1. Differential Evolution (DE) and its Variant Enhanced DE (EDE)
Presented By: Ayesha Zafar
Student of MS(CS)
Research Domain: Energy Management in Smart Grid
Supervised By: Dr. Nadeem Javaid

2. Differential evolution
Introduction
Metaheuristic
Evolutionary
Genetic algorithm
Genetic programing
Memetic algorithm
Differential evolution
Swarm Intelligence
Particle swarm optimization
Ant colony optimization
Artificial bee colony
Cuckoo search
Stochastic
Random search
Tabu search
Hill climbing DE
Population based evolutionary algorithm
Proposed by Storn and Price in 1995 [1]
Idea behind DE is a scheme for generating trial parameter vectors
Advantages of DE
Simple structure
Ease of use
Speed
[1] R. Storn and K. V. Price, ”Differential Evolution - a Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces,” ICSI, TR March 1995.

3. DE Basic Steps (1/6) Initialization Mutation Recombination / Crossover
Selection
Initialization

4. DE Basic Steps (2/6) Initialization
Randomly generate initial population
X ini = X l + rand()*(X u - X l)
Initialization

5. DE Basic Steps (3/6) Mutation Expand search space
For each vector Xi randomly select three vectors
X r1 , X r2 , X r3
r1, r2, r3 are distinct from each other
Add difference of two vectors to the third to form mutant vector
Vi = X r1 + F. (Xr2 - X r3 )
F is a constant factor range between [0 2]
Initialization

6. DE Basic Steps (4/6) Crossover or Recombination
Recombination incorporates successful solution from previous generation
Exchanges the element of mutant vector and target vector to form trial vector
Types of crossover
Exponential and Binomial
Binomial crossover is more commonly used than exponential.[2]
Initialization
[2] Arafa, M., Sallam, E. A., & Fahmy, M. M. (2014, May). An enhanced differential evolution optimization algorithm. In Digital Information and Communication Technology and it’s Applications (DICTAP), 2014 Fourth International Conference on (pp ). IEEE.

7. DE Basic Steps (5/6) Crossover or Recombination
Xi = (Xi1, X i2, X i3, X i4, X i5) Vi = (Vi1, V i2, V i3, Vi4, Vi5)
Ui = (Vi1, Xi2, Xi3, Vi4, Xi5)
Initialization

8. DE Basic Steps (6/6) Selection Survival of the fittest solution
Compare target vector with trial vector, one with the better fitness admitted to the next generation
Mutation, crossover and selection continue until some stopping criteria reaches

9. Example DE (1/5) Consider following two dimensional function
f (x, y) = x2 + y2
Start with a population of 5 candidates randomly initiated in the range [-10, 10]
Initial population
X 1,0
(2, -1)
X 2,0
(6, 1)
X 3,0
(-3, 5)
X 4,0
(-2, 6)
X 5,0
(6, -7)

10. Example DE (2/5) Randomly select three vectors X 2, X 4 and X 5
Now form the mutant vector
V 1,0 = X 2,0 + F. (X 4,0 – X5,0 )

11. Example DE (3/5)
Now form the trial vector by exchanging components of V1,0 with the target vector X1,0
Let rand(0,1) = 0.6
Set CR = 0.9
If 0.6 < 0.9 then
U1,1,0 = V1,1,0 = - 0.4
Again next time let rand(0, 1) = 0.95
0.95 > 0.9
U1,2,0 = X1,2,0 = -1
Finally the trial vector is
U1,0 = [-0.4, -1]

12. Example DE (4/5) Compare fitness of trial vector with target vector
If fitness of trial vector is better than fitness of target
Trial vector is replaced by target vector at G=1
f (x, y) = x2 + y2
Fitness of target
Fitness of trial
f (2, -1) = 22 + (-1)2
= 5
f (-0.4, -1) = (-0.4)2 + (-1)2
= 1.16

13. Example DE (5/5) Initial population G = 0 Initial fitness
Mutant vector
Trial vector
Fitness of trial vector
G = 1
Evolved population
X 1,0 =(2, -1) 5
V 1,0 = (-0.4, 10.4)
U 1,0 = (-0.4, -1)
1.16
X 1,1 = (-0.4, -1)
X 2,0 =(6, 1) 37
V 2,0 = (1.2, -0.2)
U 2,0 = (1.2, 1)
2.44
X 2,1 = (1.2, 1)
X 3,0 = (-3, 5) 34
V 3,0 = (-4.4, -0.2)
U 3,0 = (-4.4, -0.2)
19.4
X 3,1 = (-4.4, -0.2)
X 4,0 = (-2, 6) 40
V 4,0 =(9.2, 6)
U 4,0 = (9.2, 6)
120.64
X 4,1 = (-2, 6)
X 5,0 = (6, -7) 85
V 5,0 = (5.2, 0.2)
U 5,0 = (6, 0.2)
36.04
X 5,1 = (6, 0.2)

14. Enhanced DE (EDE) Introduction Developed in 2014 [2]
Improve trial vector strategy
Increase accuracy
Create five different trial vectors
First three trail vectors are obtained by taking three different crossover rates
0.3, 0.6 and 0.9 instead of one in case of DE
Fourth trial vector increase its convergence speed
Last trial vector increases diversity of search space
[2] Arafa, M., Sallam, E. A., & Fahmy, M. M. (2014, May). An enhanced differential evolution optimization algorithm. In Digital Information and Communication Technology and it’s Applications (DICTAP), 2014 Fourth International Conference on (pp ). IEEE.

15. Enhanced DE (EDE)

16. Enhanced DE (EDE) Randomly initialize the population
While termination criteria not satisfied do
Perform mutation
If (G ≤ 100) then
Perform crossover using Eqs.(1-5) , then we have five groups of trial vectors
Find the best member in each group of trial vectors
Compare trial vectors
Choose trial vector with best value
End if
4. Perform selection
set G = G+1
End while

17. Energy Optimization by EDE (1/13)
Parameter initialization
EDE parameters
Population size 30
CR1
0.3
CR2
0.6
CR3
0.9
MaxItr
100
HEMS parameters
Number of homes 1
Number of appliances 9
Pricing signal
Real Time Pricing (RTP)

18. Energy Optimization by EDE (2/13)
Appliances
Group
Appliances
Power rating (KWh)
Daily usage (hours)
Interruptible burst load
Vacuum cleaner
0.7 6
Water heater 5 12
Water pump 1 8
Dish washer
1.8 10
Base load
Refrigerator
0.225 18 AC
1.5 15
Oven
2.15
Non-interruptible load
Washing machine
Cloth dryer 4

19. Energy Optimization by EDE (3/13)
Generate initial population
for i=1:pop_size
for j=1:D
X(i,j) = xl+rand(1)*(xu-xl);
end
End
Vacum_cleaner
Water heater
Water pump
Dish washer
Washing machine
Cloth dryer
Refrigerator AC
oven 1

20. Energy Optimization by EDE (4/13)
Mutation
b1=randperm(pop_size);
b11=b1(1,1);
r11=X(b11,:);
b21=b1(1,2);
r21=X(b21,:);
b31=b1(1,3);
r31=X(b31,:);
Mutant1= r11+0.5*(r21-r31);
Vacum_cleaner
Water heater
Water pump
Dish washer
Washing machine
Cloth dryer
Refrigerator AC
oven 1

21. Energy Optimization by EDE (5/13)
Mutation
Mutant1= r11+0.5*(r21-r31);
Vacum_cleaner
Water heater
Water pump
Dish washer
Washing machine
Cloth dryer
Refrigerator AC
oven 1
Mutant vector 1

22. Energy Optimization by EDE (6/13)
Crossover
for i=1:D
if rand(1)>=0.3
Y11(1,i)=r11(1,D);
else
Y11(1,i)=Mutant1(1,D);
end
Mutant vector 1
Target vector 1
First trial vector 1

23. Energy Optimization by EDE (7/13)
Crossover
for i=1:D
if rand(1)>=0.6
Y21(1,i)=r11(1,i);
else
Y21(1,i)=Mutant1(1,i);
end
Mutant vector 1
Target vector 1
Second trial vector 1

24. Energy Optimization by EDE (8/13)
Crossover
for i=1:D
if rand(1)>=0.9
Y31(1,i)=r11(1,i);
else
Y31(1,i)=Mutant1(1,i);
end
Mutant vector 1
Target vector 1
third trial vector 1

25. Energy Optimization by EDE (9/13)
Crossover
for i=1:D
Y41(1,i) = rand(1)*r11(1,i);
end
Y51(1,i) = rand(1)*Mutant1(1,i)+(1- rand(1))*r11(1,i);
Mutant vector 1
Target vector 1
Fourth trial vector 1
Fifth trial vector 1

26. Energy Optimization by EDE (10/13)
Crossover
F11=Electricity_cost*Y1_1';
F21=Electricity_cost*Y2_1';
F31=Electricity_cost*Y3_1';
F41=Electricity_cost*Y4_1';
F51=Electricity_cost*Y5_1';
F=[F11 F21 F31 F41 F51 ];
Trial vectors
Cost 1
14.5 11 16
17.5 19
if min(F)==F11
T=Y1_1;
elseif min(F)==F21
T=Y2_1;
elseif min(F)==F31
T=Y3_1;
elseif min(F)==F41
T=Y4_1;
else min(F)==F51
T=Y5_1;
end
Final trial vector 1

27. Energy Optimization by EDE (11/13)
Selection
T11=Electricity_cost*T';
T12=Electricity_cost*r11_1';
if T12 < T11
X(b11,:)=r11_1;
else
X(b11,:)=T;
end
Target vector
Cost 1 25
Final trial vector
Cost 1 11

28. Energy Optimization by EDE (12/13)
Selection
Vacum_cleaner
Water heater
Water pump
Dish washer
Washing machine
Cloth dryer
Refrigerator AC
oven
fitness 1 25 15 18 13 12
Final trial vector
cost 1 11

29. Energy Optimization by EDE (13/13)
Selection
Vacum_cleaner
Water heater
Water pump
Dish washer
Washing machine
Cloth dryer
Refrigerator AC
oven 1
Final trial vector 1

31. Thank You Any Question