Power System State Estimation Using Phasor Measurement Units

Transcription

1 Unversty of Kentucky UKnowledge Theses and Dssertatons--Electrcal and Computer Engneerng Electrcal and Computer Engneerng 213 Power System State Estmaton Usng Phasor Measurement Unts Jaxong Chen Unversty of Kentucky, Clck here to let us know how access to ths document benefts you. Recommended Ctaton Chen, Jaxong, "Power System State Estmaton Usng Phasor Measurement Unts" (213). Theses and Dssertatons--Electrcal and Computer Engneerng Ths Doctoral Dssertaton s brought to you for free and open access by the Electrcal and Computer Engneerng at UKnowledge. It has been accepted for ncluson n Theses and Dssertatons--Electrcal and Computer Engneerng by an authorzed admnstrator of UKnowledge. For more nformaton, please contact UKnowledge@lsv.uky.edu.

2 STUDENT AGREEMENT: I represent that my thess or dssertaton and abstract are my orgnal work. Proper attrbuton has been gven to all outsde sources. I understand that I am solely responsble for obtanng any needed copyrght permssons. I have obtaned and attached hereto needed wrtten permsson statements(s) from the owner(s) of each thrd-party copyrghted matter to be ncluded n my work, allowng electronc dstrbuton (f such use s not permtted by the far use doctrne). I hereby grant to The Unversty of Kentucky and ts agents the non-exclusve lcense to archve and make accessble my work n whole or n part n all forms of meda, now or hereafter known. I agree that the document mentoned above may be made avalable mmedately for worldwde access unless a preapproved embargo apples. I retan all other ownershp rghts to the copyrght of my work. I also retan the rght to use n future works (such as artcles or books) all or part of my work. I understand that I am free to regster the copyrght to my work. REVIEW, APPROVAL AND ACCEPTANCE The document mentoned above has been revewed and accepted by the student s advsor, on behalf of the advsory commttee, and by the Drector of Graduate Studes (DGS), on behalf of the program; we verfy that ths s the fnal, approved verson of the student s dssertaton ncludng all changes requred by the advsory commttee. The undersgned agree to abde by the statements above. Jaxong Chen, Student Dr. Yuan Lao, Major Professor Dr. Cacheng Lu, Drector of Graduate Studes

3 POWER SYSTEM STATE ESTIMATION USING PHASOR MEASUREMENT UNITS DISSERTATION A dssertaton submtted n partal fulfllment of the requrements for the degree of Doctor of Phlosophy n the College of Engneerng at the Unversty of Kentucky By Jaxong Chen Lexngton, Kentucky Drector: Dr. Yuan Lao, Assocate Professor of Electrcal and Computer Engneerng Lexngton, Kentucky 213 Copyrght Jaxong Chen 213

4 ABSTRACT OF DISSERTATION POWER SYSTEM STATE ESTIMATION USING PHASOR MEASUREMENT UNITS State estmaton s wdely used as a tool to evaluate the real tme power system prevalng condtons. State estmaton algorthms could suffer dvergence under stressed system condtons. Ths dssertaton frst nvestgates mpacts of varatons of load levels and topology errors on the convergence property of the commonly used weghted least square (WLS) state estmator. The nfluence of topology errors on the condton number of the gan matrx n the state estmator s also analyzed. The mnmum sngular value of gan matrx s proposed to measure the dstance between the operatng pont and state estmaton dvergence. To study the mpact of the load ncrement on the convergence property of WLS state estmator, two types of load ncrement are utlzed: one s the load ncrement of all load buses, and the other s a sngle load ncrement. In addton, phasor measurement unt (PMU) measurements are appled n state estmaton to verfy f they could solve the dvergence problem and mprove state estmaton accuracy. The dssertaton nvestgates the mpacts of varatons of lne power flow ncrement and topology errors on convergence property of the WLS state estmator. A smple 3-bus system and the IEEE 118-bus system are used as the test cases to verfy the common rule. Furthermore, the smulaton results show that addng PMU measurements could generally mprove the robustness of state estmaton. Two new approaches for mprovng the robustness of the state estmaton wth PMU measurements are proposed. One s the equalty-constraned state estmaton wth PMU measurements, and the other s Hachtel's matrx state estmaton wth PMU

5 measurements approach. The dssertaton also proposed a new heurstc approach for optmal placement of phasor measurement unts (PMUs) n power system for mprovng state estmaton accuracy. In the problem of addng PMU measurements nto the estmator, two methods are nvestgated. Method I s to mx PMU measurements wth conventonal measurements n the estmator, and method II s to add PMU measurements through a post-processng step. These two methods can acheve very smlar state estmaton results, but method II s a more tme-effcent approach whch does not modfy the exstng state estmaton software. KEY WORDS: Weghted Least Square, Phasor Measurement Unt, Topology Error, Load Increment, Optmal Placement Jaxong Chen November 6, 213

6 POWER SYSTEM STATE ESTIMATION USING PHASOR MEASUREMENT UNITS By Jaxong Chen Dr. Yuan Lao Drector of Dssertaton Dr. Ca-Cheng Lu Drector of Graduate Studes November 1, 213 Date

7 ACKNOWLEDGEMENTS I wsh to acknowledge and thank those people who helped me complete ths dssertaton: Dr. Yuan Lao, my academc advsor, who gave me the chance to do research n ths feld and guded me through my studes by hs valuable help and encouragement. I could not have magned a better mentor than Dr. Lao. Hs nsghtful experence and assstance have always been helpful to my research. I also gratefully acknowledge Dr. Be Gou for hs nvaluable suggestons durng the research. Specal thanks are also extended to members of the commttee, Dr. Yu-Mng Zhang, Dr. Paul Dolloff, Dr. Alan Male and Dr. Zongmng Fe for ther nsghtful advce and wllngness to serve n the Dssertaton Advsory Commttee. I would also lke to thank my parents for ther endless love, support and understandng.

8 Table of Contents ACKNOWLEDGEMENTS... Lst of Tables... v Lst of Fgures... v Chapter 1 Introducton Background Phasor Measurement Unt Lessons of Northeast Blackout n Contrbuton of Ths Dssertaton Dssertaton Outlne... 8 Chapter 2 Impacts of Load Levels and Topology Error on WLS State Estmaton Convergence WLS State Estmaton Algorthm Numercal Formulaton The Measurement Jacoban Ill-condtoned Problem Overall Descrpton of the Performed Studes Results of Dvergence Characterstcs Study Analyss of the Converged cond (G) Gan Matrx Stablty Index Sngular Value Decomposton of Gan Matrx Testng Results of n Concluson Chapter 3 Convergence Property of the State Estmaton Consderng Two Types of Load Increment and PMU Measurements Introducton State Estmaton Consderng Load Increment for All the Load Buses... 4 v

9 3.2.1 Addng Conventonal Voltage Magntude Measurements n Measurement Vector Addng PMU Voltage Phasor Measurements Smulaton Results Consderng Sngle Load Increment Effect of Addng PMU Measurements on State Estmaton Accuracy Concluson Chapter 4 Convergence Property of State estmaton wth Load Increment on a Specfc Lne Introducton Formulaton of Topology Error n WLS State Estmaton Smulaton results Test on a Smple 3-bus System Test on the IEEE 118-bus system Effect of PMU measurements on WLS State Estmaton convergence Future work Concluson Chapter 5 Incorporaton of PMU Measurements n WLS State Estmaton Introducton Equalty-constraned State Estmaton wth PMU Measurements Approach Hachtel's Matrx State Estmaton Wth PMU Measurements Approach Smulaton result of the IEEE 14-bus system Chapter 6 Optmal Placement of Phasor Measurement Unts for Improvng Power System State Estmaton Accuracy Introducton State estmaton wth PMU measurements Method I: Mxng PMU Measurements wth Conventonal Measurements n the Estmator Method II: Incorporatng PMU measurements through a post-processng step Heurstc PMU placement algorthm Smulaton results Concluson v

10 Chapter 7 Concluson Appendx A Appendx B References VITA v

11 Lst of Tables 2.1 Four sets of values of measurement standard devaton Standard devaton settng of the measurement errors Smulaton results of addng voltage magntudes Comparson of the number of dvergence cases for runnng 1-tme state estmaton wth random topology errors Experment results of state estmaton accuracy mprovement Standard devaton settng of measurement errors Impacts of PMU measurements on loadablty Standard devaton settng of the measurement errors Smulaton results of the condton number of gan matrx G Smulaton results for.1% of the standard devaton of PMU measurements Smulaton results for.1% of the standard devaton of PMU measurements The correspondng data table of the above fgure Tme comparson of two methods runnng on the 4 case studes v

12 Lst of Fgures 1.1 State estmaton block dagram Functonal blocks of a generc PMU The dvergence rate versus the load ncrement The zoom-n curve of case The converged cond (G) for 2% load ncrement The converged cond (G) for 3% load ncrement The converged cond (G) for 4% load ncrement The converged cond (G) for 5% load ncrement The converged cond (G) for 6% load ncrement The converged cond (G) wth load ncrement Comparson of the converged cond (G) Comparson of teraton number The mean of the converged cond (G) Iteraton number for measurement The mean of the converged cond (G) Iteraton number for measurement The mean of the converged cond (G) Iteraton number for measurement The mean of the converged cond (G) Iteraton number for measurement Mnmum sngular value versus load ncrement wth a random topology error Overall change of the dvergence rate Zoom-n varaton of the dvergence rate Comparson of 3 case studes Change of dvergence rate for runnng 1-tme state estmaton wth dfferent number of PMUs The network wth topology error n branch Voltage magntudes of bus 98 and 1 wthout topology error v

13 3.7 Voltage angles of bus 98 and 1 wthout topology error Voltage magntudes of bus 98 and 1 wth topology error Voltage angles of bus 98 and 1 wth topology error Voltage magntudes of bus 98 and 1 wth topology error Topology error: a) shows the true stuaton; b) shows the stuaton modeled n state estmaton A smple 3-bus system Voltage magntudes of 3-bus system Voltage angles of 3-bus system Voltage magntude of bus The topology error n branch 6-61 of the IEEE 118-bus system Voltage magntudes of bus 6 and Voltage angles of bus 6 and Voltage magntude of bus 98 and Voltage angles of bus 98 and Comparson of voltage magntude of bus Comparson of voltage magntude of bus Two-port -crcut model for a network IEEE 14-bus system wth PMU measurements IEEE 14-bus system wth ntal measurements Smulaton results of 4 case studes x

14 Chapter 1 Introducton 1.1 Background Power system state estmaton s an essental tool used by system operators for real tme analyss of the power system. It s able to estmate the optmal voltage magntudes and angles at the system bus-bars based on the redundant raw measurements avalable. The dea of state estmaton appled nto power system was frst ntroduced by MIT professor Fred Schweppe n 197s, and now has been wdely appled n the energy control centers (ECCs) of electrc utltes and ndependent system operators (ISOs) [1-3]. It has consttuted the backbone of the Energy Management System (EMS), whch plays an mportant role n montorng and controllng power systems for relable operatons. The state estmaton block dagram s shown n Fg Montorng and control of power system s conducted by the supervsory control and data acquston (SCADA) system, whch collected the measurement data n real tme from the remote termnal unts (RTUs) nstalled at the substatons across the power system network. The term SCADA conssts of two parts. Supervsory control ndcates that the operators n ECC have ablty to control the RTUs. Data acquston ndcates that the data gathered by RTUs are sent to the operators for montorng purpose [4-7]. Typcal RTU measurements nclude bus voltage magntudes, lne current magntudes, power njectons and flows (both real and reactve). In addton to these measurements, RTUs also record the on/off status of swtchng devces, such as crcut breakers and transformer taps. Ths set of the measurement and status nformaton s telemetered to the energy control center through a 1

15 perodc scan of all RTUs. A typcal scan cycle s usually 2 or 4 seconds. The tradtonal types of SCADA telemetry ncludes telephone wre and mcrowave rado. A more recent development of communcaton technologes has taken advantage of fber optc cable, satelltes, spread spectrum rado, and nternet/ntranet systems, whch have mproved the communcaton relablty and speed, although the cost s stll hgher than the conventonal medums [8]. SCADA RTU RTU RTU SCADA front end Measurement Data Status of Swtchng Devces State Varables Bad Data Processng State Estmator Network Topology Network Topology Processor Fg. 1.1 State estmaton block dagram By processng the RTU status nformaton of swtchng devces, the network topology processor n EMS determnes the topology of the network, whch characterzes the connectvty between buses (nodes), the shunt elements at each bus, and whch generators and loads are connected to these buses by usng one-lne dagram. The status nformaton of swtchng devces comng to topology processor s referred to as the bus sectonbreaker-swtch data. It provdes the on/off nformaton at each substaton and how they 2

16 are connected. Dfferent bus sectons connected wth closed breakers and swtches can be recognzed as an electrcal bus. The topology processor converts the bus sectonbreaker-swtch data nto so-called bus-branch n one-lne dagram, whch s an approprate approach for modelng transmsson lne and transformer connectons at each substaton, rather than the precse bus-secton connectons at each substaton [9]. The network topology must reflect the actual network condton n order for the state estmator to determne the optmal operatng state of the current system. Unfortunately, to obtan an accurate network topology s not always avalable. Many current topology processors are not capable to acqurng the status change of crcut breakers automatcally due to the destructon of communcaton medums. Besdes, equpment status of remote substatons s usually managed manually through telephone call to report to the ECC. Hence, t s common to have topology errors occurred n the network models. 1.2 Phasor Measurement Unt The conventonal SCADA measurements do not nclude phase angle measurements of voltage and current phasors. Wth the nventon of the phasor measurement unt (PMU), the phase angle was frst drectly measured. A PMU s a dgtal devce that can provde synchronzed voltage and current phasor measurements. The phasor s a vector representaton of the magntude and phase angle of an AC waveform. Phase angles n dfferent stes can be determned when the measurements are synchronzed to a common tme source. The global postonng satellte (GPS) s capable to provde the common tmng sgnal of the order of 1 mcrosecond, whch can obtan hghly accurate PMU voltage and current phasors [1]. 3

17 Fg. 1.2 provdes the functon blocks of a generc PMU [11]. The analog nputs nclude voltages and currents obtaned from the secondary wndngs of the voltage and current transformers. The ant-alasng flter s used to attenuate the frequences that are hgher than the Nyqust frequency. The phase-locked oscllator converts GPS 1 purse per second nto a sequence of hgh speed tmng purses that wll be used n waveform samplng. The A/D converter can convert the analog voltage and current sgnals to dgtal sgnals, whch are mported nto the phasor mcroprocessor to execute the Dscrete Fourer Transform GPS Recever Analog Inputs Phase-locked Oscllator Modem Ant-alasng Flters A/D Converter Phasor Mcroprocessor Fg. 1.2 Functonal blocks of a generc PMU (DFT) phasor calculatons. The computed strng of phasors s assembled n a phasor data concentrator (PDC) and ths phasor stream s then transmtted to the modems. The IEEE standard for synchrophasor formulates real tme phasor data transmsson. In the recent years, PMUs are gradually appled n the montorng and control of power systems. The current and potental benefts are dscussed n [12-15]. The wdespread applcatons of PMUs also brng about the benefcal mpacts to the state estmaton, 4

18 whch ncludes the mprovement of network observablty and state estmaton accuracy, etc [16-21]. 1.3 Lessons of Northeast Blackout n 23 The Northeast blackout of 23 was one of the most severe power outages occurred n North Amercan hstory. It took place n eght Northeastern states of the Unted States and one Canadan provnce on August 14, 23. More than 5 mllon people lost power for up to two days. Ths severe event resulted n at least 11 death and cost the economc loss of about $6 bllon. After that, a team consstng of the natonal experts from the U.S. and Canada was bult mmedately to nvestgate the reasons of the blackout, and ther fnal report of the U.S.-Canada Power System Outage Task Force was released n Aprl, 24 [22]. Ths Task Force report helped people open the Pandora s box of electrc utlty problems. The prmary cause of the blackout was that overgrown trees came nto contact wth a straned hgh voltage transmsson lne owned by FrstEnergy Corp n the state of Oho. Cascaded outage propagated through the system and caused the wdespread blackout. To make the stuaton even worse, the montorng computer runnng the state estmaton program n Oho was not workng due to the software gltch. The operators became 'blnd' to the crss and unable to take any effectve actons at early stage to prevent the wdespread blackout. The blackout gves people a proof that how fragle the nterconnected power system really s. Each day roughly 5, Amercans encounter at least two hours wthout electrcty n ther daly lfe, and these outages cost the economy $15 bllon a year. 5

19 Although no one has nvestgated that how many of these blackouts are due to the falure of computer montorng functon, many power grd control centers acknowledge that computer gltches occur regularly -weekly or monthly [23]. State estmaton s a crtcal tool for montorng and control of power system. It s tuned to be effectve under normal load condtons and may fal under condtons of hgh transmsson lne loads. The weghted least square (WLS) state estmaton method s most commonly used n the control centers. The nherent flaws, as dscussed n next chapter, cause the soluton of state estmaton to be naccurate and unavalable. Researchers have made contnuous efforts to solve ths challenge for decades, and fortunately a lot of revolutonary approaches have been proposed and appled to mprove the relablty and robustness of the state estmator. 1.4 Contrbuton of Ths Dssertaton As dscussed n the prevous secton, the WLS state estmator dd not work properly n the blackout. When the load level becomes severe, the state estmator may not converge to a soluton. Besdes, a topology error n state estmaton model could worsen the convergence of the state estmator, whch s the motvaton for ths dssertaton. The man contrbutons of ths dssertaton are lsted as follows: The smulaton of the blackout s created to nvestgate the mpacts of the topology errors and the load levels on the commonly used WLS state estmator by usng the IEEE 118-bus system. The nfluence of the topology errors on the system s also studed from the pont vew of the condton number of the gan 6

20 matrx. Besdes, the mnmum sngular value of gan matrx G s proposed to measure the dstance between the operatng pont and state estmaton dvergence. The convergence property of WLS state estmaton under two types of load ncrement s studed, one s load ncrement of all load buses, and the other one s a sngle load ncrement. Smulaton shows that addng PMU measurements can fnally solve state estmaton dvergence problem. In addton, the effect of topology error on state estmaton when there s a sngle load ncrement s also studed. The voltage magntude of generator bus wll ncrease f there s a topology error n the state estmaton. It s also found that addng PMU measurements n state estmaton can reduce the error of voltage magntude and angle estmaton. The mpact of topology error on a lne wth ncreasng power flow on the WLS state estmator s nvestgated. It s found that the voltage magntude of the load bus wll decrease at frst and then ncrease untl the state estmator dverges. For other buses ncludng the generator buses, the voltage magntudes wll always ncrease. Besdes, the smulaton shows that PMU measurements could make the WLS state estmaton more robust when the topology error occurs. Novel approaches of ncorporatng PMU measurements nto the state estmaton are proposed. These approaches can reduce the condton number of the coeffcent matrx n state estmaton, and thus are able to mprove the robustness of the state estmaton. A heurstc PMU placement approach s proposed to mprove state estmaton accuracy. The obtaned PMU placement table and fgure could help plannng 7

21 engneers determne the optmal placement of PMUs when they have only a lmted number of PMUs to place n the system. In addton, two methods for calculatng state estmaton results are utlzed n the PMU placement approach. It s observed that the method for addng PMU measurements through a postprocessng step can sgnfcantly mprove the computaton effcency of the proposed approach. 1.5 Dssertaton Outlne The dssertaton s organzed as follows. Chapter 2 wll start wth the revew of the WLS state estmaton method, and then explore the mpact of the load level and topology errors on the convergence property of the WLS state estmaton from the pont vew of condton number of the gan matrx. Besdes, t also proposes a method of the mnmum sngular value of gan matrx to measure the dstance between the current operatng pont and state estmaton dvergence. Chapter 3 further studes the convergence property of the WLS state estmaton through consderng load ncrement of all load buses and sngle load ncrement. It also llustrates the effect of addng PMU measurements on state estmaton accuracy. Chapter 4 frstly presents the formulaton of the topology error n state estmaton, and descrbes the smulaton results usng a smple 3-bus system and the IEEE 118-bus system. It also studes the effect of PMU measurements to convergence of WLS state estmaton. Chapter 5 presents two approaches of addng PMU measurements nto state estmaton, ncludng the equalty-constraned approach and the Hachtel's matrx approach. The comparson of these two methods and conventonal WLS state estmaton method usng the IEEE 14-bus system s also llustrated. Chapter 6 proposes a heurstc optmal PMU placement approach to mprove state estmaton accuracy. The 8

22 IEEE 14-bus system s used to test the proposed approach. Chapter 7 provdes the concluson for the whole dssertaton. 9

23 1 Chapter 2 Impacts of Load Levels and Topology Error on WLS State Estmaton Convergence Ths chapter s organzed as follows. Secton 2.1 gves a revew of the weghted least square (WLS) state estmaton algorthm and the ll-condtoned problem. Secton 2.2 presents the overall descrpton of the performed studes. Secton 2.3 descrbes the results of the dvergence characterstc study. Secton 2.4 provdes the analyss of the converged (G) cond. Secton 2.5 proposes the mnmum sngular value of the gan matrx as the gan matrx stablty ndex, followed by the concluson [63]. 2.1 WLS State Estmaton Algorthm In the past decades, varous methods have been proposed to solve the power system state estmaton problem [24-26]. The WLS state estmaton algorthm s the most commonly used n the electrc utlty ndustry. We wll revew ths algorthm consstng of numercal formulaton, the measurement Jocoban and ll-condton problem n ths secton Numercal Formulaton For a gven set of measurements, the measurement equaton s gven as follows [27]: e x h e e e x x x h x x x h x x x h z z z z m n m n n m ) ( ),, ( ),, ( ),, ( (2.1) where,

24 z ] T [ z1, z2,, z m s the measurement vector (m x 1); T h x) [ h ( x), h ( x),, h m ( x)] s a vector of nonlnear functons that relate the states to ( 1 2 the measurements; x ] T [ x1, x2,, x n s the state vector (n x 1) to be estmated; e ] T [ e1, e2,, e m s the measurement error vector (m x 1). It s necessary that m n and the Jacoban matrx of has a rank of n. The optmal state estmate vector x can be determned by mnmzng the sum of weghted squares of resduals Mn T J( x) [ z h( x)] R 1 [ z h( x)] (2.2) 2 where, R s a dagonal matrx wth the measurement varance measurement ndex., wth beng the J (x) s a non-lnear functon, and thus the frst dervatve s set equal to zero to fnd a mnmum. J ( x) g( x) H x T ( x) R 1 [ z h( x)] (2.3) where H (x) s the measurement Jacoban matrx wth dmenson m by n h( x) H( x) (2.4) x The nonlnear functon h (x) s lnearzed h( x x) h( x) H( x) x (2.5) 11

25 The teratve approach s used to solve equaton (2.3) as follows: ( H T R 1 H) x H T R 1 [ z h( x)] (2.6) x k 1 x k x (2.7) where, The symmetrc matrx T 1 G H R H s called gan or nformaton matrx. k s the teraton ndex. Equaton (2.6) s the so-called normal equaton of the WLS state estmaton algorthm. A flat start for the state varables s usually utlzed, where all bus voltages are assumed to be 1. per unt and n phase wth each others. The teraton wll be termnated when the measurement msmatch reaches a prescrbed threshold, e.g. 1e-4, or the maxmum number of teratons s reached The Measurement Jacoban For a system contanng N buses, bus 1 s usually consdered as the reference bus, thus the phase angle of bus 1 s set equal to degree. The state vector x has ( 2N 1) elements, ncludng N bus voltage magntudes and ( N 1) phase angles, expressed as follows: T x 2, 3, N, V1, V2,, N V (2.8) The measurement vector usually ncludes voltage magntudes, real and reactve power njectons and flows, the structure of the measurement Jacoban H wll be as follows [28]: 12

26 H P nj Qnj P Q V mag V P nj V Q nj V P V Q V (2.9) where, V mag s the voltage magntude. P nj and P and Q nj are the real and reactve power njectons, respectvely. Q are the real and reactve power flow from bus to j, respectvely. The expressons for each partton are gven below: Elements correspondng to voltage magntude measurements V : V V V for all and j, 1, V V j (2.1) Elements correspondng to real power njecton measurements P nj : P nj P nj j N j1 VV VV j j G sn B G sn B cos cos V B 2 (2.11) (2.12) P nj V N j1 V j G cos B sn V G (2.13) P nj V j V G cos B sn (2.14) 13

27 Elements correspondng to reactve power njecton measurements Q nj: Q nj Q nj j N j1 VV VV j j G cos B G cos B sn sn V G 2 (2.15) (2.16) Q nj V N j1 V j G sn B cos V B (2.17) Q V nj j V G sn B cos (2.18) Elements correspondng to real power flow measurements P : P VV j g sn b cos (2.19) P j VV j g sn b cos (2.2) P V V j g cos b sn 2gV (2.21) P V j V g cos b sn (2.22) Elements correspondng to reactve power flow measurements Q : Q Q j VV VV j j g g cos b cos b sn sn (2.23) (2.24) Q V V j g sn b cos 2bV (2.25) 14

28 Q V j V g sn b cos (2.26) where, V and are the voltage magntude and phase angle at bus, respectvely; ; j G jb s the th row and jth column of the complex bus admttance matrx. g jb s the admttance of the seres branch connectng buses and j. Bs 2 s the lne-chargng susceptance Ill-condtoned Problem A whole complete set of state estmaton process typcally ncludes the followng functons [28]: Topology processng: Obtans the one lne dagram of network topology for state estmaton based on the nformaton of crcut breaker/swtch statuses. Observablty analyss: Tests whether or not the avalable set of measurements s suffcent to obtan the soluton of state estmaton. Identfes the observable slands and adds pseudo-measurements to make the whole network observable [29-36]. State estmaton: Solves a set of nonlnear equatons to obtan the system states based on the network model and avalable measurements. Also provdes the estmates of the lne flows, loads and generator outputs. 15

29 Bad data processng: Detects f there exsts bad data n the measurements. Identfes and removes the bad data so that the state estmaton soluton wll not be based [37-43]. There are many hot research topcs for each functon of state estmaton, and t wll be hard to cover over 4 years of actve researches n the theory and practce of power system state estmaton. Hence, we have chosen to revew the numercal problem that s drectly related to ths dssertaton. As mentoned n secton 2.1.1, the normal equaton of Equaton (2.6) s the common approach to the soluton of the WLS state estmaton. To deal wth the nverse of the gan matrx G H R T 1 H, the Choleskey decomposton s appled to factor the matrx G n the normal equaton, and then followed by forward/backward substtutons to obtan the soluton. However, the dffculty n mplementng normal equaton approach s that the gan matrx may be ll-condtoned, whch causes the state estmaton to fal to converge to a soluton. The condton number s used to represent the degree of system ll-condtonng. The more sngular a matrx s, the more ll-condtoned ts assocated system wll be. For the WLS state estmaton, the man reasons of ll-condtoned gan matrx are descrbed as follows [28]: Very accurate measurements (.e. vrtual measurements) A large number of njecton measurements Long and short lnes connected to the same bus Vrtual measurements are zero njectons at the swtchng buses, and they represents perfect measurements wth very large weghts n the WLS state estmaton, whch renders the gan matrx G ll-condtoned [44-45]. Orthogonal factorzaton, also known as QR 16

30 factorzaton, s proposed to prevent computaton of the gan matrx. Ths method s based on column-wse Householder transformaton and Gvens rotatons. It turns a zero element of a sparse gan matrx to a nonzero elements n the process of factorzaton called a fll-n. The tme-consumng process n dealng wth extensve fll-ns prevents the method from beng wdely appled [46-47]. The alternatve method, called approach of Peters and Wlknson, performs LU composton on matrx H. Although t s computatonally more expensve than the normal equaton approach, ths method s a tradeoff between speed and stablty. The mprovement of numercal condtonng compared wth the normal equaton approach s shown n [48]. 2.2 Overall Descrpton of the Performed Studes The convergence property of the WLS state estmator s a crtcal ssue for real tme montorng and control of power grds. In addton to the three reasons mentoned n the last secton that cause ll-condtoned gan matrx, the topology error can also cause the WLS state estmator to dverge wthout reachng a soluton. The Northeast blackout of 23 s a well-known example. The fact that the WLS dd not converge due to the exstence of a topology error was an ndrect factor leadng to the blackout. Besdes, t s known that the load levels became severe before the blackout. In ths secton, we wll study the mpact of topology errors on the convergence characterstcs of the WLS state estmator durng the blackout when the loads gradually ncrease. Topology errors can be broadly classfed n two categores: branch status errors and substaton confguraton errors. Branch status errors nclude branch excluson error and 17

31 branch ncluson error. Ths smulaton utlzes the branch excluson error, whch takes place when a lne, actually n servce, s excluded from the formulaton of the state estmator [49]. The IEEE 118-bus system s used as the test case [5]. Varous standard devaton settngs for measurements are utlzed, whch play a sgnfcant role n dctatng the dvergence characterstcs of the state estmaton. The detaled study procedures are descrbed below. Frst, the measurements for state estmaton are generated. The Matpower tool s used to generate the power flow results of the IEEE 118-bus system by usng the Newton Raphson load flow method [51]. Then, the power flow results are contamnated wth normal dstrbuton noses to form the measurements for state estmaton. The measurements comprse all the bus voltage magntudes and angles, and real and reactve power njectons. To obtan the measurements correspondng to prescrbed load levels, real and reactve power of all the loads and real power of generators n the 118-bus system are ncreased n proporton, e.g. 1%, and the power flow program s run to check f t can converge or not. The experments show that the maxmum load ncrement under whch the power flow program stll converges s 218%. Thus, the measurements below 218% load ncrement wll be consdered for applcaton n state estmaton. Second, branch excluson errors are appled on state estmaton. The 118-bus system has 186 branches. The Matlab functon randnt s used to generate a vector of 1 random ntegers rangng from 1 to 186. The vector s used to represent the branch error ndex, and each value n the vector represents the branch ndex that wll have branch error n state estmaton. 18

32 Each tme when the state estmator s run, a value from the branch error ndex vector s selected. For example, a number 1 s selected from the branch error ndex. Then, ths 1 th branch wll be removed from the system and wll not be consdered n constructng the bus admttance matrx. Meanwhle, the real and reactve power flow measurements of the 1 th branch are also excluded from the measurement vector. The state estmaton program s run to check convergence. In ths study, the maxmum teraton number s set to be 5; f the state estmator does not converge wthn 5 teratons, t s consdered to be dverged. In total, state estmaton s run 1 tmes for each load ncrement to check how many topology errors wll cause state estmaton to dverge under a specfc load ncrement, from whch the dvergence rate s calculated. 2.3 Results of Dvergence Characterstcs Study It has been found that the standard devaton of measurements can sgnfcantly affect the dvergence rate of state estmaton. Thus, four dfferent sets of standard devaton settngs are chosen to fnd out the dvergence rates. The notatons of standard devatons of measurements are defned as follows. V m : Standard devaton of voltage magntude measurements. V a : Standard devaton of voltage angle measurements. p : Standard devaton of real power njecton measurements. Q : Standard devaton of reactve power njecton measurements. The standard devaton settngs are shown n Table 2.1. The dvergence rates of the four cases are plotted n Fg. 2.1 and 2.2. Fg. 2.1 depcts the overall change of the dvergence 19

33 Dvergence Rate rate versus load ncrement. Fg. 2.2 llustrates the zoom-n varaton of the dvergence rate versus load ncrement between 6% and 7% for case 2. Table 2.1 Four sets of values of measurement standard devaton Standard devaton settngs Settng 1 1e 3, 1e 3, 1e 2 and 1e 2 Vm Va P Q Settng 2 1e 2, 1e 2, 1e 3 and 1e 3 Vm Va P Q Settng 3 1e 2, 1e 3, 1e 3 and 1e 2 Vm Va P Q Settng 4 1e 3, 1e 2, 1e 2 and 1e 3 Vm Va P Q 1.8 Case 1 Case 2 Case 3 Case Load Increment (%) Fg. 2.1 The dvergence rate versus the load ncrement 2

34 Dvergence Rate Load Increment (%) Fg. 2.2 The zoom-n curve of case 2 As seen n Fg. 2.1 and 2.2, the standard devaton values of the measurements can sgnfcantly affect the dvergence rate wth elevated load levels. It s observed that when both real and reactve power njecton measurements have relatvely small standard devatons, such as 1e-3 n case 2, the dvergence rate wth the exstence of topology errors wll reach 1., or 1%, when the load ncrement equals to 7%. A load ncrement of 7% means that the load level of the system ncreases by 7% compared to the base load level. For the other 3 cases, the dvergence rates are relatvely small. For example, case 1 has the mnmum value of dvergence rate, whch s.3 at 218% load ncrement, whle case 3 has less than.2 of dvergence rate at 218% load ncrement. In addton, smulaton results show that the state estmator wthout topology error n case 2 wll dverge when the load ncreases to 63%. For other 3 cases, the state estmators wll converge at any load ncrement f no topology error occurs. It s observed that when the 21

35 dvergence rate wth topology error s low, e.g. less than.2, the state estmator wll converge when no topology errors are present. If the dvergence rate reaches a hgh value, lke.7, the state estmator wll dverge even when there are no topology errors. 2.4 Analyss of the Converged cond (G) The condton number s used to measure the degree to whch a matrx s ll-condtoned, and s defned as [52] 1 cond ( A) A A (2.27) where, A s a gven matrx. represents a gven matrx norm. A large condton number ndcates an ll-condtoned matrx. Ths secton studes the nfluence of the topology errors on the condton number of the gan matrx G, denoted as cond (G). In subsecton A, a set of 1 cases wth random topology errors s used to study the varaton of the cond (G) versus load ncrement under measurement standard devaton settng 2, and certan phenomena are obtaned. In subsecton B, a set of 1 cases wth random topology errors are utlzed to further nvestgate the varaton of the cond (G). A. Test results of 1 cases wth random topology errors under measurement standard devaton settng 2 The loc of the cond (G) for the 1 cases, each of whch has a random topology error, versus load ncrement for standard devaton settng 2 are depcted n Fg. 2.3 to 2.7. When the state estmaton converges, the convergence teraton count s a known lmted number, and the cond (G) wll converge to a certan value for a specfc load ncrement. 22

36 Whle for a dverged case, the condton number mght oscllate between two values, or dverge, as shown n Fg. 2.7, smlar to what s reported n [53-54]. In ths work, the maxmum teraton number s set to be 5; f the state estmator does not converge after 5 teratons, t s consdered to be dverged. As seen from Fg. 2.3 to 2.7, as the load ncreases, the teraton numbers of certan cases sgnfcantly ncrease. For the load ncrement of 2%, 3% and 4%, all the cases converge. For the load ncrement of 5%, one case dverges, as shown n Fg For the load ncrement of 6%, one case oscllates, where the condton number oscllates around the value of.5*1e8, as shown n Fg. 2.7, and the same case that dverges under load ncrement of 5% stll dverges wth load ncrement of 6%. Fg. 2.8 plots the changes of the converged cond (G) of these two cases for the ranges of load ncrement under whch the two cases converge. It s observed that the converged cond (G) wll ncrease when the load level ncreases for the oscllated case and dverged case. 23

37 Condton number Condton number 5 x Iteraton Fg. 2.3 The converged cond (G) for 2% load ncrement 6 x Iteraton Fg. 2.4 The converged cond (G) for 3% load ncrement 24

38 Condton number Condton number 9 x Iteraton Fg. 2.5 The converged cond (G) for 4% load ncrement 3 x Iteraton Fg. 2.6 The converged cond (G) for 5% load ncrement 25

39 Condton number of converged G Condton number 2.5 x 18 2 Dverged case Oscllated case Iteraton Fg. 2.7 The converged cond (G) for 6% load ncrement 4.8 x The oscllated case The dverged case Load Increment (%) Fg. 2.8 The converged cond (G) wth load ncrement 26

40 The results for the other 8 cases wth random topology errors are llustrated as follows. For 5 of the cases, the converged cond (G) wll ncrease when the load level ncreases f the case converges. For the other 3 cases wth topology errors, the converged cond(g ) decreases wth load ncrement. The mean of the converged cond (G) for the 1 cases, each of whch has a topology error, at each load ncrement s calculated to measure the mpact of topology errors on the converged cond (G) wth load ncrement. If a case does not converge, ths case wll be excluded from the calculaton of the mean of the converged cond (G). As a comparson, the values of the converged cond (G) for cases wthout topology errors wth load ncrement are also calculated. The values of the converged cond (G) are plotted n Fg. 2.9, and the change of the curves can be explaned as follows. As mentoned earler, the converged cond (G) wll ncrease for some cases, and decrease for other cases. From load ncrement of 2% to 5%, the total ncrease of the converged cond (G) for the cases whose condton number ncreases exceed the total decrease of the converged cond (G) for the cases whose condton number decreases. As a result, the mean of the cond(g) ncreases for load ncrement of 2% to 5%. When the load ncrement goes beyond 5%, the decrease of the converged cond (G) surpasses the ncreases, and thus the mean value of the condton number decreases versus the load ncrement. Fg. 2.1 depcts the teraton number requred to reach convergence for cases wth and wthout topology errors for measurement devaton settng 2. It s shown that the cases wth topology errors requre more teratons than the cases wthout topology errors. 27

41 Convergence teraton number Converged cond(g) 3.8 x Wth topology error No topology error Load ncrement (%) Fg. 2.9 Comparson of the converged cond (G) Wth topology error No topology error Load ncrement (%) Fg. 2.1 Comparson of teraton number B. Test results of 1 cases wth random topology errors under dfferent devaton settngs In ths secton, under each set of the measurement standard devatons as shown n Table 2.1, the changes of the converged cond (G) under 1 random topology errors are 28

42 studed. The dvergence rates of the 4 case studes are plotted n Fg. 2.1 and 2.2. It s known that as load level keeps ncreasng, the dvergence rate of case 2 reaches 1., whle the other 3 cases are under.2. In the followng studes, smlar to the prevous 1 random topology error tests, the mean of the converged cond (G) under 1 random topology errors for a specfed load ncrement s calculated. For those dverged state estmaton cases, the condton numbers are not converged and unpredctable, thus are not consdered n the mean calculaton of the converged cond (G). (1) Results for measurement devaton settng 1 Fg depcts the converged cond (G) for cases wth measurement devaton settng 1. The mean of the converged cond (G) for the cases wth topology errors has a smlar trend as that for the cases wthout topology errors. They both ncrease at frst to the maxmum condton number, and then decrease wth load ncrement. Through the analyss of the curves of converged cond (G) of 1 random topology errors, 9.2% of the curves follow ths trend of the curve wthout topology error. Though 9.8% of curves of 1 random topology errors do not follow ths trend, they do not affect the trend of the curve wth topology error. Moreover, t s observed that t takes more teratons for the state estmaton to converge for the cases wth topology errors than the cases wthout topology errors, as shown n Fg

43 Iteraton Number Condton Number Wth topology error No topology error Load Increment (%) Fg The mean of the converged cond (G) 5 Wth topology error No topology error Load Increment (%) Fg Iteraton number for measurement (2) Results for measurement devaton settng 2 Fg shows the converged cond (G) for cases wth measurement devaton settng 2. As shown n Fg. 2.13, the mean of converged cond (G) for cases wth topology errors s larger than that wthout topology errors, and t oscllates durng 6% - 7% load 3

44 Condton Number ncrement. For 92.7% of the 1 cases wth topology errors, the converged cond (G) ncreases wth load ncrement; for the other cases wth topology errors, the converged cond (G) decreases wth load ncrement. Because the amount of ncrease and decrease of the condton number for each case s dfferent, the mean value of the condton number may ncrease or decrease wth dfferent load levels and thus oscllate when load level vares. It s noted that the dverged cases are excluded from the calculaton of the mean value of the converged cond (G). It s also observed that the case wthout topology error dverges at about 62% load ncrement. In addton, the topology error wll lead to a larger teraton number than that wthout errors, as shown n Fg x 17 4 Wth topology error No topology error Load Increment (%) Fg The mean of the converged cond (G) 31

45 Iteraton Number 2 15 Wth topology error No topology error Load Increment (%) Fg Iteraton number for measurement (3) Results for measurement devaton settng 3 Fg depcts the converged cond (G) for cases wth measurement devaton settng 3. Fg shows the comparson of teraton number requred to converge for the cases wth and wthout topology errors. As shown n Fg. 2.15, as the load level ncreases, the condton number frst reaches the maxmum value, and then decreases wth load ncrement. 97.4% of 1 cases wth topology errors follow ths trend. 32

46 Iteraton Number Condton Number 3.43 x Wth topology error No topology error Load Increment (%) Fg The mean of the converged cond (G) 12 1 Wth topology error No topology error Load Increment (%) Fg Iteraton number for measurement (4) Results for measurement devaton settng 4 Fg plots the converged cond (G) for cases wth measurement devaton settng 4. Fg shows the comparson of teraton number requred to converge for the cases wth and wthout topology errors. 33

47 Iteraton Number Condton Number The two curves n Fg monotoncally decrease wth load ncrement. 94.9% of the 1 cases wth topology errors follow ths trend. Fg gves the comparson of the teraton number requred to converge between the cases wth and wthout topology errors x 17 Wth topology error No topology error Load Increment (%) Fg The mean of the converged cond (G) Wth topology error No topology error Load Increment (%) Fg Iteraton number for measurement 34

48 Based on the varaton of the converged cond (G) versus load ncrement as shown n Fg. 2.11, 2.13, 2.15 and 2.17, t can been seen that the mean of the converged cond (G) for the cases wth topology errors s not necessarly larger than that for the cases wthout topology errors. The mean of the converged cond (G) also depends on the standard devaton values of the measurements. Moreover, the teraton number to reach convergence for cases wth topology error s larger than cases wth no topology error. 2.5 Gan Matrx Stablty Index The mnmum sngular value of the gan matrx s used as the gan matrx stablty ndex, ndcatng the dstance between the studed operatng pont and state estmaton dvergence. The sngular value decomposton s frst ntroduced, followed by the smulaton on the IEEE 118-bus system Sngular Value Decomposton of Gan Matrx The sngular value decomposton s an mportant orthogonal decomposton method n matrx computaton [55]. Consder the gan matrx G wth dmenson n by n, where n s the number of the state varables. Matrx G can be decomposed by usng sngular value decomposton method as follows [56]: G UV T n 1 u v (2.28) T where U and V are n by n orthonormal matrces. The sngular vectors u and v are the columns of the matrces U and V respectvely. Matrx s a dagonal matrx wth ( G) dag{ ( G)} 1,2,, n (2.29) 35

49 where for all. The dagonal elements of matrx are usually ordered so that n The smallest sngular value of the matrx G s a measure of the dstance, n the l 2 -norm, between G and the sngular matrx wth no full rank [57]. Moreover, the sngular value decomposton s well-condtoned snce the sngular values are farly nsenstve to the permutatons n the matrx elements. If the smallest sngular value n s close or equal to zero, the correspondng matrx G could be sngular. Ths property can be used n the WLS state estmaton to measure the dstance between the operatng pont to the state estmaton dvergence Testng Results of n The IEEE 118-bus system s used as the test case, and a random topology error s appled n the state estmaton. The standard devaton settng of measurement errors utlzes settng 3 of Table 2.1. The smulaton process s smlar to secton 2.3. In order to study the convergence property of state estmaton versus load ncrement, we need to obtan the measurements correspondng to prescrbed load levels. Let real and reactve power of all the loads and real power of generators n the 118-bus system ncrease n proporton, e.g. 1%, then the power flow program s run to check f t can converge or not. The experments show that the maxmum load ncrement under whch the power flow program stll converges s 218%. Thus, the measurements below 218% load ncrement wll be consdered for applcaton n state estmaton. The followng fgure plots mnmum sngular value versus load ncrement wth a random topology error. As can be seen n Fg. 2.19, the mnmum sngular value decreases 36

50 Mnmum sngular value gradually at frst, then t oscllates to a pont, wth 22% load ncrement. At ths pont, the state estmator stll converges. When the mnmum sngular value reduces below 1, the state estmator dverges. Thus, the mnmum sngular value of gan matrx can be used as convergence ndex to judge f the state estmator converge or not. If ths value s close to zero, the state estmator wll dverge X: 22 Y: Load Increment (%) Fg Mnmum sngular value versus load ncrement wth a random topology error 2.6 Concluson Ths chapter frst revews the WLS state estmaton method and reasons that cause llcondtoned gan matrx. It then nvestgates the mpacts of the topology errors and the load levels on the commonly used WLS state estmator. As the load level ncreases, the dvergence rate of the state estmaton may ncrease to 1. f the standard devatons of 37