Optimal Power Flow by a Primal-Dual Interior Point Method

Transcription

1 Optmal ower Flow y a rmal-dual Interor ont Method Amnuay augnm and Somporn Srsumrannuul Astrat One of the most mportant requrements n power system operaton, ontrol and plannng n energy management system (EMS) of modern power system ontrol enters s optmal power flow (OF). Optmal power flow, whh s haraterzed as a dffult optmzaton prolem, nvolves the optmzaton of an ojetve funton that an tae varous forms, for example, mnmzaton of total produton ost, and mnmzaton of total loss n transmsson networs, sujet to a set of physal and operatng onstrants suh as generaton and load alane, us voltage lmts, power flow equatons, and atve and reatve power lmts. Ths paper presents a method to solve an optmal power flow prolem wth the ojetve of the mnmzaton of total produton ost ased on a prmal-dual nteror pont theory. The method has een mplemented n a omputer program whh an effetvely solve a non-lnear ojetve funton wth lnear onstrans and has a good onvergene haraterst. The effeny of the methodology appled to the optmal power flow prolem s demonstrated y the IEEE -, -, 7- and -us test systems. Keywords IEEE test systems, KKT ondtons, optmal-dual nteror pont method, optmal power flow NOMENCLATURE FT = Total ost ojetve funton G = Atve power generaton at us NG = Numer of generaton uses = Atve power at us D = Atve demand at us Q = Reatve power at us QG = Reatve power generaton at us QD = Reatve demand at us ND = Numer of load uses L = System real power loss V = Voltage magntude at us Vj = Voltage magntude at us j N = Total numer of uses Yj = Element of admttane matrx at row and olumn j δ = Voltage angle at us δ j = Voltage angle at us j θj = Angle of admttane Y j T = Tap poston of a transformer at G,mn = Lower lmt of atve power generaton at us G,max = Upper lmt of atve power generaton at us QG,mn = Lower lmt of reatve power generaton at us Q = Upper lmt of reatve power generaton at us G,max Amnuay augnm and Somporn Srsumrannuul are wth the Department of Eletral Engneerng, Faulty of Engneerng, Kng Mongut s Insttute of Tehnology North Bango, Bang Sue, Bango 000, Thaland hone: Ext: 0, Fax: Ext: 0, E-mals: amnuay09@yahoo.om, spss@mtn.a.th V,mn = Lower lmt of voltage magntude at us V,max = Upper lmt of voltage magntude at us T,mn = Lower lmt of tap poston of a transformer at T,max = Upper lmt of tap poston of a transformer at Q = Quadrat ost oeffent matrx of ojetve funton x = Vetor of varales n prmal prolem A = Coeffent matrx of onstrants = Vetor of rght hand sde onstant value of onstrants w = Vetor of varales n dual prolem s = Vetor of sla varales n dual prolem X = Dagonal matrx of prmal varales S = Dagonal matrx of sla varales e = Unty vetor μ = Barrer parameter dx = Vetor of searh dreton of prmal varales dw = Vetor of searh dreton of dual varales ds = Vetor of searh dreton of sla varales t = rmal slaness vetor u = Dual slaness vetor v = Complementary slaness vetor α = rmal step length αd = Dual step length α = Atual step length = Iteraton ndex. INTRODUCTION Optmal power flow has een reeved onsderale attentons from reseahers for the past few deades as t s one of the most mportant tools n energy management system (EMS). The man ojetve of an OF prolem s to determne the optmal settng of ontrol varales n a power system networ to optmze an ojetve funton whle respetng a set of physal and operatng onstrants suh as generaton and load alane, us voltage lmts, power flow equatons, and atve and reatve power lmts. Generally, an OF rolem an e formulated as

2 []: Mnmze F( xu, ) sujet to gxu (, ) = 0 hxu (, ) 0 F( xu, ) = ojetve funton u = set of ontrol varales (e.g., generator atve power, generator voltage, transformer tap poston) x = set of dependent varales (e.g., load us voltage, phase angle) g( xu, ) = power flow onstrants hxu (, ) = set of non-lnear nequalty onstrants The ojetve funton an tae varous forms suh as fuel ost, transmsson losses, and reatve soures alloaton. The ojetve funton of nterest n ths paper s desgnated as the mnmzaton of the total produton ost of sheduled generatng unts. Suh a mnmzaton prolem s most used as t reflets urrent eonom dspath prate and mportanly, ost related aspet s always raned hgh among operatonal requrement n power systems. Varous tehnques have een proposed to solve the OF prolem for example, non-lnear programmng [], quadrat programmng [], lnear programmng []-[], and nteror pont methods [7]-[9]. Among these, the nteror pont method has een of reent nterest and wll e employed as a major tool to solve the OF prolem n ths paper. The nteror pont (I) tehnque was prososed y N.K.Karmarar []. It an solve a large-sale lnear programmng prolem y movng through the nteror, rather than the oundary as n the smplex method, of the feasle regon to fnd an optmal soluton. The I method was orgnally proposed to solve lnear programmng prolems; however, later t was mplemented to effently handle quadrat programmng prolems [0]-[]. The nteror pont tehnque starts y determnng an ntal soluton usng Mehrotra s algorthm [], whh s used to loate a feasle or near-feasle soluton. There are then two proedures to e performed n an teratve manner untl the optmal soluton has een found. The former s the determnaton of a searh dreton for eah varale n the searh spae y a Newton s method. The latter s the determnaton of a step length normally assgned a value as lose to unty as possle to aelerate soluton onvergene whle strtly mantanng prmal and dual feaslty. A alulated soluton n eah teraton wll e heed for optmalty y the Karush-Kuhn-Tuer (KKT) ondtons, whh onsst of prmal feaslty, dual feaslty and omplementary slaness. Four ase studes are performed and analysed to llustrate the effeny of a developed nterror pont ased OF omputer program, usng the IEEE -, -, 7-, and -us test system.. OF FORMULATION As has een dsussed, the ojetve funton onsdered n ths paper s to mnmze the total produton ost of sheduled generatng unts. OF formulaton onssts of three man omponents: ojetve funton, equalty onstrants, and nequalty onstrants.. Ojetve Funton N NG G T = ( G ) = ( G + G + ) = = Mn F F a (). Equalty Constrants The equalty onstrants are atve/reatve power flow equatons and generaton/load alane. where V (, δ )- + = 0 () G D Q ( V, δ )-Q + Q = 0 () G D NG ND G ) D ) L = = ( - ( - = 0 () N δ = j j δ δ j θj j = V (, ) V VYos( ) () N Q ( V, δ ) = V V Y sn( δ δ θ ) () j j j j j = Equatons () and () are nonlnear and an e lnearzed y the Taylor s expanson usng where ΔV (, δ ) J J Δδ ΔQV (, δ ) J J ΔV J J J J = s the jaoan matrx. Transmsson loss ( L ) gven n the Equaton () an e dretly alulated from the power flow.. Inequalty Constrants The nequalty onstrants onsst of generator atve/reatve power lmts, voltage magntude lmts, and transformer tap poston lmts. () G,mn G G,max Q Q Q (9) G,mn G G,max,mn,max (7) V V V (0) T T T (),mn,max. DIM METHOD The DIM method s started y arrangng a prmal quadrat programmng prolem nto a standard form as Mnmze T T xqx+ x () sujet to Ax =, x 0 () Equaton () an e transformed nto the orrespondng dual prolem havng the form. T T Maxmze xqx+ w () sujet to T Qx + A w + s =, s 0, w Free () Stoppng rtera for the algorthm s ased on three ondtons of Karush-Kuhn-Tuer (KKT): prmal feaslty, dual feaslty, and omplementary slaness; namely, these

3 three ondtons have to e satsfed and are gven n Equatons (), (7) and (). where Ax =, x 0 (rmal feaslty) () T Qx + A w + s =, s 0 (Dual feaslty) (7) XSe = μe (Complementary slaness) () T ( ) μ = x s n (9) X = dag( x,..., x n ) (0) S = dag( s,..., s n ) () From the KKT ondtons, the dretons of translaton are alulated usng the Newton s method whh yelds the followng system equatons []. A 0 0 dx Ax T T -Q A I dw = Qx + A w + s S 0 X ds X S e μ e () The rght hand sde of Equaton () s so-alled slaness vetors and an e assgned to new varales as t = Ax () T u = Qx + A w s () = From Equatons ()-(), we have v μ e X S e () x = Ad t () T x + A w + s Qd d d = u (7) Sd x + Xd s = v () Comnng and rearrangng Equatons (7) and () gve T dx + ( S + XQ ) XA dw = ( S + XQ ) ( Xu v ) (9) Wth Equatons () and (9), a dual searh dreton an e derved as ( ) ( ) ( ) T dw = A S + XQ X A A S + XQ Xu v + t () The equaton for a prmal searh dreton an e derved from Equaton (9). ( ) ( ) d T x = S + X Q w + X A d u v () Wth the prmal searh dreton and Equaton (), a sla searh dreton an e otaned y ( ) s = x d X v S d () To fnd approprate step lengths whle eepng the prmal and dual prolem feasle, Equatons ()-() are used. ( x j x ) j ( s j s ) j mn - / α = d d < 0 () xj α = mn - / d d < 0 () D sj ( D ) αmax = mn α, α () α = 0.99α () max An updated soluton an e omputed y Equatons (7)- (9). x x α d (7) + = + x w w α d () + = + w s s α d (9) + = + s. ALICATION OF DIM TO OF The DIM algorthm appled to the OF prolem s summarzed step-y-step as follows. Step Read relevant nput data. Step erform a ase ase power flow y a power flow suroutne. Step Estalsh an OF model. Step Compute Equatons ()-(). Step Calulate searh dretons wth Equatons ()-(). Step Compute prmal, dual and atual step-lengths wth Equatons ()-(). Step 7 Update the soluton vetors wth Equatons (7)-(9). Step Che f the optmalty ondtons are satsfed y Equatons ()-() and f μ ε (ε =0.00 s hosen). If yes, go to the next step. Otherwse go to step. Step 9 erform the power flow suroutne. Step 0 Che f there are any volatons n Equaton ()-(). If no, go to the next step; otherwse, go to step. Step Che f a hange n the ojetve funton s less than or equal to the prespefed tolerane. If yes, go to the next step; otherwse, go to step. Step rnt and dsplay an optmal power flow soluton.. NUMERICAL RESULTS A DIM-ased OF omputer program was developed on MATLAB and tested on a omputer wth CU entum,. GHz and RAM M. The effeny of the program s demonstrated y the four IEEE standard power systems:,, 7, uses. The data of the four power systems are provded n the appendx of ths paper. The parameters V,mn and V,max are assgned as 0.9 and.0 per unt. Note that some of the data are suppled y the authors as they are not ompletely avalale n the lterature and westes the authors ould fnd.. IEEE -Bus System The IEEE -us system (Fgure 9) [] onssts of generatng unts wth a total apaty of 9 MW. The total demand n ths system s 9 MW. Tale shows the optmal atve dspath from the generatng unts. It s oserved that the atve power of the generators loated at uses, and s ndng at ther upper lmts. Fgure shows the onvergene of the optmal soluton (.e., the dfferene n the total osts of the prmal and dual prolems s less than the tolerane) n teraton. Fgure onfrms that the us voltages are ept wthn 0.9 and.0 per unt. It an e seen n ths ase that the voltages at all uses are aove the nomnal value. Tale Atve power dspath

4 for the IEEE -us system Bus No. Bus No loss.9 F T ($/hr) Bus No. Bus No loss.79 F T ($/hr) Total ost n prmal prolem Total ost n dual prolem Total ost n prmal prolem Total ost n dual prolem Total ost($/hr) Voltage magntude (per unt) Numer of teraton Fgure Convergene of the total osts of the IEEE -us system er-unt voltage magntude Total ost($/hr) Voltage magntude (per unt) Numer of teraton Fgure Convergene of the total osts of the IEEE -us system er-unt voltage magntude Bus numer Fgure Bus voltage magntudes for the IEEE -us system. IEEE -Bus System Ths system ontans generatng unts wth a total nstalled apaty of MW, as shown n Fgure 0 []. The system demand s. MW. Atve power dspath s gven n Tale. Ths system requres teratons to onverge to the optmal soluton, as shown n Fgure. The soluton onvergene trend s very smlar to that of the IEEE -us system; namely, the program starts onvergng after a few teratons have een performed. The us voltage magntudes an e mantaned wthn the lmts and s shown n Fg., where only voltage at us s slghtly lower than per unt. Tale Atve power dspath for the IEEE -us system Bus numer Fgure Bus voltage magntudes for the IEEE -us system. IEEE 7-Bus System The algorthm has tested on a medum-szed power system: the IEEE 7-us system (Fgure ) []. The system has 7 generatng unts wth a total nstalled apaty of 0 MW, supplyng a total demand of 0. MW. The optmal soluton for the atve dspath s gven n tale. Fgure shows the onvergene of the optmal total osts n the prmal and dual prolem, ndatng that ths system requres 9 teratons to onverge. The us voltage magntudes are depted n Fgure, where all the us voltage magntudes are ept wthn the lmts. Tale Atve power dspath for the IEEE 7-us system Bus No. Bus No.

5 Total ost($/hr) Voltage magntude (per unt) loss.7 F T ($/hr).7. x Total ost n prmal prolem Total ost n dual prolem Numer of teraton..0 Fgure Convergene of the total osts of the IEEE 7-us system er-unt voltage magntude loss F T ($/hr) Bus numer Fgure Bus voltage magntudes for the IEEE 7-us system. x 0. Total ost n prmal prolem Total ost n dual prolem. IEEE -Bus System For the last ase study, a large-szed system, the IEEE -us system [] as shown n Fgure, s tested. The system s omposed of generators wth a total nstalled apaty of MW and the system demand s MW. Tale shows the atve dspath of the generators wth a total ost of 97.0 $/hr and a loss of.0 MW. The onvergene result s gven n Fgure 7 sgnfyng that the system requres teratons to onverge. All the us voltage magntudes are shown n Fgure. Tale Atve power dspath for the IEEE -us system Bus No. Bus No. Total ost($/hr) Numer of teraton Fgure 7 Convergene of the total osts of the IEEE -us system

6 .0 er-unt voltage magntude Tale 7 Generaton ost funtons for the IEEE -us system Voltage magntude (per unt) Gen. Bus No. a mn 0 0 max Bus numer Fgure Bus voltage magntudes for the IEEE -us system. COMARISON OF NUMBER OF ITERATIONS AND COMUTTATION TIME Tale summarzes the numer of teratons and omputaton tme requred for the IEEE -, -, 7- and - us power systems. It s oserved that the numer of teraton remans unhanged for the frst two systems, slghtly nreases for the thrd and notaly nreases n the last one. The man reason for the last system s that many onstrants are found ndng, ausng the varales orrespondng to those onstrants to hange ther dretons toward an optmal soluton and therefore more teratons requred. As to the omputaton tme, the IEEE -us system sees a onsderaly dfferene from the others as a onsequene of ts numer of teratons. Tale Numer of teratons and omputaton tme Case study IEEE -us IEEE -us IEEE 7-us IEEE -us Numer of teratons 9 Total CU tme (s) CONCLUSION Ths paper has presented the prmal-dual nteror pont method employed to solve the optmal power flow prolem. The DIM method ntalzes a soluton y nvong the Mehrotra s algorthm and determnes a searh dreton and orrespondng step length for a move nsde the feasle regon to an mproved soluton. The algorthm s termnated when all the KKT ondtons are satsfed. The developed omputer program was tested on the IEEE -, -, 7-, and -us power systems. The ase studes have shown that the DIM method s roust and an provde an optmal soluton wth fast omputaton tme and a small numer of teratons. 7. AENDIX Tale Voltage magntude and transformer tap poston lmts Type Mnmum Maxmum Remar Voltage magntude All uses Transformer tap 0..0 All Gen. Bus No. Gen. Bus No. 7 9 Tale Generaton ost funtons for the IEEE -us system a mn Tale 9 Generaton ost funtons for the IEEE 7-us system a Gen. Bus No. a mn Tale 0 Generaton ost funtons for the IEEE -us system mn 0 max max max 0 0 0

7 Tale 0 (ontnued). REFERENCES Gen. Bus No a mn max [] Momoh, J. A. 00. Eletr ower System Applatons of Optmzaton. New Yor : Marel Deer. [] Contaxs, G.C.; Dels, C.; and Korres, G. 9. Deoupled optmal power flow usng lnear or quadrat programmng. IEEE Transaton on ower System. [] Wood, A.J.; and Wollenerg, B. F. 99. ower Generaton Operaton and ontrol. nd ed. : John Wley and Sons. [] Krshen, D.S.; and Van Meeteren, H.. 9. MW/Voltage ontrol n a lnear programmng ased optmal power flow. IEEE Transatons on ower Systems. Vol. No. : -9. [] Muherjee, S.K.; Reo, A.; and Doulgers, C. 99. Optmal power flow y lnear programmng ased optmzaton. IEEE. : 7-9. [] Olofsson, M.; Andersson, G.; and Soder, L. 99. Lnear programmng ases optmal power flow usng seond order senstvtes. IEEE Transatons on ower Systems. Vol.0 No. : [7] Momoh, J.A.; Austn, R.F.; Adapa, R.; and Oguor, E.C. 99. Applaton of nteror pont method to eonom dspath. IEEE. : [] We, H.; Sasa, H.; and Yooyama, R. 99. An applaton of nteror pont quadrat programmng algorthm to power system optmzaton prolems. IEEE Transatons on ower Systems. Vol. No. : 0-. [9] Dng, Q.; L, N.; and Wang, X Implementaton of nteror pont method ased voltage/reatve power optmzaton. IEEE. : [0] Momoh, J.A.; Guo, S. X.; Oguor, E. C.; and Adapa, R. 99. The quadrat nteror pont method solvng power system optmzaton prolems. IEEE Transatons on ower Systems. Vol.9 No. : 7-. [] Momoh, J. A. 99. Applaton of Quadrat Interor ont Method to Optmal ower Flow. TR-0. : Eletr ower Researh Insttute (ERI). Howard Unversty [] Momoh, J.A.; and Zhu, J.Z Improved nteror pont method for OF prolems. IEEE Transatons on ower Systems. Vol. No. : -0. [] Mehrotra, S. 99. On the mplementaton of a prmal-dual nteror pont method. Sam Journal on Optmzaton. Vol.. [] Fang, Sh.C.; and uthenpura, S. 99. Lnear Optmzaton and Extensons:Theory and Algorthms. New Jersey: rente-hall Internatonal, In. [] Satpathy,.K.; Das, D.; and Dutta Gupta,.B. 00. Crtal swthng of apators to prevent voltage ollapse. Eletral ower Systems Researh 7. : -0 [] Alves da Slva, A..; and Quntana, V.H. 99. attern analyss n power system state estmaton. Eletral ower & Energy Systems. Vol.7 No. : -0 [7]

8 Fgure 9 Dagram of the IEEE -us system Fgure Dagram of the IEEE 7-us system Fgure 0 Dagram of the IEEE -us system Fgure Dagram of the IEEE -us system