AC network state estimation using linear measurement functions

Transcription

1 AC network state estmaton usng lnear measurement functons R.A. Jabr and B.C. Pal Abstract: The real/reactve ower and current magntude measurements can be accounted for n an AC network state estmator usng lnear measurement functons. The nonlnearty n the conventonal AC state estmator equatons s transferred from the measurement functons nto a system of nonlnear equalty constrants whch s ndeendent of the measurement set. The new format of equatons entals two advantages. Frst, t can be easly ntegrated n otmsaton routnes whch emloy frst- and second-order dervatve functons. Second, the lnear measurement functons can beneft from scalng technques whch are well documented n the lnear rogrammng lterature. Ths research detals the mlementaton of a least absolute value state estmator emloyng the new format of equatons. The otmsaton method s based on a rmal-dual nteror-ont method that can accurately account for zero njecton measurements and ower drectons. Numercal testng s used to valdate the aroach for networks wth measurement sets that are () conventonal and () have a hgh roorton of current magntude measurements and ower sgns. Lst of symbols b sh / 1/ chargng suscetance n the equvalent model b n seres suscetance n the equvalent model B n magnary art of ^Y n g n seres conductance n the equvalent model G n real art of ^Y n T h k row vector of the kth lnear measurement functon I n magntude of the lne ( n) current leavng bus m number of measurements n the network N number of buses n the network P real ower njecton at bus P n real ower lne ( n) flow leavng bus Q reactve ower njecton at bus Q n reactve ower lne ( n) flow leavng bus r k, s k ostvely bounded varables n the least absolute value reresentaton RMS- root mean square voltage error RMS- root mean square angle error maxmum relatve voltage error maxmum relatve angle error # The Insttuton of Engneerng and Technology 008 do: /et-gtd: Paer frst receved 11th Arl and n revsed form nd June 007 R.A. Jabr s wth the Electrcal, Comuter and Communcaton Engneerng Deartment, Notre Dame Unversty, PO Box 7, Zouk Mkhael, Zouk Mosbeh, Lebanon B.C. Pal s wth the Deartment of Electrcal and Electronc Engneerng, Imeral College, London SW7 BT, UK E-mal: rjabr@ndu.edu.lb R n T n u V x e x tr x Ŷ n z k z k mn, z k max d 1 Introducton varable n the roosed ower flow format defned as V V n cos(d d n ) varable n the roosed ower flow format defned as V V n sn(d d n ) varable n the roosed ower flow format defned as V = ffff voltage magntude at bus state vector estmated value of x true value of x comlex reresentaton of an element n the bus admttance matrx kth measurement mnmum and maxmum values of the kth measurement voltage angle at bus Governments of many ndustralsed natons have ledged to reduce ther carbon emssons n accordance wth the Kyoto agreement on clmate change [1]. For nstance, UK government has set a carbon cut target of 60% by 050 []. Reducng carbon emssons requres governments and ower utltes to nvest n renewable energy sources such as tdal, solar and wnd ower. These sources are by nature geograhcally dsersed and comaratvely small szed whch n turn necesstates ther connecton to dstrbuton networks. Consequently, there s an emergng need for ntegrated montorng and control of both transmsson and dstrbuton networks. State estmaton s the man functon for montorng ower networks. Snce the 1970s, t has been researched n the context of transmsson networks [3, 4]. In transmsson state estmaton, conventonal measurement sets IET Gener. Transm. Dstrb., 008,, (1),

2 are comosed of real/reactve ower ars and voltage magntudes. Current magntude measurements are occasonally emloyed for ncreasng the level of redundancy. Dstrbuton networks, on the other hand, have lmted conventonal measurements. In many cases, the measurement sets for such networks are domnated by current magntude measurements. These measurements are known to comlcate the state estmaton roblem because they do not contan drectonal nformaton. The lack of ower flow drectons mles that although a network may be numercally observable n the conventonal sense (measurement Jacoban s of full column rank), ts state estmaton soluton may not be unque [5]. The lterature reorts examles of networks havng a large roorton of current magntude measurements but that can be stll made unquely observable through ncororatng nto the constrant set ower njecton sgns and zero njecton seudo-measurements [6]. Therefore for AC network state estmaton emloyng current magntude measurements, the comutatonal engne should be caable of handlng functonal equalty/ nequalty constrants. The underlyng comutatonal rocedure n any state estmator s an otmsaton functon. In the state estmaton lterature, the emloyed otmsaton functons can be classfed as ether frst- or second-order methods, deendng on the order of the dervatve. The frst-order methods nclude the classcal weghted least squares [7], the teratvely re-weghted least squares [8 11] and the lnear rogrammng based least absolute value estmator [1]. The second-order methods requre the evaluaton of the Lagrangan Hessan matrx. The most recent second-order state estmaton mlementatons rely on rmal-dual nteror-ont methods. They have been mlemented n the context of a least absolute value estmator [13] and a Huber M-estmator [14]. Moreover, the nteror-ont mlementatons have been shown to be successful n enforcng zero njecton measurements and load ower lmts [14]. Reference [15] reorts a least absolute value nteroront mlementaton whch also accounts for current magntude measurements. In ractce, the ugrade of an nteror-ont based state estmaton functon to account for current magntude measurements on to of the conventonal measurement set requres sgnfcant tme and effort for codng and subsequently testng the comutaton of the Jacoban and Lagrangan Hessan matrces. Ths aer rooses a rmal dual nteror-ont mlementaton of a least absolute value AC network state estmator that uses lnear measurement functons. These functons have a constant Jacoban matrx and therefore do not contrbute to the Lagrangan Hessan. In the roosed format, the nonlnearty of the state-estmaton functon s emboded n a fxed set of equalty constrants. Ths set of nonlnear constrants s the only one whch contrbutes to the Langrangan Hessan comutaton. The aer shows that the new format of equatons can be obtaned usng smle varable substtutons. A smlar format of equatons was frst roosed for radal networks n [16] and then shown to be equvalent to a second-order cone rogram n [17]. Power flow equatons Let ^Y n G n þ jb n denote the comlex rectangular reresentaton of an element n a N N bus admttance matrx. If bus voltages are exressed n olar form ( ~V V /d ), the real and reactve njected owers at an arbtrary bus [18] are P V G þ XN n1 h V V n G n cos (d d n ) þ V V n B n sn (d d n ) Q V B þ XN n1 V V n G n sn (d d n ) h V V n B n cos (d d n ) By followng [17], defne R n V V n cos (d d n ), T n V V n sn (d d n ) and u V =. The nonlnear ower flow equatons become P (1) () ffff G u þ XN [G n R n þ B n T n ] (3) n1 Q ffff B u XN n1 [B n R n G n T n ] (4) From the above defntons of R n, T n and u, t follows that 3 State estmator Let x denote the state vector u u n R n þ T n (5) d d n tan 1 T n R n (6) x [..., u,..., R j,..., T j,..., d,...] T (7) where t s understood that u 0 and R j 0. The least absolute value state estmaton roblem can be exressed as mnmse Xm h T k x z k subject to (8) k1 equalty constrants: h T k x z k, (9) nequalty constrants: z mn k h T k x z max k, (10) feasblty constrants: equatons (5) and (6). In the above roblem, h T k s the kth row of the lnear measurement functon matrx, z k reresents the kth measurement (k 1,..., m), (9) models the zero-njecton seudomeasurements and (10) constrans the drecton of the ower njecton or flow. The nequalty constrants (10) are commonly used n conjuncton wth the current magntude measurements. The measurement functons take one of the followng forms, deendng on the measurement tye. 1. Real ower njecton measurement P h T k x G u þ XN [G n R n þ B n T n ] (11) n1 IET Gener. Transm. Dstrb., Vol., No. 1, January 008

3 . Reactve ower njecton measurement Q h T k x B u XN n1 [B n R n G n T n ] (1) 3. Real ower flow measurement P n h T k x gn u g n R n b n T n (13) where g n, b n : seres conductance and suscetance n the equvalent model. 4. Reactve ower flow measurement Q n h T k x bn þ b sh u þ b n R n g n T n (14) where b sh =: 1/ chargng suscetance n the equvalent model. 5. Current magntude measurement I n transformed nto I n [6] h T k x ffff Au þ Bun CR n þ DT n (15) where A g n þ b n þ b sh, B gn þ b n, C g n þ b n b n þ b sh, D g nb sh 6. Voltage magntude measurement V transformed nto V = ffff h T x u (16) The above lnear measurement functons can account for addtonal state varables from load-flow control devces such as ta-changng or hase-shftng transformers rovded that these devces are reresented usng the ower njecton model [19]. 4 Interor-ont solver It s well known that there are two mathematcally equvalent nonlnear rogrammng reresentatons of the drect least absolute value state estmaton roblem (5), (6), (8) (10). Both reresentatons have ther objectve and constrant functons twce contnuously dfferentable and are therefore comutatonally tractable usng nteror-ont methods [13]. The frst reresentaton relaces (8) by a lnear objectve functon and functonal nequalty constrants [13] mnmse Xm k1 s k subject to (17) s k h T k x z k s k ; k 1,..., m (18) The second reresentaton substtutes (8) wth a lnear objectve functon, functonal equalty constrants and ostvely bounded varables [13] mnmse Xm k1 (r k þ s k ) subject to (19) h T k x z k þ r k s k 0, (0) r k 0, s k 0; k 1,..., m (1) Numercal exerments reorted n [13] have shown that for least absolute value state estmaton, the second formulaton results n a more numercally robust nteror-ont mlementaton. In ths study, the drect state estmaton roblem s reresented usng (19 1) and solved va the rmal dual nteror-ont method descrbed n [13]. The nteror-ont solver requres, at each teraton, the comutaton of the Jacoban and Lagrangan Hessan matrces. The Jacoban matrx corresondng to the measurements (19) together wth the equalty (9) and nequalty constrants (10) s constant throughout all teratons. For the feasblty constrants (5), (6), the formulae for comutng the Jacoban and Hessan elements are gven n the aendx. The ntal startng vector s chosen as follows r k s k 1 for all measurements u 1= ffff and d 0 for all nodes R j 1 and T j 0 for all lnes The solver termnates when all the stong crtera are satsfed wth a tolerance of Scalng In lnear rogrammng, scalng wth ostve factors s a general condtonng transformaton that can be aled to the matrx of constrants wthout sgnfcantly changng the roblem defnton. The most wdely used method s scalng rows and columns to have unt norms [0]. The use of scalng n lnear rogrammng based least absolute value state estmaton was frst nvestgated n [1]. Accordng to the numercal results n [1], normalsaton s the most effectve rovded that () column scalng s erformed before row scalng and () the untary columns corresondng to the ostvely bounded varables (r k and s k n (0)) are reserved. Advantages of scalng nclude a reducton n the number of lnear rogrammng teratons and a decrease n the number of leverage onts [1]. The numercal exerments conducted n ths study have shown that scalng the measurement matrx such that ts rows have unt length results n mroved convergence and a reduced number of nteror-ont teratons. In fact, some roblems faled to converge wthout scalng. As n [1], the normalsaton dd not nclude the untary columns corresondng to the varables r k and s k n (0). In any case, column scalng was not mlemented because the varables n the lnear measurement equatons also aear n the nonlnear feasblty constrants. 5 Numercal results A rototye mlementaton of the roosed estmator was rogrammed n MATLAB runnng on a Pentum IV, 1.89 GHz PC wth 56 Mbytes of RAM. Testng was carred out on the IEEE 14, 30 and 118 bus systems. The lne and load data for these systems are avalable from []. For each system, the true values of the measurements were generated usng a Newton Rahson load flow algorthm. The nose was added to the true values assumng IET Gener. Transm. Dstrb., Vol., No. 1, January 008 3

4 Table 1: Measurement sets adated from [3] Measurements and constrants IEEE 14 IEEE 30 real njecton, 4, 5, 6, 10, 11, 1, 13, 14 1,, 3, 5, 8, 10, 14, 15, 17, 0, 1, 3, 4, 6 reactve njecton, 4, 5, 6, 8, 10, 11, 1, 13, 14 1,, 3, 5, 8, 10, 13, 14, 15, 17, 0, 1, 3, 4, 6 real zero njecton 7, 8 6, 9, 11, 13,, 5, 7, 8 reactve zero njecton 7 6, 9,, 5, 7, 8 real/reactve ower flow 1, 3, 4 7, 5 6, 6 11, 6 13, 7 8, 7 9, , 1 3, 5, 6, 6 10, 9 11, 4 1, 16 17, 18 19, 19 0, 4 5, 7 30, 9 30, 6 8, 13 1, 6 5. voltage magntude, 4, 5, 6, 7, 8, 10, 11, 1, 13, 14 1,, 3, 5, 6, 8, 9, 10, 13, 14, 15, 17, 0, 1, 3, 4, 6, 7, 8 Gaussan dstrbuton wth zero mean and a standard devaton of 0.01 u for ower/current measurements and u for voltage measurements. The state estmator was tested under two measurement scenaros: frst for conventonal measurement sets and second for measurement sets wth current magntudes and ower sgns. 5.1 Conventonal measurement sets The conventonal measurement sets for the IEEE 14 and 30 bus systems were adated from [3]. The observablty algorthm n [3] reles on the DC network model. The corresondng measurement sets for the AC network model are gven n Table 1. For the IEEE 118 bus test system, a smle measurement set was used. It conssted of () the voltage magntude at the slack node and () the real and reactve ower flows at one end of each lne, that s estmaton was carred out usng a lne-only state estmator. Table shows the followng erformance metrcs for each of the above systems smulated n the resence of measurement nose s P N tr 1 (V V e ) RMS voltage error (u): RMS- N sff P N 1 RMS angle error (rad): RMS- (dtr d e ) N Maxmum relatve voltage error (%): V tr V e max 100 1,..., N V tr Table : State estmator erformance conventonal measurement set wth nose IEEE IEEE IEEE Maxmum relatve angle error (%): d tr d e max 1,..., N d tr 100 CPU tme : Number of nteror-ont teratons: In the above equatons, the suerscrt tr desgnates the true value and e reresents the estmated value. Note that n order to avod dvson by zero n the equaton, the angle at the slack node was set to 1 rad. The value of 1 rad was chosen because t allows meanngful comarson between and n Table. In the absence of measurement nose, the state estmator converged to the true soluton. In fact, all values of and n Table were at 0.00%. Table 3 shows smlar results for the test systems n the resence of bad data. It s mortant to note that the error values n Tables and 3 are n agreement wth the range of errors from a revous study reortng the Table 3: State estmator erformance conventonal measurement set wth nose and bad data IEEE IEEE IEEE Table 4: True, measured and estmated values Network Quantty True value Measured value Estmated value IEEE14 P Q IEEE30 P Q IEEE118 P Q IET Gener. Transm. Dstrb., Vol., No. 1, January 008

5 Table 5: Measurement sets adated from [4] and [14] Measurements and constrants IEEE14 IEEE30 real njecton 1,, 3, 4, 6, 9, 10, 1, 13 1,, 3, 5, 14, 16, 17, 6, 9, 30 reactve njecton 1,, 3, 4, 6, 9, 10, 1, 13 1,, 3, 5, 11, 14, 16, 17, 6, 9, 30 real zero njecton 7, 8 6, 9, 11, 13,, 5, 7, 8 reactve zero njecton 7 6, 9,, 5, 7, 8 real/reactve ower flow 1, 4 7, 4 9, 7 8, 7 9 1, 1 3, 4, 5, 6, 4 6, 6 7, 6 8, 6 9, 6 10, 4 1, 1 14, 1 15, 1 16, 19 0, 10 1, 10, 15 3, 4 5, 7 9, 7 30, 6 8, 1, 4 3, 6, 19 18, 4, 4 3, 7 8, 8 8, 8 6 voltage magntude 1 1 current magntude 6 11, , 10 0 real ower njecton sgn 10 18, 0 reactve ower njecton sgn 10 13, 18, 0 mlementaton of a least squares, a lnear rogrammng based least absolute value, and an teratvely re-weghted least squares estmator [11]. All the estmators n [11] use the standard ower flow equatons format. The bad data measurements together wth ther true and estmated values are gven n Table 4. These results suggest that the roosed state estmator, lke the conventonal least absolute value estmator, s caable of rejectng bad data as long as they do not corresond to leverage ont measurements. As for the comutatonal erformance, comarsons wth revous studes ndcate that the executon tme of the roosed estmator s less than the corresondng average executon tmes of the lnear rogrammng based least absolute value and the teratvely re-weghted least squares estmators (c.f. Tables and 3 n [11]). Moreover, the executon tme s comarable wth that of the nteroront least absolute value state estmator emloyng the conventonal nonlnear measurement functons (c.f. Tables 4 and 5 n [13]). Both the estmator n ths aer and the one n [13] emloy the same nteror-ont mlementaton. 5. Unconventonal measurement sets Unconventonal measurement sets nclude current magntudes and ower sgns. These can be useful for extendng system observablty [6]. For nstance, [4] ncludes examles of measurement sets for the IEEE 14 and 30 bus test systems whch do not rovde comlete network observablty. To recover the overall system observablty, [4] rovdes canddate locatons for real/reactve ower seudo-measurements. The use of ower seudomeasurement ntervals n the measurement sets of [4] was nvestgated earler n [14]. In ths research, the Table 6: State estmator erformance unconventonal measurement set wthout nose branches used n [14] for ower seudo-measurement lacement were rovded wth current magntude measurements and the ower njecton seudo-measurements were relaced wth nequalty constrants ndcatng ther sgns. The corresondng measurement sets for the IEEE 14 and 30 bus test systems are gven n Table 5. Table 6 shows the estmaton results n the absence of nose. Table 7 shows smlar results n the resence of measurement nose. It can be nferred from the results that the current magntude measurements and ower sgns are useful for state estmaton n ths case. When the current magntude measurements consttute a very large roorton of the measurement set, state estmaton s ossble only f network observablty can be guaranteed by other means. In certan examles, observablty can be acheved by the use of zero njecton seudomeasurements and ower sgns [6]. The measurement confguratons n [6] were also used n testng the state estmator. For the IEEE 14 bus system, the measurement set conssted of () voltage magntudes at all nodes, () lne current magntudes at both ends of every lne and () zero njecton real/reactve seudo-measurements. In addton, nequalty constrants were used to enforce () the real ower njecton sgns at all nodes excet at node were both load and generaton are resent and () the reactve ower njecton sgns for load nodes. For the IEEE 30 bus system, the measurement set smlarly conssted of all voltage and lne current magntude measurements together wth the zero njecton seudomeasurements. Agan, real ower njecton sgns were secfed for all nodes and reactve ower sgns were defned for all load nodes. The estmaton statstcs n the resence of nose are shown n Table 8. It s evdent that the current magntude measurements lead to an ncreased number of nteror-ont teratons. Ths behavour has Table 7: State estmator erformance unconventonal measurement set wth nose IEEE IEEE IEEE IEEE IET Gener. Transm. Dstrb., Vol., No. 1, January 008 5

6 Table 8: State estmator erformance measurement set accordng to [6] wth nose been revously reorted n [15]. In AC networks wth such tye of measurement sets, the estmated state may be used for load estmaton by comutng the ower delvered nto each load node [5]. 6 Concluson Ths aer resented a least absolute value state estmator based on a new ower flow equatons format. In ths format, the real/reactve ower and current magntude measurements are modelled va lnear functons. The nonlnearty s accounted for by a fxed set of nonlnear feasblty constrants. The roosed format can be easly ntegrated n otmsaton functons that requre second-order dervatves, for nstance, a rmal-dual nteror-ont solver. Numercal results show that wth conventonal measurement sets, the magntude of error n the estmates s comarable to that from three standard state estmators whch are based on the least squares, the lnear rogrammng least absolute value, and the teratvely re-weghted least squares technques [11]. Even wth unconventonal measurement sets domnated by current magntude measurements and ower sgns, the new estmator was shown to be caable of mantanng an accetable level of error accuracy. 7 References IEEE IEEE Thornley, V., Jenkns, N., and Whte, S.: State estmaton aled to actve dstrbuton networks wth mnmal measurements. Proc. 15th Power Systems Comutaton Conf., 005. Avalable at: htt://www. montefore.ulg.ac.be/servces/stochastc/scc05/aers/f115.df Monbot, G.: Eco Warlord, BBC Focus, 007, 17, Abur, A., and Gómez Exósto, A.: Power system state estmaton: theory and mlementaton (Marcel Dekker, New York, 004) 4 Montcell, A.: State estmaton n electrc ower systems: a generalzed aroach (Kluwer Academc Publshers, Boston, 1999) 5 Abur, A., and Gómez Exósto, A.: Detectng multle solutons n state estmaton n the resence of current magntude measurements, IEEE Trans. Power Syst., 1997, 1, (1), Ruíz Muňoz, J.M., and Gómez Exósto, A.: A lne-current measurement based state estmator, IEEE Trans. Power Syst., 199, 7, (), Schwee, F.C., and Wldes, J.: Power system statc-state estmaton, art I: exact model, IEEE Trans. Power Aar. Syst., 1970, 89, (1), Smth, M.M., Powell, R.S., Irvng, M.R., and Sterlng, M.J.H.: Robust algorthm for state estmaton n electrcal networks, IEE Proc. C, Gener. Transm. Dstrb., 1991, 138, (4), Ml, L., Chenae, M.G., Vchare, N.S., and Rousseeuw, P.J.: Robust state estmaton based on rojecton statstcs, IEEE Trans. Power Syst., 1996, 11, (), Pres, R.C., Smões Costa, A., and Ml, L.: Iteratvely reweghted least-squares state estmaton through gvens rotatons, IEEE Trans. Power Syst., 1999, 14, (4), Jabr, R.A., and Pal, B.C.: Iteratvely reweghted least-squares mlementaton of the WLAV state-estmaton method, IEE Proc., C, Gener. Transm. Dstrb., 004, 151, (1), Irvng, M.R., Owen, R.C., and Sterlng, M.J.H.: Power system state estmaton usng lnear rogrammng. Proc. Inst. Electr. Eng., 1978, 15, Jabr, R.A.: Prmal-dual nteror-ont aroach to comute the L 1 soluton of the state estmaton roblem, IEE Proc, C, Gener. Trans. Dstrb., 005, 15, (3), Jabr, R.A.: Power system Huber M-estmaton wth equalty and nequalty constrants, Elec. Power Syst. Res., 005, 74, (), Handschn, E., Langer, M., and Klokys, E.: An nteror ont method for state estmaton wth current magntude measurements and nequalty constrants. IEEE Power Industry Comuter Alcaton Conf., 1995, Gómez Exósto, A., and Romero Ramos, E.: Relable load flow technque for radal dstrbuton networks, IEEE Trans. Power Syst., 1999, 14, (3), Jabr, R.A.: Radal dstrbuton load flow usng conc rogrammng, IEEE Trans. Power Syst., 006, 1, (3), Granger, J.J., and Stevenson, W.D. Jr.: Power System Analyss (McGraw-Hll, New York, 1994) 19 Jabr, R.A.: A rmal-dual nteror-ont method to solve the otmal ower flow dsatchng roblem, Otm. Eng., 003, 4, Tosovc, L.B.: Some exerments on sarse sets of lnear equatons, SIAM J. Al. Math., 1973, 5, (), Çelk, M.K., and Abur, A.: Use of scalng n WLAV estmaton of ower system states, IEEE Trans. Power Syst., 199, 7, (), The Unversty of Washngton Power Systems Test Case Archve, avalable at: htt:// 3 Abur, A., and Magnago, F.H.: Otmal meter lacement for mantanng observablty durng sngle branch outages, IEEE Trans. Power Syst., 1999, 14, (4), Habballah, I.O., and Irvng, M.R.: Observablty analyss for state estmaton usng lnear rogrammng, IEE Proc, C, Gener. Trans. Dstrb., 001, 148, (), Wan, J., and Nan Mu, K.: A WLS method for load estmaton n unbalanced dstrbuton networks. IEEE Power Engneerng Socety Wnter Meetng, 00, vol., Aendx To comute the Jacoban and Lagrangan Hessan matrces corresondng to (5), t s frst rewrtten as f R n þ T n u u n The Jacoban and Hessan elements n R n T Smlarly, (6) s rewrtten u f n g d d n tan 1 T n R n u, The corresondng Jacoban and Hessan elements n n T n R n þ T, n R g R n þ T, R nt n n (R n þ T n R nt n (R n þ T n R n T n (R n þ T n) 6 IET Gener. Transm. Dstrb., Vol., No. 1, January 008