An Equivalent pi Network Model for Power System State Estimation with Network Parameter Errors

Transcription

1 1 An Equivalent pi Netwrk Mdel fr Pwer System State Estimatin with Netwrk Parameter Errrs Amit Jain, Member, IEEE, and Sivaramakrishnan Raman Abstract With the rle f state estimatin in energy management systems gaining weight every day, the demand fr its accuntability is an imperative, t say the least. The reliability f the netwrk database is thus a key factr. This paper emplys the tw bus π uivalent mdel fr every line in the pwer netwrk t apprach the netwrk parameter estimatin prblem. The netwrk π mdel is knwn t cmbine the effect f fixed tap transfrmers in the uivalent s line impedance and line charging admittances. The methd aims t crrect the uivalent line parameters f the suspicius branch rather than evaluating every netwrk cmpnent individually based n whether there is a transfrmer r nt. The suspicius branch is determined by ranking the branches in rder f their respective branch indices btained in a particular way, frm nrmalized residuals f measurements invlved. The technique emplys the state vectr augmentatin apprach t crrect and stre the updated uivalent parameters f the errneus branch, which is sufficient fr an accurate state estimatin. Besides this, the crrect value f the parameters can always be estimated frm the uivalent netwrk if needed. Illustratins n the IEEE 14 and 30 bus systems have been furnished t validate the apprach fr different cases f parameter errrs n any single branch. Index Terms, measurement, parameter errr, π uivalent, residual, state estimatin, transfrmer tap. S I. INTRODUCTION TATE estimatin in pwer systems tday hlds key t mre than fr what it was utilized in the initial stages. It has becme nthing shrt f a backbne t the field f energy management studies. It caters t the basic ruirement in the system security dmain, specifically the cntingency analysis. In view f the stated, state estimatin bears the heavy respnsibility f picturing the state f the system as best and accurate as pssible. This in turn shifts fcus nt the dependability f fixed data which the state estimatin wrks n. The prcess f state estimatin places huge cnfidence n the netwrk database which is assumed t be fixed in nature. Hence, any errr in the cncerned netwrk wuld culminate in errneus interpretatin which the state estimatr wuld be cmpletely unaware f. If nt accunted fr, the errrs, even Amit Jain is Head, Pwer Systems Research Center, IIIT-Hyderabad, AP India ( amit@iiit.ac.in). Sivaramakrishnan Raman is with Pwer Systems Research Centre, IIIT, Hyderabad, India. ( sivaramakrishnan.raman@research.iiit.ac.in). small in nature wuld generate incnsistencies in pst-analysis like security assessment. Netwrk parameters are knwn t cmprise line resistance, reactance, and line charging cnductance and susceptance in majrity f the cases. Anther parameter ually r even mre significant in the netwrk is the psitin f the transfrmer tap. The lack f cmmunicatin between the substatin and the cntrl center ften leads t wrng infrmatin f the transfrmer tap. Thus, the transfrmer tap t is t be given due attentin in this case. Parameter estimatin may nt bast f abundant literature as it is still cnstantly evlving int a mature prspect. Still, the literature is nt sparse either. Different methds have cntributed twards its cause since days as early as when the cncept f the generalized state estimatr was intrduced [1] and they can be classified bradly int tw majr grups [2], [3]. The cncept f residual analysis [4]-[6] rules ne f them and the ther is based n augmenting the existing state vectr. The residual analysis technique was detailed primarily in [4], wherein, the measurement residuals prvide infrmatin f existence f grss errr. The residual analysis technique is used entirely fr the estimatin f parameters in [4], [5]. Measurement residuals are als used nly t the extent f determining suspicius branches, but nt fr parameter estimatin as such [6]. The state vectr augmentatin apprach [7]-[12] cnsiders the cnventinal state variables and the suspicius parameters as the cmbined state vectr. The state augmentatin can be carried ut by either the nrmal uatins methd [7]-[10] r the Kalman filter [11], [12], which is a recursive algrithm. The bjective in the present paper is t prvide a cmmn platfrm fr parameter estimatin n branches with r withut transfrmer taps by using the π uivalent fr every branch in the netwrk. The π netwrk s parameters cntain the effects f the fixed transfrmer tap, if any. The idea behind this is t estimate the parameters f the uivalent netwrk, which has nly the general elements and n specific tap t estimate. This is specifically useful if the aim is t attain the crrect state estimate. The uivalent parameters are updated whenever estimated and are sufficient fr accurate state estimatin. Besides this, if the riginal parameter estimates are the need f the hur, the uivalent parameters can always be cnverted back with ease as shwn in a further sectin. The uivalent netwrk des nt affect the identificatin and detectin f errneus branches because the methd

2 2 emplys measurement residuals fr the purpse. A branch index based n these residuals, described in a later sectin, is btained fr each branch. The branches are ranked in rder t determine the suspicius ne. The uivalent netwrk s parameters are augmented t the state vectr using the nrmal uatins methd t btain their estimate. The crrect estimated netwrk parameters can als be extracted frm the estimates f the uivalent. This paper addresses any type f parameter errr n a single branch. II. EQUIVALENT MODEL APPROACH A. Representatin f π Netwrk A line r branch with a transfrmer can be represented as the π netwrk. A transfrmer f rati 1:a cnnected between tw buses i and j is shwn in fig. 1. The transfrmer is cnnected t the bus i. The line admittance is given by y and the leakage admittance ual n bth sides, is given by y. Vltage and current are dented as V and I respectively. Side j: I Vy + V av y j j j i V y + y av y j i Adding and subtracting the term ay V j wuld yield in (4). ( ( 1 ) ) I V y + a y + V V ay (4) j j j i The uivalent π netwrk parameters are given by (5) with reference t (3) and (4) respectively. Fig. 2 depicts the uivalent netwrk. y ay 2 i + ( 1) j + ( 1 ) y ay aa y y y a y (5) Fig. 2. Equivalent Π Netwrk Representatin Fig. 1. Transfrmer n line cnnecting ends i and j. Transfrmer principles wuld cnfirm (1) and (2). Vt avi (1) Ii It a (2) Simple Kirchhff laws wuld supprt the vltage current relatins fr bth the sides f the line. Side i: Ii ' Ii + Ii a av V y + av y 2 i j i I a av V y + a V y 2 i i j i av y + y ayv i j Adding and subtracting the term ay V i wuld yield in (3). 2 ( ( 1) ) I V a y + a a y + V V ay (3) i i i j Equatin (5) can be split int the rectangular cmpnents fr the ruirement f use in state estimatin. Equatins (6), (7) and (8) depict the same. g ag b ab 2 i + ( 1) 2 i + ( 1) gj g + ( 1 a) g bj b + ( 1 a) b g a g a a g b a b a a b If there is n transfrmer present, a can be replaced the value f 1. If the transfrmer tap is f the frm b:1, the same prcedure can be adpted with a 1 / b. If the transfrmer is n the ther end, the ends i and j are interchanged in the same uatins. B. Significance The uivalent terms given abve are used t build the netwrk admittance matrix. The same terms are subjected t parameter estimatin prcess described later. Once estimated, the new values are stred in the database. This remves the necessity f estimating the tap psitin as a separate entity. The state estimatin can be carried ut successfully with the (6) (7) (8)

3 3 uivalent value itself. The mdel makes true sense when it cmes t transfrmer cnnected buses. It is nt always necessary t estimate all the six parameters f the uivalent shwn in (6), (7) and (8). A transfrmer is plainly used t shift the vltage level and is generally nt a part f a line having line impedance and line charging admittances. Such lines mstly have nly the transfrmer reactance in place. If this is the case, all the six terms mentined need nt be estimated. If this is the case, nly the three reactive cmpnents wuld have the transfrmer effect, and wuld have t be estimated. On the ther hand, if the line has n transfrmer, the π mdel wuld nt have different leakage admittances n either side thus reducing the number f parameters t be estimated, t fur. The estimatin f the uivalent parameters wuld yield in the same state estimate as the estimatin f the individual parameters. Hwever, if the estimatin f the individual parameter is als ruired, they can always be btained frm the uivalent anytime, as shwn in the further sectin n nrmal uatins methd f parameter estimatin. III. PARAMETER ESTIMATION A. Detectin f Parameter Errr The methd emplys the nrmalized measurement residuals [2] t detect any parameter errr. A parameter errr is analgus t the crrelated errrs in any f the measurements adjacent t the errneus branch. These measurements are the pwer flw n the errneus branch and the pwer injectins n the end ndes. Thus, the measurement residuals prvide the windw t lkut fr parameter errrs as well. The nrmalized measurement residual, when calculated t be abve a threshld value, generally 3.0 [2], indicates an indirect errr in the calculatin f that measurement due t the actual errr in ne r mre branch parameters related t that measurement. This nly detects the errr but des nt identify the branch. The errneus branch is identified by the branch index described in the immediately next sectin. The residual fr a measurement z i is given by (9). The nrmalized value is given by (10). r z h( x ) (9) i i i where x is the estimated state f the prir state estimatin, z i is the vectr f given measurement set and h i is the vectr f calculated measurements. The residuals can be determined nly after the state estimatin prcess is dne. r N i r i (10) Ω ii Ω is the cvariance matrix f the measurement residuals, given in (11). 1 T Ω R HG H (11) where R is the measurement errr cvariance matrix, H is the measurement Jacbian and G is the state estimatin gain matrix. The measurement residuals detect any errr nly when there is enugh lcal redundancy. If the measurement is critical in nature, the errr in it ges unidentified. B. Identificatin by Index The high nrmalized residuals need t be linked t the crrect branch t ensure crrect estimatin. The wrk pertaining t this paper identifies branches by ranking them in rder f their respective indices. Fr a given branch, nrmalized residuals wuld arise frm its branch flws r the pwer injectins at the end ndes. The apprach used here takes int accunt, all nrmalized residuals greater than 3.0, pertaining t a branch. The maximum amng these is the index f that branch, as shwn in (12). N Indexi max ( rj ) (12) N i [ 1, m], j { rj > 3.0} where m is the number f branches and j is crrespnding t a measurement assciated with the branch i. The branch with the maximum index is chsen t have its parameters estimated. Critical Tuples In case, tw branches end up with identical values fr their indices, the prcedure has nt been able t clearly identify any ne branch. The parameter errr culd be in either branch. This generally happens in case f critical tuples which has been discussed in [8]. The identical value may als be attributed t the nrmalized residual f a pwer injectin cmmn t bth the branches. In such cases, cunter-checking the nrmalized residuals f the branch flws s as t identify which branch has a higher value may help prvided they t are nt similar. C. Estimatin by Nrmal Equatins Methd The nrmal uatins methd f augmenting the state vectr clubs the cnventinal state variables with the suspicius parameters t be estimated tgether using the Newtn-Raphsn technique, as shwn in (13). x xv, xθ x p (13) where V, θ and p represent vltage magnitude, its angle and parameter respectively. One area f cncern with this apprach is regarding the flat start f the variables generally dne befre the start f the first iteratin. The flat start leads t Jacbian singularity n accunt f last clumn f the Jacbian turning null very ften. T cunter this, the parameters are augmented t the state vectr after the first iteratin. D. Equivalent and Individual Estimates The apprach in this paper directly gives the final estimates f the uivalent netwrk, given by y, y, y. This i j

4 4 wuld suffice fr the purpse f state estimatin t yield crrect states because at each iterative step, the netwrk admittance matrix is updated with the uivalent element estimates. Hwever, the methd is nt incapable f updating the individual parameters. The uivalent parameters are nthing but functins f these. Thus, if the uivalent netwrk estimates are given by k1, k2 and k3, the slutin f (14) in 3 variables wuld yield the individual parameter estimates. Any simple nn-linear slutin technique wuld yield the ruired results. k1 ay k a y a a y k3 y + 1 a y (14) This set f uatins can be written twice t slve fr g and b separately if ruired. Transfrmer Tap The variable estimated in (14) is a, which may nt necessarily be the tap rati. If the riginal rati was inverted t cnvert t 1:a frm fr develping this uivalent mdel, the true estimate wuld be 1 / a. IV. SIMULATION RESULTS The technique discussed in the paper thus far has been implemented n standard IEEE 14 and 30 bus systems data [13], the results f which have been furnished in detail belw. This paper currently deals with single branch errrs. Different types f errrs as depicted in the three cases given belw have been handled n single branches. The state estimatin results after uivalent parameter estimatin have been cmpared with the state estimatin results with the parameter errrs. A. IEEE 14 bus system 1) Case I: Single Errr n Single This case deals with intrductin f an errr in any ne f the branch cmpnents apart frm transfrmer tap, n a single branch. Table I shws particulars f three different branches having a single errr, but nt simultaneusly. They are three separate cases and are handled individually. The symbls r, x and bs dente line resistance, line reactance and leakage susceptance respectively. The true value and errr value are given t. TABLE I SINGLE ERROR ON SINGLE BRANCH Table II gives the ranking f the branch indices fr each f the three cases. The indices identify the branches distinctly and crrectly in each f the three cases TABLE II Rank f Highest Nrmalized Residual Crrespnding ) Case II: Multiple Errrs n Single This case deals with intrductin f errr in mre than ne cmpnent except transfrmer tap, n a single branch. Table III shws three different branches having errrs n r, x and bs at the same time. But nly ne branch is cnsidered t have the errrs TABLE III MULTIPLE ERRORS ON SINGLE BRANCH Cmpnent in Errr Errr r x bs r x bs Cmpnent in Errr Errr 2-5 r x bs r x bs Table IV gives the ranking f the branch indices fr each f the three cases. The indices identify the branches distinctly and crrectly in each f the three cases.

5 5 TABLE IV 5-6 x tap Rank f Highest Nrmalized Residual Crrespnding ) Case III: Transfrmer Tap Errr n Single This case deals with intrductin f errr in the transfrmer tap and the reactance f a single branch. Table V shws tw different branches having the tap errr. This case t pertains t ne branch at a time. TABLE V TRANSFORMER TAP ERROR ON SINGLE BRANCH 4-9 Cmpnent in Errr Errr x tap Table VI gives the ranking f the branch indices fr bth the cases. The indices identify the crrect branch clearly n bth ccasins TABLE VI Rank f Highest Nrmalized Residual Crrespnding The branches indicated by the branch index based n highest nrmalized residuals are estimated fr their uivalent parameters. Once they are estimated and updated, the state estimatin results can be cmputed. Due t space cnstraints, the state estimatin results fr nly the case f transfrmer tap errr in branch 4-9 frm Table V have been shwn. Table VII cmpares the state estimatin results btained when there is n errr with thse btained with errr and crrected uivalent parameters. TABLE VII STATE ESTIMATION RESULTS N Errr With Errr Estimated Bus V θ V θ V θ

6 6 Table VII shws clearly that the state estimatin results after the estimatin f uivalent parameters are the same as withut the errr. In additin, when (14) is slved fr these numerus cases given abve, the individual parameters match exactly with the riginal values mentined in the tables. B. IEEE 30 bus system 1) Case I: Single Errr n Single Table VIII shws particulars f three different branches having a single errr, but nt simultaneusly. They are three separate cases and are handled individually. TABLE VIII SINGLE ERROR ON SINGLE BRANCH ) Case II: Multiple Errrs n Single This case deals with intrductin f errr in mre than ne cmpnent except transfrmer tap, n a single branch. Table X shws three different branches having errrs n r, x and bs at the same time. But nly ne branch is cnsidered t have the errrs. TABLE X MULTIPLE ERRORS ON SINGLE BRANCH Cmpnent in Errr Errr Cmpnent Errr in Errr r r x bs x bs r x Table IX gives the ranking f the branch indices fr each f the three cases. The indices d nt identify the branches distinctly in the case f errr in the leakage susceptance f branch Identical Nrmalized Residuals: The maximum nrmalized residuals f bth branches and are identical. This is due t the maximum residual arising frm the cmmn pwer injectin at bus 30. Bus 30 is cnnected t nly tw buses 27 and 29. S, the pwer injectin at bus 30 is a measurement adjacent t bth buses 27 and 29. Hwever, n cmparing the nrmalized residuals f the branch flws n the tw branches, the branch flw n has a nrmalized residual greater then 8 whereas that n branch has just abve 0.5. It is clear frm this that the errneus branch is bs r x bs Table XI gives the ranking f the branch indices fr each f the three cases. The indices identify the branches distinctly and crrectly in each f the three cases. TABLE XI Rank f Highest Nrmalized Residual Crrespnding TABLE IX Rank f Highest Nrmalized Residual Crrespnding ) Case III: Transfrmer Tap Errr n Single Table XII shws tw different branches having the tap

7 7 errr. This case t pertains t ne branch at a time. TABLE XII TRANSFORMER TAP ERROR ON SINGLE BRANCH Cmpnent in Errr Errr x tap x tap XIV cmpares the state estimatin results btained when there is n errr with thse btained with errr and crrected uivalent parameters TABLE XIII Highest Nrmalized Residual Crrespnding Table XIII gives the ranking f the branch indices fr bth the cases. The indices identify the crrect branch clearly n bth ccasins. The branches indicated by the branch index based n highest nrmalized residuals are estimated fr their uivalent parameters. Due t space cnstraints, the state estimatin results fr nly the case f transfrmer tap errr in branch 4-12 frm Table XII have been shwn. Table TABLE XIV STATE ESTIMATION RESULTS N Errr With Errr Estimated Bus V θ V θ V θ

8 Table XIV shws clearly that the state estimatin results after the estimatin f uivalent parameters are very similar t thse withut the errr. In additin, when (14) is slved fr these numerus cases given abve, the individual parameters match exactly with the riginal values mentined in the tables. V. CONCLUSION The paper mtivates the idea f emplying a π uivalent netwrk fr every branch in the system fr state estimatin f a system affected by branch parameter and transfrmer tap errrs. The uivalent netwrk prvides a cmmn structure fr netwrk lines and transfrmer branches as it includes the transfrmer effect in the uivalent branches. The parameter estimatin algrithm estimates the crrect uivalent parameters rather than crrecting the riginal netwrk parameters and taps separately. Hwever, if the individual parameter estimate values are als ruired, the estimates f the uivalent can readily be used t estimate the parameters. Illustratins f the methd are prvided n IEEE 14 and 30 bus systems fr different scenaris including ne case f identical values f nrmalized residuals. This paper caters t single branch errrs. The authrs are presently wrking twards catering t errrs in multiple and adjining branches and that wrk will be published in future. VI. REFERENCES [1] O. Alsac, N. Vempati, B. Sttt, and A. Mnticelli, "Generalized State Estimatin," IEEE Transactins n Pwer Systems, Vl. 13(3), pp , August [2] A. Abur and A. Gmez-Expsit, Pwer System State Estimatin: Thery and Implementatin, New Yrk: Marcel Dekker Inc, [3] P. Zarc, and A. Gmez, "Pwer System Parameter Estimatin: A Survey," IEEE Transactins n Pwer Systems, vl. 15(1), pp , February [4] T. Van Cutsem, and V. Quintana, "Netwrk Parameter Estimatin Using Online Data with Applicatin t Transfrmer Tap Psitin Estimatin," IEE Prceedings, Vl. 135, Pt C, N. 1, pp , January [5] W. Liu, F. Wu, and S. Lun, "Estimatin f Parameter Errrs frm Measurement Residuals in State Estimatin," IEEE Transactins n Pwer Systems, Vl. 7(1), pp , February [6] M. B. D Cutt Filh, J. C. S. de Suza, and E. B. M. Meza, "Crrecting electrical netwrk parameters," Pwer & Energy Sciety General Meeting, IEEE, pp.1-7, July [7] Jun Zhu, and A. Abur, "Identificatin f netwrk parameter errrs," IEEE Transactins n Pwer Systems, vl.21, n.2, pp , May [8] Jun Zhu, and A. Abur, "Identificatin f netwrk parameter errrs using phasr measurements," Pwer & Energy Sciety General Meeting, IEEE, pp. 1-5, July [9] M. R. M. Castill, J. B. A. Lndn, and N. G. Bretas, "Identificatin and estimatin f pwer system branch parameter errr," Pwer & Energy Sciety General Meeting, IEEE, pp.1-8, July [10] P. Teixeira, S. Brammer, W. Rutz, W. Merritt, and J. Salmnsen, State estimatin f vltage and phase-shift transfrmer tap settings, IEEE Transactins n Pwer Systems, vl. 7, n. 3, pp , Aug [11] A. Debs, "Estimatin f Steady-State Pwer System Mdel Parameters", IEEE Transactins n Pwer Apparatus and Systems, vl. PAS-93, N. 5, pp , [12] I. Slutsker, and K. Clements, "Real Time Recursive Parameter Estimatin in Energy Management Systems," IEEE Transactins n Pwer Systems, Vl. 11(3), pp , August [13] Pwer System Test Case Archive, Electrical Engineering, University f Washingtn, VII. BIOGRAPHIES Amit Jain graduated frm KNIT, India in Electrical Engineering. He cmpleted his masters and Ph.D. frm Indian Institute f Technlgy, New Delhi, India. He was wrking in Alstm n the pwer SCADA systems. He was wrking in Krea in 2002 as a Pst-dctral researcher in the Brain Krea 21 prject team f Chungbuk Natinal University. He was Pst Dctral Fellw f the Japan Sciety fr the Prmtin f Science (JSPS) at Thku University, Sendai, Japan. He als wrked as a Pst Dctral Researcher at Thku University, Sendai, Japan. Currently he is heading, Pwer Systems Research Center at IIIT, Hyderabad, India. His fields f research interest are pwer system real time mnitring and cntrl, artificial intelligence applicatins, lad frecasting, pwer system planning and ecnmics, electricity markets, renewable energy, reliability analysis, GIS applicatins, parallel prcessing and nantechnlgy. Sivaramakrishnan Raman is pursuing his Masters at Pwer Systems Research Center, Internatinal Institute f Infrmatin Technlgy, Hyderabad, India. He received his B. Tech degree frm SASTRA University, Thanjavur, India in His areas f interest include pwer system mnitring and cntrl applicatins, prtectin, lad flw, state estimatin, vltage stability and reactive pwer cntrl.