senstvty for necton at k th bus as: Q k transacted k transacted J J3 Upon smplfcaton, (3.9 J J 4 k k k J k J3 J J 4 Q k transacted k transacted (3.0 Ths process s repeated by consderng each necton bus, one at a tme, and the correspondng bus voltage magntude and angle senstvtes are calculated. Alternatvely, t can be seen that each column vector obtaned from the nverse of the Jacoban matrx provdes the bus voltage magntude and angle senstvtes correspondng to the each necton bus k ac TDF/Branch MW flow senstvty calculaton Recall the ac power flow equatons (.-(.3 representng the power flow over branch (refer to Fgure., whch can be re-wrtten as: g cos( b sn( g cos( b sn( 3

45 3 cos( sn( b g Q cos( sn( b g Q The MW flow senstvty for a branch can be obtaned by consderng the partal dfferental equaton of power flow to bus voltage magntude and angle as: where, The varables θ and used n (3. are obtaned by solvng (3.9. And fnally the ac TDFs are calculated usng (3.0, (3.. and t = MW (ncremental necton Q Q Q Q Q (3. (3. sn( cos( sn( cos( cos( sn( cos( sn( b g b g b g b g (3.3 sn( cos( sn( cos( cos( sn( cos( sn( b g Q b b g Q b g Q b g Q (3.4

46 at bus k. t represents the power transacted between the source (k th bus n (3.7 and snk bus. The ac TDF matrx thus obtaned s nconsstent f used as a dc TDF matrx n all respects mentoned earler as t accounts for the nonlneartes lke losses, n the model. Ths matrx s accompaned wth followng nconsstences: The TDFs at the sendng and recevng end are not same. TDFs are nconsstent along the necton shft factor.e. sum of nectons at the non-source buses s zero. TDFs are nconsstent across the necton shft factors.e. there does not exst a unque set of reactance that can satsfy all the necton shft factors smultaneously. 3.4 TDF-BASED OTIMIZATION AROACHES Relatvely recently, bus aggregaton technques (as opposed to bus elmnaton technques such as Ward reducton have been ntroduced as an alternatve for creatng reduced network equvalents that perform better n some applcatons [5]-[7]. Bus aggregaton technques are TDF-based and rely on solvng an optmzaton problem to fnd the network seres elements of a reduced network, gven the TDF matrx of the network under study. In essence, these methods take a large consstent TDF matrx and fnd a smaller equvalent TDF matrx from whch the seres elements of a reduced network can be nferred. Ths approach can be used to advantage n the work here by applyng t wth the followng change: We take a large nconsstent TDF 33

47 matrx and, n essence, map t to a partally consstent TDF matrx of the same sze and sparsty pattern, from whch we nfer the seres elements of the dc network model. Typcally we have topology and the branch susceptance from whch we calculate the dc TDF matrx,. Our stuaton s a bt dfferent. We have an ac TDF matrx (from whch we wll calculate a dc TDF matrx and wsh to fnd the branch reactances consstent wth ths dc TDF matrx. Ths can be accomplshed n the followng way. Equaton (3.9 may be manpulated as follows, C T C T dag C dag C x x dag C dag C 0 x x (3.5 (3.6 C T I dag C 0 x (3.7 Usng the matrx algebra for further smplfcaton, T C I dag c 0, m,,, N m x (3.8 where, C = Bus-branch ncdence matrx = [c, c..., cm, cn] cm =m th column vector of bus-branch ncdence matrx dag(cm = dagonal matrx formed usng cm Alternatvely, 34

48 0 x (3.9 where, T ( C I dag ( c T ( C I dag ( c T ( C I dag ( c N NLL (3.30 Several observatons about ths over-determned set of equatons, (3.9, are mportant. Frst, the trval soluton to (3.9, (/x = 0 s of no nterest. Ths mples that all lne reactances are nfnte,.e., all branches are open crcuted. Equaton (3.9 and (3.30 descrbe -matrx whch has a rank of at most L-, and s therefore rank defcent. Because the dc power-flow problem s based on lnear angle-flow relatonshps and lnear bus-power-balance constrants, t can be thought of as a current dvder network where current s the analog of power. Snce the seres branch values of any dc network wth only seres branch resstances and shunt current nectons can be scaled by an arbtrary constant wthout affectng the current dvson propertes, (3.9 should be rank defcent by at least one degree. 3.5 CONERSION FROM AC TDFs TO DC TDFs Rank-defcency notwthstandng, a unque soluton to (3.9 exsts wth all resduals dentcally equal to zero f the (dc DTF matrx s consstent. More precsely, the TDF matrx consstency and nconsstency can be stated as: each column, or necton shft factor, s self-consstent f a network model could be created whch corresponds 35

49 exactly to the necton shft factor. If all columns, or necton shft factors, are mutually consstent (.e., the TDF matrx s fully consstent then a network model could be created whch corresponds exactly to all necton shft factors smultaneously. The ac TDF matrx s consstent n nether of these ways because of the branch power losses. A method for creatng a partally consstent dc TDF matrx from an nconsstent ac TDF matrx s ntroduced next Inecton Bus Bus Wthdrawal Loss Inecton Fgure 3.3 Loss modeled as Inectons (ostve/negatve Inherently ac TDFs are nconsstent and each column of ac TDF s,.e., necton shft factor, can be made self-consstent (but not mutually consstent f the branch power loss s modeled as a power necton/wthdrawal at one end of each branch, as shown n Fgure 3.3. Ths compensaton of losses modeled as necton/wthdrawal accounts for the nconsstency due to resstve elements n the network. Defne the necton at bus k needed to compensate for branch losses assocated wth the q th necton shft factor as : q k q k q q ( k _ sendng k _ recevng (3.3 36

50 37 where, q sendng k _ = Sum of power receved at node k from all branches where k s the sendng end node based on drecton of the power flow q recevng k _ = Sum of power receved at node k from all branches where k s the recevng end node drecton of the based on power flow Defne necton vector 'q assocated wth the q th necton shft factor can be wrtten as: Combnng all the necton vectors correspondng to each necton shft factor nto a matrx Usng ths matrx, the relatonshp between ac TDF, derved dc TDF and necton matrx can be wrtten as: T q N q k q q q,,,, ' (3.3 N N N N N N N N (3.33

51 38 where, dc = derved dc TDF matrx ac = nconsstent ac TDF matrx m = represents m th branch Ψ T = transpose of the loss necton matrx ac m = represents the ac TDF correspondng to m th branch for th ISF dc m = represents the dc TDF correspondng to m th branch for th ISF ( ( ] [ N dc m dc m dc m dc m T N N X N ac m ac m ac m ac m N N Solvng ths equaton for the dc necton shft factors yelds, (3.34 ( ( ] [ N ac m ac m ac m ac m T N N X N dc m dc m dc m dc m N N (3.35

52 39 Equaton (3.35 can be used to calculate the TDFs correspondng to every necton bus correspondng to each branch one at a tme. Lnear system of equatons (3.35 s solved usng LU factorzaton and forward/backward substtuton to calculate the dc TDFs. Usng the process each necton shft factor comprsng the dc TDFs are self-consstent but are not mutually consstent. 3.6 OBTAINING THE DC NETWORK MODEL Now, the next problem that needs to be attacked s fndng the equvalent lne reactances for each branch usng the above derved equvalent dc TDF matrx. As stated above, the necton shft factors (ISF derved correspondng to these derved dc TDFs are nconsstent among themselves.e. no sngle network reactances can satsfy all the ISFs smultaneously. ( ( ] [ N L ac L ac L ac L ac m ac m ac m ac ac ac ac ac ac N L ac N N N N (3.36 ( ( ] [ N L dc L dc L dc L dc m dc m dc m dc dc dc dc dc dc N L dc N N N N (3.37

53 The problem then s to calculate the network reactances that can ft all of these derved dc TDFs smultaneously as closely as possble. Usng the derved dc TDFs ( dc to compute the matrx n place of classcal dc TDFs ( n (3.9, s gven by: mn x [/ x] (3.38 s. t. x Z Z 0 (3.39 where, Z s an arbtrary small constant representng a lower bound on the L norm of vector (/x and s the matrx defned by (3.33. In addton to (3.38, constructng an optmzaton approach whch avods the trval soluton to ths optmzaton problem can be handled n several ways, such as an egenvalue approach as dscussed n [5]. Snce all the susceptances can be scaled proportonally, a constrant (as per (3.39 has been ntroduced for the purposes. Let us defne the vector of network susceptances as, y / x T [ y, y y, yl ] y (3.40 The constrant n (3.39 can be re-wrtten wth susceptance as: y T y Z (3.4 Alternatvely we defne (3.45 as: (3.4 f ( y y y y Z Z L 0 40

54 4 In order to solve the problem n a lnear least-squares sense, the constrant n (3.4 or (3.46 s lnearzed by usng the frst two terms of a Taylor-seres expanson around the base-pont,.e. hgher order terms are neglected. The base pont for the susceptance s selected as: The truncated Taylor-seres expanson of constrant s wrtten as: Upon further smplfcaton, ths yelds Ths lnearzed constrant can be embedded nto (3.38 as, where, ( =,, m teraton ndex Ths equaton can be restated as: T o L o o o o y y y y y ],, [ ( ( ( ( ( ( ] ( ( ( [ ( ( ( ( ( ( ( ( ( Z y y y y y y y y y y y y o L o o o L L o L o o o o (3.44 Z y y y y y y y y y o L o o L o L o o ( ( ( ( ( ( ] [ ( ] [ ( ( ( ( ( Z y y y y y L L N L (3.46 y b ] [ ' (3.47

55 whch s a set of over-determned lnear equatons; therefore the solutons s an error mnmzaton process. [ y] ( [( ' ( T ' ( ] [ ' ( ] T b ( (3.48 ( T ( Also, note that ( ' ' s a square matrx of cardnalty L L and t s a very sparse matrx, so one can explot the benefts of the sparsty technques to evaluate (3.48. Ths calculaton usng sparse LU factorzaton and forward/backward substtuton s computatonally more effcent than the egenvalue decomposton, QR factorzaton method proposed n [5]. Equaton (3.47 s solved recursvely to obtan better estmates of the network parameters (y at each teraton. Equaton (3.47 s terated untl the convergence tolerance (0-4 s met. max( abs. y y ( ( (3.49 It s worthwhle to note that the rank of the lambda ( matrx may vary wth the network topology and ths subect shall be elaborated upon more n the next chapter. 3.7 SUMMARY In ths chapter, a TDF based dc network model development procedure s ntroduced. The proposed model s a least square based evaluaton technque for dervng network parameters. It s worth mentonng, that several tests were conducted for optmal performance at both the stages.e. equvalent dc TDF calculaton and evaluaton of network parameters n the dervaton of the proposed model. In the 4

56 dervaton of dc TDFs several possbltes of ncludng weghted power flow constrants n (3.34 was explored. Whle several optmzaton technques such as L norm, least absolute value approach and L-nfnty were ntroduced for (3.38-(3.39 to derve equvalent model parameters. These varatons n optmzaton technques were evaluated for power flows at the base operatng condton and under contngency analyss. Also, the egenvalue approach descrbed n [5] was mplemented and tested for (3.38-(3.39. It was found that network parameters obtaned usng the least square approach and egenvalue approach yelded same results (although scaled wthn the precson of e -3. It can be concluded the least square approach and egenvalue approach performed better than the other optmzaton technques mentoned above when evaluated for dstrbuton of power flows n network under contngency condtons. Furthermore, t s essental to note that egenvalue approach s computatonally much more expensve (as t nvolves sngular value decomposton than performng least squares. The model development process can be summarzed n the flowchart shown n Fgure

57 START Run ac power flow on full model (Obtan and θ Calculate ac TDF s usng and θ (from dfferental Calculate the loss necton matrx Let teraton ndex m = Calculate the dc TDFs Increment m m = m+ Is m Num. branches NO A 44

58 A Set = Intalze the susceptance vector Construct the lambda (Ʌ matrx usng dc TDF s. Augment ths matrx wth lnearzed constrant. Increment = + Calculate and update susceptance vector Check for convergence NO Is YES Obtaned dc network susceptance STO Fgure 3.4 Flowchart for entre network equvalencng process 45

59 4 MODEL ALIDATION AND NUMERICAL ILL-CONDITIONING 4. INTRODUCTION In the prevous chapter, a novel network equvalence technque has been proposed. Ths model provdes the network equvalent when the lambda ( matrx (pror to augmentng t wth constrant s rank defcent by only one degree. However, dependng on the topology, the rank defcency of -matrx may be greater than one. Ths leads to a theoretcally undetermned problem, though due to mprecson n the TDF s the matrx wll appear to be of full rank, but numercally ll-condtoned, leadng to erroneous results for the value of the network susceptances. The focus of ths chapter s on valdatng the proposed model and analyzng the topologcal dependency of rank of the lambda ( matrx and on how to resolve the rank defcency ssues. The approach to dentfy topologcal rank dependency s studed emprcally. 4. MODEL ALIDATION Model valdaton of the physcal network whch ncludes resstance, phase-shftng transformers usng the proposed method s a dauntng task. Ths s attrbuted to the use of ac TDFs whch nclude the nherent nonlneartes, such as losses, typcal of such network. Thus the network parameters derved from the proposed model cannot be compared aganst any reference. Therefore, some santy-check exercses are ntroduced for algorthmc verfcaton purposes on a small 46

60 system, before the applcaton of ths technque on the larger and realstc power systems. Snce the proposed model s derved n two stages.e. frst, obtanng the equvalent dc TDFs from ac TDFs and, second, dervng network susceptances usng these derved dc TDFs, the santy check of ths proposed model s also segregated nto two ndependent parts. 4.. CHECKS ON THE EQUIALENT DC TDFs The classcal dc TDFs, whch are completely consstent, are dfferent than the derved dc TDFs, whch are consstent wthn a shft factor but nconsstent across shft factors. As an example, consder a non-branchng radal network. In ths network, the power dstrbuton s undrectonal.e. power wll flow from the source or the necton bus to the snk bus. Therefore, the derved dc TDFs and ac TDFs should converge to a same value. Ths smple valdaton experment explots the topologcal property of the radal network. ` 3 4 AC 0.05 pu 0.5 pu 0.6 pu pu 0. pu pu 9 MW 3 MAr 5 MW Fgure 4. Radal network of four buses For the sample radal network show n Fgure 4., an ac power flow s run to compute the bus voltage magntudes and angles. Based on the network parameters and ac soluton, the ac TDFs are computed as dscussed n prevous chapter. Bus 4 s 47

61 the slack bus or the snk bus. The ac TDFs at the from end for the shown network s gven by: ac TDF from Lne In. Bus L L 3 L It s worth notng that the ac TDFs can acheve TDF values greater than due to the senstvtes of the ac model or nserton of seres capactors (often used n the lne compensaton. Usng the proposed model, the dc TDFs are obtaned as: Lne In. Bus L dc TDF L 3 L Ths s what we expect the model to delver, for the radal network.e. the power from the nected bus flows to the snk bus. Note that the dc TDFs have a maxmum value equal to.0. The dc TDFs can also attan values greater than.0 n cases where negatve susceptances are ntroduced n the model.e. nserton of seres capactors for lne compensaton. 4.. TOWARD ALIDATING THE SUSCETANCE EALUATION ALGORITHM Snce the dc TDFs obtaned usng the proposed algorthm are nconsstent across the shft factors, the valdaton of the reactance/susceptance computaton algorthm becomes a thorny ssue: how does one compare the obtaned dc reactances wth the 48

62 gven network ac parameters? Thus, for the purpose of algorthm valdaton the classcal dc TDFs are used to compute the lambda ( matrx nstead of the derved dc TDFs as explaned n the proposed model. As stated earler, the classcal dc TDFs are fully consstent along and across the shft factors. Ths wll result n a unque soluton reactance vector. As stated earler, the angle of ths reactance vector s unque, though t can be scaled n magntude wthout alterng ts current dvson propertes. One way of partally valdatng the approach s to use (3.9 whle replacng by the classcal dc TDFs of the network. Then, usng the ntal estmate as per (3.43 and solvng for susceptances usng (3.46 (3.48, the soluton wll be the susceptances of the orgnal network. Consder a small 3-bus example for llustraton. 3 AC 0.06 pu 0.04 pu 0.05 pu Snk Bus Fgure 4. Three bus sample system The classcal dc TDFs are gven by (3.9: Lne In. Bus L Classcal dc TDF L 3 L

63 The correspondng lambda ( matrx derved usng (3.9 s Usng the ntal estmate of susceptances y = [,, ] T ; and augmentng the lambda ( matrx wth the lnearzed constrant, the lnear system of equatons s solved as per (3.46-(3.48. The converged soluton s gven by: y = [.997, , ] T The value of the susceptance vector based on classcal dc model reactances s gven by: y = [5, , 0] T It can be notced that the susceptance values are scaled by a factor of Therefore, f the soluton s scaled up by a common multplyng factor for all the susceptances then the orgnal susceptances are obtaned. In ths case a random ntal value was chosen whch llustrates that the robustness of ths algorthm and non-dependency of soluton on ntal pont. However, for fast convergence a more approprate ntal pont such as susceptance vector correspondng to the classcal dc reactance could be chosen. The same test was conducted on IEEE-8 bus system and ERCOT for valdaton purposes and smlar test results were obtaned. 50

64 4.3 ROBLEMS ASSOSCIATED WITH RANK DEFICIENCY In the proposed modelng process, the reactances are evaluated by solvng the over-determned set of lnear equatons whose coeffcents are descrbed by the -matrx and b vector. If the - matrx becomes theoretcally sngular then the condton number goes to nfnty. In practcal problems, due to roundoff error, theoretcally sngular matrces wll appear nonsngular but wll have very large condton numbers, whch allows the calculatons to go forward but results n an erroneous answer. In the next sectons the cause of such rank defcency n the -matrx s explored. 4.4 TOOLOGICAL DEENDENCY OF THE RANK OF Let us frst revst the mathematcal concept of the rank of the matrx. For the matrx A of cardnalty m by n Row rank: Is the maxmum number of lnearly ndependent rows n the matrx. Row rank of A m Column rank: Is the maxmum number of lnearly ndependent columns n the matrx. Column rank of A n rank (A mn(m,n Also, by lnear algebra rank (A = rank (A T Suppose two matrces A and B have rank as m and n respectvely. Then rank (A.B = mn(rank(a, rank(b 5

65 Ths property shall be exploted as the - matrx sze grows wth the ncrease n the system sze (recall that the cardnalty of = N.L L. The problem at hand nvolves the soluton to the over-determned set of equatons, whose soluton s gven by (3.48 re-wrtten here as: [ y ] ( [( ' ( T ' ( ] [ ' ( ] T b ( Fndng the rank of s much more computatonally expensve than that of T whch has cardnalty of L L. Therefore, T s used to compute rank of EMIRICAL ANALYSIS OF NETWORK TOOLOGIES Consder the followng network topologes for whch the rank of the matrx s determned and tabulated. Ths emprcal approach wll lead to an ntutve understandng of the relatonshp between the rank of and the network topology. Case : Radal network 3 4 Fgure 4.3 Four bus sample radal network rank ( = 0 An mportant concluson can be drawn from ths result s that t s mpossble to determne the network parameters for a radal network. Also, t algns wth our ntutve understandng of the radal network that the power flow s ndependent of the 5

66 network reactance. Therefore, network branch parameters need to be specfed for such a network. Case : Meshed network Let us now merge bus and bus 4 from the prevous network. 3 Fgure 4.4 Three bus meshed network rank ( = (The rank of -matrx for network shown n Fgure 4.4 s two. Thus, n a meshed network f the lnearzed constrant ((3.45-(3.46 s ncluded then the equvalent network branch parameters can be calculated usng the proposed algorthm. Case 3: Sem-meshed network For the 8-bus model shown n Fgure 4.5 the navely expected rank of the matrx s 8 (Number of branches -. However, the actual rank turns out to be 5. Ths algns wth our ntuton that radal lnes ( n number and radally-attached sub-networks (buses 5, 6, 7 and 8 reduce the rank of the matrx by degree each. 53

67 Fgure 4.5 Eght bus meshed network We observe that t s not necessary that the sub-networks be connected only through radal lnes but can be connected radally on the bus tself as shown n Fgure 4.6. Sub-network Sub-network Network Radal Lne Fgure 4.6 Generalzed network 4.5 IDENTIFICATION OF THE SUB-NETWORKS Now that some sense for the cause of numercal ll-condtonng or rank defcency due to network topology of the matrx s developed, t becomes 54

68 convenent to use a larger network to dentfy the radal lnes and sub-networks usng an algorthm. 6 Branch Branch 6 Branch 8 Branch Branch 3 Branch 4 Branch 5 Branch 7 Branch 9 8 Branch 0 Branch 7 Branch 9 0 Fgure bus network Consder the 0-bus sample network shown n Fgure 4.7 wth bus 0 servng as the snk bus. Assume that the network s lossless and all the reactances are set to per unt. The classcal dc TDF matrx s computed for the network shown n Fgure 4.7 usng (3.9 as shown n Table 4.. Ths dc TDF matrx structure can help decpher our emprcal observatons from the prevous secton. The followng key characterstcs are observed n the dc TDF matrx: Strctly radal branches have TDF values of ether 0 or (branch 4 and 5; Identfy the pattern of non-zero TDFs n non-radal branches.e. rows n the TDF matrx other than the rows of the TDF matrx representng radal branch. (.e. n essence make note of for whch necton buses there s flow over the branch. Branches correspondng to TDF matrx whch have same pattern of non-zero TDF shall form a sub-network. For example: 55

69 Branches/Lne, and 3 have the same pattern of non-zero elements. Branches/Lne 6, 7, 8 and 9 have the same pattern of non-zero elements. Branches/Lne 0, and have the same pattern of non-zero elements. From ths observaton, t can be concluded that branches (, and 3 form one sub-network, branches (6, 7, 8 and 9 form the second sub-network and branches (0, and form the thrd sub-network n the network. Also branches 4 and 5 are the radal branches n the network. For the network shown n Fgure 4.7 the rank ( = 7. For a -branch network, the -matrx s defcent by a degree of 5, whch exactly matches our concluson (3 sub-networks + radal branches. 56

70 57 Table 4. Classcal dc TDFs for network n fgure 4.7 Lne From To

71 4.6 EQUIALENT NETWORK SOLUTION WITH SUB-NETWORKS Upon dentfcaton of radal lnes and sub-networks the problem at hand s to evaluate the equvalent dc TDFs usng the rank defcent matrx. The network parameter evaluaton process can be dvded nto sub-problems, each of whose matrces are rank defcent by one: Fx the susceptance of the radal lnes arbtrarly. Solve each ndependent sub-network one at a tme. For example, n the network shown n Fgure 4.7 the subsequent steps are followed: Set the radal branches 4 and 5 to classcal dc reactance. Then calculate the -matrx usng the subset of the TDF matrx assocated wth each of the sub-networks (Sub Network :, and 3; Sub Network :, 9 and 0; Sub Network 3: 5, 6, 7 and 8. S 0 x S (4. where, S T ( S CS I dag( c T ( S CS I dag( c T ( S CS I dag( c S N S S S = Subscrpt S ndcates the values correspondng to the sub-networks. 3 The S matrx thus formed s augmented wth the constrant as explaned n secton 3.6. An ntal estmate of susceptances s chosen for branches n each sub-network. The susceptance of a network model correspondng to each of the 58

72 sub-network s calculated usng the teratve procedure descrbed n prevous chapter. 4.7 SUMMARY In ths chapter, model valdaton of the proposed model was explored and then the problems assocated wth the rank defcency of -matrx were dscussed. The followng flowchart (Fgure 4.8 schematcally summarzes the procedure to dentfy the sub-networks and evaluaton of the network parameters. It s found that even for very large networks such sub-networks are not very large n number and sze. Therefore, t s not very computatonally expensve to evaluate such sub-networks. 59

73 START Compute the classcal dc TDF Identfy the radal lnes and sub-networks Choose one sub-network one at a tme Intalze the susceptance vector y Construct the Ʌ S -matrx usng derved dc TDF s. Augment ths matrx wth lnearzed constrant. Calculate the susceptance vector untl converged NO Are all sub-networks calculated YES Obtaned dc network susceptance STO Fgure 4.8 Flowchart for sub-network reactance evaluaton 60

74 5 NUMERICAL EXAMLES 5. INTRODUCTION Ths chapter presents the mplementaton of the concepts developed through numercal examples. Snce the accuracy and performance of dc models are subect to the assumptons made durng the modelng stage and loadng levels (.e. lghtly loaded or heavly loaded of the system. Therefore, an attempt has been made to develop general nsght for these dc models. The accuracy of several parameters lke branch power flows durng contngency condtons, TDF s, branch reactance were evaluated for a 7-bus, the IEEE-8 and the 5650 bus ERCOT nterconnecton to draw comparson between dfferent models currently n use (.e. classcal dc, sngle multpler and proposed model descrbed earler. 5. CASE STUDIES AND DESCRITION Ths secton summarzes the general nformaton of the three cases llustrated: Table 5. Case Study detals S. No. Case Descrpton Buses Branches Radal Branches 7-Bus 7 0 IEEE ERCOT

75 5.. Case Study : 7-Bus Model Fgure 5. shows the 7-bus, -branch network model. The network parameters are gven n Table 5. for reference. Fgure 5. 7-Bus Network Model Table 5. Network arameters for 7-bus model Lne No. From Bus To Bus R (p.u. X (p.u. (MW Q (MAr measured at from bus 6

76 The ac TDFs for the model are derved usng (3.6 are shown n Table 5.3 and Table 5.4. Table 5.3 ac TDFs (at sendng end From To Bus Bus Table 5.4 ac TDFs (at recevng end From To Bus Bus The equvalent dc TDFs thus obtaned usng (3.35 are gven n Table

77 Table 5.5 Equvalent dc TDFs From To Bus Bus Also the classcal dc TDFs for the model are gven for reference n Table 5.6. Table 5.6 Classcal dc TDFs From To Bus Bus Snce the derved equvalent dc TDFs and classcal dc TDFs correspond to the proposed model and more common prevalent models, respectvely, ther comparson s shown n Fgure

78 0.8 Comparson of Equvalent dc TDFs vs Classcal dc TDFs Classcal dc TDFs ISF ISF ISF3 ISF4 ISF5 ISF6 Ref. Lne Equvalent derved dc TDFs Fgure 5. Comparson of Classcal dc and Equvalent derved dc TDFs ISF ISF6 n Fgure 5. represent the necton shft factors (.e. column vectors n TDF matrx correspondng to each non-snk bus for a 7-bus system. + represent each TDF values along the necton shft factors. Fgure 5. shows there s an offset n the two TDFs values from the reference lne (marked blue whch represent the slope of 45 o. Ths s attrbuted to the varaton of the reactance values thus obtaned usng the reactance evaluaton algorthm descrbed n Chapter 3 and Chapter 4. The value of loss compensaton n a sngle multpler type dc model s computed usng (.3 and for ths 7-bus system s gven by: Sngle Multpler (γ =.07 The reactances obtaned usng the proposed algorthm and the correspondng power flows at the base operatng pont are shown n Table 5.7 below: 65

79 Table 5.7 Reactance and power flow comparson at base operatng pont From To Bus Bus X Xprop FCl. dc FSM. dc Fprop FCl. dc = Classcal dc; FSM. dc = Sngle Multpler; Fprop = roposed Model The performance of varous dc models under base-case operatng condtons s compared aganst ac branch power flow soluton. For ths purposes, we defne branch power flow error as: Error Fmod el F ac (5. Whereas, F mod el = dc power flow for a gven model (n MW F ac = ac power flow for the model (n MW Fgure 5.3 shows the performance of the models at base operatng pont. 66

80 ower Flow Error (n MW Comparson of power flow error at base case Branch ID Classcal dc Sngle Multpler roposed Model Fgure 5.3 Comparson of power flow errors for dfferent models at base case It can be seen that there are large MW flow errors for both the classcal dc and sngle multpler model. However, the proposed model matches the base pont power flows to the precson of 0-3. Often, the flow errors are compared n percentage (% but ths unt becomes msleadng for branches havng small flow because f a small flow, say MW, reverses drecton the error n percent can easly be 00%. Therefore, n ths work the errors are plotted n MW only. 67

81 Reactance alue (n per unt Also the reactance values are compared for the two cases n Fgure 5.4. The 0.6 Comparson of Reactances Branch ID Classcal dc Reactance Derved Reactance Fgure 5.4 Comparson of reactance between dfferent models reactance value s scaled for the proposed model for purpose of comparson (branch has same reactance value for all models. It s nterestng to note that although small varatons appear for most of the branches,, there s a huge varaton n reactance for branch 9, whch s hgh R/X rato (R/X = branch. The next effort s to examne the dfference n the models under contngency condtons. The models are evaluated for the sngle branch outage contngency. In ths case each lne s removed, one at a tme, and the branch power flow s calculated for each of the dc models and compared to the ac model. Maxmum absolute error and root mean square (rms flow errors are used as metrcs for the analyss. a Maxmum absolute MW error It s defned as the maxmum of the absolute value of branch power flow error defned by (5.. Ths s computed for each lne contngency for the system under study. 68

82 Mathematcally, t s defned as: Max. Absolute Error ( (5. max [ ],, L where, F F ac dc F F ac dc (5.3 L L Fac Fdc F ac = ac power flow on branch F dc = dc power flow on branch = Contngency branch ID number Ths metrc gves an ndcaton of the accuracy of the dc models wth respect to ac contngency results. 69

83 80 70 Comparson of Maxmum absolute MW error vs Contngent branch roposed Model Classcal dc Model Sngle Multpler Maxmum absolute error (n MW Contngent Branch ID Fgure 5.5 Comparson of maxmum absolute MW error for contngences It can be seen from Fgure 5.5 that the proposed model gves better results as compared to the classcal dc and sngle multpler models. Sgnfcant mprovements are wtnessed on the lnes near slack bus and on hgh R/X rato branches. A summatve metrc, an rms (MW branch flow error dscussed next wll provde an overall estmate of error across all branches for each contngency. b Root mean square (rms error estmate Root mean square s defned mathematcally as: RMS Error ( L ( Fmod el Fac L (5.4 where, 70

84 L = Number of branches Fmod el = dc power flow for a gven model.e. classcal dc, sngle multpler proposed model for branch F ac = ac power flow for branch Ths measure of error provdes the rms flow error (MW across all the branches for a contngency at a partcular branch. Fgure 5.6 Comparson of rms error for contngences Table 5.8 summarzes the results for 7-bus model whch ndcates that proposed model works better than the other two models n terms of both the metrcs chosen for comparson. 7

85 Table 5.8 Summary of results for 7-bus model Max. Absolute Error (MW Max. RMS Error (MW roposed Model Classcal DC Sngle Multpler Case Study : IEEE-8 Bus Model In ths secton, the IEEE-8 bus system s examned for the accuracy of dc power flow models (classcal dc, sngle multpler and proposed model both at the base case and under sngle-branch outage contngency condtons. (These are the same numercal experments performed on the 7-bus model. It s worthwhle to note that the IEEE-8 bus system has one sub-network (shown n Fgure 5.7 that needs to be solved ndependently (as descrbed n Secton 4.5 and Secton 4.6 as a sub-problem to obtan the network model. All the branch power flows obtaned from dc models under case operatng condton (no contngency s compared aganst the ac branch power flow soluton.as per (.3 multpler for loss compensaton s computed to be: Sngle Multpler (γ =.03 Fgure 5.8 Comparson of power flow errors for dfferent models at base case, computed usng (5. for the IEEE-8 bus system. It can be seen that classcal dc model has large errors at the base case. Snce the sngle multpler accounts largely for 7

86 losses, the branch flow errors are comparatvely low. Our proposed method matches the base case wth the precson of 0-3. Fgure 5.7 IEEE-8 bus Model 73

87 ower Flow Error (n MW Comparson of power flow error at base case Classcal dc Branch ID Fgure 5.8 Comparson of power flow errors for dfferent models at base case Fgure 5.9 and Fgure 5.0 llustrate the maxmum absolute MW error (usng (5.. It becomes qute clear from the error duraton curve (Fgure 5.0 that the proposed model has sgnfcantly lower errors as compared to the contemporary methods. 74

88 Maxmum Absolute MW Error Maxmum absolute flow error (n MW Maxmum Absolute MW Error for contngences roposed Model Classcal dc Sngle Multpler Contngent Branch ID Fgure 5.9 Comparson of maxmum absolute MW error for contngences Error Duraton Curve % of Total Contngent Branches roposed Model Sngle Multpler Classcal DC Fgure 5.0 Error Duraton Curve for maxmum absolute MW error for contngences Fgure 5.0 shows that, for 0% of the branch contngences, the maxmum absolute branch flow error exceeds 4.MW, 4.35MW and 63.6MW for proposed, sngle multpler and classcal dc model respectvely. Fgure 5. llustrates the RMS 75

89 error across all the branches n the network for each specfed contngency. It s obtaned usng ( Comparson of RMS Error for Contngences roposed Model Classcal dc Sngle Multpler 0 RMS Error (n MW Contngent Branch ID Fgure 5. Comparson of RMS error for contngences Branch-flow error results for IEEE-8 bus system are summarzed n Table 5.9. Table 5.9 Summary of results for IEEE-8 Max. Absolute Error (MW Max. RMS Error (MW roposed Model Classcal DC Sngle Multpler

90 5..3 Case Study 3: ERCOT Interconnecton Electrcty Relablty Councl of Texas (ERCOT shown n Fgure 5. s one of the nterconnectons n North Amerca whch can serve as a realstc test bass for testng the performance of the dc models dscussed so far. Ths practcal system offers new challenges n terms of phase-shftng transformers (as dscussed n Secton. and Secton.. The phase-shft values for the phase-shfters was kept same as n the orgnal network durng the network equvalence process of computng the model parameters. Tests smlar to 7-bus and IEEE-8 were conducted on ERCOT nterconnecton for performance measurement. Fgure 5. ERCOT Interconnecton All the branch power flows obtaned from dc models under base-case operatng condtons (no contngency s compared aganst the ac branch power flow soluton. As per (.3, the multpler for loss compensaton s computed to be: 77