Evolutionary Computational Techniques to Solve Economic Load Dispatch Problem Considering Generator Operating Constraints

Transcription

1 Internatonal Journal of Engneerng Research and Applcatons (IJERA) ISSN: Natonal Conference On Advances n Energy and Power Control Engneerng (AEPCE-K1) Evolutonary Computatonal Technques to Solve Economc Load Dspatch Problem Consderng Generator Operatng Constrants Santosh L.Mhetre Department of Electrcal Engneerng, PVG s COET, Pune Pune, Inda Mangesh.S. Thakare Department of Electrcal Engneerng, PVG s COET, Pune Pune, Inda Abstract The economc load dspatch plays an mportant role n the operaton of power system, and several models by usng dfferent technques have been used to solve these problems. More recently, the soft computng technques have receved more attenton and were used n a number of successful and practcal applcatons. The purpose of ths work s to fnd out the advantages of applcaton of the evolutonary computng technques n partcular to the economc load dspatch problem. Here, an attempt has been made to fnd out the mnmum cost by usng PSO & GA consderng generator operatng constrants. These ncluded are contnuous prohbted zones, ramp rate lmts, and cost functons whch are non smooth or convex. The results are compared wth the tradtonal technque and GA, PSO seems to gve a better result wth better convergence characterstc. Keywords Economc Load Dspatch, Genetc Algorthm, Partcle Swarm Optmsaton. I. INTRODUCTION Electrcal energy cannot be stored, but s generated from natural sources and delvered as demand arses. A transmsson system used for the delvery of bulk power over consderable dstances, and a dstrbuton system s used for local delveres. Such confguraton apples to all nterconnected networks (regonal, natonal, nternatonal), where the number of elements may vary. The transmsson networks are nterconnected through tes so that utltes can exchange power, share reserves and render assstance to one another n tmes of need. For an nterconnected system, the fundamental problem s one of mnmzng the source expenses [1]. Economc dspatch (ED) problem s one of the fundamental ssues n power system operaton. In essence, t s an optmzaton problem and ts objectve s to reduce the total generaton cost of unts, whle satsfyng constrants. Prevous efforts on solvng ED problems have employed varous mathematcal programmng methods and optmzaton technques. These conventonal methods nclude the lambdateraton method, the base pont and partcpaton factors method, and the gradent method. In these numercal methods for soluton of ED problems, an essental assumpton s that the ncremental cost curves of the unts are monotoncally ncreasng pecewse-lnear functons. The proposed method consders the nonlnear characterstcs of a generator such as ramp rate lmts and prohbted operatng zone for actual power system operaton. The feasblty of the proposed method was demonstrated for three dfferent systems respectvely, as compared wth the real-coded GA method n the soluton qualty and computaton effcency []. In ths respect, evolutonary computatonal technques such as GAs (GAs), may prove to be very effectve n solvng nonlnear ELD problems wthout any restrctons on the shape of the cost curves [3]. These evolutonary algorthms (EAs) are search algorthms based on the smulated evolutonary process of natural selecton, varaton, and genetcs. EAs are more flexble and robust than conventonal calculus-based methods. Evolutonary Programmng dffers from tradtonal GAs n two aspects: Evolutonary Programmng uses the control parameters (real values), but not ther codng as n tradtonal GAs, and Evolutonary Programmng reles prmarly on mutaton and selecton, but not crossover, as n tradtonal GAs. Hence, consderable computaton tme may be saved n EP [3]. Chen and Chang presented a GA method that used the system ncremented cost as encoded parameter for solvng ED problems that can take nto account network losses, ramp rate lmts, and valve-pont zone [4]. The GA methods have been employed successfully to solve complex optmzaton problem, recent research has dentfed some defcences n GA performance. Partcle Swarm Optmsaton (PSO), frst ntroduced by Kennedy and Eberhart, s one of the evolutonary computatonal technques. It was developed through smulaton of a smplfed socal system, and has been found to be robust n solvng contnuous nonlnear optmzaton problems [5]. The PSO technque can generate hgh-qualty solutons wthn shorter calculaton tme and stable convergence characterstc than other stochastc methods [6]. Although the PSO seems to be senstve to the tunng of some weghts or parameters, many researches are stll n progress for provng ts potental n solvng complex power system problems [7]. The programs are developed n MATLAB and Spread Sheet, to solve economc load dspatch problem and to verfy the solutons by usng Evolutonary Computatonal Technques (GA and PSO) consderng generator operatng constrants such as Power Balance equaton,transmsson loss, Mnmum and Maxmum Power lmts,generator ramp rate lmts & Generator prohbted operatng zone by varyng above constrants ELD problems are solved. Intally to check the effectveness of PSO, ELD problems were solved by usng Conventonal Method, GAs and by PSO. Vgnan s Lara Insttute of Technology and Scence Page 30

2 Later on more focus s gven on PSO method due to ts effectveness over others. II. PROBLEM STATEMENT The Economc Load Dspatch (ELD) s generatng adequate electrcty to meet the contnuously varyng consumer load demand at the least possble cost under a seven of constrants. Practcally, whle the scheduled combnaton of unts at each specfc perod of operaton are lsted, the ELD plannng must perform the optmal generaton dspatch among the operatng unts to satsfy the load demand, spnnng reserve capacty, and practcal operaton constrants of generators. A. Objectve Functon The objectve of the economc dspatch problem s to mnmze the total fuel cost of thermal power plants subjected to the operatng constrants of a power system. The smplfed cost functon of each generator can be represented as a quadratc functon as n Mnmze Ft F ( P ) [1] Where, F t Total generaton cost, F Cost functon of generator, P Power of generator, n Number of generators A, B, C Cost coeffcents of generator, B. Constrants The Economc load Dspatch s subjected to the followng Constrants 1) Power balance equaton a) Wthout consderng transmsson loss b) Wth consderng transmsson loss For power balance, an equalty constrant should be satsfed. The total generated power should be the same as total load demand plus the total lne loss. Where 1 t ( ) F F P A P B P C 1 1 P g P [3] P D s the total system demand and P L s the total lne loss. c) Constrant Satsfacton Technque D P P P [4] g D L To satsfy the equalty constrant of equaton (3), a loadng of any one of the unts s selected as the dependent loadng Pd and ts present value s replaced by the value calculated accordng to the followng equaton, P P P P [5] d D L 1 1 n [] Where P d can be calculated drectly from the equaton (5) wth the known power demand P D and the known values of remanng loadng of the generators. Therefore the dspatch soluton wll always satsfy the power balance constrant provded that P d also satsfes the operaton lmt constrant as gven n equaton (5). An nfeasble soluton s omtted and above procedure s repeated untl P d satsfes ts operaton lmt. Because P L also depends on P d, we can substtute an expresson for P L n terms of P 1, P.P d..p n and B mn coeffcents. After substtutng t n the equaton (.5), separate the ndependent and dependent generator terms to obtan a quadratc equaton for Pd. Solvng the quadratc equaton for Pd, the power balance equalty condton s exactly satsfed. ) Transmsson Loss The total transmsson loss, ) The general form of the loss formula usng B-coeffcents s P L g j g 1 j1 The transmsson loss formula s known as the George s formula ) The accurate form of the loss formula Where P g and P gj are the real power njectons at the th and j th buses respectvely. B 00, B 0 and B j are the loss coeffcents whch are constant under certan assumed condtons. s number of generaton buses. The transmsson loss formula s known as the Kron s formula. 3) Mnmum and maxmum power lmts Generaton output of each generator should be lad between maxmum and mnmum lmts. The correspondng nequalty constrants for each generator are mn max g g g Where P g, mn and P g, max are the mnmum and maxmum output 4) Generator ramp lmts P B P [6] P B B P P B P [7] L 00 0 g g j g 1 1 j1 P P P [8] In ELD research, a number of studes have focused upon the economcal aspects of the problem under the assumpton that unt generaton output can be adjusted nstantaneously. Even though ths assumpton smplfes the problem, t does not reflect the actual operatng processes of the generatng unt. The operatng range of all on-lne unts s restrcted by ther ramp rate lmts []. Fg.1 shows three possble stuatons when a unt s on-lne from hour t-1 to t. Fg l (a) shows that the unt s n a steady operatng status. Fg l (b) shows that the unt s n an ncreasng power generaton status. Fg. l (c) shows that the unt s n a decreasng power generaton status. Vgnan s Lara Insttute of Technology and Scence Page 31

3 Fg. 1 Three possble stuatons of an on-lne unt Then the nequalty constrants due to ramp rate lmts are gven, 1) If generaton ncreases ) If generaton decreases P P UR [9] 0 P P DR [10] 0 Where P 0 s the prevous output power of unt. DR and UR are the down ramp and up ramp lmts Combnng (9), (10) and (1) the constraned optmzaton problem s modfed as follows, 0 0 Max P, P DR P Mn P, P UR [11] 5) Generator prohbted operatng zones The operatng zone of a generatng unt may not be avalable always for power generaton due to lmtatons n practcal operatng constrants as shown n Fg. Fg. Cost functons wth prohbted operatng zones l P,mn P P,1 u l P k,3,... N PZ, [1] P P k, 1 P, k u l P, N P Where P l PZ, P,k, and P u,max k, are the lower and upper boundary of prohbted operatng zone of unt, respectvely. N PZ, s the number of prohbted zones of unt. 6) Valve Pont Effects: The generatng unts wth mult-valve steam turbnes exhbt a greater varaton n the fuel-cost functons. Snce the valve pont results n the rpples as shown n Fg.3 Fg. 3 Valve Pont effects The valve-pont effect, snusodal functons are added to the quadratc const functons. Therefore, quadratc functons equaton should be replaced follows: where e and f are the coeffcents of unt reflectng valve pont effects. The generatng unts wth multvalve steam turbnes exhbt a greater varaton n the fuel-cost functons. The valve-pont effects ntroduce rpples n the heat-rate curves. F ( P ) A P B P C esn f P mn P [13] Vgnan s Lara Insttute of Technology and Scence Page 3 III. METHODS TO SOLVE ECONOMIC LOAD DISPATCH In ths work economc load dspatch problem s solved by evolutonary computatonal technques (GA and PSO) and compared wth the conventonal method.e. Lagrangan method. A. GENETIC ALGORITHM The GA s a stochastc global search method ntroduced by John Holland of Mchgan Unversty n 1970 s.that mmcs the metaphor of natural bologcal evoluton such as selecton, crossover, and mutaton. The GA combnes an artfcal prncpal wth genetc operaton. The artfcal prncpal s the Darwnans survval of fttest prncpal and the genetc operaton s abstracted from nature to form a robust mechansm that s very effectve at fndng optmal solutons to complex real world problems. GA s operates on strng structures. The strng s bnary dgts representng a codng of control parameters for a gven problem. The each parameter of the gven problem s coded wth strngs of bts. The ndvdual bt s called gene and the content of the each gene s called allele. The total strngs of such genes of all parameters wrtten n a sequence are called a chromosome so there exsts a chromosome for each pont n the search space. Here we have to know about search space. In ths approach, a GA canddate soluton s represented as a lnear strng analogous to a bologcal chromosome. The general scheme of GAs starts from a populaton of randomly generated canddate solutons (chromosomes). Each chromosome s then evaluated and gven a value whch corresponds to a ftness level n objectve functon space. In each generaton, chromosomes are chosen based on ther ftness to reproduce offsprng. Chromosomes wth a hgh level of ftness are more lkely to be retaned whle the ones wth low ftness tend to be dscarded. Ths process s called selecton. After selecton, offsprng chromosomes are constructed from parent chromosomes usng operators that resemble crossover and mutaton mechansms n evolutonary bology. The crossover operator, sometmes called recombnaton, produces new offsprng chromosomes that nhert nformaton from both sdes of parents by combnng partal sets of elements from them. The mutaton operator randomly hangs elements of a chromosome wth a low probablty. Over multple generatons, chromosomes wth hgher ftness values are left based on the survval of the fttest. B. PARTICLE SWARM OPTIMISATION: Partcle Swarm Optmsaton (PSO) s a populaton based stochastc optmzaton technque developed by Dr. Ebehart and Dr. Kennedy n 1995, nspred by socal behavor of brd flockng or fsh schoolng. PSO shares many smlartes wth evolutonary computaton technques such as GAs (GA). The

4 system s ntalzed wth a populaton of random solutons and searches for optma by updatng generatons. However, unlke GA, PSO has no evoluton operators such as crossover and mutaton. PSO smulates the behavors of brd flockng. Suppose the followng scenaro: a group of brds are randomly searchng food n an area. There s only one pece of food n the area beng searched. All the brds do not know where the food s. But they know how far the food s n each teraton. So what's the best strategy to fnd the food? The effectve one s to follow the brd, whch s nearest to the food. PSO learned from the scenaro and used t to solve the optmzaton problems. In PSO, each sngle soluton s a "brd" n the search space. We call t "partcle". All of partcles have ftness values, whch are evaluated by the ftness functon to be optmzed, and have veloctes,whch drect the flyng of the partcles. PSO s ntalzed wth a group of random partcles (solutons) and then searches for optma by updatng generatons. In every teraton, each partcle s updated by followng two "best" values. The frst one s the best soluton (ftness) t has acheved so far. (The ftness value s also stored.) Ths value s called pbest. Another "best" value that s tracked by the partcle swarm optmzer s the best value, obtaned so far by any partcle n the populaton. Ths best value s a global best and called gbest. When a partcle takes part of the populaton as ts topologcal neghbors, the best value s a local best and s called p-best. After fndng the two best values, the partcle updates ts velocty and postons wth followng equaton (14) and (15). t1 t t Vd. v C1. rand(). PbestdP gd In the above equaton, t1 t t1 P P V [15] gd gd d The term t rand 0. pbestd P s called partcle memory gd nfluence. The term t rand0. gbestd t P s called swarm gd nfluence. v (t) whch s the velocty of th partcle at teraton t must le n the range V mn V (t) V max. The parameter V max determnes the resoluton, or ftness, wth whch regons are to be searched between the present poston and the target poston.if V max s too hgh, partcles may fly past good solutons. If V mn s too small, partcles may not explore suffcently beyond local solutons. The constants C 1 and C pull each partcle towards Pbest and gbest postons. The acceleraton constants C 1 and C are often set to be 1.5 to. accordng to past experences. In general, the nerta weght ω s set accordng to the followng equaton, max mn max ter Where termax ω s the nerta weghtng factor ω max - maxmum value of weghtng factor ω mn - mnmum value of weghtng factor ter max - maxmum number of teratons t C. Rand(). gbestd t P [14] gd IV. ter - current number of teraton CASE STUDY EXAMPLES AND RESULTS PARAMETERS: The Lagrangan Method parameters are The GA Method parameters are The PSO method parameters are EXAMPLES OF GENERATI RAMP RATE LIMITS AND PROHIBITED ZONES CONSTRAINTS: EXAMPLE 1: THREE UNIT SYSTEM The cost characterstcs of the three unts are gven as F 1 = P P Rs/h F = P P Rs/h F 3 = P P Rs/h The unt operatng ranges are: 50 MW P 1 50 MW 5 MW P 150 MW 15 MW P MW Generatng unts ramp rate lmts and prohbted zones Sr. No. Convergence tolerance ε Step sze α Maxmum allowed teratons 0 Length of strng, P 0 (MW) UR (Mw/ Hr) DR (Mw/ Hr) 16 bts Populaton sze L 0 Crossover probablty, P c 0.8 Mutaton probablty, P m 0.01 Populaton sze 10 No. of Iteratons 10 Inerta weght factor ω s set by ω max = 0.9 and ω mn = 0.4 lmt of change n velocty of each member n an ndvdual was as Vd max = +0.5 Pg mn, Vd mn = -0.5 Pg mn Acceleraton constant C 1 = 1.5 and C =. Prohbted zones (MW ) [105,117] [165,177] [50,60] [9,10] [5,3] [60,67] B mn Coeffcent matrx B mn = For the system load of 300 MW the GA and PSO method s appled RESULT OF POWER GENERATION AND TOTAL FUEL COST The results for 300 MW power and fuel cost wth transmsson loss are obtaned as Vgnan s Lara Insttute of Technology and Scence Page 33

5 Ft (Rs/Hr) Total Fuel cost Power Output (MW) GA Method P 1 (MW) P (MW) P 3 (MW) Total Power output (MW) Power loss P L (MW) Total generaton cost (Rs /Hr) No. of Iteraton Table 1: Comparson of Power and fuel cost of 300 MW wth loss The comparson of results of total fuel cost for 300 MW wth loss, Prohbted zone and ramp rate between GA Method and as shown n fgure 4 Fgure 4 Total fuel cost of 300MW wth Prohbted Zone & Ramp rate lmts EXAMPLE : SIX UNIT SYSTEM Ths example adapted from [] comprses sx generatng unts wth ramp rate and prohbted zone. RESULT OF POWER GENERATION AND TOTAL FUEL COST: The results for 163 MW power and fuel cost wth transmsson loss, ramp rate and prohbted zone are obtaned as Table : Comparson of Power and fuel cost of 163 MW wth loss The comparson of results of total fuel cost for 163 MW wth loss between Conventonal method, Genetc Algorthm Method and as shown n fgure GA Method Prohbted zones & Ramp rate lmts GA Method Total Fuel Cost Seres1 Power Output(MW) GA Method P 1 (MW) P (MW) P 3 (MW) P 4 (MW) P 5 (MW) P 6 (MW) Total Power output (MW) Power loss P L (MW) Seres1 Seres Total generaton cost (Rs /Hr) No. of Iteraton Fg. 5 Comparson of Power and fuel cost of 163 MW wth loss EXAMPLE OF VALVE POINT EFFECTS Ths example adapted from [5] comprses three generatng unts wth quadratc cost functons together wth the effects of valve pont loadngs. RESULT OF TOTAL FUEL COST The results for 850 MW total fuel cost wth valve pont loadngs as Power Output(MW) GA Method Total generaton cost (Rs /Hr) No. of Iteraton Table 3: Comparson of total fuel cost of 850MW V. CONCLUSIONS Based on the observatons and results obtaned from smulaton studes carred out for three unts, systems for Lambda Iteraton, GA and PSO the followng conclusons are drawn, Lambda Iteraton method requres large number of teratons converges slowly and s not cost effectve. GA method provdes fast convergence and requres less number of teratons and s cost effectve. The PSO method requres less number of teratons s cost effectve and provdes fastest convergence among the three.pso method s effcent for both, wth and wthout loss problems and constrants. VI. REFERENCES [1] D. P. Kothar and J. S. Dhllon, Power System Optmzaton, Prentce Hall of Inda, New Delh, 004. [] Zwe-Lee. Gang, Partcle Swarm Optmsaton to solvng the economc dspatch consderng the generator constrants, IEEE Trans. Power Syst.18 (3) (003), pp [3] P. H. Chen and H. C. Chang, Large-Scale Economc Dspatch by Genetc Algorthm, IEEE on Power Systems, Vol. 10, No. 4, pp , Nov [4] J. Kennedy and R.C. Eberhart, Partcle Swarm Optmsaton, Proceedngs of the IEEE, Internatonal Conference on Neural Networks Perth, Australa (1995), pp [5] D. C. Walters and G. B. Sheble, Genetc Algorthm soluton of economc dspatch wth valve pont loadng, IEEE Trans. Power Syst., 8 (August (3)) (1993), pp [6] Sudhakaran. M, Ajay-D-Vmal Raj. P; Palanvelu T.G; Applcaton of Partcle Swarm Optmsaton for Economc Load Dspatch Problems, 007.ISAP 007.Internatonal Conference on 5-8 Nov.007 Page(s) :1-7 [7] Had Saadat Power System Analyss McGraw Hll company, Inc, [8] Kwang Y. Lee, and Jong-Bae Park, Applcaton of Partcle Swarm Optmsaton to Economc Dspatch Problem: Advantages and Dsadvantages, Internatonal conference PSCE 006. [9] Park, J.-B., K.-S. Lee, J.-R. Shn, and K. Y. Lee, A Partcle Swarm Optmsaton for Economc Dspatch wth Non-Smooth Cost Functons, IEEE Transactons on Power Systems, February 005 [10] A. Jan, I. Musrn, N. Amnudn, M. M. Othman, T. K. A. Rahman, Partcle Swarm Optmsaton (PSO) Technque n Economc Power Dspatch Problems Proc.4th PEOCO010, Shah Alam, Selangor, MALAYSIA: 3-4 June 010 Vgnan s Lara Insttute of Technology and Scence Page 34