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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