Economic Dispatch & Optimization Techniques

Dr R. N. Sharma

EED, NIT Hamirpur

Simple Economic Dispatch Problem

SEDP(Continued)

SEDP(Continued)

SEDP(Continued)

Include power generator limits

Generator limits included

Generator limits included

Generator limits included

Network losses considered

Network losses considered

OptimizationUnconstrained Constrained

Continuous Discrete

Smooth Non-smooth

Search Based

Gradient Based Method

Optimization (continued)

Search

Gradient Based Tree-Based Stochastic Heuristic

Based on Decomposition

Subproblems

Search

Gradient Based Tree-Based Stochastic Heuristic

Optimization

Min f(x)

f() continuous and differentiable scalar functionX real variable

Stationarity condition df/dx=0For min d2f/dx2>0

Global Min

Local Mindf/dx=0

df/dx=0max

Convex function has a unique global minimum

Lambda-iteration method

Lambda-iteration method

Gradient method

Gradient method (continued)

Gradient method

Newton’s method

Gradient method Limitation

Dynamic Programming

Dynamic Programming (continued)

DP Advantage

Genetic Algorithm

• GAs work with a coding of the parameter set, not the parameters themselves.

• GAs search from a population of points, not a single point.

• GAs use payoff information, not derivatives or auxiliary knowldege.

• GAs use probablistic transition rules, not deterministic rules.

Vocabulary

• Gene – An single encoding of part of the solution space.

• Chromosome – A string of “Genes” that represents a solution.

• Population - The number of “Chromosomes” available to test.

Simple Example• f(x) = {MAX(x2): 0 <= x <= 32 }• Encode Solution: Just use 5 bits (1 or 0).• Generate initial population.

• Evaluate each solution against objective.

A 0 1 1 0 1

B 1 1 0 0 0

C 0 1 0 0 0

D 1 0 0 1 1

Sol. String Fitness % of Total

A 01101 169 14.4

B 11000 576 49.2

C 01000 64 5.5

D 10011 361 30.9

1. Select parents for the mating pool

(size of mating pool = population size)

2. Shuffle the mating pool

3. For each consecutive pair apply crossover with probability pc , otherwise copy parents

4. For each offspring apply mutation (bit-flip with probability pm independently for each bit)

5. Replace the whole population with the resulting offspring

SGA reproduction cycle

Simple Example (cont.)• Create next generation of solutions

– Probability of “being a parent” depends on the fitness.

• Ways for parents to create next generation– Reproduction

• Use a string again unmodified.– Crossover

• Cut and paste portions of one string to another.– Mutation

• Randomly flip a bit.– COMBINATION of all of the above.

GA operators: 1-point crossover• Choose a random point on the two parents• Split parents at this crossover point• Create children by exchanging tails• Pc typically in range (0.6, 0.9)

N-point Crossover

• Choose n random crossover points

• Split along those points

• Glue parts, alternating between parents

• Generalisation of 1 point (still some positional bias)

GA operators: mutation

• Alter each gene independently with a probability pm

• pm is called the mutation rate

– Typically between 1/pop_size and 1/ chromosome_length

SGA operators: Selection

• Main idea: better individuals get higher chance– Chances proportional to fitness– Implementation: roulette wheel technique

» Assign to each individual a part of the roulette wheel» Spin the wheel n times to select n individuals

A C

1/6 = 17%

3/6 = 50%

B

2/6 = 33%

fitness(A) = 3

fitness(B) = 1

fitness(C) = 2

The Basic Genetic Algorithm

• [Start] Generate random population of n chromosomes (suitable solutions for the problem)

• [Fitness] Evaluate the fitness f(x) of each chromosome x in the population

• [New population] Create a new population by repeating following steps until the new population is complete

– [Selection] Select two parent chromosomes from a population according to their fitness (the better fitness, the bigger chance to be selected)

– [Crossover] With a crossover probability cross over the parents to form new offspring (children). If no crossover was performed, offspring is the exact copy of parents.

– [Mutation] With a mutation probability mutate new offspring at each locus (position in chromosome).

– [Accepting] Place new offspring in the new population

• [Replace] Use new generated population for a further run of the algorithm

• [Test] If the end condition is satisfied, stop, and return the best solution in current population

• [Loop] Go to step 2

Thanks