Genetic Algorithms

🎯 Learning Objectives

• Understand the basic idea of evolutionary optimization.
• Explain population, chromosome, gene, and fitness.
• Understand selection, crossover, mutation, and elitism.
• Formulate an optimization problem mathematically.
• Understand how fitness is connected to the objective function.
• Follow one complete GA generation step by step.
• Apply Genetic Algorithms to power-system optimization.

1. What is a Genetic Algorithm?

A Genetic Algorithm (GA) is an optimization method inspired by natural evolution.

2. Optimization Problem

A general optimization problem can be written as:

subject to:

where:

• x = decision variables
• J(x) = objective function
• gᵢ(x) = inequality constraints
• hⱼ(x) = equality constraints

3. Population, Chromosome, and Gene

4. Example of a Chromosome

Suppose we want to optimize three variables:

One chromosome may be:

5. Fitness Function

The fitness function measures how good a candidate solution is.

If we are minimizing a positive cost J(x), one simple fitness transformation is:

Therefore:

6. Why Do We Need a Population?

Classical local optimization often follows one solution trajectory. A Genetic Algorithm works with many solutions simultaneously.

7. Initial Population

The first population is usually generated randomly within the allowed bounds:

A random value can be generated as:

where:

8. Selection

Selection determines which chromosomes become parents.

Common selection methods include:

• Roulette-wheel selection
• Tournament selection
• Rank-based selection

9. Roulette-Wheel Selection

Suppose the fitness values are:

Then the selection probability of chromosome i can be:

10. Crossover

Crossover combines information from two parents to produce offspring.

For arithmetic crossover:

where:

11. Simple Crossover Example

Assume:

and:

Then:

12. Mutation

Mutation introduces small random changes into offspring.

where Δ is a small random perturbation.

13. Elitism

Elitism means copying one or more of the best chromosomes directly into the next generation.

14. Complete GA Cycle

15. Stopping Criteria

The algorithm may stop when:

• A maximum number of generations is reached.
• The best objective value is sufficiently small.
• The solution improvement becomes negligible.

16. Handling Constraints

One common approach is to add a penalty when a solution violates constraints:

where P(x) measures the constraint violation.

17. Example Optimization Problem

Consider:

The minimum is clearly:

Assume the initial population is:

18. Exploration vs Exploitation

19. Important GA Parameters

20. Power-System Application: Economic Dispatch

Economic Dispatch determines generator outputs that minimize generation cost.

subject to power balance:

and generator limits:

21. Power-System Application: PV/BESS Sizing

A chromosome may represent:

where:

• PPV = installed photovoltaic capacity
• EBESS = battery energy capacity

An objective could combine investment cost and operating cost:

22. Power-System Application: Optimal Power Flow

In an OPF problem, GA can search for variables such as:

while minimizing an objective such as:

23. MATLAB Concept

A basic GA program follows this logic:

A simplified objective evaluation may look like:

24. Advantages of Genetic Algorithms

• Do not require derivatives of the objective function.
• Can handle nonlinear and nonconvex problems.
• Can work with continuous and discrete variables.
• Search several candidate solutions simultaneously.
• Can be adapted to many engineering optimization problems.

25. Limitations

• Can require many objective-function evaluations.
• Performance depends on parameter tuning.
• No guarantee of finding the exact global optimum in finite time.
• Poor constraint handling can produce infeasible solutions.
• Results may vary between independent runs.

26. Summary

• GA is a population-based evolutionary optimization method.
• A chromosome represents one candidate solution.
• Fitness evaluates the quality of a solution.
• Selection favors better solutions.
• Crossover combines parent information.
• Mutation introduces diversity.
• Elitism preserves the best solutions.
• Penalty functions can be used to handle constraints.
• GA is useful for Economic Dispatch, OPF, and PV/BESS sizing.

📝 Review Questions

1. What is a Genetic Algorithm?
2. What is the difference between a chromosome and a gene?
3. What is the purpose of the fitness function?
4. How does roulette-wheel selection work?
5. Explain arithmetic crossover.
6. Why is mutation necessary?
7. What is elitism?
8. What is the difference between exploration and exploitation?
9. How can penalties be used for constraint handling?
10. How can GA be applied to Economic Dispatch?
11. How can GA be used for PV/BESS sizing?