Empirical Convergence Analysis Of Genetic Algorithm For Solving Unit - - PowerPoint PPT Presentation

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Empirical Convergence Analysis Of Genetic Algorithm For Solving Unit Commitment Problem

Domen Butala Coauthors:

• doc. dr. Dejan Veluˇ

sˇ cek

• 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 analysis An Upper Bound on the Convergence Speed Convergence of Homogenous Algorithm Combination of Both Approaches 3 Implementation Problem formulation Algorithm 4 Results and comparisons One-point crossover Multi-point crossover 5 Appendix 6 Authors 7 Bibliography

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Introduction Power system

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Introduction Power system Unit Commitment problem?

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Introduction Power system Unit Commitment problem? Motivation for an optimization approach

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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, Benders Decomposition, Dynamic Programming, . . .

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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 length l > 1, mutation probability 0 < pm ≤ 1

2 and let {

Xt, t ≥ 0} be the Markov chain population, π(t) distribution of tth generation of Xt and 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 of

• algorithm. If it holds

m > an + c(n − 1)∆⊗ min(a, b/2, cδ) , (1) then ∀x ∈ SN : lim

i→∞ lim t→∞ P([X i t ] ⊂ f ∗|X i 0 = 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 of parameters {(nt, pm(t)), t ≥} that it holds nt < nt+1 and pm(t) > pm(t + 1). A Genetic algorithm set like this could be called a variable-structure GA [1].

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Problem formulation

Problem formulation

min

xtype

i,t

T

• t=1

n

• i=1

(mpi,txtype

i,t

+ max{si,t − si,t−1, 0}sci)

• n
• i=1

xtype

i,t

≥ PDPt(price), ∀t (2) si,t = 1, ”if xtype

i,t

> 0, 0,

• therwise.
• , ∀t, i

(3) sti,t = (−1)1−si,t 1

I=[t−a,t+b] ∧ a,b≥0: si,t=si,¯

t ∀¯

t∈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 = 0 2: P(t) = SetInitialPopulation(P)

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Algorithm

Algorithm

1: t = 0 2: P(t) = SetInitialPopulation(P) 3: Evaluate(P(t)) 4: while not EndingCondition() do 5:

t+ = 1

6:

P(t) = Selection(P(t − 1))

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Algorithm

Algorithm

1: t = 0 2: P(t) = SetInitialPopulation(P) 3: Evaluate(P(t)) 4: while not EndingCondition() do 5:

t+ = 1

6:

P(t) = Selection(P(t − 1))

7:

P(t) = Crossover(P(t))

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Algorithm

Algorithm

1: t = 0 2: P(t) = SetInitialPopulation(P) 3: Evaluate(P(t)) 4: while not EndingCondition() do 5:

t+ = 1

6:

P(t) = Selection(P(t − 1))

7:

P(t) = Crossover(P(t))

8:

P(t) = Mutation(P(t))

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Algorithm

Algorithm

1: t = 0 2: P(t) = SetInitialPopulation(P) 3: Evaluate(P(t)) 4: while not EndingCondition() do 5:

t+ = 1

6:

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 10

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Multi-point crossover 11

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 to implementations in other languages (R, Matlab and C#).

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Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography

Domen Butala Financial Mathematics, Faculty of Mathematics and Physics, Ljubljana, Slovenia domen.butala@yahoo.com Dejan Veluˇ sˇ cek Department of Mathematics, Faculty of Mathematics and Physics, Ljubljana, Slovenia dejan.veluscek@fmf.uni-lj.si Gregor Papa Computer Systems Department, Joˇ zef Stefan Institute, Ljubljana, Slovenia gregor.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 of

Canonical Genetic Algorithms. Complexity International, Vol. 5, 1998.

• R. Cerf, Asymptotic Convergence of Genetic Algorithms.

CNRS, Universit´ e d’Orsay, Paris, 1997.

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