Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Empirical Convergence Analysis Of GeneticAlgorithm For Solving Unit Commitment

Problem

Domen Butala

Coauthors:doc. dr. Dejan Velušček

doc. dr. Gregor Papa

Ljubljana, September 13th, 2014

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

1 Introduction

2 Convergence analysisAn Upper Bound on the Convergence SpeedConvergence of Homogenous AlgorithmCombination of Both Approaches

3 ImplementationProblem formulationAlgorithm

4 Results and comparisonsOne-point crossoverMulti-point crossover

5 Appendix

6 Authors

7 Bibliography

2

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Introduction

Power system

Unit Commitment problem?

Motivation for an optimization approach

Techniques as MIP, MILP, LR, BendersDecomposition, Dynamic Programming, . . .

3

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Introduction

Power system

Unit Commitment problem?

Motivation for an optimization approach

Techniques as MIP, MILP, LR, BendersDecomposition, Dynamic Programming, . . .

3

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Introduction

Power system

Unit Commitment problem?

Motivation for an optimization approach

Techniques as MIP, MILP, LR, BendersDecomposition, Dynamic Programming, . . .

3

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Introduction

Power system

Unit Commitment problem?

Motivation for an optimization approach

Techniques as MIP, MILP, LR, BendersDecomposition, Dynamic Programming, . . .

3

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

An Upper Bound on the Convergence Speed

An Upper Bound on the Convergence Speed

Theorem

[1] Let the size of population of the GA be n ≥ 1, coding lengthl > 1, mutation probability 0 < pm ≤ 12 and let {~Xt , t ≥ 0} be theMarkov chain population, π(t) distribution of tth generation of ~Xtand π be the stationary distribution. Then it holds

||π(k) − π|| ≤ (1− (2pm)nl)k .

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Convergence of Homogenous Algorithm

Convergence of Homogenous Algorithm

Theorem

[2] Let a, b, c > 0 be constants and i intensity perturbations ofalgorithm. If it holds

m >an + c(n − 1)∆⊗

min(a, b/2, cδ), (1)

then

∀x ∈ SN : limi→∞

limt→∞

P([X it ] ⊂ f ∗|X i0 = x) = 1.

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Combination of Both Approaches

Combination of Both Approaches

Idea, to get the best algorithm possible, is to set a sequence ofparameters {(nt , pm(t)), t ≥} that it holds nt < nt+1 andpm(t) > pm(t + 1).

A Genetic algorithm set like this could be called avariable-structure GA [1].

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Problem formulation

Problem formulation

minx typei,t

{ T∑t=1

n∑i=1

(mpi ,txtypei ,t + max{si ,t − si ,t−1, 0}sci )

}n∑

i=1

x typei ,t ≥ PDPt(price), ∀t (2)

si ,t =

{1, ”if x typei ,t > 0,

0, otherwise.

}, ∀t, i (3)

sti ,t = (−1)1−si,t∑

1{ I=[t−a,t+b]∧ a,b≥0:si,t=si,t̄ ∀t̄∈I ∧ si,t−a−1=si,t+b+1=1−si,t

} (4)sti ,t ≥ tupi ∨ sti ,t ≤ −tdowni , ∀t, i (5)xi ,t = xmaxi ,t (6)

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Algorithm

Algorithm

1: t = 02: P(t) = SetInitialPopulation(P)

3: Evaluate(P(t))4: while not EndingCondition() do5: t+ = 16: P(t) = Selection(P(t − 1))7: P(t) = Crossover(P(t))8: P(t) = Mutation(P(t))9: Evaluate(P(t))

10: end while

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Algorithm

Algorithm

1: t = 02: P(t) = SetInitialPopulation(P)3: Evaluate(P(t))4: while not EndingCondition() do5: t+ = 16: P(t) = Selection(P(t − 1))

7: P(t) = Crossover(P(t))8: P(t) = Mutation(P(t))9: Evaluate(P(t))

10: end while

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Algorithm

Algorithm

1: t = 02: P(t) = SetInitialPopulation(P)3: Evaluate(P(t))4: while not EndingCondition() do5: t+ = 16: P(t) = Selection(P(t − 1))7: P(t) = Crossover(P(t))

8: P(t) = Mutation(P(t))9: Evaluate(P(t))

10: end while

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Algorithm

Algorithm

1: t = 02: P(t) = SetInitialPopulation(P)3: Evaluate(P(t))4: while not EndingCondition() do5: t+ = 16: P(t) = Selection(P(t − 1))7: P(t) = Crossover(P(t))8: P(t) = Mutation(P(t))

9: Evaluate(P(t))10: end while

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Algorithm

Algorithm

1: t = 02: P(t) = SetInitialPopulation(P)3: Evaluate(P(t))4: while not EndingCondition() do5: t+ = 16: P(t) = Selection(P(t − 1))7: P(t) = Crossover(P(t))8: P(t) = Mutation(P(t))9: Evaluate(P(t))

10: end while

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

One-point crossover

Results

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

One-point crossover

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Multi-point crossover

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Appendix

Algorithm was implemented in the programming language Python.

Numpy

Cython

Numba

On some parts of the code performance comparisons were made toimplementations in other languages (R, Matlab and C#).

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Domen ButalaFinancial Mathematics, Faculty of Mathematics and Physics,Ljubljana, Sloveniadomen.butala@yahoo.com

Dejan VeluščekDepartment of Mathematics, Faculty of Mathematics andPhysics, Ljubljana, Sloveniadejan.veluscek@fmf.uni-lj.si

Gregor PapaComputer Systems Department, Jožef Stefan Institute,Ljubljana, Sloveniagregor.papa@ijs.si

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Y. Gao, An Upper Bound on the Convergence Rates ofCanonical Genetic Algorithms. Complexity International, Vol.5, 1998.

R. Cerf, Asymptotic Convergence of Genetic Algorithms.CNRS, Université d’Orsay, Paris, 1997.

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IntroductionConvergence analysisAn Upper Bound on the Convergence SpeedConvergence of Homogenous AlgorithmCombination of Both Approaches

ImplementationProblem formulationAlgorithm

Results and comparisonsOne-point crossoverMulti-point crossover

AppendixAuthorsBibliography