A Simplified Approach for Economic Dispatch with Piecewise Quadratic Cost Functions

Transcription

1 Iteratioal Joural of Computer ad Electrical Egieerig, Vol, No 5, October, 00 A Simplified Approach for Ecoomic Dispatch with iecewise Quadratic Cost Fuctios S SUBRAMANIAN, Seior member, IEEE ad S GANESAN Abstract This paper presets a simple methodology for solvig ecoomic dispatch problems with multiple fuel optios The problem has bee solved i two stages I the first stage, the most ecoomic fuel of each uit is idetified ad i the secod stage Ecoomic Load Dispatch (ELD) for the selected fuels is performed The Composite Cost Fuctio (CCF) is used to idetify the most ecoomical fuel ad the feasible operatig regio of each uit With these details, Sequetial Approach with Matrix Framework (SAMF) is performed to determie the optimal dispatch of each uit The effectiveess of the proposed methodology is tested with a stadard te uit system Numerical simulatio results idicate that the proposed methodology has close agreemet with recet reports Idex Terms Composite cost fuctio, ecoomic load dispatch, piecewise quadratic cost fuctio, sequetial approach I INTRODUCTION The primary objective of the ecoomic load dispatch problem is to schedule the geeratios of thermal uits so as to meet the required load demad at miimum operatig cost while satisfyig the system costraits Traditioally, the cost fuctio for geeratig uits has bee approximated as a quadratic fuctio The efforts o solvig the ecoomic dispatch problems have employed various mathematical programmig methods ad optimizatio techiques The classical optimizatio methods are reported that to be highly sesitive to the selectio of the startig poits ad sometimes coverge to a local optimum The methods based o the operatioal research ad artificial itelligece cocepts such as geetic algorithm, evolutioary algorithms, fuzzy ad artificial eural etworks have bee give attetio for solvig ecoomic dispatch problems because of their ability to fid the solutio ear global optimal Simulated Aealig (SA) techique is similar to a local search techique had bee applied to determie the optimal geeratio schedule for ecoomic dispatch problem i a power system [] Geetic Algorithm (GA), a heuristic search techique has bee applied to fid the optimal geeratios of thermal uits cosiderig the prohibited operatig zoes of some geeratig uits [] A eural et topology based o multilayer perceptro has bee developed to solve ecoomic power dispatch problem [3] Mauscript received May, 00 S Subramaia is with the Departmet of Electrical Egieerig, Aamalai Uiversity, Aamalaiagar, Tamiladu, Idia ( dr_smai@yahoocoi) S Gaesa is with the Departmet of Electrical Egieerig, Aamalai Uiversity, Aamalaiagar, Tamiladu, Idia ( gaeshshriraj@gmailcom) 793 Two phase eural et model has bee developed to determie the optimal schedule of geeratig uits [4] The heuristic search techiques such as article swarm Optimizatio (SO), Evolutioary rogrammig (E), Differetial evolutio (DE) ad hybrid versios of above metioed search techiques are also proposed for solvig ecoomic dispatch problems The stochastic search techique, SO has bee developed by the ispiratio of bird flickig characteristics ad it has bee applied to solve the ecoomic dispatch problems cosiderig various geerator operatioal costraits [5] Three differet evolutioary programmig techiques such as Classical E (CE), Fast E (FE) ad Improved Fast E (IFE) have bee developed with the differet mutatio techiques ad the effectiveess are validated for solvig differet kids of ecoomic dispatch problems [6] The viability of SO has also bee tested for solvig differet types of ecoomic dispatch problems [7] Differetial Evolutio (DE) based solutios for ecoomic dispatch problem has also bee preseted [8] Variable Scalig Hybrid Differetial Evolutio (VSHDE) has bee developed which overcomes the demerit of the fixed ad radom scalig factor i differetial evolutio ad the effectiveess has bee tested for solvig medium ad large scale ecoomic dispatch problems [9] The hybrid search techiques have bee applied to obtai the ecoomic schedule of geeratios [0]-[] A approach based o a ew codig scheme for solvig this type of problems through Simulated Aealig like article Swarm Optimizatio (SA-SO) has bee suggested [0] Bacterial Foragig algorithm, a ew optimizatio techique has bee developed by the ispiratio of foragig strategy of Ecoli bacteria preset i huma itestie ad it is combied with Nelder-Mead method had also bee applied to solve ecoomic dispatch problems [] The heuristic search methods provide the optimal solutio because the efficiecy of their search process which deped the various cotrol parameters The cotrol parameters such as populatio size, umber of geeratio, etc are problem idepedet ad the solutio quality depeds o these cotrol parameters I certai fossil fired geeratig uits use differet fuels hece the cost fuctio are represeted as a segmeted piecewise quadratic fuctio The geeratig uits, particularly those uits which are supplied with multiple fuel sources are faced with the problem of determiig the most ecoomical fuel to bur The piecewise quadratic fuctio is approximated as piecewise liear fuctio ad solved by the traditioal methods A better approach is eeded to retai the assumptios of piecewise quadratic fuctio ad to solve this problem Li ad Viviai [] have reported a hierarchical method based decetralized computatios to solve the

2 Iteratioal Joural of Computer ad Electrical Egieerig, Vol, No 5, October, 00 ecoomic dispatch problem with piecewise quadratic fuctios Artificial eural etwork models such as Hopfield Neural Network (HNN) ad Adaptive Hopfield Neural Network (AHNN) models have bee preseted for the solutio of ecoomic load dispatch problems with multiple fuel optios [3]-[4] Baskar etal, reported Real Coded GA (RCGA) to solve ecoomic dispatch problems with piecewise quadratic cost fuctios [5] Recetly, Artificial Immue System (AIS) has bee applied to solve ecoomic dispatch problems [6] AIS imitates the immuological ideas ad utilizes the cloal selectio priciple ad evolutioary approach wherei cloig of atibodies is performed followed by hypermutatio AIS algorithm has also bee applied to solve the ecoomic dispatch problem with piecewise quadratic equatios I this article, a ew methodology based o composite cost fuctio ad sequetial approach with matrix framework has bee preseted for solvig ecoomic dispatch problems with piecewise quadratic cost fuctios The most ecoomic fuel of each geeratig uit is idetified by evaluatig the composite cost fuctio of each uit ad the by performig sequetial approach with a matrix framework After arrowig the search space, the ecoomic dispatch for the selected fuel is obtaied by sequetial approach with matrix framework II ROBLEM FORMULATION I the ecoomic dispatch problem, the fuel cost of each geerator is represeted by a sigle quadratic cost fuctio Owig to multiple fuel optios, the cost fuctio may become piecewise quadratic Hece, the ecoomic dispatch problem with piecewise quadratic fuctio is defied as, N mi F () i= i ( i ) Where ai i + bi i + ci, fuel, i,mi i i ai i + bi i + ci, fuel,i i i F i( i ) = () aim i + bim i + cim, fuel m,i,m i i, max Where, a im, b im ad c im are the cost coefficiets of geeratig uit i usig the fuel type m, N is the umber geeratig uits ad i is the active power geeratio of geeratig uit i i MW This costraied optimizatio problem is subjected to variety of costraits These iclude power balace costraits ad maximum ad miimum power geeratio limits of geeratig uits These costraits are discussed as follows A ower balace costrait This costrait is based o the priciple of equilibrium that the total geeratio should satisfy the total system demad i= = = i G D (3) Where, D is the total load demad of the system i MW B Geeratig capacity costraits The active power output of each geerator has a lower ad upper boud so that the geeratios of geeratig uit lies withi this limit This iequality ca be give as follows i,mi (4) i i,max Where, i,mi ad i,max are the miimum ad maximum value of active power allowed at geerator i i MW III SOLUTION METHODOLOGY A simplified solutio methodology for solvig ecoomic dispatch problem with piecewise quadratic cost fuctio has bee detailed i this sectio This methodology comprises two stages I the first stage, the most ecoomic fuel of each uit is idetified usig composite cost fuctio I secod stage, the optimal dispatches of geeratig uits are obtaied by sequetial approach with matrix framework The derivatio of composite cost coefficiets ad the methodology of sequetial approach with matrix framework are preseted i this sectio The flow of two stage methodology is show i Fig Fig Two stage methodology A Evaluatio of composite cost fuctio The icremetal productio cost of a plat is a prior requiremet for coordiatio amog plats The icremetal productio cost of the plat ca be derived by a simple realigmet of the fuel cost coefficiets of the uits Cosider a uit system ad the cost equatio of i th uit is, F = a + b + c ; i i i i i i i (5) ad the composite cost fuctio of the plat ca be writte as, F = A + B C (6) T G G + Where, G is the total geeratio i MW The composite cost coefficiets are derived as follows The total fuel cost ($/h) of the plat ca be writte as, F T = F + F + + F (7) For most ecoomical geeratio, i = ( λ bi )/ ai (8) G = ( λ B) / Ai Where, λ is the icremetal productio cost of the plat i MW The total geeratio of the plat ca be writte as, = System G G = λ = Fuel selectio usig CCF [( λ / )( / a + / a + + / a )] [ ( / )( b / a + b / a + + b / a )] [ / ( / a + / a + + / a )] G + ( b / a + b / a + + b / a ) [ / ( / a + / a + + / a )] Selected fuels ELD by SAMF Optimum dispatch (9) 794

3 Iteratioal Joural of Computer ad Electrical Egieerig, Vol, No 5, October, 00 By comparig Eq (8) ad Eq (9), A = / ( / a + / a + + / a ) (0) B = ( b / a + b / a + + b / a )A () The fuel cost ca be rewritte as, Fi =λ / 4ai bi / 4ai + ci ; () FT =λ / 4 A B / 4 A+ C ; From Eq (), C = ( c + c + + c ) (3) b / a + b / a + + b / a + B / 4 ( ) A B Ecoomic dispatch solutio usig SAMF Sequetial approach with matrix framework has bee developed for solvig ecoomic dispatch problems A maide attempt has bee made to determie the optimal geeratio dispatches of a system for all possible load demads i sigle executio The demostratio of the solutio methodology is preseted i this sectio The electric power productio i a power plat is allowed to vary from miimum techical limit ( mi ) to maximum techical limit ( max ) Iitially the i,mi of all geeratig uits i a power plat are cosidered as iitial state iput values ad is represeted by a sigle dimesioal matrix as, s = [,mi,mi 3,mi,,, mi ] (4) Based o the above sigle dimesioal matrix, a square matrix (I) is developed to idetify the ecoomic schedule of geeratio The formatio of the square matrix is as follows The process starts with a step icremet i geeratio by Δ MW i,mi by keepig the remaiig uits at its iput value This will form first row of the square matrix I = [,mi + Δ,mi 3,mi,,, mi ] (5) The icremet i geeratio is made i the secod elemet by keepig the other elemets at its iput value that leads to the developmet of secod row of the square matrix I = [,mi,mi + Δ 3,mi,,, mi ] (6) I the same logic, a icremet is made for remaiig uits oe at a time ad a square matrix with a dimesio equal to the umber of uits has bee developed For every step icremet i the operatig rage of the plat the uit oe at a time is allowed to experiece the chage i geeratio thus leads to the formatio of the square matrix I,mi+Δ,mi,mi +Δ I,mi,mi,mi I = = I,mi,mi,mi+Δ (7) Each elemet i the square matrix represets the geeratio of a uit correspodig to the colum that should satisfy the uit capacity costraits I the square matrix the uit geeratios of each row that satisfy the costraits are idetified ad total fuel cost of geeratio is evaluated The desired ecoomic schedule of geeratio is idetified by aalyzig ess of each row The ess fuctio of each row is calculated as, ( j ) FT ( j ) = j =,, (8) ( pd + Δ) Where, pd is the total of iput values The schedule with the miimum ess is chose as the successive state iput values This process is repeated till all the geeratig uits reach their maximum geeratio capacity The feasible solutios for every icremet from mi to max are obtaied ad hece the best solutio for ay load demad falls i the operatig boudary ca be easily sited The detailed computatioal flow of the proposed method is preseted i Fig The proposed methodology i the form of matrix framework to support the demostratio is as follows, mi + Δ, mi, mi, max Δ, max, max, mi, mi, mi, max, max, max + Δ Δ, mi, mi, mi + Δ, max, max, max Iitial Fial Δ () ( ) ( ) Stage Stage () ( ) ( ) Choose mi Choose mi IV IMLEMENTATION OF THE ROOSED METHOD ( ) ( ) The ecoomic dispatch problem with multiple fuel optios has bee solved i two stages The implemetatio procedure of the proposed methodology has bee detailed i this sectio A Stage I: The most ecoomical fuels idetificatio The geeratig uits are cosists of multiple fuels, each fuel comprises with differet cost fuctios ad it is importat to idetify the most ecoomic fuel of each geeratig uit for the ecoomic operatio of the system I this sectio, a ew approach has bee developed to realize the most ecoomic fuel of geeratig uit The primary search process calculates the composite cost fuctio of each geeratig uit as detailed earlier sectio ad the SAMF has bee performed to idetify the most ecoomic fuel of each uit The composite cost fuctio ad uit active power geeratio limits are cosidered ad sequetial approach with matrix framework is performed The uit geeratios are the output which idicates the most ecoomic fuel of each uit The fuel correspodig to the uit geeratio is the most ecoomical fuel The geeratio limit correspodig to the most ecoomic fuel is the feasible operatig regio of the uit This search process provides the most ecoomic fuel ad the feasible operatig regio of each uit for all possible load demads B Stage II: Evaluatio of optimal dispatch I this stage, a secodary search process has bee implemeted to refie the dispatches of geeratig uits withi the feasible operatig regio The cost fuctio of the selected fuel of each uit ad the correspodig geeratio 795

4 Iteratioal Joural of Computer ad Electrical Egieerig, Vol, No 5, October, 00 limits are cosidered Based o these details, SAMF has bee performed oce agai to determie the optimum geeratio dispatch of each geeratig uit for a particular load demad The implemetatio procedure for solvig ecoomic dispatch problem with multiple fuel optios is preseted i Fig 3 Start Read system data ad load demad ( D ) Create iitial matrix i = Create the i th row of square matrix by icreasig the geeratio of i th uit by Δ MW methodology for solvig this problem were implemeted i Matlab 70 platform ad executed with etium IV, 8 GHz persoal computer This ecoomic load dispatch problem icludes the total fuel cost fuctio as a objective with geeratio of idividual uits as variables subject to power balace costrait ad maximum ad miimum geeratio limits of geeratig uits costraits (ie equality ad 0 iequality costraits) Start Read system data ad load demad Calculate CCF of each uit i=i+ NO YES YES If i <= i,max If i = NO Compute total power geeratio, total fuel cost, ess of each row i = i,max erform sequetial approach with matrix framework Idetify the most ecoomic fuel ad the feasible operatig regio of each uit NO If G = max YES Choose the row with best ess Update the iput values erform sequetial approach with matrix framework for optimum dispatch with the selected fuel fuctios rit the optimal solutios Stop Stop Fig Computatioal flow of SAMF V SIMULATION RESULTS AND DISCUSSIONS The applicability ad viability of the proposed approach for practical applicatios have bee tested with a system cosists of te geeratig uits, each geeratig uit with two or three fuel optios [] The prime search process has bee made to idetify the most ecoomic fuel of geeratig uits by the composite cost fuctio ad SAMF After idetificatio of the most ecoomic fuel ad the feasible operatig regios of each uit, the secodary search process has bee achieved to determie the optimal dispatches of geeratig uits Sequetial approach provides the optimal schedule of geeratios for all possible load demads which is varied from miimum techical limit by a small icremet to maximum techical limit of the system The selectio of icremet is also a importat factor Too large icremet may ed up with ufeasible solutio ad too small icremet may take log executio time Based o experiece, the desired icremet is chose as MW The proposed Fig 3 Computatioal flow of the proposed methodology I the first stage, the fuel optios of te idividual uits are cosidered as the variables to be optimized The composite cost fuctio of each geeratig uit is calculated by combiig the cost coefficiets of the fuels used i the geeratig uit The geeratio limits ad composite cost fuctio of each geeratig uit are used ad the sequetial approach is performed The various combiatios of geeratio dispatches are obtaied ad the solutio quality is aalyzed by ess fuctio The ess fuctio is defied as a ratio of total composite cost to total geeratio The miimum ess is cosidered as optimal The active power geeratio of each uit is the output ad the fuel correspodig to the geeratio is the most ecoomical fuel of the uit The geeratio limit correspodig to the most ecoomic fuel is the feasible operatig regio of the uit This search process provides the most ecoomic fuel of geeratig uits ad feasible operatig limits for all possible loads I secod stage, the optimum geeratios of idividual uits are obtaied withi the feasible geeratio limits through sequetial approach The ecoomical fuel ad the 796

5 Iteratioal Joural of Computer ad Electrical Egieerig, Vol, No 5, October, 00 feasible operatig regio of each geeratig uit for a particular demad are cosidered ad sequetial approach is performed oce agai to obtai the optimal dispatches of geeratig uits The simulatio results for load demads of 500 MW, 600 MW ad 700 MW are detailed i Table The total fuel cost for above metioed load demads obtaied by this methodology, Hierarchical Method (HM) [], Hopfield Neural Network (HNN) [3], Adaptive Hopfield Neural Network (AHNN) [4], Hybrid Geetic Algorithm (HGA) [5], Classical Evolutioary rogrammig (CE) [6], Fast Evolutioary rogrammig (FE) [6], Improved Fast Evolutioary rogrammig (IFE) [6], article Swarm Optimizatio (SO) [7] ad Artificial Immue System (AIS) [6] are compared ad the compariso of results are detailed i Table The executio time of the proposed methodology for above metioed load demads is 055 s TABLE ECONOMIC DISATCH RESULTS Load demad (MW) Uit 500 MW 600 MW 700 MW Fuel Ge Fuel Ge Fuel Ge type (MW) type (MW) type (MW) Total cost ($/h) TABLE COMARISON OF TOTAL FUEL COST Method Total fuel cost ($/h) 500 MW 600 MW 700 MW HM [] HNN [3] AHNN [4] HGA [5] IFE [6] FE [6] CE [6] SO [7] AIS [6] roposed method From the compariso of results, it is clear that the proposed approach provides comparable result for ecoomic dispatch problem with piecewise quadratic fuctio The proposed method has always provided better results tha all the other methods except HGA ad AIS The optimal fuel cost obtaied through the proposed method is exactly same as HGA ad AIS However, whe compared with the earlier reported techiques, it is worth poitig out that the computatioal procedure of the proposed approach for ecoomic dispatch of uits with multiple fuel optios is miimal VI CONCLUSION This paper presets a simplified methodology, combiig the subuits to a composite fuctio followed by sequetial approach for solvig ecoomic dispatch of geeratig uits with piecewise quadratic cost fuctios The problem has bee solved i two stages I the first stage the decetralized cofiguratio of geeratios are cosidered The composite cost fuctio of each geeratig uit is calculated the prime search process is performed to obtai the most ecoomical fuel ad the feasible operatig regio of each uit I secod stage, based o the details of first stage, the optimum geeratios of idividual uits are determied by sequetial approach The developed algorithm has bee validated with te uit sample test system, each uit with two or three fuel optios The proposed approach yields the optimal geeratio dispatch with less computatioal effort The compariso of results depicts that the proposed method is efficiet to provide the global optimal or high quality solutio for ecoomic dispatch problem with piecewise quadratic cost fuctios ACKNOWLEDGMENT The authors gratefully ackowledge the maagemet the support ad facilities provided by the authorities of Aamalai Uiversity, Aamalaiagar, Idia to carry out this research work REFERENCES [] K Wog, ad CC Wag, Simulated aealig based ecoomic dispatch algorithm, IEE roc Ge Trasm Distrib, vol 40, pp , Nov 993 [] SO Orero, ad MR Ervig, Ecoomic dispatch of geerators with prohibited operatig zoes: a geetic algorithm approach, IEE roc Ge Trasm Distrib, vol43, pp59-534, Nov 996 [3] M Djukaovic, M Calovic, B Milosevic, ad DJ Sobejic, Neural-Net based real time ecoomic dispatch for thermal power plats, IEEE Tras Eergy Covers, vol, pp755-76, Dec996 [4] R Naresh, J Dubey, ad J Sharma, Two-phase eural etwork based modellig framework of costraied ecoomic load dispatch, IEE roc Ge Trasm Distrib, vol5, No3, pp , May 004 [5] ZL Gaig, article swarm optimizatio to solvig the ecoomic dispatch cosiderig the geerator costraits, IEEE Tras ower Syst, vol8, o3, pp 87-95, Aug 003 [6] T Jayabarathi, K Jayaprakash, D N Jayakumar, ad T Raghuatha, Evolutioary programmig techiques for differet kids of ecoomic dispatch problems, Elect ower Syst Res, vol 73, pp 69-76, 005 [7] DN Jeyakumar, T Jayabharathi, ad T Raghuatha, article swarm optimizatio for various types of ecoomic dispatch problems, Elect ower Eergy Syst, vol 8, pp 36-4, 006 [8] Nasimul Noma, ad Hitoshi Iba, Differetial evolutio for ecoomic dispatch problems, Elect ower Syst Res, vol 78, pp 3-33, 008 [9] Ji-yg Chiou, Variable scalig hybrid differetial evolutio for large scale ecoomic dispatch problems, Elect ower Syst Res, vol 77, pp -8, 007 [0] Cheg Chie Kuo, A ovel codig scheme for practical ecoomic dispatch by modified particle swarm approach, IEEE Tras ower Syst, vol3, o4, pp , Nov 008 [] BK aigrahi, ad V Ravikumar adi, Bacterial foragig optimizatio: Nelder Mead hybrid algorithm for ecoomic load dispatch, IET Ge Trasm Distrib, vol, o 4, pp , 008 [] CE Li, ad GL Viviai, Hierarchical ecoomic dispatch for piecewise quadratic cost fuctios, IEEE Tras ower App Syst, AS -03(6), pp 70-75, 984 [3] JH ark, YS Kim, IK Eom, ad KY Lee, Ecoomic load dispatch for piecewise quadratic cost fuctio usig Hopfield Neural Network, IEEE Tras ower 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6 Iteratioal Joural of Computer ad Electrical Egieerig, Vol, No 5, October, 00 [4] KY Lee, AS Yome, ad JH ark, Adaptive Hopfield Neural Networks for ecoomic load dispatch, IEEE Tras ower Syst, vol 3, o, pp 59-56, 998 [5] S Baskar, Subbaraj, ad MVC Rao, Hybrid geetic algorithm solutio to ecoomic dispatch problem, It J Computers Elect Egg, vol 39, o 3, pp , May 003 [6] BK aigrahi, Salik R Yadav, Shubham Agarwal ad MK Tiwari, A cloal algorithm to solve ecoomic load dispatch, Elect ower Syst Res, vol 77, pp , 007 SSubramaia is bor i Maduari, Tamiladu, Idia He received the BE degree i Electrical ad Electroics Egieerig ad the ME degree i ower Systems with distictio from Madurai Kamaraj Uiversity, Madurai, Idia i the year 989 ad 990 respectively He received the hd degree i Electrical Egieerig from Aamalai Uiversity, Aamalaiagar, Idia i the year 00 He is curretly with the Departmet of Electrical Egieerig as a rofessor He has guided 5 hd projects, 30 ME projects ad 0 BE projects He has bee pursuig research with several faculty members He has published a book chapter i Advaces i Differetial Evolutio (Spriger book series: Studies i computatioal itelligece) He published 0 research articles i various referred iteratioal jourals, atioal jourals, iteratioal cofereces ad atioal cofereces His area of iterest icludes power system operatio ad cotrol, desig aalysis of electrical machies, power system state estimatio ad power system voltage stability studies He is a Seior member i IEEE, Fellow of Istitutio of Egieers ad a member of various professioal bodies such as System Society of Idia, Idia Society for Techical Educatio ad Idia Sciece Cogress Associatio His biography has bee icluded i MARQUIS who is who i the world, MARQUIS who is who i Egieerig ad Iteratioal Biographical Cetre (IBC), UK He received the Best Teacher Award i recogitio of his research cotributios i Aamalai Uiversity SGaesa is bor i odicherry, Idia He received the BE degree i Electrical ad Electroics Egieerig from Govermet College of Egieerig, Salem, Idia i 00 ad ME degree i ower Systems with distictio i the year 008 from Aamalai Uiversity, Aamalaiagar, Idia He is curretly with the Departmet of Electrical Egieerig as a Lecturer (Seior Scale) He is curretly pursuig hd degree (art-time) i the Departmet of Electrical Egieerig, Aamalai Uiversity, Aamalaiagar, Idia His research topics iclude power systems operatio ad cotrol He is a Life member of Idia Society for Techical Educatio 798