Nonlinear Least-Square Estimation (LSE)-based Parameter Identification of a Synchronous Generator

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1 Nonliner Lest-Squre Estimtion (LSE)-sed Prmeter Identifiction of Synchronous Genertor Yngkun Xu, Student Memer, IEEE, Yin Li, Student Memer, IEEE, Zhixin Mio, Senior Memer, IEEE Astrct The ojective of this pper is to identify synchronous genertor s internl voltge, dq-xis rectnces using Phsor Mesurement Unit (PMU) dt otined t the terminl us of the genertor Our pproch is to firstly clssify the PMU dt into inputs nd outputs nd then identify the reltionship etween the inputs nd the outputs With different clssifictions nd different ssumptions on slience, we otin three different estimtion models With the the reltionship expressed y nonliner lgeric equtions given, we then pply nonliner lestsqure estimtion (LSE) nd regulted nonliner LSE to conduct prmeter estimtion Cse studies re conducted to illustrte the estimtion procedure Index Terms Nonliner lest-squre estimtion, Prmeter estimtion, Phsor Mesurement Unit (PMU), Synchronous genertor I INTRODUCTION Accurte estimtion of synchronous genertor prmeters nd stte vriles plys n importnt role in predicting the dynmic performnce of the power system tht is crucil to ensure its stility nd reliility [1], [] In recent yers, Phsor Mesurement Unit (PMU) is developed in power systems Although PMU cnnot cpture the electromgnetic dynmics properly due to its lower smpling rte (-1Hz) [3] [6], the electromechnicl dynmics cn e reflected PMU dt hs ecome more nd more populr on identifiction nd optimiztion [3], [7] [9] If PMU is instlled t the terminl of synchronous genertor, it cn provide PMU mesurements including the terminl us voltge, us voltge phse ngle, rel power, nd rective power Both of the non-slient genertor nd the slient genertor re considered in this pper Non-slient genertor is ssumed to e rectnce ehind constnt voltge (E ) These two prmeters cn e estimted sed on three mesurements from PMU t the terminl us of the genertor The mesurement dt will e treted s the system s inputs nd outputs For non-slient genertor, there re two different wys to clssify the dt into inputs nd outputs They re nmed s Model 1 nd Model in Section II Furthermore, slience of genertor is considered Model 3 is designed to estimte prmeters including the internl voltge E, d-xis rectnce nd the q-xis rectnce X q with the sme three PMU mesurements As one of mjor estimtion methods, nonliner LSE is used s the min tool in this pper First, the input/output nonliner reltionships re formulted y lgeric equtions corresponding to different estimtion models Then, nonliner Y Xu, Y Li, nd Z Mio re with Dept of Electricl Engineering, University of South Florid, Tmp FL 336 Emil: zmio@usfedu LSE is solved using Guss-Newton method to estimte the unknown prmeters of synchronous genertor The reminder of this pper is orgnized s follows Section II introduces three estimtion models in three su-sections respectively The topology of the system is lso mentioned in Section II Section III shows the cse studies The mesurement dt re otined from time-domin simultions uilt in MATLAB/Simulink With the mesurement dt, prmeter estimtion is crried out nd the estimted vlues re compred with the rel vlues used for time-domin simultions Section VI concludes the pper The prmeters used in the timedomin simultion re listed in Appendix II ESTIMATION MODELS There re three estimtion models re proposed in this pper All of them re sed on the synchronous mchine model shown in Fig 1 The synchronous genertor model includes the swing dynmics, rotor flux dynmics, utomtic voltge regultor (AVR), nd turine-governor (TG) PMU is normlly instlled t the terminl us of the genertor We cn otin the mesurements of the terminl us voltge V, terminl us ngle θ, rel power P e, nd rective power Q e of the genertor y using PMU mesurements AVR TG SG V, P e Q e PMU θ X L Power grid Fig 1: Synchronous genertor connected to infinite us through trnsmission line A Model 1: P V s inputs nd Q s output In the first two estimtion models, the slience of synchronous genertor is not considered so the d-xis rectnce is equl to the q-xis rectnce ( =X q ) In ddition, it is ssumed tht the genertor resistnce is neglected compred with its rectnce Hence, the rel nd rective power of the genertor cn e expressed y the following equtions: { Pe = EV sin( ) Q e = E V cos( ) V (1) V

2 where is phse ngle difference etween the ngles of E nd V shown in Fig I q I d V jx ' did I ' E E jxqiq ' d d j( X -X ) I Fig : Phsor digrm shows the reltion mong E, V nd In Model 1, to void estimting rotor ngle, equtions (1) cn e converted to eqution for Q e () using severl steps: (E V ) = (E V sin( )) + (E V cos( )) (E V ) = (P e ) + (Q e + V ) (E ) V = Q e = Pe V () The difficulty here is tht the internl voltge E is not possile to e mesured In the cse studies, sudden lod chnge only cuses the smll vrition of E Therefore, it is ssumed tht the E is n internl vrile nd its finl stedystte condition is considered s prmeters to e estimted In Model 1, E nd re two constnt prmeters which require to e estimted We introduce two new prmeters, nd, to replce ( E ) nd 1 Then, () is rewritten s (3) which is used to design Model 1 Q e = V P e V (3) This lgeric eqution contins two prmeters nd to e estimted The input vector is u = [ ] T P e V nd the output vector is y = Q e The nonliner LSE cn e solved itertively using Guss- Newton method The estimted rective power ˆQe cn e clculted using (3) with the input dt u, nd n ssumed x = [ ] T Let us expnd nonliner eqution in Tylor series nd stopping t the second term A liner LSE prolem is uilt up s follows: Q e (mesured) ˆQ ek + F k d k (4) where F mtrix is the Jcoin mtrix of prtil derivtives of (3) with respect to nd d vector is the Guss-Newton correction k mens the itertions step (k ) The ove prolem cn e expressed y (5) d Q e1 Q e Q en } {{ } r = Q e Q e n Q e Q e n } {{ } F [ ] d where r vector is the residul vector of the mesurement Q e nd its estimtion ˆQ e The suscripts, 1 to n, is corresponding to ech step of mesurements In this pper, the smpling rte of PMU is 1Hz so n will e 1 if 1-second mesurements re used for estimtion d cn e identified directly using the norml eqution of LSE in (6) nd updted in ech itertion (5) d k = (F T k F k ) 1 F T k r k (6) At the first itertion, x ws evluted sed on the initil vector, θ = [ E ] T for Model 1 The selection rnge of θ will e discussed in Cse Studies The lengths of oth x nd θ re dependent on d so they re different in other two modes For the next itertions, x k is updted using (7) to clculte new ˆQ e [ ] k+1 k+1 x k+1 [ ] k = k x k [ ] k + k d k After enough itertions, nd cn e estimted successfully Then, it is esy to clculte E nd sed on estimted x Remrk : Although LSE method in (6) is le to estimte d, it cnnot gurnteed tht the precise solution in the strict sense In nother word, the proposed estimtion pproch will fil to produce the ccurte estimtion if mtrix inversion, F T k F k, hs the singulrity issue To overcome the forementioned issue, we need to improve (6) y dding λi in the inverse mtrix [1] This formul first derived y Tikhonov in 1963 [11] (7) d k = (F T k F k + λi) 1 F T k r k (8) where I is the identity mtrix nd λ is clled the regulriztion prmeter which is greter thn zero The effect of this λ will e indicted in Cse Studies B Model : V s input nd P Q s outputs Compred with Model 1, Model hs one input signl V nd two output signls P e nd Q e Also, it cn e used to estimte the dynmic prmeter, rotor ngle, since Model considers in the power eqution (9) s follows: { P e = V sin( ) Q e = V cos( ) V (9) where = E nd = 1

3 Considering Model use the sme estimtion methodology, the liner LSE model is uilt s follows: P e1 P e1 P e1 δ m1 P e1 P e P e P e P e δ m1 P en P e n P e n P e n δ Q e1 = mn Q e 1 δ Q e Q e Q e mn Q en Q e n Q e n Q e n n (1) Compred to Model 1, the length of r in Model is douled since it consists of two output mesurements, P e nd Q e The length of d increses to n + ecuse the numer of estimted is dependent on how mny steps of mesurements used for the estimtion Fig 3: Mtl/Simulink simultion model C Model 3: V s input nd P Q s outputs considering slience Model 3 hs the sme input nd outputs s with Model ut it considers slient pole ( X q ) With the slience, the power equtions, (11), ecome more complex: ( ) P e = E V sin( ) + V 1 X q 1 sin( ) ( ) Q e = E V cos( ) V cos (11) ( ) + sin ( ) X q In Model 3, E nd cn e estimted, long with X q III CASE STUDIES To generte the time-domin mesurements, nonliner model of the synchronous genertor connected to infinite us system ws uilt in MATLAB/SIMULINK The screenshot of the nonliner model is shown in Fig 3 The model uilding exploits the vector feture of MATLAB The detils of the technology re descried in our previous ppers [1] [14] The vlues of nonliner model prmeters used for the timedomin simultion s well s in the theoreticl clcultion re provided in APPENDIX The durtion of the simultion ws 5 seconds nd the step size ws 1s In nother word, the smpling rte of the mesurement ws 1Hz Considering tht sudden lod chnge event is more likely hppen thn three-phse fult, 1% step chnge ws pplied to the Re-express (11) using three new prmeters,, nd c: mechnicl power reference P ref t 5 sec Corresponding { to the non-slient genertor nd the slient genertor, there P e = V sin( ) + V (c ) sin( ) ( Q e = V cos( ) V cos ( ) + c sin ( ) ) were two groups of mesurements Fig 4 shows the timedomin simultion results of slient genertor model V, P e (1) nd Q e were treted s input nd output mesurements tht where = E, = 1, nd c = 1 X q In (1), esides used for estimtion while E nd were used to verify the dynmic stte, there re three unknown prmeters Hence, corresponding estimted vlues The simultion results of nonslient genertor model were very similr with results in Fig d include,, nd c Model 3 is expressed y the new d nd F s: 4, so they were not plotted P e1 P e1 P e1 P e1 δ m1 c For the etter ccurcy of the estimtion, we should void P e1 P e P e P e P e P e c the mesurements in flt run or lrge step response Therefore, δ m1 the mesurements re cquired from s to 4s In nother P en P e n P e n P e n P e n word, the estimtion portion is seconds nd totl smpling δ Q e1 = mn c numer n of the mtrix is 1 For etter visulizing the Q e 1 c n dynmic detils of input nd output mesurements, we zoomed Q e Q e Q e Q e c in three mesurements, V, P e, nd Q e, in the estimtion Q en c Q e n Q e n Q e n Q e n n c (13) portion s shown in Fig 5 After the time-domin mesurements re otined, the vlues of prmeters cn e estimted following mny steps Tking Model 1 s n exmple, the required steps for the estimtion re summrized s follows: 1 Compute x sed on the selection of initil vector θ Injection of the collected mesurements into model in (5) 3 Execution of Nonliner LSE with GN method For k =, 1) Produce Jcoin mtrix F k using x k ) Compute the first set of estimted ˆQ e k using x k 3) Compute r k using ˆQ e k nd mesurement Q e k 4) Compute ˆd k utilizing LSE method nd check the condi-

4 V (pu) E (pu) P e (pu) Q e (pu) (degree) Estimtion Time (s) Fig 4: The time-domin simultion results plotted in the following order: sttor open circuit voltge E, terminl voltge V, electricl ctive power P e, rective power Q e nd rotor ngle V (pu) P e (pu) Q e (pu) Guss-Newton method is lrgely dependent on the selection of the initil vector, θ Thus there should e n cceptle rnge of the initil vector [] In nother word, the rnge which these estimtion models cn e pplied to is limited Therefore, we did not only test the estimtion ccurcies of three models, ut lso determined the cceptle rnges of the initil vlues of E,, X q, nd the first rotor ngle, 1 The difference etween the initil vlue nd the ctul vlue is noted s error ini error ini = θ θ ctul θ ctul 1% (14) where θ ctul is ctul vlues of stte vriles nd prmeters The errors etween the estimted vlues nd ctul vlues, error est, re used to judge the ccurcy It cn e otined s follows: error est = θ estimted θ ctul θ ctul 1% (15) To find the cceptle rnge of the initil vector, ech of estimtion model ws tested for multiple times with different initil vector Normlly, the estimtion method is considered working well if error est is elow 5% [15], [16] Hence, the initil vectors which cused one of the errors up closed to 5% were considered s the limits of the rnge Both of upper limit nd lower limit would e determined A Model 1: Q s input nd P V s outputs Fig 6 () shows tht two estimted prmeters nd in Model 1 converged fter two itertions with the regultion; then, using the estimted vlues of nd, θ est could e clculted to compre with θ ctul After testing mny initil vectors, the limits of the initil vector θ of Model 1 were found s 5% nd 6% of the ctul vlues Tle I contins the initil vlues, estimted vlues, ctul vlues, nd errors which were under the mrgin conditions Remrk : The effect of regultion ws indicted y the converging process of the estimted prmeters shown in Fig 6 In Fig 6(), the convergence ws lmost stopped fter seven itertions ut the errors of E nd were 55% nd 93% It indicted tht the estimtion ws not ccurte t ll without the regultion With the regultion λ = 1, the errors of E nd were under 5% sed on the estimted vlues of nd shown in Fig 6() Time (s) Fig 5: Three mesurements were used for the estimtion tion numer of mtrix F T k F k 5) Updte x k end 4 Repet Step 3 until ˆd is pproching zero nd chieve convergence 5 Compute θ estimted sed on the vlues of x k t the finl itertion B Model : V s input nd P Q s outputs Bsed on the estimtion results, the rotor ngle 1 ws sensitive since its initil vlue could not exceed 15% nd +% of the ctul vlue However, the rnges of the internl voltge E nd d-xis rectnce were from % to +3% All of results re tulted in Tle II Becuse 1 is too sensitive, the limits in Tle II nd Tle III presented E, nd E,, X q, respectively Fig 7 () verified tht the estimted prmeters converged t the second itertion Different thn E,, nd X q, the rotor ngle is dynmic stte, so the estimted rotor ngle nd the ctul rotor ngle were plotted in time domin shown in Fig 8

5 3 1 1 Y: 539 Y: Y: Y: 587 itertions () Without regulriztion 3 itertions () With regulriztion Fig 6: Model 1: converging process of the estimted prmeters TABLE I: Identifiction results of Model 1 of synchronous genertor Prmeters Actul error ini = -5% error ini =6% Initil Estimtion error est (%) Initil Estimtion error est (%) E % % % % TABLE II: Identifiction results of Model of synchronous genertor Prmeters Actul error ini =-% error ini =3% Initil Estimtion error est(%) Initil Estimtion error est(%) % % E % % % % Y: 1398 Y: itertions Fig 7: Model : converging process of the estimted prmeters (rdin) Estimted Signl Actul Signl Time(s) Fig 8: Model : Actul nd estimted signls for the rotor ngle with respect to the mchine terminl IV CONCLUSION The discrepncy etween the estimted vlue nd ctul vlue ws very smll (< %) Therefore, we considered tht the mtching degree ws dequtely high to show the vlidity of the estimtion of rotor ngle C Model 3: V s input nd P Q s outputs considering sliency Three converged prmeters shown in Fig 9 verified Model 3 work well The rotor ngle in Model 3 ws sensitive, too The rnge ws etween 15% nd % For rest of prmeters, E, nd X q, the lower nd upper limits ws 5% nd +1% Every vlue ws concluded in Tle III Fig 1 shows tht Model 3 hd smll difference etween the estimted rotor ngle nd the ctul rotor ngle This pper investigtes the nonliner lest squre estimtion with Guss-Newton in three different estimtion models for synchronous genertor identifiction Only three mesurements re used to do estimtion nd they re treted s inputs or outputs in different estimtion models respectively To otin the time-domin simultion results, nonliner synchronous genertor model ws uilt in MATLAB/Simulink According to the results, ll of three estimtion models cn e demonstrted to estimte genertor prmeters ccurtely Moreover, wht rnges of initil vlues these models cn e pplied for re determined The nonliner genertor model includes electromechnicl dynmics, turine-governor dynmics nd excittion dynmics Therefore, it is considered tht these estimtion models cn e pplied to PMU dt

6 TABLE III: Identifiction results of Model 3 of synchronous genertor Prmeters Actul error ini =-5% error ini =1% Initil Estimtion error est (%) Initil Estimtion error est (%) % E % % X q % c Y: 1394 Y: 581 Y: itertions Fig 9: Model 3: converging process of the estimted prmeters (rdin) Estimted Signl Actul Signl Time(s) Fig 1: Model 3: Actul nd estimted signls for the rotor ngle with respect to the mchine terminl APPENDIX Genertor model dt is listed s follows: H = 65s T do = s D = 1 = 18 X d = 3 X q = 17 X line = T g = 1s R = 4 k e = 1 All quntities, except time constnts, re in per unit [4] A T Mthew nd M N Arvind, Pmu sed disturnce nlysis nd fult locliztion of lrge grid using wvelets nd list processing, in 16 IEEE Region 1 Conference (TENCON), Nov 16, pp [5] P H Gdde, M Biswl, S Brhm, nd H Co, Efficient compression of pmu dt in wms, IEEE Trnsctions on Smrt Grid, vol 7, no 5, pp , Sept 16 [6] T A Bhtti, A Rheem, T Alm, M O Mlik, nd A Munir, Implementtion of low cost non-dft sed phsor mesurement unit for 5 hz power system, in 16 Interntionl Conference on Computing, Electronic nd Electricl Engineering (ICE Cue), April 16, pp 1 15 [7] K G Khjeh, E Bshr, A M Rd, nd G B Ghrehpetin, Integrted model considering effects of zero injection uses nd conventionl mesurements on optiml pmu plcement, IEEE Trnsctions on Smrt Grid, vol 8, no, pp , Mrch 17 [8] S Brhm, R Kvsseri, H Co, N R Chudhuri, T Alexopoulos, nd Y Cui, Rel-time identifiction of dynmic events in power systems using pmu dt, nd potentil pplictions 81;models, promises, nd chllenges, IEEE Trnsctions on Power Delivery, vol 3, no 1, pp 94 31, Fe 17 [9] E R Fernndes, S G Ghiocel, J H Chow, D E Ilse, D D Trn, Q Zhng, D B Bertgnolli, X Luo, G Stefopoulos, B Frdnesh, nd R Roertson, Appliction of phsor-only stte estimtor to lrge power system using rel pmu dt, IEEE Trnsctions on Power Systems, vol 3, no 1, pp 411 4, Jn 17 [1] A Neumier, Solving ill-conditioned nd singulr liner systems: A tutoril on regulriztion, SIAM review, vol 4, no 3, pp , 1998 [11] A N Tihonov, Solution of incorrectly formulted prolems nd the regulriztion method, Soviet Mth, vol 4, pp , 1963 [1] Z Mio nd L Fn, The rt of modeling nd simultion of induction genertor in wind genertion pplictions using high-order model, Simultion Modelling Prctice nd Theory, vol 16, no 9, pp , 8 [13] L Fn, R Kvsseri, Z L Mio, nd C Zhu, Modeling of dfig-sed wind frms for ssr nlysis, Power Delivery, IEEE Trnsctions on, vol 5, no 4, pp 73 8, 1 [14] Y Li nd L Fn, Determine power trnsfer limits of n smi system through liner system nlysis with nonliner simultion vlidtion, in 15 North Americn Power Symposium (NAPS), Oct 15, pp 1 6 [15] T G Lndgrf, E P T Cri, nd L F C Alerto, Online prmeter estimtion of synchronous genertors from ccessile mesurements, in 16 IEEE/PES Trnsmission nd Distriution Conference nd Exposition (T D), My 16, pp 1 5 [16] J C N Pntoj, A Olrte, nd H Dz, Simultneous estimtion of exciter, governor nd synchronous genertor prmeters using phsor mesurements, in 14 Electric Power Qulity nd Supply Reliility Conference (PQ), June 14, pp REFERENCES [1] F C Schweppe, Power system sttic-stte estimtion, prt iii: Implementtion, IEEE Trnsctions on Power Apprtus nd Systems, vol PAS-89, no 1, pp , Jn 197 [] P Kundur, N J Blu, nd M G Luy, Power system stility nd control McGrw-hill New York, 1994, vol 7 [3] S Toscni, C Muscs, nd P A Pegorro, Design nd performnce prediction of spce vector-sed pmu lgorithms, IEEE Trnsctions on Instrumenttion nd Mesurement, vol 66, no 3, pp , Mrch 17