Juan Miguel Lujano Rojas - Genetic Optimization Techniques for Sizing and Management of Modern Power Systems

Heuristic techniques; Particle swarm optimization; Economic dispatch; Generator prohibited operating zones; Generator valve-point loading

With the progress of science and technology, investigators from different fields have created a vast family of heuristics based on natural processes. These optimization techniques have been used in the solution of many problems in electrical engineering. In this section, we briefly describe the working principles of some of the most important algorithms. Most of the mathematical formulations presented in this section are used for illustrative purposes without following any strict format.

A popular method known as the ant system () is inspired by the behavior and aptitudes of ant colonies. Ant colony optimization (ACO) is a population-based algorithm with versatility and robustness. It can be applied to many combinatory problems with only a limited number of changes in its structure.

In the ACO, each ant is considered as an agent participating in the search process. Pheromone trails inspire information exchange among the agents. The ants use these trails in the colony to share information about the shortest route to the food source. Thus, an ant on the optimal path follows a strong pheromone trail left by the rest of the colony, while an ant following a weak trail moves randomly. It is essential to highlight that an ant on a determined route also leaves its own pheromone trail on the way. The accumulative quantity of pheromones on a specific route determines how attractive this route is for the colony.

The intensity of the trail (ij) laid by the ants when they travel from a point i to j at a determined time t after n iterations of the algorithm (every n iterations the ants complete a tour) is shown in Eq. ):

ijt+n=ijt+ij

(1.1)

where is a parameter used to model the evaporation of the trail once the ants have completed a tour. This means between the intervals t and t + n. Furthermore, ij is the cumulative amount of pheromones laid by the group between the steps t and t + n.

Regarding the agent model, this is very basic. Each ant or agent makes its next movement according to a probability determined by the distance to the next stage of the route and the trail left by other ants that previously stopped at that stage. Even when the mathematical model is simple, the work of a massive number of agents in a collective and consolidated manner makes ACO a powerful optimization tool. A positive feedback loop mathematically represents the cumulative effect of a pheromone trail over a specific route. The probability of following this route increases with the number of ants that have chosen it.

Although natural ant colonies inspire the algorithm, it employs artificial ones. Artificial ant colonies live in an environment with discrete time, have limited memory, and are not blind. Additionally, the feasibility of each route or solution to the optimization problem is forced using a tabu list.

As previously stated, the probability of an ant transitioning from one stage to another one depends essentially on the pheromone trail and the distance. To this end, the decision maker (DM) adjusts several parameters related to the trail, the visibility of the next stage, the persistence of the trail, and the amount of pheromone left by the other ants.

Eq. ) shows the probability (pijk(t)) of moving from a point i to j at time t for the ant k. This is the expression for valid transitions, which are included using a tabu list. The parameters and are used to represent the importance of the pheromone trail and route visibility (ij).

pijkt=ijtijkiktik