GAtoolbox: a Matlab-based Genetic Algorithm Toolbox for Function Optimization

Transcription

1 GAtoolbox: a Matlab-based Genetic Algorithm Toolbox for Function Optimization Code: Justo José Roberts 1,3, Agnelo Marotta Cassula 2, José Luz Silveira, Pedro Osvaldo Prado 3, José Celso Freire Junior 2 1 IPBEN-UNESP Guaratinguetá, Brazil 2 São Paulo State University (UNESP), Brazil 3 National University of Mar del Plata (UNMdP), Argentina

2 Structure I. Introduction and Objectives II. Theoretical Background III. Description of GAtoolbox IV. Implementation V. Conclusion and Future Work 12/11/2017 2

3 I. Introduction Optimization is the determination of values for design variables which minimize (maximize) the objective, while satisfying all constraints. Determine how desirable or undesirable the performance of a system or a design is every day engineers decision. It has become very popular in engineering activities, primarily because of the availability and affordability of high speed computers. Genetic Algorithms (GA) constitutes an important topic in Engineering Optimization undergraduate/postgraduate course. 12/11/2017 3

4 I. Objectives Describe the main features of the developed GAtoolbox. Present the results of its implementation in a practical study case. 12/11/2017 4

5 II. Theoretical Background Classification of optimization algorithms Algorithms Deterministic Linear programming Non-linear programming... Gradient-based Gradient-free Stochastic Heuristic Metaheuristic 11/12/2017 5

6 II. Theoretical Background Genetic Algorithm 12/11/2017 6

7 II. Theoretical Background Genetic Algorithms (GA) Probabilistic algorithms Evolutionary algorithms population based naturally inspired Metaheuristic. Conceived in 1960 by John Holland species adaptation and natural selection computers. Later, in the 1980s David E. Goldberg first success in industrial application with Gas. Chromosome DNA Gene 11/12/2017 7

8 II. Theoretical Background Simple Genetic Algorithm Start Generate first population Evaluate fitness Are optimization criteria met? Yes Best individuals No Selection Translate Generate first population Recombination End Mutation 12/11/2017 8

9 III. Description of GAtoolbox START Parameter definition Optimization problem definition Minimize fm x m 1,...,M subjected to g j x 0, j 1,..., J + K l u x x x, i 1,..., n i i i T x x, x,..., x S 1 2 n 1 Linear ranking Non-linear ranking Roulette wheel selection Stochastic universal sampling Tournament selection Normalized geometric selection Real decimal Integer decimal Binary Variables setting 2 Subroutines Discrete recombination Single-point crossover Double-point crossover Multi-point crossover Shuffle crossover Blend recombination (BLX-α) Line recombination Arithmetic recombination Flat recombination (BLX-0.0) Intermediate recombination Binary mutation Breeder mutation Boundary mutation Uniform mutation Multi-uniform mutation Non-uniform mutation Multi-non-uniform mutation Geration of initial population Number of variables Population size Evolution Fitness assignment Selection Recombination Reinsertion 3 4 Four main modules Mutation Maximum time and generations Hitting a bound Running mean Standard deviation Best-worst Phi Satisfy optimization criteria? Uniform reinsertion Worst and fitness-based reinsertion 12/11/2017 END 9

10 III. Description of GAtoolbox 1 Problem Definition Module Minimize f ( ) m x m1,2,..., M Mono-objective Multi-objective Subjected to g ( x) 0 j 1,2,..., j J h ( x) 0 k 1,2,..., k K Unconstrained xlbi xi xub i1,2,..., i x S n Constrained 12/11/

11 III. Description of GAtoolbox 1 Problem Definition Module Mono-objective min f( x) min f ( x) f ( x), f ( x),..., f ( x) Multi-objective m 1 2 M f 1 Non-dominated solution A B C Dominated solutions 3 Dominace : Pareto Optimal : xs x x Pareto Front : x S x * * PF x S x * Region with no feasible solutions D E Pareto Front F NSGA-II Nondominated Sorting Genetic Algorithm II f 2 12/11/

12 III. Description of GAtoolbox 1 Problem Definition Module Constrained problem gj ( x) 0 j 1,2,..., J hk ( x) 0 k 1,2,..., K xlb x xub i1,2,..., n i i i Approach adopted: dominance-based penalty-free f x if g j x x m fworst g j x otherwise j 0 Method of Deb Niching strategy based on Euclidean distance Method of Coello Coello and Montes Variable number of competitors and not deterministic 11/12/

13 III. Description of GAtoolbox 2 Variable Setting Module Representation critical decision in any application, namely that of deciding how best to represent a candidate solution of the algorithm. Phenotype space Encoding (representation) Decoding (inverse representation) Genotype space = {0,1} Generally accepted that it is better to encode numerical variables directly as: Integers Floating point variables 12/11/

14 III. Description of GAtoolbox 2 Variable Setting Module Variable representations supported by GAtoolbox GA works on Variable representation Need conversion? binary real decimal Yes binary integer decimal Yes real decimal real decimal No integer decimal integer decimal No Integer representation truncation method x x se r 0,5 se r 0,5 12/11/ i xi x i i se x i

15 III. Description of GAtoolbox 3 Generation of Initial Population Module First step in the GA is to create the initial population of individuals Generate a required number of individuals uniformly distributed in the desire range Binary coding x11 x pop Real representation x11 x pop /11/

16 III. Description of GAtoolbox 4 Evolution Module Fitness assignment Selection Recombination Reinsertion Mutation Satisfy optimization criteria? 12/11/

17 III. Description of GAtoolbox 4 Evolution Module Fitness assignment Linear ranking Non-linear ranking higher selective pressure 12/11/

18 III. Description of GAtoolbox 4 Evolution Module Roulette wheel selection Stochastic universal sampling Spin Selection Fitness Value Tournament method Chromosome Spin Normalized geometric selection Ranking Fitness Chromosome 1 Q 2 A A Pick the best 3 Z parent Pick K at 4 W random A 5 S 6 R R 7 F 12/11/ I I 1 2,1 Q 2 5 A 3 7,5 Z 4 10 W 5 11,2 S 6 20 R 7 2,3 F 8 24 I P rank 1 18 q i q q 1 1 rank x 1 popsize

19 III. Description of GAtoolbox 4 Evolution Module Recombination Operator Discrete recombination Single-point crossover Double-point crossover Multi-point crossover Shuffle crossover Variable representation All representations Binary and decimal integer Binary and decimal integer Binary and decimal integer Binary and decimal integer Blend recombination (BLX-α) Decimal real Line recombination Decimal real Arithmetic recombination Decimal real 12/11/

20 III. Description of GAtoolbox 4 Evolution Module Recombination Single-point crossover Double-point crossover Multi-point crossover Shuffle crossover Binary and decimal integer Binary and decimal integer Binary and decimal integer Binary and decimal integer Single-point crossover Multi-point crossover 12/11/

21 III. Description of GAtoolbox 4 Evolution Module Discrete recombination Line recombination Recombination Arithmetic recombination Blend recombination (BLX-α) All representations Decimal real Decimal real Decimal real 12/11/

22 III. Description of GAtoolbox 4 Evolution Module Operator Binary mutation Variable representation Binary Mutation Breeder GA Decimal real and integer Boundary mutation Decimal real and integer Uniform mutation Decimal real and integer Multi-uniform mutation Decimal real and integer Non-uniform mutation Decimal real and integer Multi-non-uniform mutation Decimal real and integer 12/11/

23 III. Description of GAtoolbox 4 Evolution Module Breeder GA Mutation Uniform mutation Non-uniform mutation 12/11/

24 III. Description of GAtoolbox 4 Evolution Module Maximum time or Nº of generations tmax t 0 maxgen ngen 0 Hitting bound F t F * lim obj obj Termination criteria Running mean Standard deviation t 1 1 * tlast obj i t t * F t F i obj last N 2 pop Npop 1 1 TC F pop obj i F pop obj i N N i1 i1 Best-worst F max F i i 1 N * * obj obj pop 12/11/2017 Phi * Fobj 1 1 N 1 pop N pop i1 F obj i 24

25 III. Description of GAtoolbox 4 Evolution Module Reinsertion Pure reinsertion Uniform reinsertion Elitist reinsertion Worst and fitness-based offspring (λ) = parents (μ) offspring (λ) < parents (μ) offspring (λ) < parents (μ) offspring (λ) = parents (μ) Parents Offspring Old population λ best λ worst μ best μ worst λ gengap λ (1-genGap) μ gengap μ (1-genGap) New population λ best μ best /11/

26 III. Description of GAtoolbox Parallel Computing To reduce the computational time in the execution of the optimization algorithm, GAtoolbox allows the possibility of parallel computation. Parallel Computing Toolbox (PCT) Run as many as eight Matlab workers on your local machine in addition to your Matlab client session. parfor (parallel for-loops) local workers parfor 12/11/

27 IV. Implementation Step 1: setting up the optimization model Step 2: create the objective function % OPTIMIZATION MODEL GaOptions = ga_options(); GaOptions.popSize = 50; GaOptions.maxGen = 100; GaOptions.numObj = 1; GaOptions.numVar = 2; GaOptions.numCons = 2; GaOptions.xLB = [0,0]; GaOptions.xUB = [6,6]; GaOptions.varType = [2]; GaOptions.objFun % COMPONENTS' NAME GaOptions.nameObj = {'Obj_1'}; GaOptions.nameVar = {'x_1','x_2'}; % SELECTION GaOptions.selFun = 'coelloselec'; GaOptions.coelloSr = 0.7; % GENETIC ALGORITHM OPERATORS GaOptions.recombiFun = 'blxrecombin'; GaOptions.BLXalpha = 0.5; GaOptions.pXover = 0.9; GaOptions.mutateFun = 'nonuniformmutate'; GaOptions.shapeB = 5; GaOptions.pMut = 0.2; % PARALLEL COMPUTING GaOptions.useParallel = 'no'; % PLOT RESULTS GaOptions.bestFitMeanPlot = 1; % CALL GA FUNCTION [xbest,bestobjvalue,timerun,gaoptions,result] = ga_constraint_optimization(gaoptions); #Generation 50 / 50 currentgen 50 evaluatecount 2500 totaltime 588 Nwt Twt Npv Tpv Ndg NPVc function [y, cons] = test_fun305c(x) y = [0]; cons = [0,0]; % OBJECTIVE FUNCTION % MIN problem objective function y(1) = (x(1).^2 + x(2) - 11).^2 + (x(1) + x(2).^2-7).^2; % CONSTRAINTS % Compute the constraint violation c = (x(1) ).^2 - (x(2) - 2.5).^2; cons(1) = (c<0)*abs(c); c = x(1).^2 + (x(2) - 2.5).^2-4.84; cons(2) = (c<0)*abs(c); ,99E ,00E ,00E ,01E ,01E ,03E+05 Step 3: Results 12/11/ ,03E+05 27

28 IV. Implementation Study Case Optimal design of Hybrid Power Systems (HPS) F dg (L/h) η conv (%) E pv (kwh) P dg (kw) η dg (%) DG P conv /P conv,r (%) PV t(h) P dg (kw) = CONV WT E wt (kwh) E load (kwh) t(h) L AC SOC(%); Q bt (kwh) BT t(h) Complexity AC DC t(h) Stochastic behavior of renewable resources and demand Non-linear characteristic of some components 12/11/

29 START Input parameters: component characteristics, renewable resources, load Create first generation Generate new population (recombination, mutation) Assess population N wt N pv N dg N bt N conv Compute the energy balance of the system for one year of operation: Compute: 8760 i1 i E t Select best individuals NPV c LPSP EMCO 2 Objetive/s No Stopping criteria met? Yes f ren f exess SOC min Restrictions Optimization Module (GAtoolbox) Simulation Module END Set of efficient solutions (configurations of HPS) 12/11/

30 Energy (kwh) SOC (%) IV. Implementation Study Case Simulation Module Horas (h) WT PV DG Load Unmet Load Energy Excess SOC % % 80% 70% 60% 50% 40% 30% 20% 10% 0 0% Hours (h) 12/11/

31 IV. Implementation Study Case Input data Meteorological measurements Demand load curve Weekday Weekend Weekday Weekend Power (kw) PV panel Rated power 0.11 kwp Area 1.02 m 2 Initial capital (IC) 2900 $/kw O&M 1% of IC $/year Lifetime 25 year Life cycle emissions Hour (h) HPS components kgco 2e /kwh Wind turbine Rated power 7 kw Tower height 17.5 m Initial capital (IC) 3669 $/kw 12/11/ O&M 3% of IC $/year Lifetime 20 year Life cycle emissions 0.02 kgco 2e /kwh

32 IV. Implementation Study Case Minimization of Cost and Maximization of Reliability Net Present Value of costs MIN f ( NPV, LPS) m Loss of Power Supply Probability C Restrictions LPSP adm [%] 5 % f ren.min [%] 0 % f exess,adm [%] 100 % F dg,adm [L/year] Inf EMCO 2,adm [t/year] Inf [N wt,min ; N wt,max ] [0,15] [N pv,min ; N pv,max ] [0,300] [N bt,min ; N bt,max ] [0,50] [N dg,min ; N dg,max ] [0,3] [N conv,min ; N conv,max ] [0,6] 6,677,248 combinations 12/11/

33 IV. Implementation Study Case Optimization Graphical Results MIN f ( NPV, LPS) m C Generation 1 Generation 2 Generation 3 Generation 4 Generation 5 6,0 5, Sol 1 Generation 10 Generation 30 Generation 50 6,0 5,0 LPS (kwh/year) ,0 3,0 2,0 LPSP (%) LPS (kwh/year) Sol 2 4,0 3,0 2,0 LPSP (%) , Sol 3 1,0 0 0, NPVc ($) NPVc ($) 0,0 12/11/

34 IV. Implementation Study Case Optimization Numerical Results MIN f ( NPV, LPS) m C WT (kw) PV (kw) DG (kw) BT (kwh) CONV (kw) Eexess (%) RF (%) Hdg (h/yr) Fdg (L/yr) LCOE ($/kwh) LPSP (%) NPVc ($) Sol 1 7x x4 4x , Improve reliability + 23% investment Sol 2 9x x4 4x , Sol 3 7x7 0 2x10 21x4 4x , /11/

35 V. Conclusion The GAtoolbox is a useful tool to teach the basics of GA in an undergraduate/postgraduate optimization course. Advantages of implementing the code in Matlab students become familiar with it aid to develop their final projects. The implementation of the toolbox in open architecture software such as Matlab meets the expectations of a flexible tool for researching purposes as well. 12/11/

36 V. Future Work Other functionalities will be integrated in future versions of the GAtoolbox: Friendly graphical user interphase. Other metaheuristics such as Particle Swarm Optimization, Tabu Search, Differential Evolution, among others. Help documentation. 12/11/

37 Thank you! Justo José Roberts 12/11/