Journal of Scientific & Industrial Research Vol. 74, July 2015, pp. 395-399 A Novel Approach to Solve Unit Commitment and Economic Load Dispatch Problem using IDE-OBL P Surekha 1 * and S Sumathi 2 *1,2 Department of Electrical and Electronics Engg, PSG College of Technology, Coimbatore-641004, India Received 1 September 2014; revised 6 March 2015; accepted 28 May 2015 The non-convex and combinatorial nature of the UC-ELD problems requires the application of heuristic algorithms to generate optimal schedules. In studies reported so far, the Unit Commitment and the Economic Load Dispatch problems are solved as separate problems. In the addressed work, the commitment and de-commitment of generating units is obtained using a Genetic Algorithm (GA), and the optimal load distribution of the scheduled units is obtained using Improved Differential Evolution with Opposition Based Learning (IDE-OBL). The power demand is varied for 24 hours to determine the schedule in the IEEE 30 bus system including transmission losses, power balance and generator capacity constraints. Optimal distribution of load among generating units, fuel cost per hour, power loss, total power and computational time are computed for each of the test systems using the intelligent algorithms. From the comparative analysis, it can be concluded that GA-IDE-OBL is a better approach for solving UC-ELD problems in terms of optimal solution, robustness, and computational efficiency. Keywords: Genetic Algorithm, Improved Differential Evolution, Opposition based learning, UC-ELD, optimal fuel cost, computational time Introduction Computational Intelligence (CI) techniques has attracted several research engineers, decision makers and practicing researchers in recent years for solving unlimited number of complex real-world problems particularly related to research area of optimization. Due to the non-convex and combinatorial nature of UC and ELD problems, it is difficult to obtain a solution using conventional programming methods like Lambda iteration method 1, dynamic programming 2, mixed integer programming 3, and Newton s method 4. UC and ELD are nonlinear optimization problems whose solutions can be obtained through CI paradigms since these are an efficient alternative over analytical methods that suffer from premature convergence, local optimal trapping and curse of dimensionality. In this work, Differential Evolution 5 combined with the concept of opposition based learning 7 is proposed for solving the ELD problem. The initial population is generated through the concept of opposition based learning, and the algorithm uses only one population set throughout the optimization process, thus improving the rate of convergence. A jumping factor is added to the generation phase, thus improving the stability in obtaining optimal solutions. Author for correspondence E-mail: surekha_3000@yahoo.com GA for UC Genetic algorithms are adaptive search techniques based on the principles and mechanisms of natural selection and survival of the fittest from biological evolution 5. In this application, the unit commitment problem is solved using Genetic Algorithm that generates the on/off status of the generating units. The step by step procedure involved in the implementation of GA for UC problem is explained below: Step 1: Input data Specify generator cost coefficients, generation power limits for each unit and transmission loss coefficients (B-matrix) for the test system. Read hourly load profile of the generators for the test system. Initialize parameters of GA such as number of chromosomes, population size, number of generations, selection type, crossover type, mutation type, crossover probability and mutation probability to suitable values. Step 2: Initialize GA s population Initialize population of the GA randomly, where each gene of the chromosomes represents commitment of a dispatchable generating unit. The first step is to encode the commitment space for the UC problem based on the load curve from the load profile.
396 J SCI IND RES VOL 74 JULY 2015 Step 3: Computation of total cost The total generation cost for each chromosome is computed as the sum of individual unit fuel cost. Step 4: Computation of cost function and fitness function The augmented cost function for each chromosomes of population is computed using the sum of startup cost, running cost and shut down cost. Step 5: Application of genetic operators In two point crossover, the offspring are evaluated for fitness and the best one is retained while the worst is discarded from the population. The flip-bit mutation operation is performed by selecting a chromosome with specified probability. Step 6: The algorithm terminates after a specified number of generations have reached. If the termination condition is not satisfied then go to Step 3. IDE-OBL for ELD Though Standard Differential Evolution (SDE) has emerged as one of the most popular technique for solving optimization problem, it has been observed that the convergence rate of SDE does not meet the expectations in case of multi-objective problems. Hence, certain modifications using the concept of opposition based learning are performed on the SDE. The proposed IDE-OBL varies from the basic SDE in terms of the following factors: IDE-OBL uses the concept of opposition based learning in the initialization phase while SDE uses the uniform random numbers for initialization of population. During algorithmic run, an opposition based generation phase is added in IDE-OBL with a jumping factor to determine fitter opposite points. SDE uses two sets of population current population and an advanced population for next generation individuals. IDE-OBL uses only one population set throughout the optimization process, which is updated in successive generations with the best individuals found in each generation. The steps of the proposed algorithm for implementing ELD are explained below: Initialization The basic step in the IDE-OBL optimization is to create an initial population of candidate solutions by assigning random values to each decision parameter of each individual of the population. A population P consisting of individuals is constructed in a random manner such that the values lie within the feasible bounds and of the decision variable, according to, where represents a uniform random number in the interval [0,1], and are the lower and upper bounds for the j th component respectively, D is the number of decision variables. Each individual member of the population consists of an N-dimensional vector where the i th element of the i th generating unit. An opposite population represents the power output of is constructed by, where denotes the points of population P. The new population for the proposed approach is formed by combining the best individuals of both populations P and as. Mutation Next generation offspring are introduced into the population through the mutation process. Mutation is performed by choosing three individuals from the population in a random manner. Let, and represent three random individuals such that, upon which mutation is performed during the G th generation according to whe re is the perturbed mutated individual and represents the best individual among three random individuals. The difference of the remaining two individuals is scaled by a factor F, which controls the amplification of the difference between two individuals so as to avoid search stagnation and to improve convergence. Crossover New offspring members are reproduced through the crossover operation based on binomial distribution. The members of the current population (target vector) and the members of the mutated individual are subject to crossover operation thus producing a trial vector
SUREKHA & SUMATHI: A NOVEL APPROACH TO SOLVE UNIT COMMITMENT 397, where is the crossover constant that controls the diversity of the population and prevents the algorithm from getting trapped into the local optima. The crossover constant must be in the range of [0 1]. C r =1 implies the trial vector will be composed entirely of the mutant vector members and C r =0 implies that the trial vector individuals are composed of the members of parent vector. Selection Selection procedure is performed with the trial vector and the target vector to choose the best set of individuals for the next generation. In this proposed approach, only one population set is maintained and hence the best individuals replace the target individuals in the current population. The objective values of the trial vector and the target vector are evaluated and compared. For minimization problems like ELD, if the trial vector has better value, the target vector is replaced with the trial vector as The fittest individuals are selected from the new population set as the current population. Experimental Results The effectiveness of the proposed IDE-OBL is tested on IEEE 30 bus system 9 with 6 generating units. The ON/OFF commitment status through GA is implemented in Turbo C while the optimal dispatch is executed using MATLAB R2008b on Intel i3 CPU, 2.53GHz, 4GB RAM PC. Parameters of GA The control parameters for Genetic Algorithm are initialized as follows: No. of chromosomes: No. of generators Chromosome size: 24 (Hours) x No. of generators No. of generations: 500 Selection method: Roulette wheel Crossover Type: Two point crossover Crossover rate: 0.6 Mutation Type: Flip bit Mutation Rate: 0.001 Table 1 Commitment of units using GA for six unit test system Fitness evaluation The objective function for the ELD problem based on the fuel cost and power balance constraints is framed as, (1) where k is the penalty factor associated with the power balance constraint, is the i th generator cost function for output power, N is the number of generating units, is the total active power demand and represents the transmission losses. For ELD problems without transmission losses, setting k=0 is most rational, while for ELD including transmission losses, the value of k is set to 1. Generation jumping The maximum and minimum values of each variable in current population are used to calculate opposite points instead of using the predefined interval boundaries of the variables according to Hour Demand Combination of Units P1 P2 P3 P4 P5 P6 CT (s) 1 166 ON OFF ON ON OFF ON 1.21 2 196 ON OFF ON ON ON ON 1.33 3 229 ON OFF ON ON ON ON 1.25 4 267 ON ON ON ON ON OFF 1.24 5 283.4 ON ON ON ON ON OFF 1.31 6 272 ON ON ON ON ON OFF 1.28 7 246 ON ON ON ON ON OFF 1.34 8 213 ON ON ON ON ON OFF 1.24 9 192 ON ON ON ON OFF OFF 1.26 10 161 ON ON ON OFF OFF OFF 1.29 11 147 ON ON OFF OFF OFF OFF 1.33 12 160 ON ON OFF OFF OFF OFF 1.35 13 170 ON ON OFF OFF OFF OFF 1.34 14 185 ON ON OFF OFF OFF OFF 1.26 15 208 ON ON OFF OFF OFF OFF 1.22 16 232 ON ON ON OFF OFF OFF 1.27 17 246 ON ON ON OFF OFF ON 1.22 18 241 ON ON ON OFF OFF ON 1.26 19 236 ON ON ON OFF OFF ON 1.37 20 225 ON ON ON OFF OFF ON 1.22 21 204 ON ON ON OFF OFF ON 1.24 22 182 ON ON ON OFF OFF ON 1.29 23 161 ON ON ON OFF OFF ON 1.31 24 131 ON ON ON OFF OFF OFF 1.26
398 J SCI IND RES VOL 74 JULY 2015 UC using GA The on/off status and the computational time (CT) of the six generating units for 24 hours load demand is determined using GA and tabulated in Table 1. From the results, it is clear that the unit P1 is ON (binary 1) for 24 hours because this unit generates power with minimum fuel cost as the value of coefficient A is minimum for this unit. Units P5 and P6 is OFF (binary 0) for most of the hours because the value of fuel cost coefficient is the maximum for these two units and hence the fuel cost to generate power using these units is expensive when compared to other units. Parameters of IDE-OBL The parameters of IDE-OBL with their settings are listed below. No. of members in population: [20,100] Vector of lower bounds for initial population: [-2,-2] Vector of upper bounds for initial population: [2,2] No. of iterations: 200 Dimension: Problem dependant Crossover Rate: [0,1] Step size: [1,2] Strategy parameter: DE/rand/1/bin Jumping rate: 0.37 Refresh parameter: 10 Value to Reach: 1.e -6 ELD using IDE-OBL The committed schedules obtained through GA are dispatched using IDE-OBL based on the 24 hour load demand. The results are evaluated in terms of power dispatched to the 6 units (P 1 to P 6 ), fuel cost (FC), power loss (P L ), total power generated (P T ), and computational time (CT) as shown in Table 2. Comparative Analysis The results obtained by the IDE-OBL algorithm are compared with those available in literature namely Lambda Iteration Method (LIM) 1, Evolutionary Programming (EP) 8, Pattern Search (PS) 9, Hybrid GA and PS (GA-PS) 9, GA 10, Ant Colony Optimization (ACO) 11, Artificial Bee Colony (ABC) 12, Hybrid Genetic Algorithm (HGA) 13, Slow GA 14, Fast GA 14, Self-adaptive Differential Evolution (SADE) 15, and Weight-Improved Particle Swarm Optimization (WIPSO) 16 as shown in Table 5. For a demand of 283.4 MW, the minimum fuel cost obtained by the HOUR P D Table 2 ELD results using IDE-OBL for six unit system Power generated / unit P1 P2 P3 P4 P5 P6 FC ($/hr) P L P T 1 166 127.57 0 16.81 10 0 12 437.8 0.36 166.38 0.48 2 196 146.52 0 17.97 10 10 12 534.18 0.47 196.49 0.53 3 229 177.82 0 19.90 10 10 12 643.41 0.69 229.72 0.58 4 267 178.52 49.46 19.96 10 10 0 735.8 0.91 267.94 0.58 5 283.4 200 44.64 19.84 10 10 0 774.17 1.08 284.48 0.64 6 272 200 38.05 15 10 10 0 736.77 1.04 273.05 0.59 7 246 162.04 45.83 18.90 10 10 0 657.65 0.76 246.78 0.66 8 213 135.78 40.34 17.43 10 10 0 548.69 0.54 213.56 0.59 9 192 127.41 38.29 16.79 10 0 0 483.01 0.47 192.49 0.5 10 161 114.53 31.86 15 0 0 0 361.73 0.39 161.39 0.51 11 147 115.35 32.04 0 0 0 0 350.96 0.38 147.38 0.55 12 160 126.09 34.36 0 0 0 0 381.92 0.46 160.46 0.47 13 170 134.36 36.15 0 0 0 0 401.42 0.52 170.52 0.53 14 185 146.77 38.84 0 0 0 0 449.7 0.61 185.61 0.55 15 208 165.81 42.97 0 0 0 0 531.98 0.78 208.78 0.51 16 232 170.47 43.98 18.40 0 0 0 591.27 0.84 232.84 0.5 17 246 172.02 44.35 18.50 0 0 12 632.54 0.87 246.87 0.51 18 241 168.26 43.35 18.23 0 0 12 622.02 0.84 241.84 0.5 19 236 164.48 42.48 17.84 0 0 12 604.66 0.80 236.80 0.73 20 225 155.48 40.76 17.48 0 0 12 564.19 0.72 225.72 0.59 21 204 138.88 37.18 16.52 0 0 12 508.52 0.58 204.58 0.48 22 182 121.68 33.35 15.42 0 0 12 438.59 0.45 182.45 0.58 23 161 104.62 29.72 15 0 0 12 381.28 0.34 161.34 0.56 24 131 89.75 26.50 15 0 0 0 288.44 0.25 131.25 0.48 CT (s)
SUREKHA & SUMATHI: A NOVEL APPROACH TO SOLVE UNIT COMMITMENT 399 Algorithm Table 3 Comparative Analysis Fuel Cost ($/hr) Total power Power loss CPU time LIM 808.9491 292.8948 9.48889 25.9063 EP 802.404 292.8791 9.4791 NA PS 802.015 292.7344 9.3349 NA GA-PS 802.0138 292.7287 9.3286 NA GA 803.699 292.917 9.5177 315 ACO 803.123 292.8611 9.4616 20 ABC 801.881 271.18 NA 8.94 HGA 802.465 292.9105 9.5105 NA SGA 799.384 292.6801 9.6825 0.483 FGA 799.823 292.8093 9.6897 0.125 SADE 802.404 292.8791 9.4791 NA WIPSO 799.1665 292.0591 8.66 15.453 IDE-OBL 774.17 284.48 1.08 0.64 proposed IDE-OBL is $774.17 (Table 3), is comparatively less than the optimal fuel cost produced by other methods. The minimum fuel cost reported in literature so far is 799.1665 $/hr and IDE- OBL has proved to generate optimal economic dispatch with a difference of 3.2%. Conclusion In this paper, IDE-OBL is proposed for solving the UC-ELD problem. The UC problem is solved using GA to determine the ON/OFF schedule for a 24 hour time horizon. Based on the GA committed/decommitted schedule, the power is dispatched economically for corresponding load requirement using IDE-OBL. In the IDE-OBL algorithm, the concept of opposition based learning is applied in the initialization phase to accelerate the standard differential evolution algorithm with a motive of achieving optimal solutions with faster convergence characteristics. Likewise, the concept of OBL is also applied in the generation phase to ensure stability in convergence. From the results observed, it can be concluded that GA-IDE-OBL has shown significant improvements in the perspective of optimal fuel cost, power loss and computational time. In future, new optimization techniques like stud genetic algorithm, population-based incremental learning, intelligent water drop algorithm, bio-geography based algorithm and hybrid combination of these paradigms can also be applied to obtain optimal solution of UC-ELD problems. References 1 Wadhwa C L, Electrical Power Systems, (New Age International (P) Limited Publishers, New Delhi) 2000. 2 Lowery P G, Generation unit commitment by dynamic programming, IEEE Trans Power App Syst, 102 (1983) 1218 1225. 3 Wilson J A & Muckstadt R C, An application of mixedinteger programming duality to scheduling thermal generating systems, IEEE Trans Power App Syst, 87 (1968) 1968-1978. 4 Park J B, Lee K S, Shin J R & Lee K Y, A particle swarm optimization for economic dispatch with non smooth cost functions, IEEE Trans Power Syst, 8 (1993) 1325-1332. 5 Price K V, Storn R M & Lampinen J A, Differential Evolution: A Practical Approach to Global Optimization, (Springer, Heidelberg) 2005. 6 Goldberg D E, Genetic Algorithms in Search Optimization and Machine Learning, (Addison Wesley Longman Publishing, USA) 1989. 7 Rahnamayan S, Tizhoosh H R & Salama M M A, Opposition based differential evolution, IEEE Trans Evol Comp, 12 (2008) 64-79. 8 Yuryevich J & Wong K P, Evolutionary Programming Based Optimal Power Flow Algorithm, IEEE Trans Power Syst, 14 (1999) 1245 1250. 9 Labbi Y & Ben Attous D, A hybrid GA PS method to solve the economic load dispatch problem", J Theor App Inf Tech, 15 (2010) 61-68. 10 Tarek Bouktir, Linda Slimani & Belkacemi M, A Genetic Algorithm for Solving the Optimal Power Flow Problem, Leonardo J Sci, 4 (2004) 44-58. 11 Allaoua B & Laoufi A, Optimal Power Flow Solution Using Ant Manners for Electrical Network, Adv Electri Comp Eng, 9 (2009) 34-40. 12 Sumpavakup I S & Chusanapiputt S, A solution to the Optimal Power Flow using Artificial Bee Colony algorithm, IEEE Proc Int Conf Power Syst Tech (Hangzhou), (2010) 1-5. 13 Mary N & Thenmozhi D, Economic emission load dispatch using hybrid Genetic Algorithm, IEEE Proc Region 10 Conf TENCON (Chiang Mai, Thailand), (2004) 476-479. 14 Sailaja Kumari M & Sydulu M, A Fast Computational Genetic Algorithm for Economic Load Dispatch, Int J Recent Trends Eng, 1 (2009) 349-356. 15 Thitithamrongchai B & Eua-arporn, Self-adaptive Differential Evolution Based Optimal Power Flow for Units with Non-smooth Fuel Cost Functions, J Electri Syst, 3 (2007) 88-99. 16 PhanTu Vu, DinhLuong Le, NgocDieu VO & Tlusty Josef, A novel weight-improved particle swarm optimization algorithm for optimal power flow and economic load dispatch problem, in IEEE Trans Distrib Conf Expos (LA, USA) 19-22 April 2010.