Hybrid Genetic Algorithm for Optimizing Environmental/Economic Power Dispatch

Transcription

1 Mna Unversty rom the SelectedWorks of Dr. Adel A. Elbaset Sprng June 16, 010 Hybrd Genetc Algorthm for Optmzng Envronmental/Economc Power Dspatch Dr. Adel A. Elbaset, Mna Unversty Avalable at:

2 Hybrd Genetc Algorthm for Optmzng Envronmental/Economc Power Dspatch Adel A. Elbaset Electrcal Engneerng Dept. Mna Unversty, aculty of Engneerng, EGYPT E-mal: الملخص العربي ي أ ى انمضبيب انخي ح ى انعبيهي في يدبل انبيئ في ز اال ي لضي حمهيم األ بعثبث ان ه ث ي كبف ان صبدس. خذيذ نخغزي األح بل ح صيع ب بي حذاث انخ نيذ انك شببئي ان خخهف بطشيم إلخصبدي أخزي في األعخببس يمذو زا انبحث طشيم حمهيم األ بعبثبث ان ه ث انصبدس ي حذاث انخ نيذ انخي حع م ببن ل د األحف سي. يسأن أخز حأثيش األ بعبثبث ان ه ث إلخصبديبث انخشغيم عهي ح صيع األح بل عببسة ع يسبن The generaton of electrcty from fossl fuel plays an mportant rule n atmosphere polluton phenomenon; snce t releases several pollutants, such as Sulfur Oxdes, Ntrogen Oxdes and Carbon Doxde. Recently, ths problem has attracted much attenton due to the pressng publc demand for clean ar. Snce the text of the Clean Ar Act Amendments of 1990 and smlar Acts by European and Japanese governments, envronmental constrants have forced the utltes to modfy ther desgn or operatonal strateges to reduce polluton and atmospherc emssons of the thermal power plants. Achevng only the mnmum cost can no longer be the only crteron for dspatchng electrc power due to ncreasng concern wth envronmental consderaton. Emssons can be reduced by dspatch of power generaton to mnmze emssons nstead of or as a supplement to the usual cost objectve of economc dspatch. Envronmental/Economc dspatch s a mult-objectve problem wth conflctng objectves because polluton s conflctng wth mnmum cost of generaton. Tradtonally, the cost functon and emsson for each generator has been approxmately represented by a lambda teraton method, frst-order gradent method, second-order gradent method, Newton Raphson method (NR), Lnear programmng, and dynamc programmng. In tradtonal methods, formulaton of Lagrangan functon as well as the ncremental loss s always the key pont n the soluton algorthm. All of these methods lead to Page 1 of 10 يثبني غيش خطي يخعذدة انمي د األ ذاف حى حح يه ب إني دان راث ذف احذ ع طشيك يعبيم انعم ب factor( )Penalty..)Hybrd Genetc Algorthm يمذو زا انبحث طشيم نهحم حعخ ذ عهي انطشيم ان مخشح طبمج عهي انشبك انك شببئي انميبسي رج سيبضي ببسخخذاو طشيمت انحسبة اندي ي ان خخهظ ( IEEE 30 bus أ ضحج ان خبئح يذي فبعهي انطشيم ان مخشح بعذ ا حى يمبس خ ب يع خبئح انطشق انخمهيذي. Englsh Abstract Abstract:- The conventonal economc power dspatch s a non-lnear optmzaton problem wth several constrants. The envronmental ssues concernng the pollutant emssons produced by fossl based thermal generatng unts became a matter of concern n recent years. Accordngly, mnmzaton of emssons by dspatch of power generaton s very desrable. The problem s how to supply all electrcal loads at mnmum cost takng the envronmental ssues nto account (mnmum polluton). Envronmental/Economc dspatch s a mult-objectve problem treats economc and pollutant emssons. Ths mult-objectve problem s converted nto sngle objectve functon usng a modfed prce penalty factor approach to calculate envronmental /economc power dspatch problem. A commonly used technque to solve ths problem s to apply genetc algorthm to a small number of generatons to get near optmum economc soluton for the power system dspatch. Ths paper presents an applcaton of hybrd genetc algorthm (HGA) to acheve an optmal soluton for the Combned Economc Emsson Dspatch problem (CEED). The optmum soluton obtaned by the proposed technque s faster and more effcent than that obtaned by usng both the Conventonal Optmzaton methods (CM) and smple Genetc Algorthm (GA). The proposed algorthm s tested on standard IEEE 30-bus model system. Keywords:- Optmzaton, GA, HGA, power Dspatch 1. INTRODUCTION

3 naccurate results due to the nonlnear and non-convex characterstcs of generatng unts. These methods also fal to fnd the optmal soluton n case of complex dspatch problems. Thus we are n a bad need for developng a relable, fast and effcent algorthm to solve the power dspatch problem. Recently, the economc dspatch problem has been solved usng modern heurstc optmzaton technques, such as evolutonary algorthms [1], Tabu search [], Partcle Swarm optmzaton [3], smulated annealng [4], genetc algorthms [5], Hopfeld neural networks [6,7], fuzzy [8,9] and Ant colony technques [10]. GA s probablstc heurstc procedures or optmzng algorthm, whch s based-on the prncple of natural selecton and genetcs. It has demonstrated consderable success n provdng good solutons to many nonlnear optmzaton problems. Recently, GA has been studed to solve the power system optmzaton problems. It combnes soluton evaluaton wth randomzed, structured exchanges of genetc nformaton between solutons to obtan optmalty. GA contans many computatonal advantages, such as smplcty and generalzaton. In addton, t searches multple solutons smultaneously n contrast to conventonal optmal algorthms. Therefore, the possblty of fndng global optmal soluton s ncreased. But due to the premature convergence nature of the smple GA method, there s a possblty for gettng stuck at local optmal soluton. Therefore, the objectves consdered n ths study are mnmzng both fuel cost and envronmental mpact of emsson by usng HGA to mprove optmal soluton and work more effcently than smple GA. The man advantage of HGA s that t fnds near optmal soluton n relatvely short tme compared wth other random searchng methods; conventonal methods or smple GA. The prce penalty factor that combnes the emsson costs wth the normal fuel costs s presented n ths paper. Ths paper s organzed as follows. Secton gves the mathematcal descrpton of the economc power dspatch problem. Secton 3 presents a technque for calculatng prce penalty factor. Secton 4 presents the methodology of MatPower and GA and also the mprovements mplemented for solvng the problem. Secton 5 presents a case study and smulaton results. Secton 6 summarzes the conclusons of study.. PROBLEM ORMULATION The Envronmental/Economc dspatch problem s mult-objectve, snce the two conflctng objectves, fuel cost and pollutant emsson, should be mnmzed smultaneously to satsfy the system constrants. A. Objectve unctons Objectve 1: Mnmzaton of fuel costs The classcal economc power dspatch problem s to fnd the real power generaton for each unt, whch mnmzes the total fuel cost whle satsfyng the total requred demand. The generator cost curves are represented by quadratc functons to represent the loadng effects. The objectve functon s the total producton cost measured n dollars per hour can be mathematcally defned by the followng equaton [5,10 ] t ng n g 1 Page of 10 1 ( a. P b. P c ) (1) g g Where t : Total producton cost, $/h, : Producton cost of th generator, $/h, a, b, c : The fuel cost coeffcents of the th generator, P g :The power generated by th generator, ng :The number of power generators. Objectve : Mnmzaton of Emsson The total emsson can be reduced by mnmzng the three major pollutants: oxdes of ntrogen (NO x ), oxdes of sculpture (SO x ) and carbon doxde (CO ). The total emsson of atmospherc pollutants can be expressed n a quadratc equaton as the sum of all the three pollutants resultng from generator real power P g. Measured n tons per hour. The objectve functon can be expressed as follows [1,3 ] :- E Nox ng n g 1 E nox 1 ( d. P e. P f ) () g g Where; d, e, and f are the coeffcents of generator emsson characterstc. The polluton control cost (n $/ton) can be obtaned by assgnng a penalty factor.

4 The prevous two equatons are combned together gvng the total objectve functon whch represents both the fuel cost and the total emsson. In addton, prce penalty factor (h) s used n the objectve functon to combne both fuel cost, $/h and pollutant emssons, ton/h. The combned economc and emsson dspatch problem can be formulated as follows: ng n g mn T ( a. P b. P c ) h. ( d. P e. P f ) (3) c 1 g g 1 Once prce penalty factor (h) s calculated, equaton (3) can be rewrtten as follows: n g mn T ( a h. d ). P ( b h. e ). P ( c h. f ) (4) c 1 g 3. Dvde the mum cost of each generator by ts mum NOx emsson. Page 3 of 10 g g g B. Constrants The economc/envronmental dspatch problem s subject to two types of constrants, the real power balance equalty constrant and generaton capacty nequalty constrant Constrant 1: Real power balance The total power generated must supply the total load demand plus the transmsson losses as the followng equaton: ng 1 P g P P d Loss (Equalty Constraned) (5) Where; P d : Total load demand, P L : Transmsson losses. The transmsson loss can be determned form B-coeffcent method. It can be expressed as follows: P Loss n g 1 n g j1 n g Pg. Bj. Pgj Pg. B 0 B00 1 MW Where P, : The real power generaton at the th, j th generator, g P gj B j : The transmsson loss coeffcents, B o : The dmensonless vector of lnear loss coeffcents, B 00 : The constant of loss coeffcents, MW. All B-coeffcents can be calculated based on load flow solutons. A Newton-Raphson load flow, losses calculaton as well as B loss coeffcents are mplemented n "lne flow", whch s wrtten usng MATLAB. The loss s used as an evaluaton functon n the Genetc Algorthm Optmzaton Toolbox to search the optmal CEED problem. Constrant : Generaton capacty or a stable operaton, the real power generated, power lmts as follows:- mn P P P (Inequalty Constraned) (7) g where g g mn P g and Pg (6) P g by each generator s constraned by lower and upper are the mnmum and the mum real power outputs of th generator. 3. CALCULATION O PRICE PENALTY ACTOR, H [ 9,11] The prce penalty factor, h can be calculated as follows: 1. Evaluate the mum cost of each generator at ts mum output as follows: ( P ) ( a.( P ) b.( P ) C ) (8) g g g. Evaluate the mum NOx emsson of each generator at ts mum output as follow:- E ( P g ) ( d.( P ) e.( P ) f ) Ton/h (9) g g

5 ( P E ( P g g ) a.( P ) d.( P ) g g ) b.( P e.( P g g ) C ) f (10) Recallng that ( Pg ) h (11) E ( P ) g n g 1 g 4. Sort the obtaned values of h s n ascendng order 5. Add the mum capacty of each unt, ( P ), repeatedly startng from the smallest h untl total demand s met accordng to the nequalty shown below. P g P d 6. At ths stage, h assocated wth the last unt n the process s the prce penalty factor h ($/ton) for a gven load Pd, and equaton (4) can be solved to obtan envronmental economc dspatch usng GA and HGA. 4. METHODOLOGY Our technque uses Matlab package MATPOWER and hybrd genetc algorthm to optmze the envronmental economc dspatch problem. A. MATPOWER MATPOWER s a package of Matlab m-fles for solvng the power flow and optmal power flow problems. The data fles used by MATPOWER are smply Matlab m-fles whch defne and return the varables base MVA, bus, branch, gen, and gencost. The bus, branch, and gen varables are matrces. MATPOWER has three power flow solvers. MATPOWER uses two approaches for solvng the optmal power flow problem. The frst one s based on the constr functon ncluded n Matlab s optmzaton Toolbox, whch uses a successve quadratc programmng technque wth a quas-newton approxmaton for the Hessan matrx. The second one s based on lnear programmng [1]. B. Genetc Algorthm [13] A genetc algorthm GA s a search technque used to fnd exact or approxmate solutons to optmzaton and search problems. GA s are categorzed as global search heurstcs. These algorthms are a partcular class of evolutonary algorthms that use technques nspred by evolutonary bology such as nhertance, mutaton, and crossover. The basc termnology of the GA s ftness functon. The ftness functon s the objectve functon. The GA tres to fnd the mnmum of the ftness functon. The ftness functon of the CEED s wrtten as an M-fle whch s treated as a functon handle nput argument to the man genetc algorthm functon. The ftness functon can be expressed as follows:- n g t ( P ) ( a h. d ). P ( b h. e ). P ( c h. f ) (13) g 1 g g A Hybrd GA s an optmzaton functon to mprove the value of the ftness functon. The hybrd GA uses the fnal pont from the genetc algorthm as ts ntal pont. HGA s a robust approach because no restrctons on the soluton space are made durng the search process. Although the bnary representaton s usually appled to power optmzaton problems, n ths paper, we use the real valued representaton scheme for soluton. The use of real valued representaton n the HGA s used n ths paper. 5. CASE STUDY AND SIMULATION RESULTS To assess the feasblty of the HGA method, t has been appled to solve the emsson, economc and CEED problem on power systems wth 6 unts. Every test case was solved for approxmately more than 40 ndvdual trals by Intel Core(TM) Duo CPU, T8300@.4 GHz, Wth 4GB RAM under Wndows Vsta Ultmate. (1) Page 4 of 10

6 Prce Penalty actor, $/ton A. CASE STUDY The proposed method has been appled on the power system IEEE 30-bus system. The 30-bus system contans sx generators wth total generaton capacty 335MW, 4 load buses and 41 transmsson lnes wth 4 tap changng transformers. The cost and emsson coeffcents are gven n Appendx A. In normal operaton of the system, the loss coeffcents B matrces wth the 100 MVA base capacty are gven n Appendx B. The computed values of proposed prce penalty factor for power generaton of IEEE-30 bus are shown n Table I and g. 1. Accordng to the results of many experments, Table II shows the control parameters for HGA algorthm after runnng a number of smulatons. Table I Prce penalty factor for each power generated Pg h Generated Power gure 1. Relaton between Prce Penalty factor and Generated power Table II Parameter Values for HGA Populaton Type Double Vector Populaton Sze 60 Elte Count 1 Crossover racton 0.9 Mgraton Interval 0 Generatons 100 Tme Lmt 60 Stall Generaton Lmt 50 Stall Tme Lmt 400 Tolerance uncton e-006 Intal Penalty 10 tness Scalng Selecton Crossover Mutaton Hybrd Page 5 of 10

7 B. SIMULATION RESULTS our methods (MatPower, NR, GA and HGA algorthms) were employed to test the system under study. In the case study, each ndvdual P g contans sx generator power outputs: P g1, P g, P g3, P g4, P g5 and P g6, whch are generated randomly under constrants as shown n Appendx A. The ftness functon for 189. MW load demand wth h=.96 s defned as follows:- n g Tc mn mn( ) 1 (14) Where, P g P $/h g P g Pg P g Pg P g Pg P g Pg $/h $/h $/h $/h P g Pg $/h The economc, emsson and CEED problems are solved by usng CM, GA and HGA. The control parameters for the HGA are shown n Table II. The followng power loads and ther correspondng percentages at each mum generaton capacty are consdered n the smulaton, 89. MW (56.48%), 39 MW (71.40%), 55 (76.1%), 56 (76.4%) and 83.4 MW (84.60%). g. shows the resultng best ftness plot after 55 generatons usng HGA for 189.MW load demand. g. -a shows the best and mean values of the populaton n every generaton. g. -b shows the current best ndvdual for each varable. rom ths gure, t can be shown that, the results of usng HGA can mprove the accuracy of the soluton effcently. g. 3 shows a comparson of fuel cost obtaned from conventonal method, smple GA and HGA for varous power demands. On the other hand, g. 4 shows a comparson of emsson generaton (ton/h) from generators for each mplemented methods under varous power demands. gure 5 shows a comparson of losses n transmsson lnes for each mplemented methods under varous power demands. gure 6 shows the best generator settng obtaned from conventonal method, smple GA and HGA for varous power demands. or accurate results, Table III shows the results of the proposed method and the results of the classcal method and GA when the load values are 189. MW, 39 MW, 55MW, 56 MW and 83.4 MW. It can be seen,from fgures and Table III, that HGA algorthm gves global or near global optmal soluton, hence t provdes better solutons than those provded by the conventonal technque and smple GA. Also, we can observe, from Table III and gs 3-5, that the soluton obtaned by the conventonal method s not an optmal one. g. 7 shows total fuel and emsson cost for each mplemented method under varous power demands. It can be seen also from g. 1 and g. 7 that, f the load ncreases from 55 MW (76.1%) to 56 MW (76.4%) the generaton cost wll be very hgh. So, t s not economc to operate the power system above 76.1% of ts capacty. Page 6 of 10

8 gure. HGA Smulaton under 189. MW load demand gure 3. Comparson of fuel cost for each methodology under varous loadng condton gure 4. Comparson of emsson for each methodology under varous loadng condton Page 7 of 10

9 gure 5. Comparson of Losses for each methodology under varous loadng condton Total Load 189. MW h=.96 $/ton Total Load 39 MW h= $/ton Total Load 55MW h=.3378 $/ton Total Load 56 MW h= $/ton Total Load 83.4 MW h= $/ton Page 8 of 10 g. 6 Comparson of best generator settng for HGA under varous loadng condton Table III Comparson of test results for dfferent algorthms under some Loadng Condtons Unts Methods Pg 1 Pg Pg 3 Pg 4 Pg 5 Pg 6 uel Cost,$/hr Emsson ton/hr Total Cost, $/hr Losses. MW NR MatPower GA HGA NR MatPower GA HGA NR MatPower GA HGA NR MatPower GA HGA NR MatPower GA HGA

10 gure 7. Comparson of total generaton cost for each methodology under varous loadng condton 6. CONCLUSION Economc load dspatch alone s not suffcent to reduce the pollutant emssons caused by fossl burnng for power generaton. So, ths paper has been nvestgated CEED problem. The CEED problem s consdered as a multobjectve optmzaton problem that s can be transformed nto a sngle objectve one by usng a modfed prce penalty factor technque. A determnstc model of CEED whch mnmzes both fuel cost and emsson smultaneously has been formulated and mplemented on IEEE-30 bus power system as a case study. The followng conclusons can be drawn from the study: 1- The fuel costs as well as the emsson characterstcs of generatng unts are represented by ther respectve equvalent characterstc n terms of power generatons. - Transmsson losses are expressed n terms of B-coeffcents and then the total generaton s also represented by total load demand and transmsson losses. 3- NR, Matpower, GA and HGA algorthm as a soluton to the CEED problem of the IEEE-30 bus test system have been presented. 4- The paper explores loadablty and ts mpacts on economc analyss. 5- Cost and Emsson for each load and losses have been calculated for dfferent load condtons. 6- The NR, Matpower and smple GA produced the hghest operaton cost. 7- The man advantages of HGA over NR, Matpower,and GA methods are: modelng flexblty, more stable convergence characterstcs and the soluton qualty. 8- The valdaton of the HGA algorthm was demonstrated by comparng the CEED results of IEEE 30 bus system wth NR, MatPower and GA. The optmal solutons were obtaned for each loadng condton wthn approxmately 60 teratons. 9- The results show that the HGA s applcable and effectve n the soluton of any economc/emsson power dspatch problems that consder nonlnear characterstcs of power systems wth dfferent objectve functons. 10- The proposed HGA algorthm can be easly extended to solve any CEED problem. 11- The results show that the cost hghly ncreases f the load s 75% of mum generaton. Appendxes Appendx ( A) Cost and emsson coeffcents of sx unts system [11] Bus P gmn P g No. MW MW a b c d e f Page 9 of 10

11 Appendx (B) Loss coeffcents B matrces of sx generatng unts B =[ ] B0 = [ ]; B00 =[ e-004]; REERENCES [1] M.A. Abdo, Multobjectve Evolutonary Algorthms for Electrc Power Dspatch Problem IEEE Tran. on Evol. Com., Vol. 10, No. 3, 006 [] S. Pothya, I. Ngamroo, W. K., Applcaton of multple Tabu search algorthm to solve dynamc economc dspatch consderng generator constrants, Energy Converson and Management 49, pp , 008. [3] Park, J., Lee, K., & Shn, J., A partcle swarm optmzaton for economc dspatch wth nonsmooth cost functons, IEEE Trans. on Power Syst, 0(1),pp. 34 4, 005. [4] Pangrah C., et. al., Smulated annealng technque for dynamc economc dspatch, Electrc Power Components and Systems, 34,pp , 006. [5] N. Ruangpayoongsak, Constraned Economc Dspatch by Combned Genetc and Smulated Annealng Algorthm, Electrc Power Components and Systems, 30,pp , 00 [6] T. Yalcnoz, B. J. Cory and M.J. Short, Hopfeld neural network approaches Ro economc dspatch problems, Electrcal power and energy System, 3,435-44, 001. [7] A. Y. Abdelazz, et al., Economc Dspatch Usng an Enhanced Hopfeld Neural Network, Electrc Power Components and Systems, 36, pp , 008. [8] Attavryanupap, P.,et.al., A fuzzy optmzaton approach to dynamc economc dspatch consderng uncertantes, IEEE Trans. Power Syst., Vol. 19, No. 3, pp , 004. [9] K. Teerth Chaturved, M. Pandt and L. Srvastava, Hybrd neuro-fuzzy system for power generaton control wth envronmental constrants, Energy Converson and Management 49, pp , 008. [10] L. Slman and T. Bouktr, Economc Power Dspatch of Power System wth Polluton Control usng Multobjectve Ant Colony Optmzaton, Int. J. of Computatonal Intellgence Research. ISSN Vol.3, No., pp , 007. [11] R. Gnanadass, N. Prasad, K. Manvannan, Assessment of avalable transfer capablty for practcal power systems wth combned economc emsson dspatch, Electrc Power Systems Research 69,pp , 004 [1] Ray D. Zmmerman and Carlos E. Murllo-Sánchez,: MATPOWER A MATLAB Power System Smulaton Package, User s Manual, School of Electrcal Engneerng, Cornell Unversty, 007, avalable: [13] Genetc algorthm Toolbox for Use Wth MATLAB, The Mathworks, Inc., Natck, MA, 007 Adel A. Elbaset Mohammed was born n Nag Hamad, Qena-Egypt, on October 4, He receved the B.Sc., M.Sc., and Ph.D. from aculty of Engneerng, Department of Electrcal Engneerng, Mna Unversty, Egypt, n 1995, 000 and 006, respectvely. He s a staff member n aculty of Engneerng, Electrcal Engneerng Dept., Mna Unversty, Egypt. Dr. A. Elbaset nterests n the area of power electroncs, power system, neural network, fuzzy systems and renewable energy, Optmzaton. Page 10 of 10