Capturing Dynamics in the Power Grid: Formulation of Dynamic State Estimation through Data Assimilation

Transcription

1 PNNL-2323 Prepared for the U.S. Department of Energy under Contract DE-AC5-76RL83 Capturng Dynamcs n the Power Grd: Formulaton of Dynamc State Estmaton through Data Assmlaton N Zhou Z Huang D Meng S Elbert S Wang R Dao March 24

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3 PNNL-2323 Capturng Dynamcs n the Power Grd: Formulaton of Dynamc State Estmaton through Data Assmlaton N Zhou** Z Huang* D Meng* S Elbert* S Wang* R Dao* March 24 Prepared for the U.S. Department of Energy under Contract DE-AC5-76RL83 *Pacfc Northwest Natonal Laboratory Rchland, Washngton **Electrcal and Computer Engneerng Department, Bnghamton Unversty, Bnghamton, NY 392

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5 Eecutve Summary Wth the ncreasng complety resultng from uncertantes and stochastc varatons ntroduced by ntermttent renewable energy sources, responsve loads, moble consumpton of plug-n vehcles, and new maret desgns, more and more dynamc behavors are observed n everyday power system operaton. To operate a power system effcently and relably, t s crtcal to adopt a dynamc paradgm so that effectve control actons can be taen n tme. The dynamc paradgm needs to nclude three fundamental components: dynamc state estmaton; loo-ahead dynamc smulaton; and dynamc contngency analyss (Fgure.). These three components answer three basc questons: where the system s; where the system s gong; and how secure the system s aganst accdents. The dynamc state estmaton provdes a sold cornerstone to support the other 2 components and s the focus of ths study. Dynamc states (e.g., rotor angle and generator speed) are the mnmum set of varables that can determne the status of a dynamc system. A dynamc model wth accurate states can fathfully reveal system responses. Therefore, dynamc state estmaton can provde a full dynamc vew of a power grd and generate crtcal nputs for other operatonal tools. To estmate the dynamc states of a power grd n real tme, we developed and evaluated data assmlaton methods to fuse phasor measurement unt (PMU) data wth power system dynamc models. In ths study, we defned a general dynamc state estmaton problem for a power system and performance evaluaton crtera. A problem s formulated for estmatng the dynamc states of synchronous generators. As an ntal effort, the followng four data assmlaton algorthms are developed, mplemented and appled to estmate the dynamc states of a synchronous generator: Ensemble Kalman flter (EnKF) Etended Kalman flter (EKF) Unscented Kalman flter (UKF) Partcle flter (PF) By comparng ther performance under statstcal framewor usng Monte Carlo methods, t was found that The EnKF algorthm outperforms other algorthms when the typcal PMU samplng rate s used for estmaton. Measurement nterpolaton methods can mprove the estmaton accuracy of the EKF, UKF, and PF. The nterpolaton does not show sgnfcant nfluence on the performance of the EnKF. Increasng the number of samples can mprove the estmaton and convergence of the PF. All four algorthms are robust to mssng data. The outlers cause some sgnfcant errors for all algorthms f the outlers are processed as normal data. The EKF, UKF, EnKF are more robust to the outlers than the PF. It taes more tme for a PF to regan accurate state tracng after the outlers dsappear.

6 Based on these prelmnary results, we wll carry out the followng studes to further enhance practcablty by ncorporatng realstc condtons: Improve algorthm robustness aganst modelng and low data qualty to mprove estmaton accuracy. Develop a realstc medum-sze system for applyng and testng the EnKF and other methods. Speed up computaton and reduce computatonal requrement wth acceptable estmaton accuracy. Develop a framewor for buldng a fleble, modular, etensble software sute that can be used n a range of envronments. The ultmate goal of these studes s to push forward for a real-world applcaton. v

7 Acronyms and Abbrevatons EKF EnKF IEEE MSE PF PMU PST TVE UKF WECC etended Kalman flter ensemble Kalman flter Insttute of Electrcal and Electroncs Engneers mean squared error partcle flter phasor measurement unt Power System Toolbo total vector error unscented Kalman flter Western Electrcty Coordnatng Councl v

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9 Contents Eecutve Summary... Acronyms and Abbrevatons... v. Bacground Introducton Based on Lterature Revew Statc and Dynamc State Estmaton Objectve Statement Current practce on dynamc state estmaton Narratve Problem Statement Epected Outputs Avalable Inputs Evaluaton Crtera Problem Formulatons Transent Models for Estmaton PMU Data Problem Formulaton Soluton Methods Usng Data Assmlaton Technques Overvew of Data Assmlaton n Other Domans Overvew of Kalman Flter (UKF, EKF, EnKF, PF) EKF UKF EnKF Basc PF Prelmnary Results A Smple Scenaro A Scenaro wth Measurement Interpolaton A Realstc Scenaro Consderng Model Inadequacy A Realstc Scenaro wth Mssng Data and Outlers Prelmnary Conclusons Scope of Further Study and Epected Outcome Improve Algorthm Robustness aganst Modelng and Low Data Qualty Develop a Realstc Model for Applyng and Testng EnKF usng Feld Measurement Data Speed Up Computaton and Reduce Computatonal Requrement wth Acceptable Estmaton Accuracy References v

10 Fgures. Integrated Dynamc Paradgm for Future Power Grd Operaton Enabled by Dynamc State Estmaton, Loo-Ahead Dynamc Smulaton and Real-Tme Dynamc Contngency Analyss PMUs n the North Amercan Power Grd Problem Statement of Dynamc State Estmaton Tmescale of Power System Dynamcs Illustraton of Smoothng, Flterng and Predcton Methods The Two-Area, Four-Machne System Comparson of MSEs from the EKF, EnKF, PF, and UKF for Sets of Monte Carlo Smulatons of a Smple Scenaro Estmated States from the EnKF wth 2 Samples of sets of Monte Carlo Smulatons for a Smple Scenaro Comparson of MSEs from the EKF, UKF, EnKF, and PF of sets of Monte Carlo Smulatons for the Scenaros wth Measurement Interpolaton and wthout Interpolaton Comparson of MSEs from the EKF, UKF, EnKF, and PF of Sets of Monte Carlo Smulatons for the Realstc Scenaro Comparson of MSEs from the EKF, UKF, EnKF, and PF of Sets of Monte Carlo Smulatons for the Realstc Scenaro wth and wthout Mssng Data and Outlers Estmated States from the EnKF of Sets of Monte Carlo Smulatons for the Realstc Scenaro wth Mssng Data and Outlers v

11 . Bacground The electrc power grd has been evolvng over the last 2 years from a sngle power lne to today s large networs. The evoluton wll contnue at an accelerated rate wth etensve smart grd development worldwde. In the net 5 years, a sgnfcant percentage of electrcty wll come from ntermttent renewable sources, a large number of cars wll be plugged nto power grds, and a vast number of loads wll actvely respond to grd condtons and ncentve sgnals. Ths development s largely drven by envronmental and economc factors, such as reducng carbon emssons and savng electrcty cost for consumers. The result s new stochastc behavors and dynamcs that the grd has never seen nor been desgned for. Operatng such a dynamc grd wth suffcent relablty and effcency s a monumental challenge. Tradtonally, a quas-steady-state assumpton s appled to operaton studes and decson mang to smplfy operaton models and reduce computatonal complety. Today s operaton s prmarly based on a model that largely gnores dynamcs n the power grd. Electromechancal nteracton of generators and dynamc characterstcs of loads and control devces are not ncluded n operatonal models. The quassteady-state assumpton reduces the computaton complety by several orders of magntude and maes operaton studes on seral computers feasble wthn the requred operatonal tme ntervals. Ths assumpton was reasonable and hstorcally has contrbuted sgnfcantly to the development of power grd computatonal methods and tools. However, wth the rapd evoluton of grd requrements and computng technologes, t s mportant to reeamne ths assumpton for mprovng grd operaton. Because of the quas-steady-state assumpton, many dynamc studes cannot be performed n an operatonal envronment. Smart grd development maes the grd much less quas-steady-state and more dynamc, compared to the tradtonal power grd. For future power grds t s essental to establsh a dynamc operaton paradgm relatve to today s steady-state model. The dynamc paradgm needs to nclude three fundamental components: dynamc state estmaton, loo-ahead dynamc smulaton, and dynamc contngency analyss (Fgure.). These three components answer three basc questons: where the system s, where the system s gong, and how secure the system s aganst accdents. The dynamc state estmaton provdes a sold cornerstone to support the other two components and s the focus of ths study. Fgure.. Integrated Dynamc Paradgm for Future Power Grd Operaton Enabled by Dynamc State Estmaton, Loo-Ahead Dynamc Smulaton and Real-Tme Dynamc Contngency Analyss.

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13 2. Introducton Based on Lterature Revew State estmaton a central functon n power grd operatons generates crtcal nputs for other operatonal tools, such as contngency analyss, automatc generaton control, and optmal power flow (Arbur 24, Montcell 2). Any naccuracy or defcency ntroduced n the state estmaton process wll be propagated and possbly eaggerated through these tools and greatly affect system operaton effcency. 2. Statc and Dynamc State Estmaton Tradtonal state estmators receve telemetered data from a supervsory control and data acquston (SCADA) system, whch are sampled n the tme nterval of several seconds. The SCADA data are used wth a steady-state power flow model to estmate a set of statc state varables,.e., bus voltages and phase angles. Because the power flow model gnores the dynamc transtons, only the statc states of the power system are estmated,.e., only a seres of snapshots of the system condtons are generated and the dynamc transton between the snapshots s not consdered. Dynamc states (e.g., rotor angle and generator speed) are the mnmum set of varables that can determne the status of a dynamc system. A dynamc model wth accurate states can fathfully reveal system responses. Therefore, Dynamc state estmaton provdes a full dynamc vew of a power grd, whch further enables real-tme dynamc smulaton and dynamc contngency analyss and wde-area control. Dynamc state estmaton s made possble by the avalablty of hgh qualty phasor measurement unt (PMU) data and ncreasng computatonal capabltes. PMU data has a typcal samplng rate of 3 or 6 samples per second, s well synchronzed wth the Global Postonng System cloc, and can contnuously capture the dynamc response of a power system under normal and abnormal condtons; thus t can enable dynamc state estmaton. In the North Amercan grd, almost 7 PMUs had been deployed by 23 (Slversten 23) (Fgure 2.). The PMU data provdes a sold cornerstone for dynamc state estmaton. Computng hardware and software technologes have been sgnfcantly advanced n the last decade. Computatonal throughput s beng drven by the large-scale use of multcore processors, wth hundreds of thousands of cores n a large-scale hgh performance computer. The challenge s how to assmlate PMU data nto a dynamc model for estmatng dynamc states and how to utlze advanced computng technologes to perform analyss n real tme. In our wor, we use the parallel flter method to address these challenges. 2.

14 Fgure 2.. PMUs n the North Amercan Power Grd (Slversten 23) 2.2 Objectve Statement The objectve of ths study s to develop fundamental technologes of dynamc state estmaton for power grd operatons. Methods to estmate the real-tme dynamc states of the power grd wll be developed, mplemented n a hgh performance computng envronment and evaluated n a practcal power system. The performance of the developed methods wll be evaluated n terms of estmaton accuracy and computng effcency for helpng real-tme operatonal decsons. 2.3 Current Practce on Dynamc State Estmaton Some ntal studes have been carred out to eplore the feasblty of the dynamc state estmaton. Huang et al. (29, 23) and Fan and Wehbe (23) used an etended Kalman flter (EKF) for onlne dynamc state estmaton. Ghahreman and Kamwa (2a) used an EKF to smultaneously estmate the generator states and unnown nputs. Ghahreman and Kamwa (2b) used an unscented Kalman flter (UKF) to estmate the dynamc states of a sngle-machne nfnte bus system. Zhou et al. (22) proposed an ensemble Kalman flter (EnKF) method to smultaneously estate the states and parameters. Zhou et al. (23) proposed an etended partcle flter (PF) to estmate the dynamc states. These studes demonstrated the value and feasblty of estmatng dynamc states usng PMU data wth Bayesan-based flterng approaches. However, how to address model sze, model errors, lmted data avalablty, and statstcal performance ssues for a real-world applcaton s stll a bg challenge. Frst, robust estmaton methods that perform well over a wde range of nose scenaros are needed. Estng wors usually focus on performance of a smple case, for eample, sngle nose nstance has been used to evaluate performance (Huang et at. [27], Huang et al. [29], Fan and Wehbe [23] and Ghahreman and Kamwa [2a,b]). Second, to mplement effectve controls n real tme for a real-world system, states need to be estmated for a realstc-sze system n real tme, whle the current studes focus on small problems wth two machnes, four machnes, or 6 machnes usng seral computers. For the real-world applcaton, the performance scalablty of data assmlaton methods needs to be evaluated 2.2

15 wth a medum-sze realstc system wth modern parallel computaton capablty. Thrd, the above studes assume that PMU data are avalable at any desred locatons, whle n practce the PMUs have been deployed and the locaton cannot be readly changed. Therefore, before estmatng a state, a data assmlaton method needs to be evaluated under a practcal PMU setup. Fourth, n a real-world applcaton, a dynamc model s only a smplfed descrpton of a real system. A practcal dynamc state estmaton method must tolerate the model nose to be appled relably. Yet, n most studes, the models for estmatng states have smlar structure to the models for generatng smulaton data. Ths practce does not gve enough consderaton to how model noses may nfluence the estmaton. Wth these techncal gaps n mnd, we defne a dynamc state estmaton problem for a power system. 2.3

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17 3. Narratve Problem Statement In ths secton, we defne a dynamc state estmaton problem for a power system. In the problem statement, we clarfy the epected outcome of dynamc state estmaton, avalable nput, and evaluaton crtera. The structure of the problem statement s llustrated by Fgure 3.. The problem statement s used to develop and evaluate potental solutons. Inputs PMU Data Dynamc State Estmaton Estmated States Dynamc Model True States Performance Evaluaton 3. Epected Outputs Fgure 3.. Problem Statement of Dynamc State Estmaton The goal of dynamc state estmaton s to estmate the electromechancal dynamc states of a power system. Dynamc states are the mnmum set of varables that can determne the status of a dynamc system (DeRusso et al. 998) for a tme scale of nterest. For the electromechancal dynamcs whch we are nterested n, the tme scale s at a level between mllseconds and second (Fgure 3.2). The electromechancal dynamc states can be used to enhance the rotor angle stablty of a power system by enablng dynamc contngency analyss and state predcton. Note that states are dfferent from parameters n that parameters often reman a constant value whle states may eep changng. Fgure 3.2. Tmescale of Power System Dynamcs Dependng on the requrement of an applcaton, the states can be estmated for past, current and future tme (Fgure 3.3). A smoothng method estmates the states of past tme and can be used to reproduce past events for forensc studes. A flterng method estmates the states at the current tme and can be used to drve control sgnals to mprove system stablty. A predcton method estmates the states 3.

18 of future tme and can be used to gude proactve remedal actons. The ultmate goal of ths study s to enhance power system stablty n real tme, and therefore the focus s placed on the flterng method. Flterng (Current States) Avalable Data States Smoothng (Past States) Predcton (Future States) -5mn -2ms -9 ms ms -3ms Current Tme 3ms 6ms 9ms Tme 3.2 Avalable Inputs Fgure 3.3. Illustraton of Smoothng, Flterng and Predcton Methods To estmate the current dynamc states, only the data and models that are avalable up to the current tme can be used as the nputs of a flterng method. All the PMU data avalable up to the current tme can be used. Dynamc models that are avalable may also be used. Studes shall be carred out wth proper consderatons of the practcal constrants of models and data.. PMU data: a. PMU data have measurement nose. The measurement noses are defned as total vector errors (TVEs). For a PMU that follows Insttute of Electrcal and Electroncs Engneers (IEEE) Standard C37.8, the devce error shall be smaller than %, n addton to the nose ncurred by the PMU devce (Martn et al. 28). Addtonal nose s often ntroduced by current transformers (CT) and potental transformers (PT). b. Because of the cost of PMU nstallaton, there are lmted numbers of PMUs avalable. Compared to the sze of a power grd, PMU data are sparse and manly avalable on hgh voltage lnes. c. Because of unepected sensor and data communcaton falures, outlers and mssng data are not uncommon n PMU data. 2. Dynamc models a. Dynamc models are often avalable to descrbe the dynamc features of a power system. The dynamc models are only an appromate descrpton of a power system. Therefore, the system responses and model responses to a stmulaton are often dfferent at varous levels. The response dfference reveals the modelng noses and defcences, and must be consdered for a real applcaton. 3.3 Evaluaton Crtera There are many algorthms for estmatng dynamc states of a power system. Evaluaton crtera are needed to evaluate and compare the estmaton performance and select a proper estmaton algorthm. The followng measurements can be appled to quantfy the performance of a dynamc state estmaton algorthm under a statstcal framewor. 3.2

19 . Accuracy: Accuracy s defned as the dfference between estmated states and true states. However, because the true states are unnown n a real power system, several ndees based on measurement resduals wll be used to evaluate state estmaton accuracy (Guo et al. 23, Al-Othman and Irvng 26). 2. Robustness aganst modelng errors and measurement errors: The robustness of an algorthm can be measured by ts tolerance of nose n a model and data. In practce, the robustness of an algorthm s evaluated by the mamum modelng errors or measurement errors under whch the algorthm can stll acheve the requred state estmaton accuracy. 3. Speed of computaton: The speed of an algorthm can be measured by the capablty of an algorthm to eep up wth the speed of nflowng data. Ideally, the results are avalable before the net set of data arrves. Ths s not currently possble for large real-world systems. In ths case t may be necessary to use multple computng systems to mantan a constant latency of results n order to process all the data. 4. Scalablty: Scalablty of an algorthm refers ts effectveness when t s appled a larger system. There are multple aspects that need to be consdered, ncludng seral scalablty, parallel scalablty, and accuracy. Seral scalablty determnes how much longer a calculaton wll tae when the problem sze s ncreased but the amount of computaton s not. For eample, f the problem sze doubles but the soluton taes four tmes as long, the algorthm s sad to scale as the square of the problem sze. Algorthms that scale lnearly or less, e.g., logarthmcally, are hghly desrable. Parallel scalablty s often consdered n terms of strong and wea scalng. In strong scalng, the tme requred to solve a problem s drectly proportonal to the amount of computaton beng used,.e., two processors can solve the problem twce as fast as one processor, ten processors ten tmes as fast, etc. Amdahl s law, whch states that parallel speedup s lmted by the amount of seral computng requred by an algorthm, maes the range of strong scalng generally very lmted. Wea scalng, on the other hand, allows the use of more processors on bgger problems because the seral sectons scale dfferently than the parallel sectons. Eventually, however, addng more processors actually slows the rate at whch results are generated. Algorthms that slow down later rather than sooner are obvously preferred. An algorthm that performs fewer arthmetc operatons (multples and adds) may not scale as well as the one that does more operatons f t has to move more data between processors, so the scalablty of an algorthm s compute-to-communcate rato s mportant. Fnally, a robust, numercally stable algorthm may be able to solve larger problems than one that s computatonally more effcent but numercally unstable, e.g., one that taes the dfferences of small numbers a process that can destroy numercal precson. Scalablty at all levels s crtcal because methods that wor for small problems may fall apart when appled to large real-world problems. 3.3

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21 4. Problem Formulatons In ths secton, we formulate a specfc dynamc state estmaton problem to estmate dynamc states of a synchronous generator. We wll dscuss the dynamc models of a generator, dynamc states to be estmated, avalable PMU data, and objectve functons. 4. Transent Models for Estmaton Ths subsecton ntroduces estmaton models used by flterng algorthms for estmatng dynamc states of a synchronous machne. Also, to apply a flterng method for dscrete measurements, a modfed Euler method s appled to dscretze the contnuous model (Kundur 994).. Contnuous-tme models for generators A fourth-order model () can be used to model a synchronous machne n local d-q reference frame to estmate the states (Kundur 994). T m Te K D H 2 e q E fd eq ( d d ) d T d e d ed ( q q ) q T q (.a) (.b) (.c) (.d) In (), δ s the rotor angles n radans, by whch the local q as leads the global R as (Kundur 994); Δω s rotor speed devaton; and e d ' and e q ' are the transent voltages along q and d aes. The parameter T m s the mechancal torque; T e s the electrc ar-gap torque; and E fd s the nternal feld voltage. The parameter ω s the rated value of the angular frequency; H s the nerta constant, and K D s the dampng factor. The parameters T d ' and T q ' are the open crcut tme constants n the drectons of the d and q aes respectvely; d and q are the synchronous reactance at the d and q aes respectvely; and d ' and q ' are the transent reactance at the d and q aes respectvely. To facltate the notaton, () s transformed nto a general state space model as gven n (2) and (3). f c (, u) w y hc (, u) vc E E T w w Q c c T v v R c c c (2.a) (2.b) (2.c) (2.d) 4.

22 T e q e d (3.a) u T m E fd R T I (3.b) y e R e I T (3.c) In (2) (3) s the state vector, u s the nput vector, and y s the output vector. Functons f c (*) and h c (*) are the state transton and output functons, respectvely. The subscrpt c ndcates the contnuous-tme model. The vectors w and c v c represent the process and output nose, respectvely. They are modeled as Gaussan whte nose whose covarance matrces are defned by (2.c) and (2.d) as Q and R. E[*] represents statstcal epectaton. To transform () nto f c (*) n (2.a), d, q, and T e were wrtten as functons of and u usng (4). d R sn I cos (4.a) q e sn cos I t d q R T P ( e ) ( e ) q d q d d q (4.b) (4.c) Smlarly, to mplement output functon h c (*) n (2.b), e I and e R were wrtten as functons of and u usng (5). Note that d and q n (5) are the functons of R and I as n (4.a) and (4.b). The model defned by (), (4), and (5) s then dscretzed and used for estmatng states. e R e )sn ( e ) cos (5.a) ( d q q q d d e I e )sn ( e ) cos ( q d d d q q (5.b) 2. Dscretzaton method and dscrete tme models To estmate states usng the dscrete measurements, the contnuous-tme model n (2) was dscretzed nto a dscrete tme model (6), where the subscrpt ndcates the tme at t. y f h(, u ) w (6.a) (, u ) v (6.b) More specfcally, the state transton functon (2.a) was dscretzed by applyng the modfed Euler method (Zhou et al. 23) usng (7). In (7), ~ f can be calculated by (8). When t s small enough, the c dscrete process nose w - can be appromated by (9). Because the contnuous-tme process nose w c s defned by (2.c), the mean of w - s and the covarance of w - can be calculated as (). Equaton () ndcates that the varance of process nose can be ncreased proportonally wth the samplng nterval t durng the state estmaton. ~ f t w c (7) 4.2

23 4.3 ), ( ~ 2 / ), ( ), ( ~ ~ c c c c u f t u f u f f (8) t t c d w w ) ( ) ( (9) t Q d d Q d d w E w w w E Q t t t t t t t t T c c T d 2 ) ( ) ( 2 2 ) ( ) ( 2 ) ( ) ( ) ( () Measurement equaton (2.b) can be dscretzed nto (). Here, v s the dscrete tme output nose. Because the contnuous-tme output nose v c s defned by (2.b), the mean value of v s. The covarance of v depends on how measurement nstruments are set up. To smplfy the study, ths report assumes no preflter. Therefore, the covarance of the v may be computed usng (2) (Schnel et al. 23). c v u h y ), ( () R v E v R T d (2) 4.2 PMU Data PMUs are deployed n power systems to measure the voltage and the current phasors at major transmsson lnes. PMU data have a typcal samplng rate of 3 or 6 samples per second, are well synchronzed wth the Global Postonng System cloc, and can contnuously capture the dynamc response of a power system under normal and abnormal condtons. Accordng to IEEE Standard C37.85 (Martn et al. 28), TVEs are used to quantfy the accuracy of a PMU. Measurement nose may also nclude the nose from current transformers and potental transformers. In addton to normal measurement noses, t s not uncommon for PMU measurements to be corrupted by mssng data and outlers. Mssng data may be caused by temporary communcaton falures, and s often dentfed and mared out by error detecton schemes assocated wth communcaton protocols. Outlers are measurements wth sgnfcantly large measurement errors that may be caused by etraordnary dsturbances or temporary sensor falures. By checng the resduals, an outler detector may be able to detect some outlers. Yet, because such an outler detector needs to mantan a balance between correct and erroneous determnatons (false postves and false negatves), t wll not be able to detect all outlers. Undetectable outlers (false negatves) carry msleadng nformaton and present a major challenge to state estmaton algorthms. Typcally there are two categores of data that are often used n estmatng dynamc states of a power system: smulaton data and feld measurement data

24 . Smulaton data: Smulaton data are generated by performng dynamc smulaton usng a dynamc power system model. Note that the smulaton model may be dfferent from the estmaton model dscussed n the prevous subsecton. A smulaton model s for mmcng system behavors whle an estmaton model s for estmatng states. There are several advantages of usng smulaton data. ) For a smulaton data set, the true states are avalable. Therefore, t s easy to evaluate the accuracy of an estmaton. ) It s easy to set up a large number of representatve scenaros to study the statstcal performance of an estmaton method. Often, Monte Carlo methods are used to evaluate the average performance. ) It s easy to test the applcablty of an estmaton method on dfferent systems because swtchng among dfferent systems s easy for smulaton studes. v) Measurements can be generated at any desred ponts for smulaton data. Therefore, the nfluence of certan measurements on the estmaton results can be readly evaluated. Because of these advantages, smulaton data have been used n almost all the studes to evaluate algorthm performance. One caveat s that the qualty of smulaton data depends on how well a model descrbes a system. The responses from a smulaton model are dfferent from a system s responses to some degree. Therefore, feld measurement data shall be used to test the applcablty of a state estmaton method n a real-world applcaton. 2. Feld measurement data Feld measurement data are collected from sensors deployed nto a system. The measurement data carry nvaluable nformaton about the status of a real system. Informaton and nowledge etracted from feld measurement data can mprove stuatonal awareness of a power system operator and drectly support real-tme decson mang n power system operatons. Unle smulaton data, there are many more uncertantes n feld measurement data. For eample, s the nose n feld measurement Gaussan or non-gaussan? How do these uncertantes mpact dynamc estmaton methods? Feld measurement data requres that state estmaton methods be hghly robust aganst nose. Ultmately, a dynamc state estmaton method needs to be tested usng real-world data to evaluate ts applcablty. 4.3 Problem Formulaton We formulate a real-tme dynamc state estmaton problem for synchronous generators as follows. Assumng a samplng nterval of t seconds, gven measurements of voltage phasors, current phasors at some locatons for tme nstance of t = t, 2t, 3t, t, estmate the synchronous machne s states δ(δt), Δω(Δt), e' q (Δt) and e' d (Δt). A user may choose a dynamc model for the estmaton. 4.4

25 5. Soluton Methods Usng Data Assmlaton Technques The Kalman flter s the most wdely used Bayesan-based data assmlaton method. It was named after Rudolf Kalman, who publshed hs famous recursve method to estmate dynamc states of a lnear system (Kalman 96). Assumng Gaussan nose, the Kalman flter provdes mnmum-varance estmates of states through a recursve approach. In addton to ts orgnal successful applcatons n lnear systems, there are many publcatons etendng the Kalman flter to nonlnear systems. The major dfference among dfferent nonlnear Kalman-flter methods s ther approach to propagatng the mean and covarance of the dynamc states. The EKF (Welch and Bshop 26, Shh and Huang 22) lnearzes the state space model usng a frstorder appromaton. The mean and covarance of states are propagated usng Jacoban matrces. The UKF (Wan and van der Merwe 2) propagates the mean and covarance of states usng a determnstcsamplng approach to acheve a second-order appromaton. The EnKF propagates the mean and covarance of states usng a Monte Carlo samplng approach (Evensen 994). In the EnKF, the dstrbuton of the states s represented by a collecton of samples, referred to as ensembles. All the above Kalman flters assume the jont Gaussan dstrbuton of both measurements and states, and use the Bayesan approach to derve the Kalman gan. In contrast, the PF (Arulampalam et al. 22) s a more general Bayesan approach, whch does not rely on Gaussan nose assumpton. Smlar to the EnKF, the PF also uses the samples (also nown as partcles) to represent the probablty dstrbuton of random varables. Dfferent from the EnKF, the PF drectly corrects the states wthout assumng Gaussan dstrbuton. Ths general approach s more applcable to hghly nonlnear systems. However, the PF usually requres a very large number of samples and therefore s dffcult to apply to hgh-dmensonal systems. 5. Overvew of Data Assmlaton n Other Domans The goal of data assmlaton s to fuse model and data together to get better results than usng model or data alone. Data assmlaton has been wdely studed n many scentfc felds, especally n the felds of geoscences, hydrology and weather forecastng. For eample, data assmlaton has been used to ntegrate atmospherc models wth radosonde and satellte observatons (Houteamer and Mtchell998). It s also used n oceanography to ntegrate ocean model wth radar data. (Hotet and Pham 24). Models used n data assmlaton can be classfed as determnstc or stochastc accordng to whether randomness s consdered n the system. They may also be classfed as statc or dynamc models dependng on whether contnuous tme-dependent change s accounted for. Dfferent methods have been developed for data assmlaton for dfferent types of models. In the least squares method and ts varants, for eample, generalzed least squares are commonly used to deal wth determnstc data assmlaton problems. As for statstcal estmaton, Fsher s mamum lelhood technques and the Bayesan approach, ncludng mamum a posteror (MAP) estmate and mnmum-varance estmate, form two basc framewors of assmlaton methods. Under the Bayesan framewor, the Kalman flter, a recursve verson of a mnmum-varance estmator, s becomng a standard method for data assmlaton (Houteamer and Mtchell 998). 5.

26 5.2 Overvew of Kalman Flter (UKF, EKF, EnKF, PF) Under a Bayesan framewor, the mplementatons of theses algorthms have a smlar structure. After ntalzaton, all the flterng algorthms assmlate one snapshot of data at every tme step. For one snapshot of data, there are two steps: a predcton step and a correcton step. In the predcton step, the mean and covarance of states at tme step are predcted based on the states at step. In the correcton step, the predcted mean and covarance are corrected based on new measurements obtaned at tme step. The algorthms for mplementng these flterng methods are detaled as follows EKF The EKF lnearzes the system at the current operatng pont usng the Jacoban matrces as n (3), (4) and (5) (Welch and Bshop 26). EKF Predcton: f (, u ) (3.a) P F EKF Correcton: T P F Q (3.b) d K ~ y (4.a) K P H T H P H T R d (4.b) ~ y z h(, u ) (4.c) P ( I K H ) P (4.e) where and P are nown as the a pror mean and covarance (of the states) respectvely. They are estmated from the data up to tme step. The symbols and P are nown as the a posteror mean and covarance of the states, respectvely, whch are derved by addng the nformaton from z to and P. The symbol K s the Kalman gan. The symbol y~ s the resdual between estmate h (, V ) and measurement z. F and H are Jacoban matrces defned by (5). A perturbaton approach s used to numerally derve the Jacoban matrces n ths report. f F (5.a) h H (5.b) 5.2

27 UKF The UKF uses an unscented transform to pc a set of samples to represent the probablty dstrbuton of states and propagates these samples through the nonlnear functons f and h to reconstruct the mean and covarance. The UKF estmaton method s summarzed as (6) and (7) (Wan and van der Merwe 2). UKF Predcton n u f 2,,, ), ( (6.a) (6.b) n P n,, ) ( (6.c) n n P n,,, ) ( (6.d) n W 2 (6.e) d n Q T W P 2 (6.f) UKF Correcton y K ~ (7.a) d T T R H P H H P K (7.b) n T y y W H 2 ~ ~ (7.c) ~ z z y (7.d) n z W z 2 (7.e) n u z h 2,,, ), ( (7.f) P H K I P ) ( (7.g) where and W are 2 n sgma ponts and ther correspondng weghts. s a scalng parameter that controls the postons of the sgma ponts.

28 5.2.3 EnKF The EnKF uses samples (also nown as ensembles) to represent and propagate the probablty dstrbutons of the states. By usng a large number of samples, the probablty densty can be appromated wth hgh accuracy. The EnKF can be summarzed by (8) and (9) (Evensen 994). EnKF Predcton f, n (, u ) w,, enkf (8.a) z h (, u ),,, n enkf (8.b) n enkf n enkf (8.c) z n enkf z (8.d) n enkf EnKF Correcton K z z (9.a) K P H T H P H T R d (9.b) P H T n enkf n enkf T z z (9.c) H P H T n enkf n enkf z z z z T (9.d) z z v (9.e) where n enkf s the total number of samples, whch are used to represent the dstrbuton. The varable w s a sample generated accordng to the Q d to smulate process nose. The symbol stands for the samples of a posteror states. Note that usng (9.c) and (9.d), the covarance matr P does not need to be epressvely calculated Basc PF The PF can be appled to systems wth Gaussan and other dstrbutons. A basc PF appromates a probablty dstrbuton functon by a set of weghted dscrete samples, as shown n (2). p n PF W ( ) u:, y: (2) After processng several data snapshots, a PF often suffers from a degeneracy problem (.e., the weght of only one partcle tends to whle weghts of all other partcles tend to ). To reduce the degeneracy problem, a resamplng step s often added to re-dsperse the dscrete samples by generatng a new set of partcles accordng to the dscrete dstrbuton of (2). To detect degeneracy, the effectve 5.4

29 samplng sze N eff s defned by (23). In the followng smulaton tests, resamplng s ntated when N eff <.N pf. The basc PF process s descrbed by the followng equatons (Arulampalam et al. 22). PF Predcton u w f PF Correcton ~ W W p W, z h(, u v ~ W npf j ~ W j ) (2) (22.a) (22.b) n PF W (22.c) PF Resamplng f degeneracy s detected usng (23) N eff n PF ( W ) 2 (23) where states W ~ s pror weghts of the th state sample. v z h(, u ) and nputs p s the lelhood of z gven the pror u. The lelhood functon s determned by the measurement nose model (.e., R d ). Symbol n PF s the total number of samples that are used to represent the probablty dstrbuton of a state. 5.5

30

31 6. Prelmnary Results In ths secton, dynamc smulaton s carred out to compare the performance of the EKF, UKF, EnKF, and PF for the purpose of estmatng dynamc states of a power system. The Power System Toolbo (PST) (Chow and Cheung 992) was selected to generate smulaton data that mmc the responses of a real system. The two-area, four-machne test system shown n Fgure 6. (stored as d2asbeghp.m n PST) s used to generate the system dynamc responses to a three-phase fault. The fault s appled to Bus 3 on the lne between Buses 3 and at. seconds. The fault s cleared at.5 seconds at Bus 3 and at.2 seconds at Bus. To capture the dynamcs and reduce ntegraton errors, the smulaton tme step s chosen to be. second. The smulaton durng s set to be 5seconds. G G3 G Fgure 6.. The Two-Area, Four-Machne System (Chow and Cheung992) Assume that PMUs are avalable at the generator Bus to measure the voltage and the current phasors. Flterng algorthms are set up to estmate the dynamc states of Generator. To mmc the measurements from PMUs, the system responses are down-sampled to a rate of 25 samples per second. Accordng to IEEE Standard C37.85 (Martn et al. 28), a certan percentage of TVEs are added to the system responses to mmc measurement nose. Because of the randomness of the measurement and process nose, the Monte Carlo methods are appled to generate M = sets of smulaton data to represent varous nstances of random nose. The mean squared error (MSE) defned n (24) s used as a metrc for comparng estmaton accuracy. Here, the symbol, represents the true state at the th tme True step, whle ˆ s the correspondng estmated state n the m th Monte Carlo test case., m 2 2 G4 M ˆ, m, True MSE ˆ (24) M m 2 To ncrease the dynamc range, log (MSE) n db s used to compare the algorthm performance under the followng four scenaros. 6. A Smple Scenaro The goal of ths scenaro (Scenaro A) s to set a benchmar for comparson. The smulatons are set up as follows. To smulate the generators dynamc responses, a fourth-order transent model as shown n () s used. Governors and ecters are ncluded. Sub-transent dynamcs and feld saturaton effects are not modeled. For the PMU measurements,.% measurement nose n TVE s added to the voltage and current phasors. T m and E fd are recorded wth.% measurement nose. The PMU samplng rate of 25 samples per second used for generatng the measurement data are also used for estmaton. 6.

32 For all the algorthms, the ntal states are estmated by settng n (2) and then solvng (2) usng the Gauss-Sedel method. To reflect uncertanty of the ntal states, covarance P s set to be tmes of the largest changes of the states,.e., P = (ma{abs[dff( :N )]}) 2. The output varance R d s set to be. 2 correspondng to the.% of errors added, whch s dag([.,.]) 2 for ths study. Q d s set to be.% of the largest changes of the states,.e., Q d =.% ma{abs[dff( :N )]}, whch s dag ([.474,.42,.289,.37]) 2 for ths study. The MSEs of the four states from EKF, UKF, EnKF, and PF are summarzed n Fgure 6.2. Both EnKF and PF use samples to represent state probablty dstrbuton. To evaluate the nfluence of the sample number on MSEs, 2 and 2 samples were used for testng these algorthms. It can be observed from Fgure 6.2 that the EnKF has the smallest MSE. Increasng the number of samples n EnKF does not sgnfcantly nfluence ts estmaton accuracy. In contrast, the MSEs of the PF notceably decrease when the sample number s ncreased from 2 to 2. UKF and PF have larger MSEs than the other methods. Fgure 6.3 shows the EnKF s estmaton results for all sets of Monte Carlo smulaton wth n pf = 2 samples. All of the EnKF estmates converge to the true states wthn.5 seconds. Note that to help the llustraton, the true value of Generator 4 s rotor angle s used as the reference angle to generate the frst plot of Fgure 6.3. For the PF wth n pf = 2 samples, 8 sets out of the PF estmates converged and the other 2 sets dverged. When the sample number of the PF s reduced from 2 to 2, the number of converged sets decreases to 47. For the UKF, only 67 sets of estmates converged. For the EKF, all estmates converged, but they have larger MSEs than the EnKF. Rotor Angle (rad) 2 MSE n db EKF UKF EnKF(2) EnKF(2) PF(2) 5 PF(2) 5 Speed Dev (pu) Ed' (pu) Eq' (pu) 5 5 Tme (sec) Fgure 6.2. Comparson of MSEs from the EKF, EnKF, PF, and UKF for Sets of Monte Carlo Smulatons of a Smple Scenaro 6.2

33 Relatve Rotor Angle (rad).5 States tracng (Basc EnKF) Speed Dev (pu) -3 True Mean 5 Mean+/-3*Std MC Ests 5 5 Eq' (pu) Ed' (pu) Tme (sec) Fgure 6.3. Estmated States from the EnKF wth 2 Samples of sets of Monte Carlo Smulatons for a Smple Scenaro (All sets converge) 6.2 A Scenaro wth Measurement Interpolaton The goal of ths scenaro (Scenaro B) s to evaluate how the nterpolaton method (Huang et al. 23, 29) may nfluence the algorthms performance. The nterpolaton method nserts the addtonal pseudo-measurement ponts between two consecutve measurement samples. Introducng addtonal measurements ncreases the effectve samplng rate, and reduces the lnearzaton errors. In ths scenaro, the samplng rate s ncreased from 25 samples/s to 2 samples/s by addng seven addtonal pseudomeasurements between every two measurement ponts through lnear nterpolaton. Note that because of the nterpolaton, the samplng tme nterval t n (7) s reduced from 4 ms to 5 ms. Followng (), the process nose covarance Q d s reduced to /8 of that n Scenaro A. The R d remans the same as n (2). All the rest of the setup remans the same as that n Scenaro A. The MSEs from the EKF, UKF, EnKF and PF are compared n Fgure 6.4 between the cases wth nterpolaton as n ths scenaro and the cases wthout nterpolaton as n Scenaro A. It can be observed that the MSEs of the EKF, UKF and PF are sgnfcantly reduced wth the nterpolaton method. In comparson, changes of MSEs for the EnKF are less sgnfcant. In addton, wth nterpolaton, the MSEs of the EKF, UKF and EnKF are sgnfcantly smaller than those of the PF. Wth the nterpolaton method, all sets of EKF, UKF, EnKF and PF estmates converge. 6.3

34 Rotor Angle (rad) EKF UKF EnKF(2) EnKF(2) PF(2) PF(2) Speed Dev (pu) Ed' (pu) Eq' (pu) Tme (sec) Tme (sec) Tme (sec) Tme (sec) Tme (sec) Tme (sec) w/o Interpolaton wth Interpolaton Fgure 6.4. Comparson of MSEs from the EKF, UKF, EnKF, and PF of sets of Monte Carlo Smulatons for the Scenaros wth Measurement Interpolaton and wthout Interpolaton 6.3 A Realstc Scenaro Consderng Model Inadequacy In ths scenaro (Scenaro C), the smulaton model for generatng measurement data are more comple than the estmaton model used for estmatng state. The goal s to mmc the more common realty where an avalable model s often a smplfed representaton of a real system. In addton, the measurement nose level s ncreased to nclude transformer nose. Fnally, some model nputs (e.g., T m and E fd ) have to be estmated because they are not usually measured by PMUs. To smulate generator responses, a sub-transent model s used. Feld saturaton effects are modeled by addng S =.654, S 2 =.4786 (Chow 28). To ncrease oscllatory dynamcs, the power system stablzers are ntentonally removed. By addng the sub-transent model and saturaton effects, the smulaton model s more comple than the estmaton model shown n (). For the PMU measurements, 5% of measurement nose n TVE s added to the voltage and current phasors. Note that addtonal nose s added to nclude measurement nose ntroduced by current transformers and potental transformers. In addton, because PMUs may not be avalable near all generators, E fd and T m are not measured n ths scenaro. E fd s estmated as a specal state. T m s estmated by low-pass flterng P e. For all the algorthms, the setup s same as that n Scenaro B (e.g., the data samplng rate s ncreased to 2 samples/s through lnear nterpolaton), ecept that the output varance R d s set to be dag([.5,.5]) 2 because the 5.% measurement nose was added. 6.4

35 The MSEs from the EKF, UKF, EnKF, and PF are summarzed n Fgure 6.5. It can be observed that the EnKF, EKF, and UKF have smlar MSEs. After appromately 4 seconds, the MSEs of the PF wth 2 samples converge to smlar levels. In contrast, the MSEs of the PF wth 2 samples are persstently the largest among all the algorthms, ndcatng performance degradaton. All estmates from EKF, UKF and EnKF converge. All estmates from PF wth 2 samples converge. In contrast, there are fve sets of estmates that dverge for the PF usng 2 samples. Rotor Angle (rad) MSE n db EKF UKF EnKF(2) EnKF(2) 5 PF(2) 5 PF(2) Speed Dev (pu) Ed' (pu) Eq' (pu) Tme (sec) Fgure 6.5. Comparson of MSEs from the EKF, UKF, EnKF, and PF of Sets of Monte Carlo Smulatons for the Realstc Scenaro 6.4 A Realstc Scenaro wth Mssng Data and Outlers The goal of ths scenaro (Scenaro D) s to evaluate all the algorthms when the PMU measurements are corrupted by mssng data and outlers. To smulate mssng data, all (smulated) measurement data between the fourth and ffth seconds s chosen to be mssng and mared out. The mssng data are then patched through lnear nterpolaton. To mmc outlers, errors on the order of tmes the standard devatons of the voltage magntudes and angles are added to the voltage phasor measurements between. and.5 seconds. These outlers are assumed to be undetected (false negatves) and are processed as normal data by all the flterng algorthms. The remanng setup s the same as that for Scenaro C. The MSEs from the EKF, UKF, EnKF and PF are compared n Fgure 6.6 between the cases wth mssng data and outlers as n ths scenaro, and the cases wthout mssng data and outlers as n Scenaro C. It can be observed that the MSEs of all the algorthms are very smlar for cases wth and wthout mssng data. Ths ndcates that all algorthms are robust aganst mssng data that last for one second. 6.5

36 EKF UKF EnKF(2) EnKF(2) PF(2) Rotor Angle (rad) Speed Dev (pu) Ed' (pu) Eq' (pu) Tme (sec) Tme (sec) Tme (sec) Tme (sec) Tme (sec) Fgure 6.6. Comparson of MSEs from the EKF, UKF, EnKF, and PF of Sets of Monte Carlo Smulatons for the Realstc Scenaro wth and wthout Mssng Data and Outlers On the other hand, the outlers at the tenth second cause sgnfcant ncreases n the MSEs of all the algorthms. After the outlers dsappear at.5 seconds, the MSEs of the EKF, UKF, and EnKF return to the orgnal value qucly, whle t taes more tme for the MSEs of the PF (wth 2 samples) to return to the orgnal values. Ths observaton ndcates that the EKF, UKF, EnKF are more robust to outlers than s the PF. Fgure 6.7 shows that all sets of EnKF estmates converge to the true states. Also, the estmates of the EKF and UKF for all sets of data converge w/o mssng data and outlers wth mssng data and outlers 6.6

37 Relatve Rotor Angle (rad) States tracng (Basc EnKF).5.5 Speed Dev (pu) True Mean Mean+/-3*Std 5 MC Ests Eq' (pu) Ed' (pu) Tme (sec) Fgure 6.7. Estmated States from the EnKF of Sets of Monte Carlo Smulatons for the Realstc Scenaro wth Mssng Data and Outlers (All sets converge) 6.7

38

39 7. Prelmnary Conclusons Accurate nformaton about dynamc states s crtcal to effcent control of a power system, especally wth the ncreasng complety resultng from uncertantes and stochastc varatons ntroduced by ntermttent renewable energy sources, responsve loads, moble consumpton of plug-n vehcles, and new maret desgns. Usng a statstcal framewor, ths report compares the performance of an EKF, a UKF, an EnKF, and a PF for the purpose of estmatng dynamc states from real-tme phasor measurements. The followng are shown through the smulaton usng a two-area, four-machne test system:. The EnKF algorthm outperforms other algorthms when the typcal PMU samplng rate s used for estmaton. 2. Measurement nterpolaton methods can mprove the estmaton accuracy of the EKF, UKF, and PF. The nterpolaton does not show sgnfcant nfluence on the performance of the EnKF. 3. Increasng the number of samples can mprove the estmaton and convergence of the PF. 4. All of the algorthms are robust to mssng data. The outlers cause some sgnfcant errors for all algorthms f the outlers are processed as normal data. The EKF, UKF, and EnKF are more robust to the outlers than the PF. It taes more tme for a PF to regan accurate state tracng after the outlers dsappear than for the EKF, UKF, and EnKF. 7.

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41 8. Scope of Further Study and Epected Outcome Because of the robustness, accuracy and potental scalablty of the EnKF algorthm, we choose to focus on developng EnKF algorthms whle contnung to use EKF and other methods as a benchmar to compare ther performance. Also, based on the prelmnary study results, we propose contnung some engneerng studes to eplore the potentals of ncreasng estmaton accuracy and mprovng eecuton effcency n the areas that follow. 8. Improve Algorthm Robustness aganst Modelng and Low Data Qualty Ths study s to etend Kalman flter technques by developng algorthms to mprove estmaton robustness aganst modelng nose and low data qualty. The scope of the study s detaled as follows. ) Modelng nose: A dynamc model s only a smplfed descrpton of a real system. Ideally, a dynamc model can appromate the correspondng system behavors reasonably well. The dfference between the model responses and real-system responses s reasonably small and can be captured by nose models. To acheve optmal performance, Kalman flters use state transton models to predct the net states and measurement models to correct the pror states. A statonary Gaussan nose model s used to descrbe the dfference between the model and system responses. In a real-world applcaton, t s not uncommon to encounter a large modelng nose that cannot be well descrbed by a statonary Gaussan nose model. The planned study wll mprove algorthm performance by quantfyng and mprovng modelng nose. The goal s to mprove the robustness of a Kalman flter aganst modelng nose. ) Low data qualty: In theory, PMUs could be deployed at every bus and provde data wth mnmal nose n real tme at a hgh speed. In real-world applcatons, only a lmted number of PMUs can be deployed. IEEE Standard C37.85 (IEEE 25) requres mamum % TVE for PMU measurements. In addton, the nose from eternal dsturbances, temporary sensor falure, temporary communcaton falure, current transformers and potental transformers cannot be gnored n the real-world PMU measurements. As a result, mssng data, outlers, and burst noses are often observed n PMU measurements. To gve an accurate state estmaton, the proposed study wll quantfy measurement noses and mprove algorthm robustness aganst measurement nose. 8.2 Develop a Realstc Model for Applyng and Testng EnKF usng Feld Measurement Data To gan credblty for real-world applcaton, a dynamc state estmaton method needs to be tested and evaluated usng feld measurement data from a real system. To acheve ths goal, we wll collect PMU data from a real system and establsh a dynamc model correspondng to the real system. In order to fuse the data wth t, a dynamc model must be establshed to descrbe a power system. PMU data from the Western Electrcty Coordnatng Councl (WECC) are streamed nto the Electrcty Infrastructure Operatons Center n real tme and archved regularly at Pacfc Northwest Natonal Laboratory. The data are readly accessble to the team and therefore are chosen to carry out the proposed study. 8.