4. NSGA II and MOPSO algorithms for solving CEED problem

Several Evolutionary Multi-objective (EMO) algorithms like NSGAII, MOPSO, SPEA 2 (Strength Pareto Evolutionary Algorithm), GDE 3 (Generalized Differential Equation) have been designed and used in solving numerous complex real word problems involving two or more objectives. All these algorithms can find the multiple Pareto-optimal solutions in a single run. Out of all these available algorithms, two of the widely used reliable methods for solving bi-objective optimization problems are the NSGA II and MOPSO. This section provides the review of these two EMO algorithms.

NSGA II was proposed in [22] as an improvement of the NSGA proposed in [25]. This NSGA II algorithm was the revised version of NSGA to overcome the following criticisms:


The NSGA II algorithm is very efficient for solving multi-objective optimization problems since it incorporates an efficient elitism preserving technique using non-domination sorting. The population is ranked based on non-domination sorting before the selection is performed. All non-dominated individuals are classified into one category. Another layer of non-dominated individuals are considered after the group of classified individuals are ignored. This process is continued until all individuals in the population are classified. NSGA II also uses a mechanism for preserving the diversity and spread of the solutions without specifying any additional parameters (NSGA uses fitness sharing). This crowding distance operator guides the selection process towards a uniformly spread out Pareto-optimal front. The NSGA II algorithm for solving the CEED problem is stated below:

	- The total demand of the power system Pd
	- Fuel cost and emission coefficients for each generating unit
	- B matrix coefficients for transmission loss calculations
	- Number of decision variables nVar
	- Lower bounds of the decision variables VarMin
	- Upper bounds of the decision variables VarMax
	- Population Size nPop

The main purpose of this constraint handling mechanism is to increase the flexibility and diversity of the algorithm and to make sure that the candidate solution generated at any point

Several Evolutionary Multi-objective (EMO) algorithms like NSGAII, MOPSO, SPEA 2 (Strength Pareto Evolutionary Algorithm), GDE 3 (Generalized Differential Equation) have been designed and used in solving numerous complex real word problems involving two or more objectives. All these algorithms can find the multiple Pareto-optimal solutions in a single run. Out of all these available algorithms, two of the widely used reliable methods for solving bi-objective optimization problems are the NSGA II and MOPSO. This section provides the review of these

NSGA II was proposed in [22] as an improvement of the NSGA proposed in [25]. This NSGA II

The NSGA II algorithm is very efficient for solving multi-objective optimization problems since it incorporates an efficient elitism preserving technique using non-domination sorting. The population is ranked based on non-domination sorting before the selection is performed. All non-dominated individuals are classified into one category. Another layer of non-dominated individuals are considered after the group of classified individuals are ignored. This process is continued until all individuals in the population are classified. NSGA II also uses a mechanism for preserving the diversity and spread of the solutions without specifying any additional parameters (NSGA uses fitness sharing). This crowding distance operator guides the selection process towards a uniformly spread out Pareto-optimal front. The NSGA II algorithm for

algorithm was the revised version of NSGA to overcome the following criticisms:

• Computational complexity associated with non-dominated sorting.

• Lack of maintaining diversity among obtained solutions.

4. NSGA II and MOPSO algorithms for solving CEED problem

of the algorithm always lies within the decision space.

88 Particle Swarm Optimization with Applications

two EMO algorithms.

• Lack of elite-preserving strategy.

solving the CEED problem is stated below:

• Specify the parameters for the CEED problem

• Number of decision variables nVar

• Specify the parameters for NSGA II Algorithm

• Population Size nPop

• The total demand of the power system Pd

• Fuel cost and emission coefficients for each generating unit

• B matrix coefficients for transmission loss calculations

• Lower bounds of the decision variables VarMin • Upper bounds of the decision variables VarMax

	- Evaluate the fuel cost objective function E and emission objective function F
	- Create offspring population
		- Selection, Crossover and Mutation
	- Apply Constraint Handling Mechanism
	- Evaluate the fuel cost objective function E and emission objective function F
	- Merge the parent and offspring population
	- Perform non domination sorting
	- Calculate crowding distance and rank based on non-domination fronts
	- Select solutions
		- Each front is filled in ascending order
		- Last front-descending order of crowding distance
	- Store the non-dominated solutions in list Ϝ<sup>1</sup>
	- Plot the non-dominated solutions in list Ϝ<sup>1</sup>
	- Increment generation count

In order to handle multiple objectives Pareto dominance is incorporated into PSO algorithm and the MOPSO algorithm is proposed in [23]. The algorithm proposed in [23] uses an external repository of particles to keep a record of the non-dominated vectors found along the search process. At each generation, for each particle in the swarm, by using Roulette wheel selection, a leader is selected from the external repository. This leader then guides other particles towards better regions of the search space by modifying the flight of the particles. A special mutation operator is applied to the particles of the swarm and also to the range of each design variable of the problem to be solved to improve the explorative behavior of the algorithm. The value of the mutation operator is decreased during the iteration. To produce well spread Pareto fronts the MOPSO algorithm in [23] uses an adaptive grid. The MOPSO algorithm for solving the CEED problem is stated below:

• For each generation do

• Select leader from external repository

• Update particle position and velocity

• Apply Constraint Handling Mechanism

• Apply Constraint Handling Mechanism

• Determine Domination

• Update grid and grid index

• Modify inertia weight

• If repository is full delete members

• Plot the members in the external repository

• Update pBest

• End for

• End for

• Apply Mutation and calculate new solutions

• Add non dominated particles to the repository

• Determine domination of new repository members

• Keep only the non-dominated members in the repository

• Evaluate the fuel cost objective function E and emission objective function F

Solution of Combined Economic Emission Dispatch Problem with Valve-Point Effect Using Hybrid NSGA II-MOPSO

http://dx.doi.org/10.5772/intechopen.72807

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5. Hybrid NSGA II and MOPSO algorithm for solving CEED problem

The mechanism of the proposed hybrid approach for solving the CEED problem is to integrate the desirable features of NSGA II (retaining the elitism feature) and MOPSO (exploitation capability) while curbing the individual flaws (NSGAII––does not have an efficient feedback mechanism, PSO overutilization of resources). The mechanism to explore the search space differs in both the algorithms. GA uses mutation and crossover operators which will enhance the exploration task of the hybrid algorithm. The particles in PSO are influenced by their own knowledge and information shared among swarm members. PSO enhances the exploitation task of the hybrid algorithm by finding better solutions from the good ones by searching the neighborhood of good solutions. In this hybrid algorithm at every generation, the Pareto dominance of the population is computed and based on these values non dominated sorting is performed [19]. In order to avoid premature convergence, the elite upper half of the population are enhanced by NSGA II algorithm while the lower

• For each particle do
