Entropy Enhanced Covariance Matrix Adaptation Evolution Strategy (EE_CMAES)

Transcription

1 1 Entropy Enhanced Covariance Matrix Adaptation Evolution Strategy (EE_CMAES) Developers: Main Author: Kartik Pandya, Dept. of Electrical Engg., CSPIT, CHARUSAT, Changa, India Co-Author: Jigar Sarda, Dept. of Electrical Engg., CSPIT, CHARUSAT, Changa, India 2018 Grid Optimization Competition Test bed A: Stochastic OPF in presence of renewable energy and controllable loads 2018 IEEE PES General Meeting, August 5-9, 2018, Portland, OR, USA

2 Table of Contents 2 Methodological Approach EE method for Optimization CMA-ES method for Optimization Simulation Results References

3 3 Methodological Approach Sequential Combination of two optimization methods Entropy Enhanced (EE) Method for exploration. Covariance Matrix Adaption Evolution Strategy (CMAES) for exploitation.

4 4 EE Method for Optimization Cross Entropy method is a versatile heuristic tool for solving difficult estimation and optimization problems based on Kullback- Leibler minimization [1]. Cross Entropy method was motivated by Rubinstein, where an adaptive variance minimization algorithm for estimating probabilities of rare events for stochastic networks was presented.

5 5 EE Method Cross Entropy method involves two iterative phases: 1. Generation of a sample of random data according to a specified random mechanism. 2. Updating the parameters of the random mechanism, typically parameters of pdfs, on the basis of the data, to produce a better sample in the next iteration.

6 EE Method Select μμ 0 and σσ 0 2, the number of samples per iteration N, the rarity parameter ρ, the smoothing parameter α, k := 0. kk = kk + 1 Generate a sample of XX 1,, XX NN from the sampling 2 distribution N(μμ kk 1, σσ kk 1 ). Compute S(XX 1 ),, S(XX NN ) and order the samples from the worst to the best performing ones, i.e. S(XX 1 ) <. < S(XX NN ). Compute γ kk as the ρ th quantile of the performance values and select NN eeeeeeeeee =ρ*n; let ψψ be the subset from the ordered set of samples that contains all the samples, i.e., the samples S(X)< γ kk. 6

7 For j = 1 to n End For XX kk,jj EE Method μμ kkkk = ii ψψ (1) σσ 2 kkkk = ii ψψ ---(2) NN eeeeeeeeee Apply smoothing μμ kk = αα μμ kk + 1 αα μμ kk (3) (XX iiii μμ kkkk ) 2 σσ 2 kk = αα σσ 2 2 kk + 1 αα σσ kk (4) Until kk < kk MMMMMM NN eeeeeeeeee For mean we shall use the same smoothing parameter αα (0.5 αα 0.9). For variance we shall use the dynamic smoothing, ββ tt = ββ ββ(1 1 tt )qq (5) qq= Integer (5 qq 10), ββ= Smoothing constant (0.8 ββ 0.99) 7

8 8 CMAES method for Optimization [2] Two main principles for the adaption of parameter of the search distribution are exploited in the CMAES algorithm. 1. Maximum-likelihood principle, based on the idea to increase the probability of successful candidate solutions and search steps. 2. Two path of the time evolution of the distribution mean of the strategy are recorded, called search or evolution paths. (i) One path is used for covariance matrix adaption procedure (ii) Second path is used to conduct an additional step-size control.

9 9 CMAES method Candidate solution is calculated using following equation x = m + σ * N(0, C ) i k k k (1) Where, mk= distribution mean and current favorite solution σ k C k = step-size = covariance matrix

10 10 CMAES method The new mean value is computed as μμ mm kk+1 = ww ii xx ii=1:λλ mm kk (2) ii=1 Where, ww ii = recombination weights λλ= number of samples per iteration μμ= λλ/2= number of parents/ points for recombination The step-size σ k is updated using cumulative step-size adaption (CSA), sometimes also denoted as path length control.

11 11 CMAES method The evolution path is updated using following equation pp σσ = 1 cc σσ pp σσ + 1 (1 cc σσ ) 2 μμ ww CC kk 1/2 mm kk+1 mm kk σσ kk ---(3) discount factor complement for discounted variance displacement of m New step-size is updated using following equation σσ kk+1 = σσ kk eeeeee cc σσ dd σσ pp σσ EE NN(0,II) (4) The step-size is increased if and only if pp σσ is larger than the expected value and decreased if it is smaller.

12 12 CMAES method Evolution path is updated using following equation pp cc = 1 cc cc pp cc + 1 0,αα nn pp σσ 1 (1 cc σσ ) 2 μμ ww mm kk+1 mm kk σσ kk (5) Covariance matrix is updated using following equation CC kk+1 = 1 cc 1 cc μμ + cc ss CC kk + cc 1 pp cc pp TT xx cc + cc μμ ww ii:λλ mm kk ii=1 ii -(6) μμ σσ kk xx ii:λλ mm kk σσ kk TT

13 Test System:- IEEE 57 bus 13

14 14 Objective Function: Minimize the total fuel cost of traditional generators (buses: 1, 3, 8, 12) plus the expected uncertainty cost for renewable energy generators (buses: 2, 6, 9) plus the compensation cost for controllable loads (buses: 8, 12, 18, 47).

15 15 Equality constraints: 1. Power balance equations Problem Constraints Inequality constraints: 1. Nodal Voltages V i : V Min i V i V Max i, i= 1,2,3..,NL..(7) 2. Allowable Branch Power Flows P ij : P Min ij P ij P Max ij, i=1,2,3..,nb..(8) 3. Generator Reactive Power Capability Q C : Q Min Ci Q Ci Q Max Ci, i= 1,2,3..,N C..(9) 4. Maximum Active Power Output of slack generator P G : P Gi P Gi Max..(10) 5. Minimum and maximum levels of optimization variable

16 16 Constraint Handling Method Select the maximum of the average sum of deviations at iteration T [3] T-1 T-1 T-1 T max ΔP, ΔV, ΔQ, ΔS T-1 T-1 T-1 T-1 t-1 t-1 t-1 t-1 t=1 t=1 t=1 t=1

17 17 Simulation Results: Test bed 1 MATLAB 2014a, Intel core i CPU with 8.00 GB RAM Case Study 1: Stochastic OPF for IEEE 57 Bus System Considering Wind Energy Generators and Controllable Loads Statistics Case:1 f_best o@fbest g@fbest fworst fmean fmedian

18 18 Simulation Results cont. Case study 2: Stochastic OPF for IEEE 57 Bus System Considering Wind and Solar Energy Generators and Controllable Loads Statistics Case:2 f_best fworst fmean fmedian

19 19 Simulation Results cont. Case study 3: Stochastic OPF for IEEE 57 Bus System Considering Wind, Solar and Small-Hydro Generators and Controllable loads Statistics Case:1 f_best fworst fmean fmedian

20 20 Simulation Results cont. Case study 4: OPF using an Analytical Uncertainty Cost Function for IEEE 57 bus system considering Wind generators and Controllable loads Statistics Case:1 f_best fworst fmean fmedian

21 21 Simulation Results cont. Case study 5: OPF using an Analytical Uncertainty Cost Function for IEEE 57 bus system considering Wind and Solar generators (Cases 5) and Controllable loads Statistics Case:1 f_best fworst fmean fmedian

22 22 References 1. G Rubinstein, R. Y. (1999), The cross-entropy method for combinatorial and continuous optimization, Methodology and Computing in Applied Probability, 2, N. Hansen. (2005 Nov.). The CMA Evolution Strategy: A Tutorial [Online]. Available: hansen/cmatutorial.pdf 3. V. Miranda and Leonel Carvalho (2014), DEEPSO Evolutionary Swarms in the OPF challenge, [online] Available opf-problems/, pp. 16.