Kalman-Filter Based Recursive Regression for Three- Phase Line Parameter Estimation using Synchrophasor Measurements

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1 Klmn-Filter Bsed Reursive Regression for Three- Phse Line Prmeter Estimtion using Synhrophsor Mesurements Chetn Mishr, StMIEEE Virgilio A. Centeno, SMIEEE Brdley Deprtment of Eletril nd Computer Engineering, Virgini Polytehni Institute nd Stte University, Blksurg, Virgini-24061, USA Emil: ; Anmitr Pl, StMIEEE Network Dynmis nd Simultion Siene Lortory, Virgini Bioinformtis Institute, Virgini Polytehni Institute nd Stte University, Blksurg, Virgini-24061, USA Emil: Astrt In this pper estimtion of three-phse trnsmission line prmeters is done with the help of synhrophsor mesurements y using reursive regression tehnique sed on the Klmn filter. The errors in the regression vetor due to presene of noise in the synhrophsor dt re lso ounted for while estimting the prmeters. The performne of this tehnique is demonstrted for medium length trnsmission line. The results indite tht the proposed pproh is le to suessfully ompute three phse trnsmission line prmeters of non-trnsposed, non-symmetri lines. Index Terms--Klmn filter, Phsor mesurement units (PMUs), Reursive regression, Synhrophsors, Trnsmission line prmeter estimtion I. INTRODUCTION An urte knowledge of trnsmission line prmeters is required for urte power flow modeling, trnsient stility nlysis, fult lotion/identifition, et. In the pst, lultions sed on ondutor dimensions, line length, nd sg were used to estimte these prmeters. However, due to the ssumptions nd pproximtions involved, for rel-time deision mking the estimtes were not very relile 1]. A good prmeter estimtor is one tht urtely trks the hnges in the prmeters ourring with usge, time nd loding nd this pper shows how tht n e done effiiently using phsor mesurements. With the development of synhrophsor sed-smrt grid tehnology, PMUs hve een extensively used for wide re monitoring nd protetion of power systems 2]. When pled in the power grid, these devies provide synhronized phsor mesurements over lrge trnsmission networks. The use of these mesurements for stte estimtion, trnsient stility nlysis, ontrol of interre osilltions, et. hs lredy een proposed 3]-5]. The primry dvntge of synhrophsor mesurements over trditionl mesurements is its higher ury 6], nd it is this feture of PMUs whih mkes it very useful for rel-time line prmeter estimtion. Different pprohes hve een previously suggested for estimting line prmeters using PMUs 1], 7]-9]. In 1], estimtion of positive sequene line prmeters using moving totl lest squres (TLS) window nd PMU dt hd een proposed. An optiml estimtor using positive sequene voltge nd urrent mesurements from oth ends of the line ws proposed in 7]. Their method involved initilly identifying nd removing the d mesurements efore performing the estimtion. The method presented in 8] utilized voltge nd urrent phsors from one end of line ssuming the other end s open or short-iruited. The identifition of short trnsmission line prmeters ws proposed in 9]. However, n ssumption tht ws mde in most of these ppers ws tht trnsmission lines were ssumed to e fully trnsposed nd lned nd/or symmetri. Thus the prmeters were estimted for single phse. Although 8] onsidered three phse lines, their lgorithm required the knowledge of the modl trnsformtion mtrix whih would not e known for generl three-phse trnsmission line. Thus no sustntil work hs een done on three phse nontrnsposed, non-symmetri line prmeter estimtion while ounting for rndom errors in PMU mesurements of the individul phses. The im of this pper is to ddress this issue. The noise in PMU mesurements is primrily due to the instrument trnsformers whih provide the nlog input tht is stepped down, digitized nd proessed to get the phsor output. In our work, the instrument trnsformers hve een ssumed to e urtely lirted. One wy for doing tht using PMUs hs lredy een developed in 10]. The rest of the pper is strutured s follows. The three phse line model tht hs een used in this pper is presented in Setion II. The prolem formultion is done in Setion III. Setion IV gives n overview of the solution methodology. The effetiveness of the solution strtegy is demonstrted in Setion V in whih the results otined using the proposed pproh re ompred with those otined using weighted reprinting/repulishing this mteril for dvertising or promotionl purposes, reting new olletive works, for resle or redistriution to servers or lists, or reuse of ny opyrighted omponent of this work in other works. The pulished version of this rtile n e found t: /PESGM

2 lest squres (WLS) 11]. The onlusions re drwn in Setion VI. II. TRANSMISSION LINE MODEL The pi-model of two-us, three phse medium length trnsmission system is given in Fig. 1. The instrument trnsformers present in the different phses is lso shown. The prmeters of the line etween the two uses (P nd Q) tht re to e estimted re given in (1). = Fig. 1: Three-phse model of two-us system 10] Y = Z ] 1 = = j = j B B + B + + B + B B ] (1) + B + + B + ] (1) + B + + B + ] (1) ] (1d) In (1), is the series impedne mtrix nd is the dmittne mtrix. The dmittne mtrix is symmetri with eh entry hving rel nd imginry prt. Thus, there re 12 unknown prmeters in it. The shunt ondutnes of the les re negleted nd so the suseptne mtries (on the P nd Q side) re given y nd B, respetively. These re symmetri nd eh entry is purely imginry, resulting in 12 (= 6 on P side + 6 on Q side) unknowns. To summrize, the totl numer of prmeters to e estimted for three phse non-trnsposed trnsmission line re 24. If the three phse urrent vetors injeted from the P side nd the Q side re given y I nd I qp, respetively, nd the three phse voltge vetors for P nd Q us re given y V nd V, respetively, then using KCL we get, Y V = Y V + B V I Y V = Y V + B V I qp Adding (2) nd (2), we n write (3). (2) (2) I + I qp = B V + B V (3) As (3) is independent of the unknown prmeter Z, it n e used to estimte suseptne mtries on oth the sides. Then, one the shunt suseptne mtrix is ompletely known, the series impedne mtrix n e found out using (4). V V = Z (I B V ) (4) It is to e noted here tht three phse mesurement is equivlent to three single phse mesurements nd therefore onsists of 6 (= 3 rel + 3 imginry) mesurements. Thus, minimum of two sets of mesurements of voltges nd urrents on oth ends of the line re required to estimte the suseptne mtries. Then, one they re estimted, the impedne mtrix n e omputed using (4). III. PROBLEM FORMULATION From (3) nd (4) it n e relized tht the estimtion prolem eing delt with here is liner one. The formultion of the regression eqution for shunt suseptnes nd series impednes re done y expnding (3) nd (4) s shown elow in (5) nd (6), respetively. V V V z se V V V In (6), h 1 = I h 2 = I h 3 = I H se h h 2 0 h 3 ] = 0 h 2 0 h 1 h 3 0] 0 0 h 3 0 h 2 h 1 jb + B + j j B + B + θ se Z Z Z Z Z Z ] ] ] + B + B ] V V V V V V V V V ] ] ] (6) (6) Sine voltges nd urrents re phsors, (5) nd (6) re omplex equtions nd hve rel nd imginry omponents emedded in them. The rel nd imginry omponents n e expressed seprtely s shown elow in (7) nd (8). reprinting/repulishing this mteril for dvertising or promotionl purposes, reting new olletive works, for resle or redistriution to servers or lists, or reuse of ny opyrighted omponent of this work in other works. The pulished version of this rtile n e found t: /PESGM

3 z sus rel(z sh ) img(z sh ) ] I I I z sh + I qp H sus = img(h sh ) rel(h sh ) ] V 0 0 V + I qp ] = 0 V 0 V V + I qp 0 0 V 0 V z imp rel(z se ) img(z se ) ] θ sus θ sh ] (7) V 0 V V V V H imp = rel(h se ) img(h se ) img(h se ) rel(h se ) ] θ imp rel(θ se ) img(θ se ) ] H sh V 0 0 V V 0 V V V 0 0 V 0 V V V V 0 0 V 0 V V (8) In the ove equtions, (7) orresponds to the omputtion of shunt suseptnes, while (8) orresponds to the omputtion of series impednes. IV. SOLUTION METHODOLOGY The proposed method reursively estimtes the line prmeters. However, it does so under ertin ssumptions whih re desried s follows. The noise in eh PMU mesurement is ssumed to e Gussin with zero men nd stndrd devitions of σ V for voltges nd σ C for urrents. This is then dded to the ext per phse mesurements of voltges nd urrents so s to emulte field mesurements. Also the noises injeted in the different phses s well s in the different mesurement sets re ssumed to e independent/un-orrelted. In the senrio tht these ssumptions hold true, the trditionl liner regression model n e desried y (9). z(k) = H(k)θ(k) + ε(k) (9) In (9), θ(k) is the vetor of prmeters to e estimted, z(k) is the vetor orresponding to the k th mesurement set, H(k) is the regression oeffiient mtrix (mesurement mtrix) nd ε(k) is the zero-men Gussin noise in the mesurements. In presene of errors in the mesurement mtrix, (9) n e reformulted s shown in (10). V V 0 V V ] j z(k) = (H true (k) + δ(k))θ(k) + ε(k) θ sh B ] (5) = H true (k)θ(k) + ε new (k) (10) In (10), ε new (k) = ε(k) + δ(k)θ(k) is the new error hving new symmetri ovrine mtrix given y R new = ovr(ε new ). The elements of R new for shunt suseptne nd series impedne n e otined from (7) nd (8), respetively. Finlly, (10) n e reursively solved in the Klmn filter sense s shown in (11). In (11), θ (k k 1) is the priori-estimte of the prmeters, θ (k k) is the orreted estimte of the prmeters fter inorporting k th set of mesurements, P(k k 1) is the ovrine mtrix of errors ssoited with the estimted prmeters, nd K(k) is the Klmn gin. By reursively solving (11), the prmeters n e orretly estimted. Equtions (9)-(11) re initilly pplied to (7) for omputing the vlues of the shunt suseptnes. One tht is done, (9)- (11) is pplied to (8) to solve for the series impednes. The reson for following this order is tht the vlues of the shunt suseptnes re used in the omputtion of the series impednes. The flowhrt summrizing the omplete proess is given in Fig. 2. V. RESULTS The test system used for the study is the pi-model of twous, three phse medium length trnsmission line system s shown in Fig. 1. Multiple mesurements re mde on oth ends of the line for different loding onditions. The trnsmission lines re lso ssumed to e un-trnsposed. The nlysis is grouped under two lsses one in whih ll mesurements re urte, nd the seond in whih they hve errors in them. The results otined re desried s follows. reprinting/repulishing this mteril for dvertising or promotionl purposes, reting new olletive works, for resle or redistriution to servers or lists, or reuse of ny opyrighted omponent of this work in other works. The pulished version of this rtile n e found t: /PESGM

4 θ (k k 1) = θ (k 1 k 1) P(k k 1) = P(k 1 k 1) K(k) = P(k k 1)(H true (k)) T H true (k)p(k k 1)(H true (k)) T + R new ] 1 θ (k k) = θ (k 1 k 1) + K(k) (z(k) H true (k)θ (k k 1)) P(k k) = I K(k)H true (k)]p(k k 1) (11) reprinting/repulishing this mteril for dvertising or promotionl purposes, reting new olletive works, for resle or redistriution to servers or lists, or reuse of ny opyrighted omponent of this work in other works. The pulished version of this rtile n e found t: /PESGM

5 Set k=1 Define: θht sus (0 0)=Zeros(12X1) P sus (0 0)=Identity(12X12) Use kth mesurement to ompute z nd H sed on (7) Use (11) to ompute K(k), θht sus (k k) nd P sus (k k) Set k=k+1 Fig. 2: Flowhrt of proposed pproh A. For perfetly urte dt The prmeters re estimted using the proposed method for ten sets of idel mesurements (hving no errors). The results re shown in Figs. 3 nd 4. From the figures it is oserved tht the error vrine of the estimtion onverges to zero y the fifth mesurement set in se of shunt suseptnes nd y fourth mesurement set in se of series impednes. This indites tht y using the proposed pproh the orret vlue is rehed very quikly. However, it must lso e noted tht the numer of mesurement sets required to estimte the prmeters orretly depends on the initil vlue of the estimtion error ovrine mtrix P(k k). Closer it is to the true vlue, lesser the numer of mesurement sets required, nd vie-vers. Is differene etween two suessive estimtes > predefined tolerne Yes No Otin B nd B from θht sus Fig. 3: Estimtion of error vrine of shunt suseptnes vs numer of mesurement sets proessed Set k=1 Define: θht imp (0 0)=Zeros(12X1) P imp (0 0)=Identity(12X12) Use kth mesurement to ompute z nd H sed on (8) Use (11) to ompute K(k), θht imp (k k) nd P imp (k k) Is differene etween two suessive estimtes > predefined tolerne No Otin Z from θht imp Yes Set k=k+1 Fig. 4: Estimtion of error vrine of series impednes vs numer of mesurement sets proessed B. For dt with noise in mesurements Phsors reported y PMUs hve n error in mgnitude of less thn 1% nd re time tgged etter thn 1 miroseond. However the min soure of errors in the field mesurements hppens due to unised rndom noise nd errors oming from the instrumenttion hnnel. The d dt typilly ppers s spikes in the mesurements. These n e removed or lened up to ertin extent using d dt detetion/lening lgorithms 12], ut their omplete removl is not prtilly possile. Therefore, they will ffet the outome of the estimtion. For the nlysis done here, in order to test the performne of the proposed lgorithm, rndomly distriuted Gussin errors with zero men nd stndrd devition of 10-3 for voltges (σ V = 10 3 ) nd 10-2 for urrents (σ C = 10 2 ) re dded to the perfet mesurements. reprinting/repulishing this mteril for dvertising or promotionl purposes, reting new olletive works, for resle or redistriution to servers or lists, or reuse of ny opyrighted omponent of this work in other works. The pulished version of this rtile n e found t: /PESGM

6 The result otined using the proposed method nd the one otined using weighted lest squres (WLS) estimtion is shown in Tle I. From the tle, it n e oserved tht for most of the prmeters, the proposed method is more roust to the mesurement errors in omprison to WLS. Thus, the proposed method gives stisftory results in presene of rndom errors in the mesurements even if they nnot get deteted through d dt detetion lgorithms. TABLE I: COMPARISON OF ACTUAL AND ESTIMATED PARAMETERS USING WLS ESTIMATION AND THE PROPOSED APPROACH Prmeter Using WLS Using Proposed Approh Atul Vlue (In p.u.) Estimted Vlue % Error Estimted Vlue % Error (In p.u.) (solute vlue) (In p.u.) (solute vlue) j j j j j j j j j j j j j j j j j j VI. CONCLUSION In this pper, the estimtion of three-phse trnsmission line prmeters is performed with the help of synhrophsor mesurements y using reursive regression tehnique sed on the Klmn Filter. The novelty of this pproh in omprison to previous studies is tht it works for nontrnsposed three phse trnsmission lines. The errors in regression vetor orresponding to noise in the PMU mesurements re lso ounted for during the estimtion proess. This tehnique is lso omputtionlly more effiient euse the new sets of mesurements required to updte the estimted prmeter re dded reursively insted of repeting the whole proess with the ugmented dt set. The performne of this tehnique in omprison to WLS estimtion hs lso een demonstrted for given medium length trnsmission line. The results indite tht it is le to give urte results even in the presene of noisy mesurements. REFERENCES 1] L. Ding, T. Bi, nd D. Zhng, Trnsmission line prmeters identifition sed on moving-window TLS nd PMU dt, in Pro. IEEE Int. Conf. Adv. Power Syst. Automtion Protetion (APAP), Beijing, Chin, vol. 3, pp , Ot ] A. G. Phdke, nd R. M. de Mores, The wide world of wide-re mesurements, IEEE Power Energy Mg., vol. 6, no. 5, pp 52-65, Sep./Ot ] F. Go, J. S. Thorp, A. Pl, nd S. Go, Dynmi stte predition sed on Auto-Regressive (AR) model using PMU dt, in Pro. IEEE Power Energy Conf. Illinois (PECI), Chmpign, IL, pp. 1-5, Fe ] M. Li, A. Pl, A. G. Phdke, nd J. S. Thorp, Trnsient Stility Predition Bsed on Apprent Impedne Trjetory Reorded y PMUs, Int. J. Elet. Power Energy Syst., vol. 54, pp , Jul ] A. Pl, J. S. Thorp, S. S. Ved, nd V. A. Centeno, Applying roust ontrol tehnique to dmp low frequeny osilltions in the WECC, Int. J. Elet. Power Energy Syst., vol. 44, no. 1, pp , Jn ] P. Komrniki, C. Dzienis, Z. A. Styzynski, J. Blumshein, nd V. A. Centeno, Prtil experiene with PMU system testing nd lirtion requirements, in Pro. IEEE Power Energy So. Generl Meeting, Pittsurgh, PA, pp. 1-5, Jul ] Y. Lio, nd M. Kezunovi, Online optiml trnsmission line prmeter estimtion for relying pplitions, IEEE Trns. Power Del., vol. 24, no. 1, pp , Jn ] S. Kurokw, J. Pissolto, M. C. Tvres, C. M. Portel, nd A. J. Prdo, A new proedure to derive trnsmission-line prmeters: Applitions nd restritions, IEEE Trns. Power Del., vol. 21, no. 1, pp , Jn ] S. Di, D. J. Tylvsky, N. Logi, nd K. M. Koellner, Identifition of short trnsmission-line prmeters from synhrophsor mesurements, in Pro. 40 th North Amerin Power Symp., Clgry, AB, Cnd, pp. 1-8, Sep ] Z. Wu, K. Thoms, R. Sun, V. A. Centeno, nd A. G. Phdke, Threephse instrument trnsformer lirtion with synhronized phsor mesurements, in Pro. IEEE Power Eng. So. Conf. Innovtive Smrt Grid Tehnol., Wshington D.C., pp. 1-6, Jn ] C. L. Lwson, nd R. J. Hnson, Solving Lese Squres Prolems, Englewood Cliffs, NJ, Prentie Hll, ] K. D. Jones, A. Pl, nd J. S. Thorp, Methodology for Performing Synhrophsor Dt Conditioning nd Vlidtion, epted for pulition in IEEE Trns. Power Syst. reprinting/repulishing this mteril for dvertising or promotionl purposes, reting new olletive works, for resle or redistriution to servers or lists, or reuse of ny opyrighted omponent of this work in other works. The pulished version of this rtile n e found t: /PESGM