Reactive Power Allocation Using Support Vector Machine

Transcription

1 Reactve Power Aocaton Usng Support Vector Machne M.W. Mustafa, S.N. Khad, A. Kharuddn Facuty of Eectrca Engneerng, Unverst Teknoog Maaysa Johor 830, Maaysa and H. Shareef Facuty of Eectrca Engneerng and But Envronment, Unverst Kebangsaan Maaysa, Bang 43600, Maaysa ABSTRACT Ths paper proposes a new modfed noda equatons (MNE) method to dentfy the reactve power transfer between generators and oad. t further focuses on creatng an approprate support vector machne (SVM) n whch support vector regresson s used as an estmator to sove the same probem n a smper and faster manner. Amost a system varabes obtaned from oad fow soutons are utzed as nput to the SVM. The actua 5-bus equvaent power system of south Maaysa s utzed as a test system to ustrate the effectveness of the SVM technque compared to that of the modfed noda equatons method. Keywords: oad fow, modfed noda equatons method, rada bass functon network, reactve power and support vector machne.. NTRODUCTON The reactve power provson becomes an mportant ssue under compettve envronment. mpementng transparent rues that aocate transmsson use fuf ths concept of farness n the ndustry. Farness can ony be acheved by adoptng a far and transparent usage aocaton methodoogy acceptabe to a partes. n vew of market operaton, t s vta to know the roe of ndvdua generators and oads to transmsson wres and power transfer between ndvdua generators to oads. Ths s necessary for the restructured power system to operate economcay, effcenty and ensure guaranteed open access to a system users []. Severa schemes have been deveoped to sove the aocaton probem n the ast few years. Methods based on the -bus or Z-bus system matrces have recenty receved great attenton snce these methods can ntegrate the network characterstcs and crcut theores nto ne usage and oss aocaton. The method reported n reference [] s based on Krchhoff's current aw (KC), equvaent near crcut transformaton and superposton prncpe. n genera t assumes that the current at each network njecton pont may reach a nes and oads. Another crcut concept method was proposed by Chang and u [3]. t was based on the system -bus matrx and Z-bus modfcaton. Ths agorthm utzes the network decomposton concept as proposed n reference [4] by Zoban and c whch determnes the use of transmsson network by ndvdua batera contracts. Teng [5] proposed a systematc method, very smar to as presented n reference [3], to aocate the power fow and oss for dereguated transmsson systems. The tracng methods [, 6-9] based on the actua power fows n the network and the proportona sharng prncpes are effectvey used n transmsson usage aocaton. Baek [6] proposed a nove power tracng method to aocate the rea and reactve power fow, however the man drawback of ths method s that t requres nvertng a arge matrx. F.F Wu et a. [7] proposed a graph theory to cacuate the contrbuton factor of ndvdua generators to ne fows and oads and the extracton factor of ndvdua oads from ne fows and generators, whch s theoretcay effcent. Ths method cannot hande oop fows and osses must be negected ntay. Reference [0] s based on the concept of generator domans, common and nks. The dsadvantage of ths method s that the share of each generator n each common (.e. the set of buses supped from the same set of generators) s assumed to be same. n a reated work based on support vector machne technques as gven n reference [], a dynamc votage coapse ndces s proposed usng support vector machne (SVM). The SVM gves faster and more accurate resuts for dynamc votage coapse predcton. The MNE methodoogy n ths paper s based on current operatng pont computed by the usua oad fow code and basc equatons governng the oad fow n the network. The method starts wth parttonng of system -bus matrces to decompose the current of the oad

2 buses as a functon of the generators current and oad votages. Then t uses the oad votages from oad fow resuts and decomposed oad currents to determne reactve power contrbuton from each generator to oads. The next goa of ths research s to ncorporate the SVM to cacuate reactve power output of ndvdua generators to system oads. The new method based on modfed noda equaton has been chosen as a traner to tran the SVM. t can be expected that the appcaton of SVM to the deveoped methodoogy w further contrbute n mprovng the computaton tme of reactve power aocaton methodoogy for dereguated system.. MODFED NODA EQUATONS METHOD The dervaton, to decompose the oad reactve powers nto components contrbuted by specfc generators starts wth basc equatons of oad fow. Appyng Krchhoff s aw to each node of the power network eads to the equatons, whch can be wrtten n a matrx form as n equaton () []: where: = V () V: s a vector of a node votages n the system : s a vector of a node currents n the system : s the -bus admttance matrx The noda admttance matrx of the typca power system s arge and sparse, therefore t can be parttoned n a systematc way. Consderng a system n whch there are generator nodes that partcpate n seng power and remanng = n- nodes as oads, then t s possbe to re-wrte equaton () nto ts matrx form as shown n equaton (): V = () V Sovng equaton () for, the oad currents can be presented as a functon of generators current and oad votages as shown n equaton (3): ( ) V = (3) Now, n order to decompose the oad votage dependent term further n equaton (3), nto components of generator dependent terms, the foowng dervatons are used. A possbe way to deduce oad node votages as a functon of generator bus votages s to appy superposton theorem. However, t requres repacng a oad bus current njectons nto equvaent admttances n the crcut. Usng a ready avaabe oad fow resuts, the equvaent shunt admttance j of oad node j can be cacuated usng the equaton (4): S j = j V j Vj S j s the oad compex power on node j and V j s the oad bus votage on node j. After addng these equvaences to the dagona entres of -bus matrx, equaton () can be rewrtten as n equaton (5): where ' = 0 ' V V (4) (5) s the modfed sub matrces n equaton (). Next, adoptng the ower haf of equaton (5) and sovng for V t s possbe to obtan the oad votages as a functon of generator votages as n equaton (6): V = ( ) V (6) ' Now, t s a smpe matter to obtan requred reatonshp as a functon of generators votage and currents. By substtutng equaton (6) nto equaton (3), the decomposed oad currents can be expressed as depcted n equaton (7): = ( ' ' ) V Ths equaton shows that the current of each oad bus conssts of current contrbuted by ndvdua generators. The frst term reates drecty the generators current and the second term corresponds to ther votages. Fnay the tota reactve power Q of a oads can be expressed as n equaton (8): Q V { } { V ( ) V ( ' = m = m (8) ' ) V } Wth further smpfcaton of equaton (8), the reactve power contrbuton that oad j acqures from generator s as shown n equaton (9): Q j = n = n V Q Q j = where: Q j : current dependent term of generator to Q j j (7) (9)

3 V Q j : votage dependent term of generator to Q j Vector Q j s used as a target n the tranng process of the proposed SVM. 3. SUPPORT VECTOR MACHNE Support Vector Machne (SVM), generay caed as Kerne machne s a more recent powerfu technque for sovng cassfcaton and regresson probems []. Unke neura network, whch tres to defne compex functons of the nput feature space, SVM performs a nonnear mappng of the data nto a hgh dmensona feature space. Then SVM uses smpe near functons to create near decson boundares n the new space. The probem of choosng an archtecture for a neura network s repaced by the probem of choosng a sutabe kerne for the SVM. n support vector regresson, the basc dea s to map the data x of the nput space nto a hgh dmensona feature space F va a nonnear mappng Φ and to perform near regresson n ths space [3]: where: ( x)3 f n ( x) w, Φ( x) > b wthφ: R F,w F =< (0) f : output functon w : weght vector x : nput b : bas threshod <.,. > : dot products n the feature space Thus, near regresson n a hgh dmensona feature space F corresponds to nonnear regresson n the ow dmensona nput space R n. Snce Φ s fxed, thus w s determned from the fnte sampes { x, y } (=,,3,,N) by mnmzng the sum of the emprca rsk R emp [f] and a compexty term w, whch enforces fatness n feature space: R reg ε [ f ] R [ f ] λ w = ( y, f ( x, w ) λ w 3 = emp = where denotes the sampes sze, λ s reguarzaton ε constant, s the ε - nsenstve oss functon whch s gven by, ( y, f ( x, w ) ( x) () 0 for f y < ε = 3 () f ( x) y ε otherwse The target functon () can be mnmzed by sovng quadratc programmng probem, whch s unquey sovabe. t can be normazed as foows: Φ ( w, ξ ) = w C ( ξ ξ ) y w, Φ( x ) b to w, Φ( x ) b y subject ξ, ξ where : C : a pre-specfed vaue ξ, ξ ε ξ ε ξ 0 (3) : sack varabes representng upper and ower constrants on the outputs of the system The frst part of ths cost functon s a weght decay whch s used to reguate weght sze and penazes arge weghts. Due to ths reguaton, the weght converges to smaer vaues. arge weghts deterorate the generazaton abty of SVM because, usuay, they can cause excessve varance. The second part s a penaty functon whch penazes errors arger than ± ε usng a ε so caed ε - nsenstve oss functon for each of the tranng ponts. The postve constant C determnes the amount, up to whch devatons from ε are toerated. Errors arger than ± ε are denoted wth the so-caed sack varabes representng vaues above ε ( ξ ) and ε ξ, respectvey. The thrd part of the equaton beow ( ) represents constrants that are set to the vaues of errors between regresson predcton f ( x) and true vaues y. The souton s gven by, max Wα,α α, α ( )= max ( α α )( αj αj ) Φ( x ), Φ( xj) α,α Wth constrants, = = = j= ( y ε ) α ( y ε ) α (4) 0 α, α C, α α = 0 =,..., (5) By sovng equaton (4) wth constrants of equaton (5), the agrange mutpers α, α and the weght can be determne as n the regresson functon of equaton (0), whch s gven by, w = ( α ) x and b = w, ( xr xs ) = α (6) The Karush-Kuhn-Tucker condtons that are satsfed by the souton are, α α = 0, =,..., (7)

4 Therefore, the support vectors are ponts where exacty one of the agrange mutpers are greater than zero (on the boundary), whch means that they fuf the Karush- Kuhn-Tucker condton [3]. Tranng ponts wth nonzero agrange mutpers are caed support vectors and gve shape to support vector regresson. When ε = 0, ε oss functon and the optmzaton probem s smpfed as, 4. APPCATON OF SVM TO REACTVE POWER AOCATON The proposed aocaton method s eaborated by desgnng an approprate SVM for the 5-bus equvaent system of south Maaysa as shown n Fg mn β ββ j x, x j = j= = β y (8) Wth constrants, C β C, = β = 0 =,..., (9) and the regresson functon gven by equaton (0), where w = = β x and b = w, ( xr xs ) (0) A non-near mappng can be used to map the data nto a hgh dmensona feature space where near regresson s performed. The Kerne approach s agan empoyed to address the curse of dmensonaty. The non-near support vector regresson souton, usng an ε nsenstve oss functon, max Wα,α α, α ( )= max α ( y ε) α ( y ε) α,α = ( α )( α j α j ) K( x, x j ) = j= α () Wth constrants resembes as equaton (5). Sovng equaton () wth constrants as n equaton (5), determnes the agrange mutpers, α, α and the regresson functon whch s gven by, ( ) f x = α α K( x, x) b () SVs n () the Kerne functon, k( x x ) = Φ( x ), Φ( x j ),. Severa Kerne functons namey, aussan rada bass functon (RBF) Kerne, near Kerne and mutayer perceptron Kerne are avaabe. The commony used Kerne functon s the aussan RBF Kerne whch s as shown n equaton (3): k ( x y) x y, e σ j = (3) Fg.. Snge ne dagrams for the 5- bus equvaent system of south Maaysa Ths system conssts of generators ocated at buses 4 to 5 respectvey. They dever power to 5 oads, through 37 nes ocated at buses,, 4, 5, and 6 respectvey. The data for tranng s assembed usng the day oad curve and performng oad fow anayss for every hour of oad demand. Smary the target vector for the tranng s obtaned from the proposed method usng MNE. nput data (D) for deveoped SVM contans ndependent varabes such as reactve oads (Q, Q, Q 4 to Q 6 ), generator reactve power (Q 4 to Q 5 ), oad bus votage magntude (V to V 6 ), generator bus votage magntude (V 4 to V 5 ), and the target/output parameter (T) whch s reactve power transfer between generators and oads paced at bus to 6. Ths s consdered as 6 outputs of SVM for reactve power transfer aocaton. a. Tranng After the nput and target for tranng data s created, t can be made more effcent to scae (preprocessng) the network nputs and targets so that they aways fa wthn a specfed range. n ths case the mnmum and maxmum vaue of nput and output vectors s used to scae them n the range of - and. The tranng output data of SVM s mpemented separatey for each target n ascendng sequence of generator ke generator connected wth bus 4 to 5 wth the same tranng nput data of one week oad pattern. Next step s to tune the reguarzaton parameter γ and Kerne parameter σ through expermentaton. For ow vaue of γ, mnmzng the compexty of mode s emphaszed, whe for arge vaue, a good fttng of the tranng data ponts s

5 stressed. ntay, the number of tras wth dfferent number of γ keepng the σ constant and vce versa s set. Then, the number of γ s taken as 5 and the number of σ as, resutng n reasonabe accuracy of the output of the SVM wth the target. Fg. shows the performance of the tranng for the RBF Kerne functon estmaton of SVM for seected oad at bus. Dstrbuton of Reactve Power (p.u) at bus x 0-3 estmaton by SVM tranng data ponts γ=5,σ =, RBF enerator Bus No. Fg.. Tranng nput and output data for oad at bus keepng γ =5 and σ =. The attractve mean square error n ths case s equa to whch s due to the estmaton by SVM and the tranng data ponts havng smar characterstcs. b. Pre-Testng and Smuaton After the SVM has been traned usng MATAB, next step s to smuate the SVM. The entre sampe data s used n pre testng. After smuaton, the obtaned resut from the traned SVM s evauated wth a near regresson anayss. The regresson anayss for the traned network that referred to contrbuton of a generators to oad at bus s shown n Fg. 3. Dstrbuton of Reactve Power (p.u) at bus x 0-3 SVM Output Target enerator Bus No. Fg. 3. Regresson anayss between the SVM output and the correspondng target keepng γ =5 and σ =. The SVM output s ndcated wth ne havng crces whereas the target s ndcated by the sod ne. 5. RESUT AND ANASS The case scenaro s that for each hour the rea and reactve power at each oad s assumed to decrement by 5% from hour to 68, from the nomna traned pattern. Besdes t aso assumed that a generators aso decrease ther producton proportonay accordng to ths varaton n the oad demands. Ths assumpton s beng made to ensure that a reactve power generaton of generator at buses 4 to 5 vares n respond to the varyng day oad patterns. The aocaton of reactve power to oads usng proposed SVM on hours 33 out of 68 hours s presented n Tabe aong wth the resut obtaned through MNE method n Tabe. TABE ANASS OF REACTVE POWER AOCATON ON HOUR 33 B THE SVM Supped by SVM Output oad bus no. (MVAr) en en en en en en en en en en en en Tota oad Actua oad TABE ANASS OF REACTVE POWER AOCATON ON HOUR 33 B THE MODFED NODA EQUATONS METHOD Supped by Modfed Noda Equatons Method oad bus no. (MVAr) en en en en en en en en en en en en Tota oad Actua oad Note that the resut obtaned by the SVM output s compared we wth the resut of Modfed noda equatons method. The contrbuton from a generators to a snge oad at bus 4 gves the argest dfference.e MVAr as compared to 06.5 MVAr of actua oad. Ths may be due to that SVM needs optma arrangement of nput and output data for tranng. The mean square error of SVM output s very sma whch are ess than n ths case, the RBF Kerne

6 functon type wth σ = s chosen as the parameter for the SVM. Moreover, ths SVM smuaton computes wthn 3 msec whereas the MNE method took 738 msec for the cacuaton of same reactve transfer power aocaton. Therefore t can be concuded that the SVM s more effcent n terms of computaton tme. The oad fow resuts for the test system are gven n Tabes. TABE BUS DATA FOR THE 5-BUS SSTEM ON HOUR 33 Bus Votage eneraton oad no. Magntude Ange Rea Reactve Rea Reactve (p.u) (p.u) (MW) (MVAr) (MW) (MVAr) t can be observed that the sum of the reactve power contrbuted by each generator obtaned from modfed noda equatons s n conformty wth the actua power fow. 6. CONCUSON n ths paper, a new modfed noda equatons method has been deveoped to dentfy the reactve power transfer between generators and oad. The robustness of the proposed method has been demonstrated on the 5-bus equvaent system of south Maaysa. The deveoped SVM adopts reactve power aocaton outputs determned by MNE technque as an estmator to tran the SVM. Better computaton tme s cruca to mprove onne appcaton. For ths the SVM output provdes the resuts n a faster and convenent manner wth very good accuracy. n future smar aocaton method can be used for both batera contract mode and power dstrbuton n the network. 8. REFERENCES [] H.Shareef and M.W. Mustafa, Rea and Reactve Power Aocaton n a Compettve Market, WSEAS Transactons on Power Systems, Vo., No. 6, 006, pp [] Reta. R, and Vargas. A, Eectrcty Tracng and oss Aocaton Methods Based on Eectrc Concepts, EE Proceedngs eneraton, Transmsson and Dstrbuton, Vo. 48, No. 6, 00, pp [3] Chang.. C, and u. C. N, An Eectrcty Tracng Method wth Appcaton to Power oss Aocaton, nternatona Journa of Eectrca Power and Energy Systems, Vo. 3, No., 00, pp [4] Zoban, A. and c. M. D, Unbundng of Transmsson and Ancary Servces Part : Technca ssues, EEE Transacton on Power Systems, Vo., No., 997, pp [5] Teng. J. H, Power Fow and oss Aocaton for Dereguated Transmsson Systems, nternatona Journa of Eectrca Power Energy Systems, Vo. 7, No. 4, 005, pp [6] J. Baek, Tracng the fow of eectrcty, EE Proceedngs eneraton Transmsson & Dstrbuton, Vo. 43, No. 4, 996, pp [7] F. F Wu, N, and P We, Power transfer aocaton for open access usng graph theory fundamentas and appcatons n systems wthout oop fows, EEE Transactons on Power Systems, Vo. 5, No. 3, 000, [8] M.W. Mustafa, H. Shareef, and M.R Ahmad, An mproved usage aocaton method for dereguated transmsson system, Proc. nt. Conf. Power Engneerng, Sngapore, 005, pp [9] S.Abde. Effcent computaton agorthm for cacuatng oad contrbutons to ne fows and osses, EE Proc. On eneraton, Transmsson and Dstrbuton, Vo. 53, No. 4, 006, pp [0] D. Krschen, R. Aan,. Strbac Contrbutons of ndvdua enerators to oads and Fows, EEE Trans Power Systems, Vo., No., 997, pp [] Nzam. Muhammad Mohamed Azah, A-Dabbagh.Majd, and Hussan.An, Dynamc votage coapse predcton n a practca power system wth support vector machne, TENCON 008 EEE Regon 0 Conference, 008, pp.-6. [] J.H. Chow, F.F Wu and J.A.Momoh, Apped mathematcs for restructured eectrc power systems, New ork:sprnger, 005. [3].Musrn and T.K.A. Rahman, Votage stabty based weak area custerng technque n power system, Natona Power & Energy Conference (PECon 004), Kuaa umpur, 004, pp [4] A.J. Smoa and B. Schokopf, On a kerne-based method for pattern recognton, regresson, approxmaton an operator nverson, Agorthmca, Vo., 998, pp. -3. [5] K. Peckmas, J.A.K. Suykens, T. Van este, J. De Brabanter,. ukas, B. Hamers, B. De Moor and J. VAndewae, S-SVMab Toobox User s ude, ESAT-SCD-SSTA Technca Report 0-45, Kathoeke Unverstet euven, 003. [6] D.Z.ang, N.n and Z.J. uo, Appcaton of support vector regresson mode based on phase space reconstructon to power system wde-area stabty predcton, nternatona Power and Energy Conference (PEC), Sngapore, 007, pp ACKNOWEDMENT The authors wsh to acknowedge the Mnstry of Scence, Technoogy and nnovaton (MOST) of Maaysa for the fnanca fundng of ths project.