entropy enhanced covariance matrix adaptation evolution
TRANSCRIPT
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
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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
Table of Contents
β’ Methodological Approach
β’ EE method for Optimization
β’ CMA-ES method for Optimization
β’ Simulation Results
β’ References
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Methodological Approach
β’ Sequential Combination of two optimization
methods
Entropy Enhanced (EE) Method for exploration.
Covariance Matrix Adaption Evolution Strategy
(CMAES) for exploitation.
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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.
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EE Method β’ 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.
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EE Method Select ππ0and ππ02, the number of samples per iteration N, the rarity
parameter Ο, the smoothing parameter Ξ±, k := 0.
ππ = ππ + 1 Generate a sample of ππ1,β¦, ππππfrom the sampling
distribution N(ππππβ1, ππππβ12 ).
Compute S(ππ1), β¦, S(ππππ) and order the samples from the worst to
the best performing ones, i.e. S(ππ1) < .β¦< S(ππππ).
Compute Ξ³ππ as the Οth quantile of the performance values and select
ππππππππππππ =Ο*N; let ππ be the subset from the ordered set of samples that
contains all the samples, i.e., the samples S(X)< Ξ³ππ.
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EE Method For j = 1 to n
ππππππ= β ππππ,ππ
ππππππππππππππβππ βββ(1) ππππππ2 = β (ππππππβππππππ)2
ππππππππππππππβππ ---(2)
End For Apply smoothing
ππππ = πΌπΌ β ππππ + 1 β πΌπΌ β ππππβ1-------------(3) ππππ2 = πΌπΌ β ππππ2 + 1 β πΌπΌ β ππππβ12 -------------(4)
Until ππ < ππππππππ For mean we shall use the same smoothing parameter πΌπΌ (0.5 β€ πΌπΌ β€ 0.9). For variance we shall use the dynamic smoothing ,
π½π½ππ = π½π½ β π½π½(1 β 1ππ)ππ-------------------------(5)
ππ= Integer (5 β€ ππ β€ 10), π½π½= Smoothing constant (0.8 β€ π½π½ β€ 0.99)
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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.
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CMAES method β’ Candidate solution is calculated using following
equation ------- (1) Where, = distribution mean and current favorite solution = step-size = covariance matrix
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* (0, )ki k k N Cx m Ο= +
kmkΟ
kC
β’ The new mean value is computed as --------------(2) Where, π€π€ππ= recombination weights ππ= number of samples per iteration ππ= ππ/2= number of parents/ points for recombination
β’ The step-size is updated using cumulative step-size adaption
(CSA), sometimes also denoted as path length control.
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ππππ+1 = οΏ½π€π€ππ
ππ
ππ=1
π₯π₯ππ=1:ππ β ππππ
CMAES method
kΟ
β’ The evolution path is updated using following equation
ππππ = 1 β ππππ β ππππ + 1 β (1 β ππππ)2 β πππ€π€ β πΆπΆππβ1/2 β ππππ+1βππππ
ππππ---(3)
discount factor complement for discounted variance displacement of m
β’ New step-size is updated using following equation
ππππ+1 = ππππ β πππ₯π₯ππππππππππ
πππππΈπΈβ ππ(0,πΌπΌ)
β 1 ---------------(4)
β’ The step-size is increased if and only if ππππ is larger than the expected value and decreased if it is smaller.
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CMAES method
β’ Evolution path is updated using following equation ππππ = 1 β ππππ β ππππ + 1 0,πΌπΌ ππ β ππππ 1 β (1 β ππππ)2 β πππ€π€ β
ππππ+1βππππππππ
--------(5)
β’ Covariance matrix is updated using following equation
πΆπΆππ+1 = 1 β ππ1 β ππππ + πππ π β πΆπΆππ + ππ1 β ππππ β ππππππ + ππππ β β π€π€πππ₯π₯ππ:ππβππππ
ππππ
π₯π₯ππ:ππβππππππππ
ππππππ=1 -(6)
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CMAES method
Test System:- IEEE 57 bus
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Objective Function:
β’ 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).
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Problem Constraints Equality constraints: 1. Power balance equations
Inequality constraints:
1. Nodal Voltages Vi :
ViMin β€ Vi β€ Vi
Max , i= 1,2,3β¦β¦..,NL β¦..(7)
2. Allowable Branch Power Flows Pij :
PijMin β€ Pij β€ Pij
Max , i=1,2,3β¦β¦..,NB β¦..(8)
3. Generator Reactive Power Capability QC :
QCiMin β€ QCi β€ QCi
Max , i= 1,2,3β¦β¦..,NC β¦..(9)
4. Maximum Active Power Output of slack generator PG :
PGi β€ PGiMax β¦..(10)
5. Minimum and maximum levels of optimization variable
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Constraint Handling Method β’ Select the maximum of the average sum of deviations at
iteration T [3]
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T-1 T-1 T-1 T-1
t-1 t-1 t-1 t-1t=1 t=1 t=1 t=1
1 1 1 1max ΞP , ΞV , ΞQ , ΞST-1 T-1 T-1 T-1
ββ ββ ββ ββ
Simulation Results: Test bed 1 β’ MATLAB 2014a, Intel core i7-2600 CPU with 8.00 GB RAM
β’ Case Study 1: Stochastic OPF for IEEE 57 Bus System
Considering Wind Energy Generators and Controllable Loads
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Statistics Case:1
f_best 81430.175
o@fbest 81430.168
g@fbest 0.007
fworst 80594.105
fmean 81382.615
fmedian 81413.841
Simulation Results cont.β¦ β’ Case study 2: Stochastic OPF for IEEE 57 Bus System
Considering Wind and Solar Energy Generators and Controllable Loads
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Statistics Case:2
f_best 68522.975
o@fbest 68522.965
g@fbest 0.010
fworst 68861.470
fmean 68519.132
fmedian 68579.969
β’ Case study 3: Stochastic OPF for IEEE 57 Bus System Considering Wind, Solar and Small-Hydro Generators and Controllable loads
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Statistics Case:1
f_best 55720.550
o@fbest 55720.547
g@fbest 0.005
fworst 56316.686
fmean 56032.936
fmedian 56043.483
Simulation Results cont.β¦
β’ Case study 4: OPF using an Analytical Uncertainty Cost Function for IEEE 57 bus system considering Wind generators and Controllable loads
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Simulation Results cont.β¦
Statistics Case:1
f_best 84342.953
o@fbest 84342.950
g@fbest 0.003
fworst 84347.422
fmean 84348.353
fmedian 84347.287
β’ 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
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Simulation Results cont.β¦
Statistics Case:1
f_best 71030.542
o@fbest 71030.540
g@fbest 0.002
fworst 71034.542
fmean 71033.364
fmedian 71033.226
References
1. G Rubinstein, R. Y. (1999), βThe cross-entropy method for combinatorial and continuous
optimizationβ, Methodology and Computing in Applied Probability, 2, 127-190.
2. N. Hansen. (2005 Nov.). The CMA Evolution Strategy: A Tutorial [Online]. Available:
http://www.lri.fr/βΌhansen/cmatutorial.pdf
3. V. Miranda and Leonel Carvalho (2014), βDEEPSO Evolutionary Swarms in the OPF
challengeβ, [online] Available http://sites.ieee.org/psace-mho/panels-and-competitions-
2014-opf-problems/, pp. 16.
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