AN EVOLUTIONARY PROGRAMMING APPLICATION TO OPTIMAL REACTIVE POWER DISPATCH

Transcription

1 U.P.B. Sc. Bull., Seres C, Vol. 72, Iss. 1, 2010 ISSN x AN EVOLUTIONARY PROGRAMMING APPLICATION TO OPTIMAL REACTIVE POWER DISPATCH Florn IONESCU 1, Constantn BULAC 2, Ioana PISICĂ 3, Ion TRISTIU 4, Lucan TOMA 5 În cadrul lucrăr se prezntă utlzarea programăr evolutve (PE) pentru repartţa optmă a puter reactve (ROPR). Obectvul ROPR îl consttue mzarea perderlor de putere actvă ş menţnerea tensunlor ş a puterlor reactve generate în lmtele admsble. În acest sens, varablele de control sunt tensunle mpuse la nodurle generatoare ş ploturle transformatoarelor. Metoda propusă a fost testată pe reţeaua test IEEE cu 30 de nodur modfcată. Ths paper presents an applcaton of evolutonary programg (EP) to optmal reactve power dspatch (ORDP). The objectve of ORDP s to mze the real power losses and keep the voltages and reactve power generatons n ther operatng lmts. In ths matter the control varables are generator bus voltages and transformer taps. The proposed EP approach was evaluated on the modfed IEEE-30 bus system. Keywords: Evolutonary Programg, Optmal Reactve Power Dspatch, Mnmzng the Real Power Losses Lst of Symbols N B set of number of total buses; N PQ, N PU set of number of PQ and PU buses; N B-S set of number of total buses excludng slack bus; L,T set of number of transmsson lnes and transformers; jθ U = Ue voltage at bus ; Yk = Gk + jb k (,k) element of nodal admttance matrx; P sch sch, Q scheduled actve and reactve power at bus ; P,k actve power loss n branch -k; Q g reactve power generaton at bus ; 1 Eng., Power Engneerng Faculty, Unversty POLITEHNICA of Bucharest, Romana 2 Prof., Power Engneerng Faculty, Unversty POLITEHNICA of Bucharest, Romana 3 Eng., Power Engneerng Faculty, Unversty POLITEHNICA of Bucharest, Romana, e-mal: oanapsca@gmal.com 4 Asoc. Prof., Power Engneerng Faculty, Unversty POLITEHNICA of Bucharest, Romana 5 Eng., Power Engneerng Faculty, Unversty POLITEHNICA of Bucharest, Romana

2 92 Florn Ionescu, Constantn Bulac, Ioana Pscă, Ion Trstu, Lucan Toma N,k turn raton of (,k) transformer ; U, U lower and upper voltage lmts; Q g, Q g lower and upper reactve power generaton lmts; N k, N k lower and upper turn rato lmts; 1. Introducton Power system economcal operaton conssts of two aspects: actve power regulaton and reactve power dspatch. In a real large scale power system ths s a complex problem and s consdered conventonally as two separate problems [1-6]. The purpose of optmal reactve power dspatch (ORDP) s to control the generator bus voltages and tap-settngs of the under-load tap changng (ULTC) to mze the network power loss and mprove voltage profles. Solvng ths problem s subject to a number of constrants such as lmts on bus voltages, ULTC settngs, reactve power of the generators, etc. Many methods based on lnear and non-lnear programg have been proposed to solve ths problem. These optmzaton methods are based on successve lnearzaton and use the frst and second dervatves of objectve functon and ts constrant equatons as the search drectons. Such treatments qute often lead to a local mum pont and sometmes result n dvergence. Some new methods based on artfcal ntellgence have recently been used n ORDP and reactve power plannng to solve local mum problems and uncertantes [7,8,9]. Ths paper presents an applcaton of evolutonary programg (EP), nstead of the conventonal methods, to solve an optmzaton reactve power dspatch problem. EP does not requre the mathematcal assumptons appled n the conventonal methods and offers a powerful global search over the control varables space. 2. Problem formulaton The objectve of ORPD s to mze the network real power loss n the transmsson network wtch can be descrbed as follows: [ LOSS ] = Δ, MIN P P (1) (, k) L T wth equalty constrants: k

3 An evolutonary programg applcaton to optmal reactve power dspatch 93 n sch = k ( k cos( θ θk) + k sn ( θ θk) ) B S k = 1 P U U G B N n sch = k ( k sn ( θ θk) k cos( θ θk) ) PQ k = 1 Q U U G B N (2) and nequalty constrans: Q Q Q N g g g PU N N N (, k) T k k k U U U N B (3) For mplementng the evolutonary programg algorthm the nequalty constrans are added as a quadrc penalty terms to the objectve functon and the actve power losses are expressed from the equaton of actve power balance: U θ PLOSS = Pg Pl = PS (, ) P const (4) N N B B where PS ( U, θ) s the njected actve power at slack-bus and P const ncludes all unchanged generated and load actve power n the system. Thus the optmzaton problem can be redefned as follows: lm lm = ( U, θ) + λ ( ) + λ ( q S U ) Qg g g NU lm 2 2 MIN f P U U Q Q (5) NQg lm where λ and U λq are the penalty factors that can be ncreased n the g lm lm optmzaton procedure and U, respectvely Q g are defned n the followng equatons: lm U f U < U U = U f U > U (6) < lm Qg f Qg Qg Qg = Qg f Qg > Qg

4 94 Florn Ionescu, Constantn Bulac, Ioana Pscă, Ion Trstu, Lucan Toma It can be seen that the generalzed objectve functon f q s a non-lnear and non-contnuous functon. Gradent based conventonal methods are not good enough to solve ths problem. 3. Evolutonary Programg Algorthm for solvng the mzaton problem Evolutonary Programg (EP) s dfferent from conventonal optmzaton methods. It does not need to dfferentate cost functons and constrants. It uses probablty transton rules to select generatons [8-11]. Each ndvdual competes wth some other ndvduals n a combned populaton of the old generaton and the mutated old generaton. The competton results are valued usng a probablstc rule. Wnners are selected to consttute the next generaton and the number of wnners s the same as that of ndvduals n the old generaton. The procedure of EP for ORDP s brefly lsted as follows [8,10]: Intalzaton The ntal control varable populaton s selected by randomly selectng p =, UPV N k, =1,2,,m, where m s the populaton sze, from the sets of unform dstrbutons rangng over U, U and N, k N k. The ftness 1 value, f =, of each ndvdual, p, s obtaned by runnng the 1 + PS ( U, θ ) Newton-Raphson method. Statstcs The values of the mum ftness, mum ftness, sum of ftness and average ftness of ths generaton are calculated as follows: avg { j j, 1, 2,..., } { j j, 1,2,..., } f = f f f f j = m f = f f f f j = m f f Σ = m = 1 fσ = m f (7)

5 An evolutonary programg applcaton to optmal reactve power dspatch 95 Mutaton Each equaton: p s mutated and assgned to p + accordance wth the followng f p+ m, j= p, j+ N 0, β ( xj xj), j = 1,2,..., n f (8) In practcal applcatons a small fxed mutaton probablty can only result n a premature convergence, whle a search wth a large fxed mutaton probablty wll not converge. To solve ths problem an adaptve mutaton scale s gven to change the mutaton probablty: β(k) βstep f f (k) unchanged β (k + 1) = β(k) f f (k) decresed ; β(0) = β βfnal f β(k) βstep < βfnal nt (9) Competton Each ndvdual p n the combned populaton has to compete wth some other ndvduals to get ts chance to be transcrbed to the next generaton. After all ndvduals had competed wth some others ndvduals, they wll be ranked n descendng order of ther correspondng ftness value. The frst m ndvduals of the 2m ndvduals formed are transcrbed along wth ther correspondng ftness value f to be the bass of the next generaton. 5. Numercal Results 5.1. IEEE 30-bus system In ths secton the modfed IEEE 30-bus system (Fg. 1) s used to show the effectveness of the EP algorthm. The network conssts of sx generatorbuses, 21 load-buses and 43 branches, of wtch four branches are ULTC transformer (6,9), (6,10), (4,12), (28,27). The branch parameters and loads are gven n [3] whle the parameters and varable lmts are lsted n Table Intal condtons The ntal condtons are defned as follows: generator bus voltages and transformer taps are set to 1.0 p.u.; total actve and reactve loads are P LOAD =2.834 p.u. and Q LOAD =1.262 p.u. respectvely.

6 96 Florn Ionescu, Constantn Bulac, Ioana Pscă, Ion Trstu, Lucan Toma After runnng the power flow program for the ntal condtons, one notces: there are sx buses wth voltages less then mum admssble value (see also Fg. 2): U 1 = p.u.; U 19 = p.u.; U 24 = p.u.; U 25 = p.u.; U 26 = p.u.; U 29 = p.u.; the generated actve and reactve power and network power losses are: P G = p.u. ; Q G = p.u.; P LOSS = p.u. Fg. 1. The modfed IEEE 30-bus test system Table 1 Parameters and varable lmts Reactve Power Generaton Lmts Bus Q g Q g Voltage and Tap Settngs Lmts U PU U PU U PQ U PQ N k N k

7 An evolutonary programg applcaton to optmal reactve power dspatch Optmal soluton obtaned usng EP In order to use EP for ORDP problem the control varables of the transmsson network (voltages at PU buses and turn rato of ULTC) are arranged as elements of an ndvdual n populatons used durng evolutonary search. p = Ug2 Ug5 Ug8 Ug11 Ug13 Ug 30 N6,9 N6,10 N4,12 N28,27 The lower and upper lmts are consdered as follows: U = 0.9 p.u. and U =1.1 p.u. for PU buses and slack bus; U = 0.95 p.u. and U =1.05 p.u. for PQ buses; N k = 0.9 p.u and N k =1.1 p.u. The populaton sze s chosen to be 50, whle the number of compettors s 20. In the ntal populaton each element p (0) j of ndvduals p (0), = 1,50 s ntalzed wth a random value between the lower and upper lmt. The ftness values f used for mutaton, competton, and reproducton are obtaned for each ndvdual by runnng the power flow program based on Newton Raphson method. After successful search usng the EP, we obtan the optmal values of control varables (PU bus voltages, slack bus voltage and transformer tap N k ). These are the elements of the best ndvdual n the last populaton: ,9 6,10 4,12 28,27 = U g U g U g U g U g U g p N N N N opt After computaton of power flow wth these optmal values, voltages at PU buses and turn rato of ULTC we obtan the new values for bus voltages (Fg. 2), total actve end reactve generated power and actve power loses as follows: P G = p.u.; Q G = p.u. and P LOSS = p.u. To be noted that after the optmzaton process we obtan a better reactve power dspatch n the system (Fg. 3) and a power savng of: P save ntal opt PLOSS PLOSS % = 100 = 100 = 14.28% ntal P LOSS (10)

8 98 Florn Ionescu, Constantn Bulac, Ioana Pscă, Ion Trstu, Lucan Toma Also as a consequence of ORPD n optmal operaton condton all voltage values are n ther lmts unlke the ntal condton where there are some buses wth voltage values less then the lower lmt (Fg. 2). 1.1 Voltage level U[p.u.] Buses Intal condtons Optmal condtons Fg.2. Voltage level Intal condtons / Optmal Condtons Reactve power generaton Q[p.u.] Buses Intal condtons Optmal condtons Fg.3. Reactve power generaton Intal condtons / Optmal Condtons

9 An evolutonary programg applcaton to optmal reactve power dspatch Conclusons Optmal Reactve Power Dspatch s an optmzaton problem of a noncontnuous and non-lnear functon wth uncertantes arsng from large-scale power systems. Evolutonary Programg wth the technques developed n ths paper s a sutable algorthm to solve such a problem. The EP does not requre the mathematcal assumpton appled n the conventonal methods and offers a powerful global search over the control varable space. Smulaton studes were carred out on the modfed IEEE-30-bus system. The smulaton results shows that the applcaton of EP to ORPD could acheve very attractve power savngs and better operatng condtons, cheepng the voltages and reactve generated powers nto ther lmts. REFERENCES [1]. H.W. Dommel, W.F.Tnney, Optmal Power Flow Solutons, IEEE Transactons on Power Apparatus and Systems, 87, 1968, pp [2]. O Alsac, B Stott, Optmal Load Flow Wth Steady State Securty, Power System Laboratory, Unversty of Manchester, Insttute of Scence and Technology Manchester, U.K. 1974, pp [3]. K.Y. Lee, Y.M. Park, J.L. Ortz, A Unted Approach to Optmal Real and Reactve Power Dspatch, IEEE Transactons on Power Apparatus and Systems, 104, 1985, pp [4]. R.R. Shoults, D.T. Sun, Optmal Power Flow Based upon P-Q Decomposton, IEEE Transactons on Power Apparatus and Systems, 101, 1982, pp [5]. K.R.C. Mamandur, R.D.Chenoweth, Optmal Control of Reactve Power Flow for Improvements n Voltage Profles and for Real Power Loss Mnmzaton, IEEE Transactons on Power Apparatus and Systems, vol. PAS-100, no. 7, July 1981, pp [6]. M. Erema, C. Bulac, H. Crscu, B. Ungureanu Analza asstată de calculator a regmurlor sstemelor electroenergetce. Metode. Algortm. Aplcaţ (Computer Aded Analyss of the Electrc Power Systems), Edtura Tehncă. Bucureşt 1985 [7]. K.H. Abdul-Rahman, S.M. Shahdehpour, A Fuzzy Based Optmal Reactve Power Control, IEEE Transactons on Power Systems, vol. 8, no. 2, May 1993, pp [8]. L L La, Intellgent System Applcatons n Power Engneerng, John Wley & Sons, New York, 1998 [9]. Q.H. Wu, J.T. Ma, Power System Optmal Reactve Power Dspatch Usng Evolutonary Programg, IEEE Transactons on Power Systems, vol. 10, no. 3, August 1995, pp [10]. D. Dumtrescu, Algortm Genetc ş Strateg Evolutve aplcaţ în ntelgenţa artfcală ş în domen conexe (Genetcs Algorthm and Evolutonary Strateges applcatons n artfcal ntellgence and other domans), Edtura Albastă, Cluj Napoca, 2006

10 100 Florn Ionescu, Constantn Bulac, Ioana Pscă, Ion Trstu, Lucan Toma [11]. M. Erema, C. Bulac, G. Cârţână, I. Trştu, Tehnc de Intelgenţă Artfcală Aplcată în Conducerea Sstemeleor Electroenergetce (Artfcal Intellgence Technques Appled n Power Systems), Edtura AGIR, Bucureşt, 2001.