Profit Based Unit Commitment in Deregulated Electricity Markets Using A Hybrid Lagrangian Relaxation - Particle Swarm Optimization Approach

Transcription

1 Profit Based Unit Commitment in Deregulated Electricity Markets Using A Hybrid Lagrangian Relaxation - Particle Swarm Optimization Approac Adline K. Bikeri, Cristoper M. Maina and Peter K. Kiato Proceedings of te Sustainable Researc and Innovation Conference, Abstract In deregulated electricity markets, individual generation companies (GENCOs) carry out independent unit commitment based on predicted energy and revenue prices. Te GENCOs unit commitment strategies are developed wit te aim of maximizing profit based on te cost caracteristics of teir generators and revenues from predicted prices of energy and reserve subject to all prevailing constraints in wat is known as Profit Based Unit Commitment (PBUC). A tool for carrying out PBUC is an important need for te GENCOs. Tis paper demonstrates te development of a solution metodology for te PBUC optimization problem in deregulated electricity markets. A ybrid of te Lagrangian Relaxation (LR) and Particle Swarm Optimization (PSO) algoritms is used to determine an optimal UC scedule in a day-aead market using te expected energy and reserve prices taking advantage of te strengts of bot algoritms. Te PSO algoritm is used to update te Lagrange multipliers giving a better quality solution. An analysis of te PSO algoritm parameters is carried out to determine te parameters tat give te best solution. Te algoritm is implemented in MATLAB software and tested for a GENCO wit 54 termal units adapted from te standard IEEE 118-bus test system. Keywords Deregulated Electricity Market, Lagrangian Relaxation, Particle Swarm Optimization, Profit Based Unit Commitment. I. INTRODUCTION Over te last few decades te electric energy sub-sector as been undergoing significant canges. Probably te biggest cange as been deregulation of many power systems especially in te developed world; toug aspects of deregulation are also beginning to take root in developing nations. Deregulation refers to te unbundling of vertically integrated power systems into Generation Companies (GENCOs), Transmission Companies (TRANSCOs) and Distribution Companies (DISCOs) [1]. Te main aim of deregulation is to create competition among GENCOs and ence provide different coices of generation options at lower prices to consumers [1], [2]. Unit Commitment as always been a significant optimization task in power systems [3], [4]. However, te approac in te deregulated environment is significantly different form tat in te regulated environment. Here, te GENCO is not te system operator. Tis means tat, unlike te regulated market were te objective of te utility in unit commitment A. K. Bikeri, Scool of Electrical, Electronic, and Information Engineering, JKUAT (pone: ; adlinebikeri@gmail.com). C. M. Maina, Department of Electrical and Power Engineering, TUK ( cmainamuriiti@gmail.com). P. K. Kiato, Scool of Electrical, Electronic, and Information Engineering, JKUAT ( pkiato@jkuat.ac.ke). is te minimization of operating cost, in te deregulated environment, te objective of te GENCO is te maximization of profit. Tis as led to wat is now referred to as Profit Based Unit Commitment (PBUC) in deregulated markets [5]. Numerous metodologies for solving bot te traditional UC and PBUC problems ave been proposed in literature. Tese metodologies can be classified as classical metods and non-classical metods. Classical metods include Priority Listing, Dynamic Programming, Branc and Bound, Mixed Integer Programming, and Lagrangian Relaxation (LR) [5], [6]. Non-classical metods include Genetic Algoritms, Particle Swarm Optimization, Artificial Bee Colony, Muller metod among oters [7], [8]. Tere ave also been proposals for ybridization of some of tese metods taking advantage of te strengts of two or more metods to provide a more effective solution algoritm [9] [11]. A compreensive review of tese metods can be found in [3], [4], [12] Despite te numerous efforts to solve wat is a very complex optimization problem over te past few years, a number of researc gaps still exist [4]. Tis paper formulates te PBUC problem incorporating reserve payments and as well as spot market energy prices. A solution metodology for te PBUC optimization problem is ten developed. Here, a ybrid of te Lagrangian Relaxation (LR) and Particle Swarm Optimization (PSO) algoritms is used to determine an optimal UC scedule including constraints of aving to meet bilaterally agreed energy supply commitments. Lagrangian Relaxation (LR) is cosen since is currently te most commonly used approac in te solution of te UC problem. However, several metods for updating te Lagrange multipliers ave been proposed. Te Particle Swarm Optimization (PSO) algoritm is one suc metod and is implemented in tis paper. Te biggest callenge wit te PSO algoritm lies in te proper selection of te various weigting factors tat largely determine te algoritm s performance. Tus, apart from just implementation of te PSO algoritm, parameters selection is also addressed in tis paper. Te rest of te paper is organized as follows: Section II introduces te LR and PSO procedures and teir application in te solution of optimization problems. Section III outlines te PBUC problem formulation wile Section IV explains te proposed solution metodology. In Section V, simulation results on a test IEEE system are presented including section of PSO parameters, analysis of te obtained optimal solution, algoritm convergence performance, and computation time. Finally, paper conclusions are given in Section VI.

2 Proceedings of te Sustainable Researc and Innovation Conference, II. LAGRANGIAN RELAXATION AND PARTICLE SWARM OPTIMIZATION Te Lagrangian relaxation (LR) metod for solving an optimization problem works by incorporating complicated constraints of te problem into te objective function using penalty terms known as Lagrange multipliers [13]. Te Lagrange multipliers penalize violations of te corresponding constraints and by systematically updating tese penalty factors, an optimal solution of te original problem can be determined. In practice, te modified (relaxed) optimization problem is usually simpler to solve tan te original problem ence te application of te metod. Te quality of te solution obtained via LR strongly depends on te algoritm used to update te Lagrangian multipliers. Traditionally, gradient based metods ave been used but more recently, one or more of te euristic metods ave been applied in an effort to improve te quality of te solution [13], [14]. Particle Swarm Optimization (PSO) is a population based stocastic optimization tecnique simulating te natural animal s beavior to adapt to te best of te caracters among entire populations like bird flocking and fis scooling [15]. Since it s inception in te mid 90 s, PSO as been widely applied by researcers in various optimization applications including te solution of te UC problem [7]. In simple terms, a population (swarm) of processing elements called particles, eac of wic representing a candidate solution forms te basis of computation in te PSO algoritm. A possible solution to te existing optimization problem is represented by eac particle in te swarm. A population of random solutions is used to initialize te PSO algoritm and optima are searced by updating te solution in eac iteration (epoc). During a PSO iteration, every particle moves towards its own personal best solution tat it acieved so far (pbest), as well as towards te global best (gbest) solution wic is best among te best solutions acieved so far by all particles present in te population. Tis is done in a random manner ensuring tat te algoritm trougly searces te solution space. After a certain pre-set number of iterations (generations), te particle wit te global best solution is stored as te optimal solution to te optimization problem. III. PBUC PROBLEM FORMULATION Te PBUC problem is formulated as as a maximization of a GENCO s profit by deciding an optimal unit commitment scedule based on expected energy and reserve prices. Te GENCO s bilateral contract commitments are also considered. Te objective function and te operational constraints are given in te following subsections. Te variables in te various equations are sown in Table I. A. Objective Function Profit (P F ) is defined as te difference between revenue (RV ) obtained from sale of energy and reserve and te total Revenue from oter ancillary services could be included in a similar manner. TABLE I NOMENCLATURE our index i generator index j PSO particle index k iteration number index H number of sceduling ours J number of PSO particles K maximum number of PSO algoritm generations N total number of generators P F GENCO Profit RV GENCO Revenue RVp revenue from energy sales (MW) at our RVr revenue from reserve sales at our T C GENCO Costs F Ci fuel cost of generator i at our SCi start up cost of generator i at our a i, b i, c i constant for fuel cost curve of generator i γ i start up cost of generator i α s unit price for spot market energy sales at our α b unit price for bilateral contracts energy sales at our α r unit price for reserve capacity sales at our Pb power supply for bilateral contracts at our Pi power output from generator i at our Pi min, Pi max minimum and maximum outputs of generator i respectively RU i, RD i ramp up and ramp down limits of generator i respectively κ factor for contract of differences Ui state of generator i at our λ j,k Lagrange Multiplier for particle j at our for iteration k Λ j,k Set of Lagrange Multipliers for particle j at iteration k vj,k velocity of particle j at our for iteration k V j,k Set of velocities for particle j at iteration k pbest j Personal best solution of particle j gbest Global best solution for all particles w 1, w 2, w 3 weigting factors corresponding to te particle s previous velocity, personal best position and global best position respectively r 1, r 2 random numbers in [0 1] operating cost (T C) of te GENCO. Te objective function of te PBUC problem is ten given as: Maximize P F = RV T C (1) 1) GENCO Revenue: RV is given by: H ( RV = RV p + RVr ) =1 Revenue from te energy market at a given our RVp calculated as: ( N ) RV p = α b P b + α s P i P b (2) is + κ ( α s α b ) P b (3) Te first term in (3) represents revenue from bilateral contracts, te second term represents revenue from te energy sold at te spot market, wile te tird term represents revenue from contracts of differences. Contracts of differences (cfds) are usually included in bilateral contracts to compensate suppliers and consumers

3 for differences between te bilaterally agreed prices and te prevailing market price. A cfd factor of κ = 0 would mean tat te GENCO sells power in te bilateral market at te bilaterally agreed price even if te market price is iger (no compensation) wile a cfd factor of κ = 1 essentially means tat te GENCO sells power in te bilateral market at te prevailing market price (full compensation). A value of κ = 0.5 is adopted in tis paper. Revenue from sale of reserve at our is given by: RV r = α r N ( P max i Pi ) 2) GENCO Costs: T C is a sum of fuel costs (F C) and start up costs (SC) for all generators over te entire sceduling period. Tis is given as: were T C = H =1 N ( F C i + SCi ) (4) (5) F Ci = a i + b i Pi ( ) + c i P 2 i (6) SCi ( ) = γ i 1 U 1 i U i (7) Fig. 1. Proceedings of te Sustainable Researc and Innovation Conference, PBUC solution algoritm using LR-PSO B. Operational Constraints GENCO operational constraints are given as: (a) Power balance for bilateral contracts N Pi Pb (8) (b) Generation limit constraints U i P min i U i P i U i P max i i, (9) (c) Ramp up constraints P i (d) Ramp down constraints (e) Minimum up time P 1 i RU i i, (10) P 1 i P i RD i i, (11) U i = 1 if U t i U t 1 i = 1, for = t,..., t + MUT 1 (12) (f) Minimum down time U i = 0 if U t 1 i U t i = 1, for = t,..., t + MDT 1 (13) Constraints (9)-(13) are similar to te traditional UC formulation [3]. However, constraint (8) indicates tat te GENCO s total generation must be greater tan its bilateral contracts commitments. Tis is in contrast wit te traditional case were generation must equal total system demand and losses. Unlike te traditional UC formulation, tere is no spinning reserve constraint as tis is not te GENCO s responsibility. Te GENCO only gets payments for supplying part of te reserve. Revenue from reserve sales is terefore added to te objective function. IV. SOLUTION METHODOLOGY A. PBUC Solution Algoritm Te basic structure of te solution algoritm for solving te PBUC problem using LR and PSO is sown in Fig. 1. Basically, a Lagrangian function is formed by relaxing constraint (8) into te objective function. Tis is because it is te only constraint tat couples te units. Possible solutions to te relaxed problem are ten randomly generated and iteratively solved using a two-step process. Te first step involves solving te relaxed problem for eac possible solution (sets of Lagrange multipliers). Wit te relaxation, optimal scedules of individual generation units can be easily determined by breaking down te relaxed function into subproblems for eac unit. A 2-state dynamic programming code is implemented to find an optimal UC scedule for eac unit given a set of Lagrange multipliers. Te second step involves updating of te possible solutions (particles) using te PSO algoritm. Tis is done iteratively for a number of pre-set iterations (maximum number of PSO generations). Te two steps are outlined in te following subsections. B. Solution of te Relaxed Problem Constraint (8) te power balance for bilateral contracts is te only constraint tat couples te generating units and is terefore relaxed by being included in te objective function to form te Lagrangian function L as: ) H N L = RV T C λ (P b (14) =1 Te relaxed problem is terefore te maximization of L subject to constraints (9) to (13). P i

4 Proceedings of te Sustainable Researc and Innovation Conference, i.e. To maximize L wit respect to P i in (14): ence L P i L P i = 0 i, (15) = ( αs αr ) ( bi + 2c i Pi ) + λ = 0 (16) P i = α s α r + λ b i 2c i (17) Te following procedure is tus used to solve te relaxed PBUC problem for a set of Lagrange multipliers: Λ = {λ 1, λ 2,..., λ H }. Step 1: Get input data (generator cost data, ourly price data, Lagrangian multipliers) Step 2: Set i = 1 Step 3: Set = 1 Step 4: calculate Pi from (17) Step 5: ceck for generator limit constraints if Pi > Pi max set Pi = Pi max if Pi < Pi min set Pi = Pi min Step 6: ceck te ramp up and ramp down constraints and cange Pi accordingly Step 7: ceck te minimum up time and minimum down time constraints and cange Pi accordingly Step 8: determine te optimal UC scedule using 2-state dynamic programming Step 9: = + 1. If H go to Step 4. Else go to Step 10 Step 10: i = i + 1. If i N go to Step 3. Else go to Step 11 Step 11: Calculate total revenue, costs and profits Step 12: Store te results (UC status for all generators, sceduled power, profit) C. Lagrange Multipliers Update via Particle Swarm Optimization Te PSO algoritm is used to update te Lagrange Multipliers to determine te set tat provides te best results. A particle represents a candidate solution wic is a set of Lagrange Multipliers one for eac our of te sceduling orizon. For a sceduling period of H ours, te j t particle after k iterations Λ j,k = {λ 1 j,k, λ2 j,k, λ3 j,k,..., λh j,k } represents a position in te H-dimension solution space. Te particle also as an associated velocity V j,k = {v 1 j,k, v2 j,k, v3 j,k,..., vh j,k } wic represents a direction in wic te particle is moving in te solution space. Te PSO algoritm moves te particles around te solution space after eac iteration in a searc for te best possible solution. Te particle position update follows two best positions: pbest and gbest. pbest j is te j t particle s personal best solution found so far wile gbest is te entire population s global best solution (te best amongst te various pbests). At eac iteration, te velocity of eac particle is updated using V j,k+1 = w 1 V j,k + w 2 r 1 (pbest j Λ j,k ) + w 3 r 2 (gbest Λ j,k ) (18) see variable definitions on te nomenclature list in Table I. Te position is ten updated using te move equation: Λ j,k+1 = Λ j,k + V j,k+1 (19) Te following procedure is used to solve te PBUC problem updating candidate solutions (sets of Lagrange Multipliers) using te PSO algoritm: Step 1: Randomly initialize J particles (candidate solutions) Step 2: set k = 1 Step 3: set j = 1 Step 4: Solve te relaxed PBUC problem for te j t particle and determine te corresponding GENCO profit P F j,k Step 5: If k = 1, set pbest j = P F j,k else if P F j,k > pbest j; set pbest j = P F j,k Step 6: j = j + 1. If j < J go to step 4. Else go to step 7 Step 7: Determine gbest as: gbest = max{pbest 1, pbest 2,..., pbest J} Step 8: set j = 1 Step 9: Update te velocity of particle j using (18) Step 10: Update te position of particle j using (19) Step 11: j = j + 1. If j < J go to Step 9. Else go to Step 12 Step 12: k = k + 1 If k K go to Step 3. Else STOP A. Test System V. SIMULATION RESULTS Te algoritm is tested for a GENCO wit 54 termal units. Te generator data is adapted from te IEEE 118- bus test system and obtained from ttp://motor.ece.iit.edu/data/ PBUCData.pdf. Te GENCO s own load (bilateral market commitment) is assumed to be constant at 3,500 MW wit PEAK and OFF-PEAK prices as sown in Table II. B. Selection of PSO Parameters Te quality of te solution obtained from te PSO algoritm is largely dependent on te values of te parameters used. Parameter selection is done in tis paper by trying various combinations of te weigting factors w 1, w 2, and w 3 in (18). w 1 was varied from 0.25 to 1.0 in steps of 0.25 wile w 2 was varied from 1.0 to 3.0 in steps of 0.5. w 3 was set using te formula: w 2 + w 3 = 4 as suggested in literature [15]. Tese settings give 20 different combinations of te PSO parameters TABLE II PRICE DATA Hour Energy Reserve Bilateral Hour Energy Reserve Bilateral Price Price Price Price Price Price

5 Proceedings of te Sustainable Researc and Innovation Conference, as sown in table III. In eac case, te number of particles was set to J = 20 and te number of PSO iterations was set to K = 500. Te Lagrange multipliers were initialized to take random values ranging from 0 to 50. Te velocity was owever not restricted so tat te final value of te Lagrange multipliers could be any positive real number. For eac combination of PSO parameters, 10 different trials of te PSO algoritm were run and te solutions analyzed. Te maximum profit, average profit, and minimum profit from eac combination of PSO parameters was determined and te results are sown in Fig. 2. From Fig. 2, it is seen tat te 12 t combination of PSO parameters (w 1 = 0.75; w 2 = 1.5; and w 3 = 2.5) provides te best results. Hence, for te simulations in tis paper tese values are cosen as te PSO parameters. C. Optimal Solution 1) Unit Commitment: Te Unit Commitment scedule for te best solution amongst all te trials carried out is sown in Fig. 5. Te orizontal axis represents te sceduling our wile te vertical axis refers to te unit number. A single box in te grid terefore indicates weter a unit is ON (sown in red) or OFF (sown in wite). Te results sow tat some of te units e.g. 27 and 45 are ON trougout te day wile oters suc as 33 and 46 are OFF trougout te day. Most of te units are ON or OFF depending on te market price at a given our. 2) Optimal Power Scedule: Fig. 3 sows te total committed generation for te 24 ours and te GENCO s own load from te UC scedule of Fig. 5. It also indicates te day s total profit as $2,355,259. From Fig. 3, te total sceduled power from te LR-PSO algoritm is always greater tan te TABLE III PSO PARAMETER SETS Set No. w 1 w 2 w 3 Set No. w 1 w 2 w GENCO s load. Tere is no deficiency in meeting te bilateral contract agreements ence te value of te Energy Not Served (ENS) is indicated as zero. Sould tere be a deficiency in meeting te total committed scedule, te value of ENS will be greater tan zero. Te value of ENS = 0 is ensured by penalizing a result in wic ENS > 0 wen determining te pbest and gbest value in te PSO algoritm. 3) Optimal Values of Lagrange Multipliers: Fig. 4 sows te resulting values of te Lagrange Multipliers corresponding to te scedule sown in Fig. 5. It is observed tat te LMs are larger for durations of low market price (rs 0 to 8) and wen te market price is lower tan te bilateral contract price (r 14, 20, 21). In tese cases, it is relatively expensive to participate in te spot market but it is necessary to generate power to meet bilateral contract commitments. During te periods of relatively ig spot market price and wen te spot market price is iger tan te bilaterally agreed price, constraint (8) is met and tere is no need to add a penalty factor ence te value LM = 0. 4) Solution Convergence and Computation Time Analysis: Fig. 6 sows te evolution of te best solution (value of gbest) as well as te computation time against te algoritm iteration number. It is seen tat after about 300 iterations, te optimal solution does not cange muc ence it is sufficient to say tat 500 iterations are enoug for te current problem size. Te solution time increases linearly wit te number of iterations ence increasing te number of iterations would only increase te computation time witout significantly improving te best solution. VI. CONCLUSION A solution metodology tat combines te Lagrangian relaxation tecnique wit te euristic particle swarm optimization tecniques to solve te profit based unity commitment problem for GENCOs in deregulated markets as been POWER [ 10 3 MW] PROFIT = $2,355,258 ENS = 0 MW 1 Scedulded power Bilateral market commitment HOUR Fig. 3. Optimal GENCO Total Power Generation Scedule Profit [USD] 2.4 x Best solution Average solution Worst solution Lagrange Multipliers PSO parameters Fig. 2. PSO Parameter Sets Performance HOUR Fig. 4. Lagrange Multipliers Corresponding to te Optimal Solution

6 Proceedings of te Sustainable Researc and Innovation Conference, Fig. 5. Unit Commitment Scedule Corresponding to te Optimal Solution Profit [$ 10 6 ] Optimal Solution Simulation Time Iteration Fig. 6. Analysis of solution convergence and computation time proposed in tis paper. Te problem as been formulated including a constraint setting te minimum GENCO output at a given our as te bilaterally committed generation for te our. Te parameters w 1 = 0.75, w 2 = 1.5, and w 3 = 0.25 ave been cosen based on an assessment of te performance of various combinations of PSO parameters in te solution of te PBUC problem. An implementation for a GENCO wit 54 termal units sows te effectiveness of te proposed metodology. REFERENCES [1] M. Saidepour and M. Alomous, Restructured Electrical Power Systems, Operation, Trading, and Volatility. New York: Marcel Decker, 1st ed., [2] L. Pilipson and H. L. Willis, Understanding Electric Utilities and De- Regulation. Florida, USA: Taylor & Francis Group, 2nd ed., [3] B. Saravanan, S. Das, S. Sikri, and D. P. Kotari, A solution to te unit commitment problem a review, Frontiers in Energy, vol. 7, no. 2, pp , Time [sec] [4] A. K. Bikeri, C. M. Muriiti, and P. K. Kiato, A review of unit commitment in deregulated electricity markets, in Scientific Researc and Innovation (SRI) Conference, 2015, pp. 1 5, May [5] T. Li and M. Saidepour, Price-based unit commitment: a case of Lagrangian relaxation versus mixed integer programming, Power Systems, IEEE Transactions on, vol. 20, pp , Nov [6] A. Rajan, C. Cristober, P. Sundarajan, V. Jamuna, R. Madusubas, and B. Udayakumar, Multi-area unit commitment in deregulated electricity market using DP approac, Inter. Journal on Recent Trends in Engineering and Tecnology, vol. 3, pp , May [7] J. Raglend, C. Raguveer, G. R. Avinas, N. Pady, and D. Kotari, Solution to profit based unit commitment problem using particle swarm optimization, Applied Soft Computing, vol. 10, no. 4, pp , Sept [8] H. S. Madraswala and A. S. Despande, Genetic algoritm solution to unit commitment problem, in 2016 IEEE 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPE- ICES), pp. 1 6, July [9] S. Selvi, M. Moses, and C. Rajan, LR-EP approac for solving profit based unit commitment problem wit losses in deregulated markets, Przeglad Elektrotecniczny, vol. 11, pp , [10] K. Laksmi and S. Vasantaratna, Hybrid artificial immune system approac for profit based unit commitment problem, J. Electrical Eng. Tecnology, vol. 8, no. 1, pp , Marc [11] A. Sudakar, C. Karri, and A. J. Laxmi, Profit based unit commitment for {GENCOs} using lagrange relaxationdifferential evolution, Engineering Science and Tecnology, an International Journal, pp., [12] N. Pady, Unit commitment problem under deregulated environment - A review, in Power Engineering Society General Meeting, 2003, IEEE, vol. 2, pp , July [13] C. Lemarecal, Computational Combinatorial Optimization - Optimal or Provably Near-Optimal Solutions, c. Lagrangian relaxation, pp Berlin: Springer-Verlag, [14] S. Gao, Bio-Inspired Computational Algoritms and Teir Applications. InTec, [15] J. Kennedy and R. Eberart, Particle swarm optimization, in Neural Networks, Proceedings., IEEE International Conference on, vol. 4, pp vol.4, Nov 1995.