IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., Robust Frequency Divider for Power System Online Monitoring and Control

Transcription

1 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 Robust Frequency Dvder for Power System Onlne Montorng and Control Junbo Zhao, Student Member, IEEE, Lamne Ml, Lfe Fellow, IEEE, Federco Mlano, Fellow, IEEE Abstract Accurate local bus frequency s essental for power system frequency regulaton provded by dstrbuted energy sources, flexble loads and among others. Ths paper proposes a robust frequency dvder (RFD) for onlne bus frequency estmaton. Our RFD s ndependent of the load models, and the knowledge of swng equaton parameters, transmsson lne parameters and local PMU measurements at each generator termnal bus s suffcent. In addton, t s able to handle several types of data qualty ssues, such as measurement nose, gross measurement errors, cyber attacks and measurement losses. Furthermore, the proposed RFD contans the decentralzed estmaton of local generator rotor speeds and the centralzed bus frequency estmaton, whch resembles the structural of the decentralzed/herarchcal control scheme. Ths enables RFD for very large-scale system applcatons. Specfcally, we decouple each generator from the rest of the system by treatng metered real power njecton as nputs and the frequency measurements provded by PMU as outputs; then a robust unscented Kalman flter-based dynamc state estmator s proposed for local generator rotor speed estmaton; fnally, these rotor speeds are transmtted to control center for bus frequency estmaton. Numercal results carred out on the IEEE 39-bus and 45- bus systems demonstrate the effectveness and robustness of the proposed method. Index Terms Frequency estmaton, frequency control, robust statstcs, decentralzed estmaton, dynamc state estmaton, unscented Kalman flter, power system dynamcs and stablty. I. INTRODUCTION WITH the ncreasng penetraton of renewable energybased generatons, the total nerta of the synchronous power system s reduced sgnfcantly. As a result, the tradtonal capacty-based requrements for prmary reserve defntons may not be able to satsfy the frequency RoCoF and nadr lmts [], []. To tackle ths potental frequency nstablty ssue, t s expected by the transmsson system operators (TSOs) that the wnd farms [3], flexble loads [5], and energy storage devces [6] would provde frequency regulatons. However, to enable effectve frequency regulatons, relable and accurate knowledge of the local bus frequency s a prerequste. In the lterature, the numercal dervatve of the voltage phase angle provded by Phasor Measurement Unt (PMU) through a washout flter s used to defne the local bus Junbo Zhao and Lamne Ml are supported n part by the U.S. Natonal Scence Foundaton under Grant ECCS-79. Federco Mlano s supported n part by the European Commsson under the RESERVE Consortum (grant No. 7748) and EC Mare Skłodowska-Cure CIG No. PCIG4-GA He s also funded by the Scence Foundaton Ireland, under Investgator Programme, Grant No. SFI/5/IA/374. Junbo Zhao and Lamne Ml are wth the Bradley Department of Electrcal and Computer Engneerng, Vrgna Polytechnc Insttute and State Unversty, Falls Church, VA 43, USA (e-mal: zjunbo@vt.edu, lml@vt.edu). Federco Mlano s wth the School of Electrcal and Electronc Engneerng, Unversty College Dubln, Ireland. (emal: federco.mlano@ucd.e). frequency [7], [8]. However, as shown by Radman et al. [9], t may produce physcally mplausble spkes n frequency due to the numercal dervatves and consequently may exhbt nstabltes f controls are taken based on them. An alternatve way to obtan local bus frequency s through the phaselocked loop (PLL) technque []. But t may be unrelable n presence of large step-nput speed changes, and has problems n presence of harmoncs, unbalance, etc. Furthermore, only a few load buses/substatons have PMUs or PLLs nstalled, whch may prevent the system from takng full advantages of local frequency controls. Fnally, measurements provded by PMUs and PLLs are always subject to nose or even gross errors, communcaton losses, etc [4]. For example, t s shown n [5] that the measurement nose affects the performance of the frequency regulator sgnfcantly, not to menton the gross errors, measurement losses, etc. To address these ssues, a model dependent analytcal expresson of bus frequency s proposed n [] assumng comprehensve and accurate models of the system. Mlano and Ortega [] mproved that approach and proposed a transent stablty smulatonbased frequency dvder. The man dea underlyng ths method s to solve a steady-state boundary value problem, where the boundary condtons are gven by synchronous generator rotor speeds. However, both methods assume accurate power system dynamc models for tme-doman smulatons, whch s dffcult to acheve n practce. In addton, the tme-doman smulatons of large-scale power systems are computatonal demandng, whch may prevent them for the onlne control applcatons. Ths paper proposes a robust frequency dvder (RFD) for onlne bus frequency estmaton. Our RFD s not dependent on load models, and the knowledge of swng equaton parameters, transmsson system lne parameters and local PMU measurements s suffcent. In addton, t s able to flter out measurement nose, suppress gross measurement errors, handle cyber attacks as well as measurement losses. To develop the proposed RFD, t s observed that to obtan an accurate estmaton of bus frequency, accurate generator rotor speeds are requred. In the meantme, complcated generator and load models should be avoded. To ths end, we propose to dvde RFD nto two subproblems, namely the decentralzed estmaton of local generator rotor speed usng local measurements and the centralzed bus frequency estmaton. To address the frst problem, we decouple each generator from the rest of the power system by treatng metered real power njecton as model nputs and the frequency measurements provded by PMU, local meter devces or Frequency montorng Network devces [3] as outputs. As a result, only the swng equa-

2 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 ton s requred for rotor speed estmaton and no detaled generator model s assumed. Snce the local measurements can be subject to data qualty ssue when mplementng an estmator, a robust unscented Kalman flter-based dynamc state estmator s proposed. Next, the local estmates are transmtted to the control center for bus frequency estmatons, yeldng two benefts: ) the requrement of communcaton bandwdth s decreased notably as only estmated rotor speeds are communcated nstead of voltage and current phasors; ) the wde-area generator rotor angles are avalable for operator to acheve better stuatonal awareness and carry out other applcatons, such as oscllatory modes montorng, rotor angle stablty analyss, etc. Last but not the least, thanks to the decentralzed and centralzed estmaton scheme, the proposed method s sutable for desgnng controllers of very large-scale power systems. The remander of the paper s organzed as follows. Secton II presents the problem formulaton. Secton III descrbes the proposed RFD n detal and Secton IV shows and analyzes the smulaton results. Fnally, Secton V concludes the paper. II. PROBLEM FORMULATION In ths secton, the analytcal relatonshp between bus frequency and generator rotor speeds wll be presented frst; then the lmtatons of ths approach wll be dscussed thoroughly, and fnally the problem statement wll be declared. A. Analytcal Relatonshp between Bus Frequency and Generator Rotor Speeds When a dsturbance occurs, such as transmsson lne faults, load sheddng or generator trppng, power msmatch appears between the mechancal torque and electrcal power at the generator termnal buses. As a consequence, the generator rotor speeds wll devate from ther nomnal values. To resynchronze generator wth the rest of the power system, an ncrease or a decrease n the rotor speed s actuated, whch causes rotor angle oscllatons as well. Due to such oscllatons, the voltage phase angles of the buses that are adjacent to generators wll encounter changes, whch n turn causes a power msmatch. In ths way, the electro-mechancal oscllatons wll be propagated throughout the entre power system wth lmted speed. Snce we are nterested n electromechancal oscllatons and the propagaton speed of such oscllatons s much lower than that of the wave, the transent effects of wave propagaton are therefore neglected. Based on the analyss above, t s clear that the spatal varatons of the system frequency are characterzed by synchronous generator rotor speeds. Those frequency varatons at each bus of the system are of vtal mportance for desgnng local controllers to enhance the frequency regulaton capablty of a power system wth hgh penetraton renewable energy ntegratons. Note that electro-mechancal oscllatons can be characterzed by the magntude and phase angle modulatons of voltages and currents as well snce they are correspondng to the movement of rotors of electrc machnes around the synchronous speed [4], [5]. Thus, to estmate bus frequency of a transmsson system, we frst need to analyze the relatonshp of the voltage or current phasors between generators and system buses. Ths relatonshp can be expressed by the balanced current njecton formula shown as follows: [ ] [ ][ ] IG YGG Y = GB VG, () I B Y BG Y BB + Y B V B where I G are generator current njectons; V G are generator nternal electromotve forces (emfs); I B and V B are current and voltage njectons of the network buses, respectvely; Y BB s the power network admttance matrx; Y GG, Y GB and Y BG are admttance matrces calculated by ncludng the nternal mpedances of the synchronous generators; Y B s a dagonal matrx, whch takes nto account the nternal mpedances of synchronous generators at the generator buses. Snce the load current njectons are neglgble compared wth that of the synchronous generators [], [8], () can be rewrtten as: [ IG ] = [ ][ ] YGG Y GB VG. () Y BG Y BB + Y B V B By takng smple algebrac operatons on the second row of (), we can derve the relatonshp between bus voltage vector V B and the emfs of generators as follows: V B = (Y BB + Y B ) Y BG V G = DV G. (3) Takng tme dervatves on both sdes of ( 3) n rotatng reference frame,.e., dq frame, we get dv B dt + jω V B = D dv G dt + jω DV G, (4) where ω s the nomnal rotor speed. For more detals of dervng (4), please see []. Defne ω B = ω B ω, ω G = ω G ω, the analytcal relatonshp between bus frequency and generator rotor speeds can be derved from ( 4), whch s expressed as follows []: ω B = ω + D (ω G ω ). (5) It can be observed from (5) that the bus frequency s correlated wth each synchronous generator n the system, but the degree of partcpaton of each generator on bus frequency s determned by the transmsson parameters. To speed up the calculaton of bus frequency from ( 5) wthout a relevant loss of accuracy, the conductances of transmsson lnes utlzed to calculate D can be neglected []. B. Lmtatons and Problem Statement When mplementng (5) to estmate bus frequency, there are two possble ways: ) the dynamc smulaton based-approach and ) the measurement-based approach. In the former approach, the transent stablty program s used to obtan the rotor speed of each generator, followed by the calculatons of the bus frequences through (5). However, to obtan good tmedoman smulaton results, accurate and detaled generator and load models are requred, whch may be dffcult to acheve n practce. In addton, the tme-doman smulatons of largescale power systems are computatonal demandng, whch may not be sutable for the onlne control applcatons. By contrast, the measurement-based approach does not have such ssues, but the metered generator rotor speeds and frequences by PMUs or Frequency montorng Network devces are assumed

3 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 3 to be of hgh qualty. However, ths assumpton may not hold true for practcal power systems as the PMU measurements are usually subject to nose or even mpulsve nose, gross errors, cyber attacks, communcaton losses, etc. Under those condtons, ths approach wll produce sgnfcantly based results and subsequently the control actons based on them may make the system even worse. Furthermore, t should be noted that the frequency of each bus s correlated wth most generators, thus f just one rotor speed measurement s corrupted, ts error may propagate to many other bus frequency estmatons. It s thus ndspensable to make sure that every generator rotor speed s of good accuracy. Problem statement: gven a lmted number of PMUs nstalled at the termnal bus of each generator, a robust frequency dvder s developed to address the data qualty ssues of PMU measurements; n the meantme, t should be model ndependent so as to mtgate the strong assumptons on the generator and load models, and fnally, t should be fast to calculate and sutable for large-scale system onlne control applcatons. Note that n ths paper, the rotor speed and frequency of each generator at ts termnal bus are assumed to be montored by PMUs. Ths s a reasonable assumpton due to several reasons: ) onlne montorng of generators plays a major role n power system operaton and control, thus t s gven a hgh prorty for PMU placement accordng to the NERC PMU placement standard [6]; ) t s requred by NERC standard [7] to have PMUs nstalled at the pont of nterconnecton for power plant model valdaton and verfcaton. Fg. : Proposed decentralzed-centralzed bus frequency estmaton framework. III. PROPOSED ROBUST FREQUENCY DIVIDER By lookng at (5), one may easly come up wth the dea that ths equaton can be formulated as a regresson problem, where the generator rotor speeds are measurements provded by PMUs whle the frequency of each bus s the unknown state vector to be estmated. However, snce the number of rows of the matrx D s larger than that of the columns, (5) cannot be treated as an estmaton problem. Therefore, alternatve approaches should be developed. Note that to obtan an accurate estmaton of bus frequency, accurate generator rotor speeds are requred. Thus, f measurement qualty ssues can be addressed locally at the generator bus through a local robust estmator, we are able to obtan accurate bus frequency estmates. To ths end, we propose a decentralzed-centralzed bus frequency estmaton framework shown n Fg.. Partcularly, we frst perform the robust unscented Kalman flter-based dynamc state estmator at the local generator substaton or phasor data concentrator (PDC) level by usng the proposed model decouplng approach; then these local estmates (generator rotor speeds and angles) are transmtted to control center for bus frequency estmaton through (5). Note that f the decentralzed dynamc state estmaton at each generator substaton s costly, the data of those generators assocated wth PMU measurements wll be transmtted to the PDC or regonal system operatng center for decentralzed estmaton. After that, local controls can be ntated f requred, otherwse, they wll be further communcated to the control center for bus frequency estmaton and coordnated control. In the followng subsectons, we wll elaborate on each of the block. A. Generator Model Decouplng Approach Due to power unbalance and other control actons, the rotor of the generator wll accelerate or deaccelerate. Those electromechancal dynamcs can be captured by the swng equatons shown as follows [8]: H ω dδ dt = ω ω, (6) dω dt = T M P e D p (ω ω ), (7) where δ s the rotor angle; H s the generator nerta constant; T M and P e are the generator mechancal power and electrcal power outputs, respectvely; t s assumed that T M remans the same value as that n steady-state condton durng transent process, whch s reasonable as the tme constant of the governor s very large; D p s the generator dampng constant. Note that the generator frst swng dynamcs are affected by P e sgnfcantly. In other words, the generator swng equaton s coupled wth ts dq wndngs and the rest of the system through P e. For example, n the two axs generator model, P e = E d I d + E q I q +(X q X d )I di q, where E d and E q are the d-axs and q-axs transent voltages, respectvely; X d and X q are generator d-axs and q-axs transent reactance, respectvely; I d and I q are the d and q axs currents, respectvely. For more detaled model, the expresson s dfferent. Motvated by the aforementoned analyss, f P e s measured and taken as the model nput whle the measured frequency by PMUs s treated as outputs, the swng equaton can be decoupled from the rest of the system. Furthermore, no nformaton of the d and q damper wndngs s requred. By dong that, no complex generator model s assumed. The physcal mean of ths decouplng approach can be explaned as follows: when there s a dsturbance at one pont of the power system, synchronous generators wll response to t through actons on the rotors; these responses reveal themselves n ther real power njectons and frequency. In other words, the generator swng dynamcs are coupled wth the rest of the system at the pont of connecton, and ts nteractons wth the rest of the system are through the real power njectons and frequency. If

4 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 4 real power njectons and frequency are measured by a PMU, ts swng responses to the dsturbance are captured completely and no other system nformaton s requred. B. Proposed Robust Decentralzed Dynamc State Estmator The model decouplng approach enables a generator swng equaton to be decoupled from the rest of the system model, whch n turn allows us to rely only on local measurements to estmate the rotor speed and angle of a generator. The dscretetme state representaton of the th synchronous generator s x k = f ( x k, u k) + w k, (8) z k = h ( x k, u k) + v k, (9) where x k s the state vector, ncludng the generator rotor speed ω and rotor angle δ ; zk s the measurement vector that contans rotor speed z k provded by PMUs and frequency z k by local meterng devces or frequency montorng network devces [3]; f ( ) represents the dscrete-tme form of (6) and (7) whle zk = [z k z k ] T and z k = ω + v k, z k =(+ ω )f +v k ; f s the nomnal system frequency; h ( ) =Hk x k and H k s a constant matrx that can be derved drectly from the measurement equatons of z k and z k ; wk and vk =[v k v k ] T are the process and observaton nose, respectvely; they are assumed to be Gaussan wth zero mean and covarance matrces Q k and R k, respectvely; u k s the nput that contans the real power njecton of the th generator. Based on the derved dscrete-tme state space equatons (8) and (9), the dynamc state estmator (DSE) can be used to estmate the generator rotor speed and angle usng local measurements. In ths paper, the UKF s chosen as the basc DSE as t acheves a more balanced performance between computatonal effcency and ablty to cope wth strong system nonlneartes than the extended Kalman flter, or the partcle flter [9]. However, UKF has been proved to be senstve to gross errors, cyber attacks and loss of measurements, etc []. To handle these ssues, a robust Generalzed Maxmumlkelhood-type UKF (GM-UKF) s proposed. It conssts of four major steps, namely a batch-mode regresson form step, a robust pre-whtenng step, a robust regresson state estmaton step, and a robust error covarance matrx updatng step. In the followng subsectons, we wll dscuss them n detal. Note that the ndex s neglected for smplcty but wthout the loss of generalty. ) Derve Batch-Mode Regresson Model: Gven a state estmate at tme step k-, x k k R n, havng a covarance matrx gven by Pk k xx, ts statstcs are captured by n weghted sgma ponts defned as [9] ( ) χ = x k k k k ±, () np xx k k wth weghts w = n, =,..., n, where n s the number state varables for each generator. Then, each sgma pont s propagated through the nonlnear system process model ( 8), yeldng a set of transformed samples expressed as ) χ (χ = f. () k k Next, the predcted sample mean and sample covarance matrx of the state vector are calculated by n P xx = = x = n = w χ, () w (χ x )(χ x ) T +Q k. (3) We defne x = x k k, where x k s the true state vector; k s the predcton error and E [ ] k T k = P xx. By processng the predctons and observatons smultaneously, we have the followng batch-mode regresson form: [ ] [ ] [ ] z k Hk v = x x I k + k (4) k whch can be rewrtten n a compact form z k = H k x k + ẽ k, (5) and the error covarance matrx s W k = E [ [ ] ẽ k ẽ T ] Rk k = P xx = S k Sk T, (6) where I s an dentty matrx; S k s calculated by the Cholesky decomposton technque. ) Perform Robust Pre-whtenng: Before carryng out a robust regresson, the state predcton errors of the batch-mode regresson form need to be uncorrelated. Ths can be done by pre-multplyng S k on both sdes of (5), yeldng S k z k = S H k k x k + S k ẽk, (7) whch can be further organzed to the compact form y k = A k x k + ξ k, (8) where E[ξ k ξ T k ]=I. However, f outlers occur, the use of S k for prewhtenng wll cause negatve smearng effect. To handle ths ssue, we frst detect and downweght the outlers by means of weghts calculated usng the projecton statstcs (PS) [] and a statcal test appled to them. Those weghts contrbute to the robust prewhtenng and ther functonals wll be shown later n the objectve functon. Specfcally, we apply the PS to a -dmensonal matrx Z that contans serally correlated samples of the nnovatons and of the predcted state varables. Formally, we have [ ] zk H Z = k x k k z k H k x, (9) x k k x where z k H k x k k and z k H k x are the nnovaton vectors whle x k k and x are the predcted state vectors at tme nstants k- and k, respectvely. The PS values of the predctons and of the nnovatons are separately calculated because the values taken by the former and the latter are centered around dfferent ponts. The mplementaton of PS can be found n []. Once the PS values are calculated, they are compared to a threshold to dentfy outlers. Accordng to our prevous work [], [], Z can be shown to follow a bvarate Gaussan probablty dstrbuton and the calculated PS values usng Z follow a ch-square dstrbuton wth degree of freedom. As

5 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 5 a result, the outlers can be flagged f ther PS values satsfy PS > χ,.975 at a sgnfcance level 97.5% n the statstcal test, and are further downweghted va [ ] ϖ = mn (,d / PS ), () where the parameter d s set as.5 to yeld good statstcal effcency at Gaussan dstrbuton. Remark: Except for the occurrence of outlers n rotor speed measurements, outlers may occur n the measured real power njectons as well, and consequently, yeldng ncorrect predcted states, called nnovaton outlers [ 3]. In such condton, the predcted state correspondng to ncorrect real power njecton wll be flagged as outlers. Snce the local measurement redundancy s not hgh, we wll not downweght t drectly, nstead we propose to replace the current real power njecton by ts prevous value and obtan the new state predctons. By dong that, we can acheve a better statstcal effcency. 3) Carry out Robust Regresson: To address the data qualty ssues, we develop a robust GM-estmator that mnmzes the followng objectve functon: J (x k )= l ϖ ρ (r S ), () = where l = m+n and m s the number of measurements; ϖ s calculated by (); r S = r /sϖ s the standardzed resdual; r = y a T x s the resdual, where at s the th row vector of the matrx A k ; s =.486 b m medan r s the robust scale estmate; b m s a correcton factor; ρ( ) s the convex Huber-ρ functon, that s { ρ (r S )= r, for r S S < λ λ r S λ /, (), elsewhere where the parameter λ s typcally chosen to be between.5 and 3 to acheve hgh statstcal effcency n the lterature [ 4]. To mnmze (), the followng necessary condton must be satsfed J (x k ) x k = l = ϖ a ψ (r S )=, (3) s where ψ (r S )= ρ(r S )/ r S s the so-called ψ-functon. By dvdng and multplyng the standardzed resdual r S to both sdes of (3) and puttng t n a matrx form, we get A T k Λ (y k A k x k )=, (4) where Λ =dag(q (r S )) and q (r S )=ψ (r S )/r S. By usng the IRLS algorthm [], the state estmates at the j teraton can be updated usng ( ) A x (j+) = A T k Λ (j) T A k k Λ (j) y k, (5) where x (j+) = x (j+) x (j). The algorthm converges when x (j+). 4) Update Error Covarance Matrx: After the convergence of the algorthm, the estmaton error covarance matrx P xx of the GM-UKF needs to be updated so that the state predcton at the next tme sample can be performed. Followng the work from [], we derve the estmaton error covarance matrx of our GM-UKF as P xx = EF [ψ (r S )] ( ) A T ( )( ) {E F [ψ (r S )]} k A k A T k Q ϖ A k A T k A k (6) where Q ϖ = dag ( ) ϖ. Remark: the proposed robust DSE s supposed to be performed for each generator substaton. It can be mplemented at the control center as well f all the generator data and PMU measurements are transmtted from local substatons to t. However, there exst several concerns by dong so, such as ncreased communcaton burden that s dscussed n the next subsecton and the delayed local control, etc. Indeed, f all the calculatons are performed at the control center whle some local controls are requred at ths perod, the estmated bus frequency for those local controls can be delayed; by contrast, our robust DSE s frst conducted locally usng local PMU measurements and ts estmated rotor speeds and angles can be used for local controls; n the meantme, they can be transmtted to control center for bus frequency estmaton and coordnated control. In practce, f t s costly to mplement the decentralzed DSE for each generator substaton, we can do t at the local phasor data concentrator (PDC) level or regonal system level. Ths can stll save a lot of communcaton burden and enable the tmely local control actons compared wth the fully centralzed strategy. C. Bus Frequency Estmaton When the rotor speed and rotor angle of each generator are obtaned, they need to be communcated to the control center for bus frequency estmaton. Dependng on the applcatons, there are two ways to communcate the estmaton results. If the control center s only nterested n montorng and regulatng the system frequency, the rotor speed of each generator s transmtted and the bus frequency s estmated usng (5). Compared wth the conventonal strategy, that s, all the measured voltage magntudes and angles, current magntudes and angles, and frequency by PMUs are communcated, the communcaton burden of our proposed approach s only % of t; If the control center are nterested n both frequency and rotor angle stablty montorng and control, rotor speed and angle estmates are communcated. In ths case, t only requres 4% communcaton burden of the conventonal strategy. Note that the total computng tme of the proposed approach conssts of two parts: the decentralzed DSE and the projecton of the rotor speed to bus frequency through ( 5). Snce each robust DSE s performed locally and ts estmates are communcated to the control center through ts communcaton lnk, the proposed DSE s ndependent of the sze of the power system. The only concern of the proposed RFD for very large-scale power system onlne applcatons s that D becomes rather dense, whch requres a lot of computer

6 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 6 memory. However, ths s not a problem f we use the sparse matrces B BB, B G and B BG nstead of D for the projecton of rotor speeds to bus frequences (see equaton () n [ ]). As a result, the computatonal burden of ths step s neglgble. Frequency at bus 34 [pu] D method Generator rotor speed Fg. : Comparng the estmated frequency at bus 34 by, and D-method wth normal measurement nose n the IEEE 39-bus system. Root Mean Squared Error [per unt] D-Method Fg. 3: RMSE of, and D-method wth normal measurement nose n the IEEE 39-bus system. Remark: The reason that the complcated generator model s not requred has been dscussed n the model decouplng secton. We dscuss here how the proposed RFD s not dependent on load model. It s well-known that the power system dynamcs are dfferent f dfferent load models are assumed. However, although system dynamc behavors are dfferent, they are reflected on the varatons of the rotor speeds of synchronous generators. Snce the proposed RFD s based on such varatons, load models have been mplctly taken nto account. Ths concluson has also been verfed by extensve smulaton results n []. Root Mean Squared Error [per unt] D-Method Fg. 4: RMSE of, and D-method wth large measurement nose n the IEEE 39-bus system, where the nose covarance matrx s changed from 6 I to 4 I. IV. NUMERICAL RESULTS In ths secton, extensve smulatons on the IEEE 39-bus test system wll be carred out to demonstrate the effectveness and robustness of the proposed RFD. Specfcally, at t=.5s, the Generator 4 connected to bus 33 s trpped to smulate system dsturbance. The transent stablty smulatons are performed to generate measurements and true state varables usng the Matlab-based software PST wth some revsons [5]. The fourth order Ruger-Kutta approach s adopted wth ntegraton step t=/s to solve dfferental and algebrac equatons. The measured real power njecton of each generator s taken as model nput, whle the measured generator rotor speed and frequency by PMU are treated as outputs/measurements. A random Gaussan varable wth zero mean and varance equal to 6 s assumed for system process nose. The generator model assumed for transent smulaton s the detaled two-axs generator model, whose parameter values are taken from [ 6]. The root-mean-squared error (RMSE) of all bus frequences s used as the performance ndex whle the estmated frequency at bus 34 s taken for llustraton. Note that, Generator 5 s connected to bus 34. The proposed non-robust UKF based method wll be called, and the proposed robust UKF based method s called whle the orgnal proposal [] that works on D matrx drectly wll be called the D- method. A. Estmaton Results wth Nosy Measurements All the methods are tested wth nosy measurements. Normal and large nose are consdered; ther nose covarance matrces are assumed to be 6 I and 4 I wth approprate dmensons, respectvely. The results are shown n Fgs. - 4, where n Fg. the rotor speed of Generator 5 s shown. From Fg., we observe that the rotor speed of Generator 5 s dfferent from ts termnal bus frequency. Ths dfference s caused by two factors: () generator nternal mpedance and () severty of the transent (e.g., how much rotor speeds dffer from each other). On the other hand, by observng both Fgs. and 3, t s found that the D-method s one of the most senstve method to measurement nose whle our and

7 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 7 Frequency at bus 34 [pu] D method Root Mean Squared Error [per unt] 8 x D Method Fg. 5: Estmated frequency at bus 34 by, and D-method wth observaton outlers n the IEEE 39-bus system, where the measured rotor speed of Generator 5 s contamnated wth % error from t=4s to t=6s. 5 5 Fg. 6: RMSE of, and D-method wth observaton outlers n the IEEE 39-bus system, where the measured rotor speed of Generator 5 s contamnated wth % error from t=4s to t=6s. approaches are able to flter them out. Ths s because n D-method, the nose s drectly propagated through the rotor speed measurements to the bus frequency. By contrast, our methods adopt the UKF to flter out the nose, yeldng better performance. When we ncrease the measurement nose level,.e., the covarance matrx s changed from 6 I to 4 I, the results of D-method become even worse whle our methods can acheve comparable performance as those n the former case (see Fg. 4). Frequency at bus 34 [pu] D method B. Impact of Observaton and Innovaton Outlers Due to cyber attacks, mperfect phasor synchronzaton, the saturaton of meterng current transformers or by meterng Couple Capactor Voltage Transformers (CCVTs), to name a few, gross errors can occur n the PMU measurements []. As for our decentralzed DSE-based bus frequency estmaton problem, there are two ways to nduce outlers: ) the measured rotor speed by PMUs s contamnated wth gross error, whch s called observaton outler; ) snce the real power njecton measured by PMUs can be contamnated wth gross error, treatng t as model nput can yeld ncorrect rotor speed predctons, whch s called nnovaton outler. Note that, as D- method s workng drectly wth rotor speed measurements, t s affected by observaton outlers whle beng ndependent of the nnovaton outler caused by ncorrect real power njecton measurements. To ths end, two cases are consdered: Case : the measured rotor speed of Generator 5 s contamnated wth % error from t=4s to t=6s to smulate observaton outler. Case : the measured real power njecton of Generator 5 s contamnated wth 3% error from t=3s to t=6s to smulate nnovaton outler. The test results for Case are shown n Fgs. 5 and 6. From these two fgures, we fnd that the estmaton results of the and the D-method are sgnfcantly based n the presence of observaton outlers. s less senstve to the observaton outler compared wth the D-method. By Fg. 7: Estmated frequency at bus 34 by, and D-method wth nnovaton outlers n the IEEE 39-bus system, where the measured real power njecton of Generator 5 s contamnated wth 3% error from t=3s to t=6s. contrast, our s able to suppress the observaton outlers thanks to the robustness provded by PS and the GMestmator, yeldng neglgble bas of the estmaton. It should be noted that due to the smearng effect of applyng ( 5) for bus frequency estmaton, the estmated frequences at many buses are affected by the ncorrect rotor speed of the generator 5. Ths however does not happen n our. The test results for Case are shown n Fgs. 7 and 8. As expected, the results of are based n the presence of nnovaton outlers whle the D-method s not affected. Due to the robustness of the proposed DSE, ths nnovaton outler has been suppressed. By comparson, stll outperforms D- method, yeldng the best results. C. Loss of PMU Measurements Due to the falures of communcaton lnks between PMU and phasor data concentrator or cyber attacks, the PMU placed at Bus 34, where Generator 5 s connected, s assumed to

8 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 8 Root Mean Squared Error [per unt] x 4 D Method 5 5 Fg. 8: RMSE of, and D-method wth nnovaton outlers n the IEEE 39-bus system, where the measured real power njecton of Generator 5 s contamnated wth 3% error from t=3s to t=6s. Root Mean Squared Error [per unt] 8 x D Method 5 5 Fg. : RMSE of, and D-method wth measurement losses from t=3s to t=6s n the IEEE 39-bus system. Frequency at bus 34 [pu] D method Root Mean Squared Error [per unt] D-Method Fg. 9: Estmated frequency at bus 34 by, and D-method wth measurement losses from t=3s to t=6s n the IEEE 39-bus system. loss ts measurements from t=3s to t=6s. Therefore, the measurement set becomes unavalable durng ths tme nterval and ther values are set equal to zero for smulaton purpose. Please note that n ths extreme case, both predcted and measured rotor speeds wll be flagged as outlers. To enable the estmaton of bus frequency by the proposed method, we advocate to ether recover the mssng data usng [ 7] or perform short-term forecastng of the PMUs usng ther spatal and temporal correlatons [8]. The test results are presented n Fgs. 9 and. It can be seen from these two fgures that the estmated bus frequences of both the and the D-method are based sgnfcantly. But the approach s less senstve to the measurement losses than the D-method. As a result, the proposed can always track the bus frequency relably and accurately. It should be noted that the two mtgaton approaches [7], [8] can be used n the stuaton that the measurements are temporally lost (a few seconds). For longer perod, they may not be vald. Otherwse, Fg. : RMSE of, and D-method wth normal measurement nose n the IEEE 45-bus system. the operator wll be warned and the decentralzed DSE stops before further careful nvestgatons are done. D. Results on Large-Scale Systems To demonstrate the applcablty of the proposed robust frequency dvder for large-scale system, the 5-machne IEEE 45-bus system s used. The dynamc data can be found through [9]. The generator located at bus 6 s trpped at t=.5s to smulate the system dsturbance. The followng three scenaros are consdered and tested: Scenaro : only normal Gaussan nose s added to the smulated data lke the test done n Secton IV-A; Scenaro : % of the measured generator rotor speeds s contamnated wth % error from t=s to t=5s; Scenaro 3: 5% of the measured generator rotor speeds are lost from t=3s to t=6s. The test results of all three scenaros are dsplayed n Fgs. -3, where the RMSE of scenaro and the estmated frequences of bus 73 under scenaros and 3 are represented

9 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 9 Frequency at bus 73 [pu] D-method Fg. : Estmated frequency at bus 73 by, and D-method wth outlers n the IEEE 45-bus system, where % of the measured generator rotor speeds s contamnated wth % error from t=s to t=5s. Frequency at bus 73 [pu] D-method Fg. 3: Estmated frequency at bus 73 by, and D-method wth measurement losses n the IEEE 45-bus system, where 5% of the measured generator rotor speeds are lost from t=3s to t=6s. accordngly. Note that bus 73 s close to the trpped generator. Based on these fgures, we conclude that our proposed robust frequency estmator stll outperforms the other two alternatves n a larger-scale system. These results are consstent wth those for the IEEE 39-bus system. It s nterestng to fnd that even wthout usng mssng data recoverng approach [7], [8], our proposed method s able to rely on good PMU measurements and the predcted dynamc state varables for flterng, yeldng good estmaton results. E. Computatonal Effcency To valdate the capablty of the proposed method for onlne estmaton, that s, to be compatble wth PMU samplng rate, ts computatonal effcency s analyzed. All cases and scenaros smulated n the prevous sectons are consdered. All the tests are performed on a PC wth Intel Core 5,.5 GHz, 8GB of RAM. The average computng tme of each TABLE I: Average Computng Tmes of The Three Methods At Each PMU Sample Scenaros D-method Secton IV-A.ms.5ms.37ms Secton IV-B Case.ms.4ms.45ms Secton IV-B Case.ms.6ms.46ms Secton IV-C.ms.5ms.46ms Scenaro.8ms.66ms.6ms Scenaro.9ms.78ms.ms Scenaro 3.85ms.79ms.4ms method for every PMU sample s dsplayed n the Table I. We observe from ths table that all methods have comparatve computatonal effcency and ther computng tmes are much lower than the PMU samplng perod, whch are 6.7ms and 8.3ms for 6 samples/s and samples/s, respectvely. On the other hand, although s the most tme consumng method compared wth and D-method, ts computng tme s neglgble for practcal applcatons consderng the PMU samplng speed. Note that the decentralzed DSE for each generator can be carred out ndependently and n a parallel manner, whch are very fast to calculate and ndependent of the scale of the power system. For very large-scale power systems, D becomes rather dense and a numercal stable and computatonal effcent approach proposed n [ ] s used to project the rotor speeds to bus frequences. The computatonal burden of ths step has been shown to be neglgble []. In concluson, the proposed method s sutable for large-scale power system onlne applcatons. V. CONCLUSION A robust frequency dvder (RFD) s proposed to estmate the frequency of each bus n a power system. Our RFD conssts of two steps, namely the estmaton of generator rotor speeds through a robust decentralzed UKF usng local measurements, and the projecton of all generator rotor speeds to bus frequency. The proposed RFD s model ndependent, and the knowledge of local PMU measurements at each generator termnal bus and transmsson system lne parameters s suffcent. Furthermore, the proposed RFD s able to flter out measurement nose, suppress gross measurement errors, handle cyber attacks as well as measurement losses. Extensve results carred out on the IEEE 39-bus and 45- bus systems demonstrate the effectveness and the robustness of the proposed method. A possble ssue of the proposed RFD s that the decentralzed and centralzed scheme may produce some delays for the estmated bus frequences used for controllers. However, thanks to the advancement of control technques, the tme delays can be effectvely mtgated [ 3], [3]. In the future work, we wll desgn ths type of robust frequency regulator based on our RFD. REFERENCES [] Natonal Grd Frequency Response Workng Group, Frequency Response Techncal Sub-Group Report, Nov., Tech. Rep. [Onlne]. [] J. Van de Vyver, J. D. M. De Koonng, B. Meersman, L. Vandevelde, T. L. Vandoorn Droop control as an alternatve nertal response strategy for the synthetc nerta on wnd turbnes, IEEE Trans. Power Syst., vol. 3, no., pp. 9-38, 6.

10 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL., NO., 7 [3] H. Ye, P. Pe, Z. Q, Analytcal modelng of nertal and droop responses from a wnd farm for short-term frequency regulaton n power systems, IEEE Trans. Power Syst., vol. 3, no. 5, pp , 6. [4] J. B. Zhao, G. Zhang, M. L. Scala, Z. Wang, Enhanced robustness of state estmator to bad data processng through mult-nnovaton analyss, IEEE Trans. Industral Informatcs, vol. 3, no. 4, pp. 6-69, 7. [5] C. Zhao, U. Topcu, S. H. Low, Optmal load control va frequency measurement and neghborhood area communcaton, IEEE Trans. Power Syst., vol. 8, no. 4, pp , 3. [6] Y. Wen, W. L, G. Huang, X. Lu, Frequency dynamcs constraned unt commtment wth battery energy storage, IEEE Trans. Power Syst., Vol. 3, no. 6, pp , 6. [7] IEEE Task Force on Load Representaton for Dynamc Performance, Load representaton for dynamc performance analyss of power systems, IEEE Trans. Power Syst., vol. 8, no., pp , 993. [8] L. Wang, D. Z. Fang, T. S. Chung, New technques for enhancng accuracy of EMTP/TSP hybrd smulaton algorthm, n Proc. IEEE Int. Conf. Electr. UtltyDeregulaton, Restructurng Power Technol., pp , 4. [9] G. Radman, M. A. Tabrz, Smulaton of wde area frequency measurement from phasor measurement unts (PMUs) or frequency dsturbance recorders (FDRs), Oct. [Onlne]. [] M. La, M. Nakano, G. Hseh, Applcaton of fuzzy logc n the phase-locked loop speed control of nducton motor drve, IEEE Trans. Industral Electroncs, vol. 43, no. 6, pp , 996. [] J. Nutaro, V. Protopopescu, Calculatng frequency at loads n smulatons of electro-mechancal transents, IEEE Trans. Smart Grd, vol. 3, no., pp. 33 4,. [] F. Mlano, A. Ortega, Frequency dvder, IEEE Trans. Power Syst., Vol. 3, no., pp , 7. [3] Y. Zhang, P. Markham, et al., Wde-area frequency montorng network (FNET) archtecture and applcatons, IEEE Trans. Smart Grd, vol., no., pp ,. [4] A. G. Phadke, B. Kasztenny, Synchronzed phasor and frequency measurementunder transent condtons, IEEE Trans. Power Delv., Vol. 4, no., pp , 9. [5] E. Ahad, M. Kezunovc, Impact of electromechancal wave oscllatons propagaton on protecton schemes, Electrc Power Systems Research, Vol. 38, pp. 85-9, 6. [6] NERC Relablty Gudelne, PMU Placement and Installaton, 6. [7] NERC Relablty Gudelne, Power Plant Dynamc Model Verfcaton usng PMUs, 6. [8] P. Sauer, M. A. Pa. Power system dynamcs and stablty. Urbana, 997. [9] S. Juler, J. K. Uhlmann, Unscented flterng and nonlnear estmaton, Proceedngs of the IEEE, Vol. 9, no. 3, pp. 4 4, 4. [] J. B. Zhao, M. Netto, L. Ml, A robust terated extended Kalman flter for power system dynamc state estmaton, IEEE Trans. Power Syst., vol. 3, no. 4, pp , 7. [] M. Gandh, L. Ml, Robust Kalman flter based on a generalzed maxmum-lkelhood-type estmator, IEEE Trans. Sgnal Processng, vol. 58, no. 5, pp. 59 5,. [] J. B. Zhao, L. Ml, Robust unscented Kalman flter for power system dynamc state estmaton wth unknown nose statstcs, IEEE Trans. Smart Grd, 7. [3] J. B. Zhao, L. Ml, Power System Robust Decentralzed Dynamc State Estmaton Based on Multple Hypothess Testng, IEEE Trans. Power Syst., 7. [4] P. J. Huber, Robust Statstcs. New York: Wley, 98. [5] J. H. Chow, K. W. Cheung, A toolbox for power system dynamcs and control engneerng educaton and research, IEEE Trans. Power Syst., vol. 7, no. 4, pp , 99. [6] IEEE PES TF on Benchmark System for Stablty Controls, Benchmark systems for small-sgnal stablty analyss and control, Aug. 5. [7] P. Gao, M. Wang, et. al, Mssng data recovery by explotng lowdmensonalty n power systems synchrophasor measurements, IEEE Trans. Power Syst., vol. 3, no., pp. 6-3, 6. [8] Y. Chakhchoukh, V. Vttal, G. T. Heydt, PMU based state estmaton by ntegratng correlaton, IEEE Trans. Power Syst., vol. 9, no., pp , 4. [9] [3] S. Prasad, S. Purwar, N. Kshor, H-nfnty based non-lnear sldng mode controller for frequency regulaton n nterconnected power systems wth constant and tme-varyng delays, IET Generaton, Transmsson & Dstrbuton, vol., no., pp , 6. [3] T. Ramachandran, M. H. Nazar, S. Grjalva, M. Egerstedt, Overcomng communcaton delays n dstrbuted frequency regulaton, IEEE Trans. Power Syst., Vol. 3, no. 4, pp , 6. Junbo Zhao (S 3) s fnshng hs Ph.D. degree n the Sprng semester of 8 at Bradley Department of Electrcal and Computer Engneerng, Vrgna Polytechnc Insttute and State Unversty (Vrgna Tech). Before that, he receved the Bachelor degree from Southwest Jaotong Unversty n electrcal engneerng n. He dd the summer nternshp at Pacfc Northwest Natonal Laboratory from May-August, 7. He s now the char of IEEE Task Force on Power System Dynamc State and Parameter Estmaton, the secretary of the IEEE Workng Group on State Estmaton Algorthms and the IEEE Task Force on Synchrophasor Applcatons n Power System Operaton and Control. He has wrtten book chapters, publshed more than 3 peer-revewed journal and conference papers and 9 Chnese patents. Hs research nterests nclude power system real-tme montorng, operatons and securty that nclude power system state estmaton, dynamcs and stablty, statc and dynamc load modelng, power system cyber attacks and countermeasures, bg data analytcs and robust statstcs wth applcatons n the smart grd. Lamne Ml (LF 7) receved the Electrcal Engneerng Dploma from the Swss Federal Insttute of Technology, Lausanne, n 976, and the Ph.D. degree from the Unversty of Lège, Belgum, n 987. He s a Professor of Electrcal and Computer Engneerng, Vrgna Tech, Blacksburg. He has fve years of ndustral experence wth the Tunsan electrc utlty, STEG. At STEG, he worked n the plannng department from 976 to 979 and then at the Test and Meter Laboratory from 979 tll 98. He was a Vstng Professor wth the Swss Federal Insttute of Technology n Lausanne, the Grenoble Insttute of Technology, the École Supéreure D électrcté n France and the École Polytechnque de Tunse n Tunsa, and dd consultng work for the French Power Transmsson company, RTE. Hs research has focused on power system plannng for enhanced reslency and sustanablty, rsk management of complex systems to catastrophc falures, robust estmaton and control, nonlnear dynamcs, and bfurcaton theory. He s the co-founder and co-edtor of the Internatonal Journal of Crtcal Infrastructure. He s the charman of the IEEE Workng Group on State Estmaton Algorthms. He s a recpent of several awards ncludng the US Natonal Scence Foundaton (NSF) Research Intaton Award and the NSF Young Investgaton Award. Federco Mlano (S, M 4, SM 9, F 6) receved from the Unv. of Genoa, Italy, the ME and Ph.D. n Electrcal Eng. n 999 and 3, respectvely. From to he was wth the Unv. of Waterloo, Canada, as a Vstng Scholar. From 3 to 3, he was wth the Unv. of Castlla-La Mancha, Span. In 3, he joned the Unv. College Dubln, Ireland, where he s currently Professor of Power System Control and Protectons. He s an IEEE Fellow and IET Fellow. Hs research nterests nclude power system modelng, stablty analyss and control. He s or has been edtor of the IEEE Transactons on Power Systems and IET Generaton, Transmsson & Dstrbuton.

11 Ths Artcle s part of a project that has receved fundng from the European Unon's Horzon research and nnovaton programme under grant agreement Nº7748