Presented By:- Nishant Chaturvedi CONTENTS Introduction Literature Review Objective Methodology Problem Formulation Implementation Results & Discussion Conclusion Future Scope of Work References In power generation our main aim is to generate the required amount of power with minimum cost. Economic load dispatch means that the generators real and reactive power are allowed to vary within certain limits, so as to meet a particular load demand with minimum fuel cost. This allocation of load are based on some constraints Equality Constrain Inequality Constrain Rani C. et al (2013) A chotic local search operator is introduced in the proposed algorithm to avoid premature convergence. Two different chaotic maps are alternatively used as pseudorandom number generator and switch over during the run of chaos driven PSO algorithm. Park Jong-Bae et al (2010) An improved PSO framework employing chaotic sequence combined with conventional linearly decreasing inertia weights and adopting a cross over operation scheme to increase both exploration and exploitation capability of the PSO.Literature Survey Pluhacek michal et al (2013) a new approach for chaos drive particle swarm optimization (PSO) algorithm is suggested. . . More specifically. Tao Zhang et al (2009) A modified tent-map-based chotic PSO (TCPSO) to solve the ELD problem. a novel dynamic inertia weight factor was incorporated with the modified hybrid tent-map-based chaotic PSO which balance the global and local search better.Literature Survey Jaini et al (2010) A particle swarm optimization algorithm (PSO) with one of the accelerating coefficient being constant are propose to solve the economic power dispatch problem. Literature Survey Chaturvedi K. . Leandro dos Santos Coelho et al (2008) The use of combining of particle swarm optimization. Et al (2008) A novel self organizing hierarchical particle swarm optimization ( SOH_PSO) for the non. T. Gaussian probability distribution function and chaotic sequence. Araujo Ernesto et al (2008) Particle swarm optimization approach intertwined with lozi map chaotic sequence to obtain Takagi.Sugeno (TS) fuzzy model for representing dynaic behavior are proposed.convex economic dispatch to handle the problem of premature convergence. R2 and self. Chuanwen J. Logistic map chaotic sequence to generate the random number R1.Bae et al (2006) A novel and efficient method for solving the economic dispatch problem with valve point effect by integrating the particle swarm optimization with the chaotic sequences. . et al (2005) Suggested a self – adaptive chaotic particle swarm optimization is used to solve the ELD problem in deregulated environment.Literature Survey Park Jong.adaptive inertia weight scale in original PSO to improve the performance. To maximize the power generation by proposing a PSO algorithm to obtain the optimum scheduling of generator .OBJECTIVE The main objective of study is to minimize generation cost using partical swarn optimization (PSO) algorithm for the economic load dispatch (ELD) problem. To integrate PSO method with Chaotic map for solving ELD problem having generated unit with non smooth cost function and multi-fuel. The purpose of the economic load dispatch (ELD) problem is to control the committed generator’s output such that the total fuel cost is minimized. while satisfying the power demand and other physical and operational constraints. METHODOLOGY Particle swarm optimization • Proposed by james kennedy & russell eberhart in 1995 • Inspired by social behavior of birds and fishes • Combines self-experience with social experience • Population-based optimization . Each particle in search space adjusts its “flying” according to its own flying experience as well as the flying experience of other particles. Update velocity and position of each particle 5. Update individual and global bests 4. PSO ALGORITHM Basic algorithm of PSO 1.Concept of PSO Uses a number of particles that constitute a swarm moving around in the search space looking for the best solution. Initialize the swarm form the solution space 2. and repeat until termination condition . Go to step 2. Evaluate the fitness of each particle 3. velocity update equation Vi k 1 Vi c1rand1 Pbesti X i c2 rand 2 Gbest k xi k k k k Where.c2 rand1.rand2 Xik Pbestik Gbestk k = Velocity of individual i at iteration k = Inertia weight factor = Acceleration Coefficient = Random number between 0 and 1 = Position of individual i at iteration k = Best position of individual a until iteration k = Best position of the group until iteration k Position update equation Xi k 1 X i Vi k k 1 (2) (1) . Vi c1. (2) No If iteration Completed Yes Stop . and Dimension (D) Initialize particles with random Position (P) and Velocity vector (V) Calculate fitness for each Population Update the Population local best Update best of local bests as gbest Upadate Particle velocity using eq. FLOW CHART OF BASIC PSO Start Define the parameter of PSO Constants C1. C2 Particle (P). (1) and Postion using eq. i) Logistic Map f k . Deterministic chaos: irregular motion generated by nonlinear dynamical systems whose laws determine the time evolution of a state of the system from a knowledge of its previous history. It describes many physical phenomena with complex behavior by simple laws.1 f k 1 .CHOTIC THEORY Chaos: a state of disorder and irregularity. Dynamical systems: systems that develop in time in a nontrivial manner. f k 1. The behaviour of the system represented by above equation is greatly changed with the change of μ. fk-1. 4]. . the solution shows a rich variety of behaviours. Despite the apparent simplicity of the equation.Where μ is a control parameter and has a real value between [0. or react chaotically in an unpredictable pattern. is a number between Zero and One. The value of μ determines whether f stabilizes at a constant size. oscillates between bounded sequences of sizes. 7 and b=0. a quadratic term in the latter is replaced with a piecewise linear contribution in the former. This allows one to rigorously prove the chaotic character of some attractors.5. The lozi map is depicted in fig.ii) Lozi map Lozi introduced in a short note. Simply. The parameters used in this work are: a=1. a two-dimensional map the equations and attractors of which resemble those of the celebrated h´enon map. X n 1 1 a | X n | bYn Yn 1 X n . The map equations are given below. the lozi map may present both regular and chaotic behaviours. the lozi map is switched over to logistic map. . The new proposed algorithm utilizes lozi map for the first part of the optimization process. When pre-defined number of iterations is achieved. Inside the region where the orbits remain bounded.Where a and b are the real non-vanishing parameters. In present work the goal is to minimize the generation cost of committed generating unit i. and ten which are given below N FT Fi Pi i 1 ai Pi bi Pi ci 2 Where.ci: Cost Function of Generator i Pi: Output Power of Generator i N: Number of Generator .e three. It is expressed in term of design variable and other problem parameter. FT: Total Generating Cost Fi: Cost Function of ith Generating Unit ai.Problem formulation An objective function expresses the main aim of the model which is either to be minimized or maximized. forty.bi. n P P i 1 i load Ploss Where Pload is the total system load. an equality constraint should be satisfied.Equality and inequality constraints Active power balance equation: for power balance. Ploss is a function of the unit power outputs that can be represented using B coefficients as follows: n n n Ploss Pi Bij Pj B0i Pi B00 i 1 j 1 i 1 . The total transmission network loss. The total generated power should be the same as the total load demand plus the total line loss. Ploss is the total line loss. However. the transmission loss is not considered in this research work for simplicity (i.min and Pi.Here. 2) Minimum and maximum power limits: power output of each generator should be within its minimum and maximum limits. Ploss = 0). Pi . Corresponding inequality constraints for each generator is.max are the minimum and maximum output of generator i.m in Pi Pi .e. ..m ax Where Pi. respectively. Non-Smooth Cost function with Multi Point Fuel Since the dispatching unit are practically supplied with multi fuel sources. a piecewise quadratic function is used to represent the input output curve of a generator with multi fuel and described as.min Pi | 2 Where ei and fi are the coefficients of generator i reflecting valve-point effects. Since the valve point result in the ripples. Here the sinusoidal functions are thus added to the quadratic cost function as follows: Fi Pi ai bi Pi ci Pi | ei sin fi Pi .Non-Smooth Cost Function with Valve-Point effects The generating units with multi valve steam turbine exhibit a greater variation in the fuel cost function. each unit should be representing with several piecewise quadratic function reflecting the effect of fuel type change. a cost function constraints higher order nonlinearity. In general. . ...... ...ai1 bi1 Pi ci1 Pi 2 if Pi min Pi Pi1 a b P c P 2 if Pi1 Pi Pi 2 i 2 i 2 i i 2 i .cin are the cost coefficients of generator i for the p-th power level... .. FiPi .. ain..bin.......... 2 ain bin Pi cin Pi if Pin 1 Pi Pi m ax Where... c. Configure the PSO running parameters population size (Psize) = 100 and total iterations (itermax) = 50 Initialize the values of fk=0.63 and mu ( ) = 4 for logistic map Initialize the initial position and velocity matrix to zero For iter = 1:iter_max .9. Provide the upper bound (UB) and lower bound (LB) constrains on generators Initialize the PSO coefficients c1 = 2. wmin =0. and b_lozi = 0.IMPLEMENTATION Pseudo Code for ELD Input required power (Pd) Initialize the coefficients a.7.5.1. Select the optimization technique Initialize the value of a_lozi = 1. e and f of all generators. wmax = 0. c2 = 1. b. For i = 1:pop_size For j=1:nvars If iter = = 1 Generate random number for initial positions (Pij) and velocities (Vij) Check for upper and lower bond and modified accordingly else assign lastly calculated Pij and Vij endif Endfor Endfor Endfor . j). .j)). update the value of w If (the technique is standard) calculate w normally else if (the technique is previous) calculate the next value from logistic map and use it to modify the w fk = * fkpre * (1-fkpre). end w = w_max – ((w_max – w_min)/iter_max) * iter.:)~=Pd)) temp_Pij = Pd – (sum(init_positions(i.Now update the variables to satisfy the Pd constrain while (sum(init_positions (i. temp_vij = init_velocity (i.:)) – init_positions(i. else calculate the next value from lozi map and use it to modify the w lozi_X = 1 – a_lozi * abs (lozi_X_pre) + b_lozi * lozi _Y_pre. end .wnew = w* fk . wnew = w * lozi _ X. else (the technique is proposed) if (iter is odd the) calculate the next value from logistic map and use it to modify the w fk = * fkpre * (1-fkpre). wnew = w * fk . fit_val(i) = obj_fun (x). :). G_best = P_best. end if P_val < G_val G_val = P_val. check for the Pbest and compare it with previous gbest end .for i = 1: pop_size calculate the fitness values for all the population x = init_positions (i. init_velocity = w_new * init_velocity + c1 * rand * (Pbest – init_positions)+c2 * rand * (G_best – init_positions). calculate the new velocity and positions for all the population and repeat . FLOW CHART Start Take Initialization Parameters Define Objective Function Define Objective Constrains Set Iter = 1 Generate Initial Population Evaluate Objective Function Use Logistic Map If mod (Iter.2) ==1 Update Velocity and Positions If iter == ter_max Select Best Solution End Use Lozi Map . RESULTS & DISCUSSION Test System 1: This system comprises of 3 generating unit and the input data of 3-generating system are given in Here. The standard PSO 8700 8650 8600 8550 Cost 8500 8450 8400 8350 8300 8250 8200 0 5 10 15 20 25 Iterations 30 35 40 45 Figure 1: Operating Cost of 3 generating unit using standard PSO 50 . the total demand for the system is set to 850MW. 2422e3 .400 350 Operating Power 300 250 200 150 100 50 0 1 2 Generator Number 3 Figure 2: Operating Power of 3 generating unit using standard PSO Figure 3: Result window of 3 generating unit using standard PSO Table 1: Minimum cost of 3 generating unit using standard PSO Technique Minimum Cost PSO 8. •The PSO with single chaotic operation 8700 8650 8600 8550 Cost 8500 8450 8400 8350 8300 8250 8200 0 5 10 15 20 25 Iterations 30 35 40 45 Figure 4: Operating Cost of 3 generating unit using PSO 1 50 . 400 350 Operating Power 300 250 200 150 100 50 0 1 2 Generator Number 3 Figure 5: Operating Power of 3 generating unit using PSO 1 Figure 6: Result window of 3 generating unit using PSO 1 Table 2: Minimum cost of 3 generating unit using PSO 1 Technique Minimum Cost PSO 1 8.2416e3 . •The PSO with double (alternative) chaotic operation 8700 8650 8600 8550 Cost 8500 8450 8400 8350 8300 8250 8200 0 5 10 15 20 25 Iterations 30 35 40 45 Figure 7: Operating Cost of 3 generating unit using PSO 2 50 . 2341e3 .400 350 Operating Power 300 250 200 150 100 50 0 1 2 Generator Number 3 Figure 8: Operating Power of 3 generating unit using PSO 2 Figure 9: Result window of 3 generating unit using s PSO 2 Table 3: Minimum cost of 3 generating unit using PSO 2 Technique Minimum Cost PSO 2 8. Minimum Operational Cost by all Three Techniques 8700 PSO PSO 1 PSO 2 8650 8600 8550 Cost 8500 8450 8400 8350 8300 8250 8200 0 5 10 15 20 25 Iterations 30 35 40 45 50 Figure 10 : Comparison of cost minimization vs. PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps. . iterations for PSO. 400 PSO PSO 1 PSO 2 350 Operating Power 300 250 200 150 100 50 0 1 2 Generator Number 3 Figure 11: Comparison of optimum operational condition for 3 generator units for PSO. . PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps. 2341e3 . Table 4: Minimum Operational Cost for 3 generating unit by all Three Techniques Technique Minimum cost PSO 8. PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps.2416e3 PSO 2 8.Figure 12: Result window for comparison of 3 generating unit for PSO.2422e3 PSO 1 8. 335 1.Test System 2: In this case the test system consists of 40generating units and the input data are given.345 Cost 1.355 x 10 1. The total demand is set to 10500 MW.35 1.32 1. •The standard PSO 5 1.33 1.315 0 5 10 15 20 25 Iterations 30 35 40 45 Figure 13: Operating Cost of 40 generating unit using standard PSO 50 .34 1.325 1. 600 500 Operating Power 400 300 200 100 0 0 5 10 15 20 25 30 Generator Number 35 40 Figure 14: Operating Power of 40 generating unit using standard PSO 45 Figure 15: Result window of 40 generating unit using standard PSO Table 5: Minimum cost of 40 generating unit using standard PSO Technique Minimum Cost PSO 1.3195e5 . 325 1.•The PSO with single chaotic operation 5 1.315 1.305 0 5 10 15 20 25 Iterations 30 35 40 Figure 16: Operating Cost of 40 generating unit using PSO 1 45 50 .355 x 10 1.33 1.32 1.35 1.345 1.335 1.34 Cost 1.31 1. 3093e5 .600 500 Operating Power 400 300 200 100 0 0 5 10 15 20 25 30 Generator Number 35 40 Figure 17: Operating Power of 40 generating unit using PSO 1 45 Figure 18: Result window of 40 generating unit using PSO 1 Table 6: Minimum cost of 40 generating unit using PSO 1 Technique Minimum Cost PSO 1 1. 27 0 5 10 15 20 25 Iterations 30 35 40 45 Figure 19: Operating Cost of 40 generating unit using PSO 2 50 .36 x 10 1.31 1.28 1.35 1.32 1.•The PSO with double (alternative) chaotic operation 5 1.34 Cost 1.29 1.3 1.33 1. Operating Power 600 500 400 300 200 100 0 0 5 10 15 20 25 30 Generator Number 35 40 45 Figure 20: Operating Power of 40 generating unit using PSO 2 Figure 21: Result window of 40 generating unit using PSO 2 Table 7: Minimum cost of 40 generating unit using PSO 2 Technique Minimum Cost PSO 2 1.2717e5 . 31 1.29 1. PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps .32 1.36 x 10 PSO PSO 1 PSO 2 1.35 1.28 0 5 10 15 20 25 Iterations 30 35 40 45 50 Figure 22 : Comparison of cost minimization vs.34 Cost 1. iterations for PSO.3 1.Minimum Operational Cost by all Three Techniques 5 1.33 1. 600 PSO PSO 1 PSO 2 500 Operating Power 400 300 200 100 0 0 5 10 15 20 25 30 Generator Number 35 40 45 Figure 23: Comparison of optimum operational condition for 40 generator units for PSO. PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps. . 3017e5 1. PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps. Table 8: Minimum Operational Cost for 40 generating unit by all Three Techniques Technique PSO PSO 1 PSO 2 Minimum cost 1.2839e5 .Figure 24: Result window for comparison of 40 generating unit for PSO.2932e5 1. •The standard PSO 1000 900 800 Cost 700 600 500 400 300 0 5 10 15 20 25 Iterations 30 35 40 45 50 Figure 25: Operating Cost of 10 generating unit using standard PSO . The total system demand is set to 2700 MW.Test System 3: Multi-Fuels with Valve-Point Effect The test system consists of 10-generating units considering multi-fuels with valve-point effects. 500 450 400 Operating Power 350 300 250 200 150 100 50 0 1 2 3 4 5 6 7 8 Generator Number 9 10 Figure 26: Operating Power of 10 generating unit using standard PSO Figure 27: Result window of 10 generating unit using standard PSO Table 9: Minimum cost of 10 generating unit using standard PSO Technique Minimum Cost PSO 318.4248 •The PSO with single chaotic operation 1000 900 800 Cost 700 600 500 400 300 200 0 5 10 15 20 25 Iterations 30 35 40 45 Figure 28: Operating Cost of 10 generating unit using PSO 1 50 500 450 400 Operating Power 350 300 250 200 150 100 50 0 1 2 3 4 5 6 7 8 Generator Number 9 10 Figure 29: Operating Power of 10 generating unit using PSO 1 Figure 30: Result window of 10 generating unit using PSO 1 Table 10: Minimum cost of 10 generating unit using PSO 1 Technique Minimum Cost PSO 1 294.1963 •The PSO with double (alternative) chaotic operation 1000 900 800 Cost 700 600 500 400 300 200 0 5 10 15 20 25 30 Iterations 35 40 Figure 31: Operating Cost of 10 generating unit using PSO 2 45 50 . 350 300 Operating Power 250 200 150 100 50 0 1 2 3 4 5 6 7 8 Generator Number 9 10 Figure 32: Operating Power of 10 generating unit using PSO 2 Figure 33: Result window of 10 generating unit using PSO 2 Table 11: Minimum cost of 10 generating unit using PSO 2 Technique PSO 2 Minimum Cost 239.8838 . iterations for PSO.Minimum Operational Cost by all Three Techniques 1000 PSO PSO 1 PSO 2 900 800 Cost 700 600 500 400 300 200 0 5 10 15 20 25 Iterations 30 35 40 45 Figure 34: Comparison of cost minimization vs. PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps 50 . 500 PSO PSO 1 PSO 2 450 400 Operating Power 350 300 250 200 150 100 50 0 1 2 3 4 5 6 7 Generator Number 8 9 10 Figure 35: Comparison of optimum operational condition for 10 generator units for PSO. PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps. . PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps. Table 12: Minimum Operational Cost for 10 generating unit by all Three Techniques Technique PSO PSO 1 PSO 2 Minimum Cost 317.Figure 36: Result window for comparison of 10 generating unit for PSO.6402 .5348 302.1667 247. CONCLUSION AND FUTURE SCOPE This work presents an efficient approach for enhancing the performance of standard PSO algorithm by alternative use of two different chaotic maps for velocity updation and applied to the ELD problem and tested for three different systems and objectives. . The simulation results shows the superiority of the proposed algorithm over the previously proposed single chaotic map based PSO algorithm and support the idea that switching over of chaotic pseudorandom number generators in the PSO algorithm improves its performance and the optimization process. CONCLUSION AND FUTURE SCOPE The results for three different experiments are collected with different settings and results compared with other methods which show that the proposed algorithm improves the results by at least 10% for all three cases. Although the result has improved we can further develop the algorithm by utilizing multiple maps and optimizing the chaotic maps parameters however these considerations are leaved for future enhancements. . 405410. . 2444-2449. ICPEC IEEE 2013. D. [2] Michal Pluhacek. Congress on Evolutionary Computation Cancun Mexico.REFERENCES [1] J. “Chaotic Self Adaptive Particle Swarm Approach for Solving Economic Dispatch Problem with Valve-Point Effect. in Proc. Rani. and K. Evolutionary Computation. 1997. Kothari. pp. Roman Senkerik and Ivan Zelinka. Busawon. June 20-23. P. IEEE 2013. 303-308. pp. 4th IEEE Cong. Kennedy. 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Vol. International Journal of scientific and engineering Research. 188-193. 5. ISO 9001: 2008. “ A Survey on Economic Load Dispatch Problem using Particle Swarm Optimization technique” International Journal of emerging Technology and Advanced Engineering. Walkey and N. pp. 292-300. March 2014. 4. 24-31. ISSN: 2250-2459. Vol. Nishant Chaturvedi and A. April 2014. ISSN: 2248-9622. A. “ A Modified PSO Based Solution Approach for Economic Load Dispatch Problem in Power System. S. March 2014. S.pp. pp. Issue 4. Patidar. S.
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