Differential Evolution Algorithm with a Modified Archiving-based Adaptive Tradeoff Model for Optimal Power Flow

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1 1 Dfferental Evoluton Algorthm wth a Modfed Archvng-based Adaptve Tradeoff Model for Optmal Power Flow

2 2 Outlne Search Engne Constrant Handlng Technque Test Cases and Statstcal Results

3 3 Roots of Dfferental Evoluton Developed n 1995 by Raner Storn and Kenneth Prce as a contnuous optmzaton method Man dea: use a mutaton/recombnaton operator based on dfference(s between pars of elements Smlartes wth older drect search methods: Pattern search (Hooke-Jeeves, 1961 Smple methods (Nelder-Mead,1965 Other populaton based methods nvolvng dfferences: Partcle Swarm Optmzaton (Kennedy & Eberhart, 1995 DE webpage storn/code.html Books: K.V. Prce, R.M. Storn, J.A. Lampnen; Dfferental Evoluton. A Practcal Approach to Global Optmzaton, 2005 U. Chakraborty, Advances n Dfferental Evoluton, 2008

4 4 Standard Dfferental Evoluton Problem to be solved: mnmze: f Intalzaton: = (,,..., D (, = (,..., D R 1, = U ( lb, ub, = 1,2,..., NP, 1,2,... { = D, 0,1, 0, 2, 0, D, 0,, 0 } populaton sze dmensonalty F scale factor CR crossover rate

5 5 Standard Dfferental Evoluton Problem to be solved: mnmze: f Intalzaton: = (,,..., whle NOT termnaton do Mutaton: v = + F, 1,2,... NP r r r ( = 1 2 3,, D ( = (,..., D R 1 { = U ( lb, ub, = 1,2,..., NP, = 1,2,... D, 0,1, 0, 2, 0, D, 0,, 0 } populaton sze dmensonalty F scale factor CR crossover rate

6 6 Standard Dfferental Evoluton Problem to be solved: mnmze: f Intalzaton: = (,,..., whle NOT termnaton do Mutaton: v = + F, 1,2,... NP r r r ( = D (, = (,..., D R 1, = U ( lb, ub, = 1,2,..., NP, 1,2,... { = D, 0,1, 0, 2, 0, D, 0,, 0 } populaton sze dmensonalty F scale factor CR crossover rate

7 7 Standard Dfferental Evoluton Problem to be solved: mnmze: f Intalzaton: = (,,..., whle NOT termnaton do Mutaton: v = + F, 1,2,... NP r r r ( = D (, = (,..., D R 1, = U ( lb, ub, = 1,2,..., NP, 1,2,... { = D, 0,1, 0, 2, 0, D, 0,, 0 } Crossover: v f rand(0,1 < CR or =, u =, otherwse, 0 populaton sze dmensonalty F scale factor CR crossover rate

8 8 Standard Dfferental Evoluton Problem to be solved: mnmze: f Intalzaton: = (,,..., whle NOT termnaton do Mutaton: v = + F, 1,2,... NP r r r ( = D (, = (,..., D R 1, = U ( lb, ub, = 1,2,..., NP, 1,2,... { = D, 0,1, 0, 2, 0, D, 0,, 0 } Crossover: v f rand(0,1 < CR or =, u =, otherwse, Selecton: u f f ( u f (, g =, g+ 1 f f ( u > f (, g, g 0 populaton sze dmensonalty F scale factor CR crossover rate

9 9 Fleblty of DE DE taonomy: DE/base element/ no. of dfferences/ crossover type Base element: random (rand: DE/rand/*/* best (best: DE/best/*/* Combnaton of current and best elements (λbest + (1- λ: DE/current-to-best/*/* combnaton of random and best elements (λr1 + (1- λ: DE/current-to-rand/*/* Number of dfferences: usually 1 (DE/*/1/* or 2 (DE/*/2/* Crossover type: bnomal (DE/*/*/bn or eponental (DE/*/*/ep

10 10 Fleblty of DE The other face of fleblty: whch varant to choose? Recommendatons No specfc knowledge on the problem: use DE/rand/1/* Need for an eplotatve method: use DE/best/1/* Need for a more eploratve method: use DE/rand/2/* Remark: dfferent varants could be approprate n dfferent stages of the optmzaton process Books: K.V. Prce, R.M. Storn, J.A. Lampnen; Dfferental Evoluton. A Practcal Approach to Global Optmzaton, 2005 U. Chakraborty, Advances n Dfferental Evoluton, 2008

11 (1+3 DE 1 parent ndvdual generates 3 offsprng ndvduals by 3 dfferent mutaton strateges: DE/rand/1: DE/rand/2: DE/current-to-best(rand/1: ( r r r F v + = ( ( r r r r r F F v + + = ( ( 2 r 1 r best F F v + + = ( ( r r r F F v + + = 11

12 12 (1+3 DE The current generaton number (denoted as current_gen s compared wth a threshold generaton number (denoted as threshold_gen threshold _ gen = k total _ gen If current_gen > threshold_gen, the DE/current-to-best/1 s actvated Journal: G. Ja, Y. Wang, Z. Ca, and Y. Jn. An mproved (μ+λ-constraned dfferental evoluton for constraned optmzaton. Informaton Scences, vol. 222, pp , 2013.

13 Constraned Optmzaton 13

14 Constraned Optmzaton 14

15 15 Constraned Optmzaton In general, for constrant optmzaton, the populaton may nevtably eperence three stuatons: The populaton contans nfeasble ndvduals only (nfeasble stuaton The populaton conssts of both feasble and nfeasble ndvduals (sem-feasble stuaton The populaton s entrely composed of feasble ndvduals (feasble stuaton

16 16 CH for Infeasble Stuaton Goals: Promptly motvate the populaton toward the feasble regon The dversty n the populaton should be kept Solutons: Truncaton technque Multobectve optmzaton-based technque

17 17 Truncaton Technque Journal: Y. Wang and Z. Ca. Constraned evolutonary optmzaton by means of (μ+λ-dfferental evoluton and mproved adaptve trade-off model. Evolutonary Computaton, vol. 19, no. 2, pp , 2011.

18 18 Multobectve optmzaton-based technque A MOP can be formulated as follows: mnmze f ( = ( f (,... f ( 1 m n where m s the number of obectve functons, = ( 1,... n S R s the decson vector whch contans n decson varables and S s the search space. Defnton 1. (Pareto domnance: Consder two decson vectors a = ( a 1,... a n and b = ( b 1,... b n, a s sad to Pareto domnate b (denoted as a b, f and only f { 1,... m}, f ( a f ( b and {1,... m}, f ( a < f ( b In ths case, b s sad to be Pareto domnated by a. If Pareto domnance does not hold between a and b, they are consdered nondomnated wth each other.

19 19 Multobectve optmzaton-based technque Defnton 2. (Pareto optmalty: a S only f b S satsfes b a. s sad to be Pareto optmal n S, f and Defnton 3. (Pareto optmal set: The Pareto optmal set (denoted as the set of all the Pareto optmal solutons: P* = { a S b S, b a} P * s Defnton 4. (Pareto front: The Pareto front (denoted as PF* = { f ( a a P*} PF * s defned as:

20 20 Multobectve optmzaton-based technque

21 21 CH for Sem-feasble Stuaton Goals: Feasble ndvduals wth small obectve functon values should reman n the net populaton Infeasble ndvduals wth slght constrant volatons and small obectve functon values should reman n the net populaton Soluton: An adaptve ftness transformaton

22 22 CH for Sem-feasble Stuaton Feasble group: Infeasble group: Z { G( = = 1 0} Z = { G( > 0} 2 The best and worst feasble solutons (denoted as and are found from the best worst feasble group. The converted obectve functon value: f (, f ( = ma{ ϕ f ( + (1 ϕ best f ( worst, f ( }, Z Z 1 2 where ϕ denotes the feasblty proporton of the populaton.

23 CH for Sem-feasble Stuaton Normalzaton of obectve functon value: Normalzaton of constrant volaton value: The fnal ftness functon: '( mn '( ma '( mn '( ( Z Z Z Z Z Z nor f f f f f = ( ( ( nor nor fnal G f f + = = 2 1, ( mn ( ma ( mn ( 0, ( Z G G G G Z G Z Z Z nor 23

24 24 Statstcal Results Case Prob. 57 bus 118 bus 300 bus ORPD Best Medan E+03 Worst E+04 Mean E+03 Std E+04 OARPD Best E E E+05 Medan E E E+05 Worst E E E+05 Mean E E E+05 Std E E+03

25 25 Statstcal Results Scenaro Best Medan Worst Mean Std

26 26 Statstcal Results Scenaro Best Medan Worst Mean Std

27 27 Statstcal Results Scenaro Best Medan Worst Mean Std

28 28 Statstcal Results Scenaro Best Medan Worst Mean Std

29 29 Statstcal Results Scenaro Best Medan Worst Mean Std

30 30 Statstcal Results Scenaro Best Medan Worst Mean Std

31 31 Statstcal Results Scenaro Best Medan Worst Mean Std

32 32 Statstcal Results Scenaro Best Medan Worst Mean Std

33 THANK YOU 33