Under the guidance of Dr. Abraham T. Mathew Submitted by: Avdesh Mehra M170250EE ICS Objective Automatic Voltage Regulator System Modeling and Simulation PID tuning Algorithms Genetic Algorithm Particle Swarm Optimization Results Work to be done References To model and analyze the synchronous power generator in MATLAB. To design an optimum PID controller for Automatic voltage regulator (AVR) system. Applying Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) for the specified problem statement. To analyze, apply and compare the PSO and GA for best performance of the system. To test different Fitness Functions and compare for best results. It is used with the synchronous generators in the power plants. Controls the voltage of the system nearer to the steady state stability. The automatic voltage regulators reduce the over voltages which occur because of the sudden loss of load on the system. It increases the excitation of the system under fault conditions so that the maximum synchronizing power exists at the time of clearance of the fault. Power transmission: SIMULINK MODEL OF AVR SYSTEM: STEP RESPONSE WITHOUT PID: Genetic Algorithm (GA): GA is a search based optimization technique based on the principle of natural genetics and natural selection. It finds the optimal solution Genetic algorithms are a particular class of evolutionary algorithms that use techniques inspired by evolutionary biology such as inheritance, mutation, selection, and crossover (also called recombination). 1. A 50x3 gain chromosome population is generated. 𝐾𝑝1 𝐾𝑖1 𝐾𝑑1 ⋮ ⋮ ⋮ 𝐾𝑝50 𝐾𝑖50 𝐾𝑑50 50×3 2. 3. 4. Fitness value is calculated for each gain vector using the fitness function. 𝑧 = 1 − 𝑒 −𝑏 × (𝑀𝑝 +𝐸𝑠𝑠 ) + 𝑒 −𝑏 × 𝑡𝑠 − 𝑡𝑟 where b=1 Best individuals will be selected according to The Roulette Wheel Selection Method. Crossover and Mutation will occur and the original population will be updated. Selected Best Parent Gain Chromosomes (A and B): A 0.1 0.2 0.3 B 0.4 0.5 0.6 C 0.4 0.2 0.3 0.9 0.3 Crossover Gain Chromosome (C) D 0.4 Mutated Gain Chromosome (D) SELECTION: • Roulette Wheel Selection Method is used. • Gains having the minimum fitness value will get selected more number of times. • A new 50x3 gain matrix will be formed using the best gain values. Kp Ki Kd Fitness Function A 0.231 0.123 0.215 0.02 0.5 A 0.231 0.123 0.215 0.02 0.365 0.2 A 0.231 0.123 0.215 0.02 0.211 0.341 0.6 C 0.159 0.247 0.365 0.1 0.917 0.782 1.2 C 0.159 0.247 0.365 0.1 Kp Ki Kd Fitness Function A 0.231 0.123 0.215 0.02 B 0.516 0.314 0.786 C 0.159 0.247 D 0.412 E 0.952 Initial Population Selected Best Gains CROSSOVER: Child1 = beta1*Parent1+(1-beta1)*Parent2 Child2 = beta2*Parent1+(1-beta2)*Parent2 Where beta value will be decided by: betas=rand(2,1)*(1+2*alpha)-(0.5+alpha) And alpha=2. Crossover is responsible for the convergence of the algorithm. Kp Ki Kd Fitness Function A-A 0.231 0.123 0.215 0.02 A-A 0.231 0.123 0.215 0.02 A-C 0.231 0.123 0.365 0.1 A-C 0.231 0.123 0.365 0.1 C-C 0.159 0.247 0.365 0.2 MUTATION: Mutation will modify the gain chromosomes randomly and generate a new population with new fitness function values. Kp Ki Kd P 0.231 0.399 0.215 Q 0.231 0.123 0.563 R 0.832 0.123 0.365 S 0.231 0.412 0.365 T 0.741 0.247 0.365 TERMINATION OF LOOP : Termination can be achieved in 3 ways: • When there has been no improvement in the population for X iterations. • When we reach an absolute number of generations. • When the objective function value has reached a certain pre-defined value. The second type of termination condition is being used in the program. 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑧𝑒 × 20 + 10 In this case population size =50, maximum generations=70. Fitness function value for each iteration: GA algorithm flow chart: Particle Swarm Optimization Algorithm (PSO): PSO is a search based optimization technique based on the principle of swarm intelligence. Inspired from the natural social behavior and dynamic movements with communications of insects, birds and fish. 1. A 50x3 gain particle population is generated, initialised and their fitness value is calculated. 𝐾𝑝1 𝐾𝑖1 𝐾𝑑1 ⋮ ⋮ ⋮ 𝐾𝑝50 𝐾𝑖50 𝐾𝑑50 50×3 For Example: Kp Ki Kd Fitness Function A 0.231 0.123 0.215 0.8 B 0.516 0.314 0.786 0.5 C 0.159 0.247 0.365 0.2 D 0.412 0.211 0.341 0.6 E 0.952 0.917 0.782 1.2 Where Kp,Ki,Kd are the uniform random values generated within the specified range of [0 0 0] to [1.5 1 1] 2. Initialize the Global Best and Personal Best fitness function values to infinite and Direction Matrix to zeros. gBest=0 and pBest=0 Example of direction Matrix: A 0 0 0 B 0 0 0 C 0 0 0 D 0 0 0 E 0 0 0 3. Update the Gains and Direction of the 𝑡 𝑡ℎ particle according to the following equations. particle(t).Direction = w*particle(t).Direction + c1*rand(VarSize).*(particle(t).Best.Gain - particle(t).Gain) + c2*rand(VarSize).*(GlobalBest.Gain - particle(t).Gain) particle(t).Gain = particle(t).Gain + particle(t).Direction; where, c1,c2=2, maximum weight=0.9, minimum weight = 0.4 After 𝑡 𝑡ℎ iteration: Kp Ki Kd Fitness Function A 0.231 0.123 0.215 1.500 B 0.516 0.314 0.786 0.200 C 0.396 0.610 0.186 0.077 D 0.412 0.211 0.341 0.121 E 0.952 0.917 0.782 1.200 gBest=0.2,pBest=0.2 Best Gains will be 𝐾𝑝 = 0.396 , 𝐾𝑖 = 0.610 , 𝐾𝑑 = 0.186 TERMINATION OF LOOP : Termination can be achieved in 2 ways: • When we reach an absolute number of generations. • When the objective function value has reached a certain pre-defined value. The second type of termination condition is being used in the program. 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 100 Fitness function value for each iteration: PSO algorithm flow chart: Original objective function: 𝑊 𝑘 = 1 − 𝑒 −𝛽 × 𝑀𝑃 + 𝐸𝑆𝑆 + 𝑒 −𝛽 (𝑡𝑠 − 𝑡𝑟 ) Where 𝛽 is constant. Modified Objective Function 1: 𝐹 𝑘 = 𝛼 × 𝑀𝑃 + 𝐼𝑆𝐸 + 𝛽 𝑡𝑠 + 𝑡𝑟 Where, 𝛼 𝑎𝑛𝑑 𝛽 are constants and 𝐼𝑆𝐸 = Integral time squared error: 𝐼𝑇𝑆𝐸 = Integral Time Absolute Error: 𝐼𝑆𝐴𝐸 = Integral Absolute Error: 𝐼𝐴𝐸 = 𝑡𝑠𝑖𝑚 0 𝑡𝑠𝑖𝑚 𝑡 0 𝑡𝑠𝑖𝑚 2 𝑒 (𝑡)𝑑𝑡 0 × 𝑒 2 (𝑡)𝑑𝑡 𝑡𝑠𝑖𝑚 𝑡 0 𝑒(𝑡) 𝑑𝑡 × 𝑒(𝑡) 𝑑𝑡 Coding and general analysis of PSO and GA algorithms. Modification of algorithms for single objective PID controller for the AVR system. Genetic algorithm is applied with the following fitness functions: Original, ITSE, ITAE, ISE, IAE, Modified PSO algorithm is applied for the following objective functions: Original, ITSE, ITAE, ISE, IAE. Results are obtained and compared. FITNESS FUNCTION Rise Time Settling Time % Peak Overshoot Steady State Error GA (Original) 0.3232 0.5022 0 0.0038 GA (ITSE) 0.1238 0.7189 21.7142 0.0033 GA (ITAE) 0.1348 0.7529 19.1245 0.0053 GA (ISE) 0.0904 3.5078 24.5994 0.0002277 GA (IAE) 0.1257 0.7264 21.5095 0.0117 GA (Modified) 0.2034 1.9503 1.000 0.0042 Table 1: For different Fitness functions with GA algorithm FITNESS FUNCTION Rise Time Settling Time % PeakOvershoot Steady State Error PSO (Original) 0.3335 0.5213 0 0.0038 PSO (ITSE) 0.1065 1.7390 42.3452 0.0091 PSO (ITAE) 0.1196 1.2043 32.7725 0.0044 PSO (ISE) 0.1000 1.6650 44.7595 0.0013 PSO (IAE) 0.1046 1.5689 43.5791 5.9365e-04 Table 2: For different Fitness functions with PSO algorithm GA bar graphs: Rise Time Settling Time Rise Time Settling Time 3.5078 0.3232 0.1238 GA 0.1348 GA (ITSE) GA (ITAE) 0.0904 GA (ISE) 0.1257 GA (IAE) (Original) 0.5022 0.7189 0.7529 GA GA (ITSE) GA (ITAE) GA (IAE) Steady State Error % PeakOvershoot 21,7142 24,5994 19,1245 Steady State Error 0.0117 21,5095 0.0038 0.0033 0.0053 0.000227 7 0 (Original) GA (ISE) (Original) % Peak Overshoot GA 0.7264 GA (ITSE) GA (ITAE) GA (ISE) GA (IAE) GA (Original) GA (ITSE) GA (ITAE) GA (ISE) GA (IAE) GA ALGORITHM RESULT PLOTS: GA STEP RESPONSE Fitness Function: Original GA STEP RESPONSE Fitness Function: ITSE GA STEP RESPONSE Fitness Function: ITAE GA STEP RESPONSE Fitness Function: ISE GA STEP RESPONSE Fitness Function: IAE GA STEP RESPONSE Fitness Function: Modified PSO ALGORITHM RESULT PLOTS: PSO STEP RESPONSE Fitness Function: Original PSO STEP RESPONSE Fitness Function: ITSE PSO STEP RESPONSE Objective Function: ITAE PSO STEP RESPONSE Objective Function: ISE PSO STEP RESPONSE Objective Function: IAE SIMULINK MODEL FOR GA-PSO COMPARISION SIMULINK OUTPUT: Applying new optimization algorithms like Fire fly algorithm, Krill herd algorithm, Fish for hash algorithm, Cuckoo search algorithm for better results. 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