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# Multi-Objective Optimization Techniques to Solve the Economic Emission Load Dispatch Problem Using Various Heuristic and Metaheuristic Algorithms

By Jorge de Almeida Brito Júnior, Marcus Vinicius Alves Nunes, Manoel Henrique Reis Nascimento, Jandecy Cabral Leite, Jorge Laureano Moya Rodriguez, Carlos Alberto Oliveira de Freitas, Milton Fonseca Júnior, Edson Farias de Oliveira, David Barbosa de Alencar, Nadime Mustafa Moraes, Tirso Lorenzo Reyes Carvajal and Haroldo Melo de Oliveira

Submitted: October 20th 2017Reviewed: March 21st 2018Published: July 18th 2018

DOI: 10.5772/intechopen.76666

## Abstract

The main objective of thermoelectric power plants is to meet the power demand with the lowest fuel cost and emission levels of pollutant and greenhouse gas emissions, considering the operational restrictions of the power plant. Optimization techniques have been widely used to solve engineering problems as in this case with the objective of minimizing the cost and the pollution damages. Heuristic and metaheuristic algorithms have been extensively studied and used to successfully solve this multi-objective problem. This chapter, several optimization techniques (simulated annealing, ant lion, dragonfly, NSGA II, and differential evolution) are analyzed and their application to economic-emission load dispatch (EELD) is also discussed. In addition, a comparison of all approaches and its results are offered through a case study.

### Keywords

• optimization techniques
• heuristic
• metaheuristic algorithms
• power plants

## 1. Introduction

The main problem of economic-emission load dispatch (EELD) is to reduce the emission level and total cost of generation at the same time accomplishing the demand for electricity from the power plant. Thermal power plants are among the maximum significant sources of contamination by sulfur dioxide (SO2), carbon dioxide (CO2), and nitrogen oxides (NOx), which create atmospheric pollution [1].

Some authors developed three approaches to solve the EELD problem [2, 3, 4]. The first one is using a single objective, considering emissions and pollution as restrictions with permissible limits [5]; the second one combines cost and emission functions into a single objective function with different weights, where cost and emission are minimized simultaneously [2]; and the third one uses the separated cost and emission functions in a multi-objective optimization [1, 6].

The solution of EELD problem is to minimize the total cost of fuel consumption and carbon emissions [7], considering power demand and operational restrictions [8]. Several techniques such as particle swarm optimization [9, 10], linear programming [11, 12], ant colony optimization [13], biogeography-based optimization [14], genetic algorithms (GA) [15, 16], Tabu search algorithm [17], simulated annealing (SA) [1], neural networks [18], differential evolution (DE) [19], harmony search algorithm [20], Lagrange functions [7], and others [19] have been used to fix the problem of EELD. In spite that all of them have been used, few are used with cost and emission functions in a multi-objective optimization.

This chapter, six multi-objective optimization approaches to solve the EELD problem are going to be presented, and a brief comparison among them is done.

In addition, the “shutdown” of the most inefficient generators is included in all multi-objective optimization approaches used [1, 8, 21].

## 2. Materials and methods

To solve a problem of EELD, two important objectives in an electrical thermal power system must be considered; they are environmental and economy impacts [22].

The parameters of objective functions are determined by curve fitting techniques based on tests of engine performance.

The multi-objective optimization problem is defined as:

E1

where F1(P) and F2(P) are the objective functions to be minimized over the set of permissible decision vector P, as follow in the next Subsections 2.1 and 2.2.

### 2.1. Cost function (F1)

The fuel cost is considered as an essential criterion for economics analysis in thermal power plants. The cost function of each generator can be assumed to be approximated by a quadratic function of generator power output Pi [8, 22]:

E2

where ai, bi, and ci are the fuel cost coefficients of the ith unit generating, n the number of generators, and Pi the active power of each generator (Table 2) .

### 2.2. Economic emission function (F2)

Emissions can be represented by a function that links emissions with power generated by each unit [23]. The emission function in ton/h, which normally represents the emission of SO2 and NOx, is a function of the power output of the generator, and it can be expressed as follows [1]:

E3

where di, ei, and fi are emission coefficients.

### 2.3. Economical load dispatch constrains

The restrictions used in the problem were of three types as follows.

#### 2.3.1. Equality power balance constrain

The real power of each generator is limited by the lower and upper limits. The following equation is the equality restriction of power balance [24, 25]:

E4

where Pi is the output power of each i generator, PD is the load demand, and PL are transmission losses; in other words, the total power generation has to meet the total demand PD and the actual power losses in transmission lines PL, i.e.

E5

The calculation of power losses PL involves the solution of the load flow problem, which has equality constraints in the active and reactive power on each bar as follows [26]:

E6

A simplification is applied to model the transmission losses, setting them as a function of the generator output through Kron’s loss coefficient derivatives of the Kron formula for losses [21]:

E7

where Bij, B0i, and B00 are the energy loss coefficients in the transmission network and n is the number of generators. A reasonable accuracy can be obtained when the actual operating conditions are close to the base case, where the B coefficients were obtained [26].

#### 2.3.2. Production capacity constraint

The power capacity total generated from each generator is restricted by the lower limit and by the upper limit, so the constrain is [1]

E8

where Pi is the output power of the i generator, Pmin.i, is the minimal power of the i generator, and Pmax.i, the maximal power of the i generator.

#### 2.3.3. Fuel delivery constraint

At each time interval, the amount of fuel supplied to all units must be less than or equal to the fuel supplied by the seller, i.e., the fuel delivered to each unit in each interval should be within its lower limit Fmin,i and its upper limit Fmax,i so that [21].

E9

where Fi,m is the fuel supplied to the engine i at the interval m, Fi,min is the minimum amount of fuel supplied to i generator, and Fmax,i is the maximum amount of fuel supplied to i generator.

## 3. Multi-objective optimization techniques to solve EELD

There are a lot of multi-objective optimization approaches that can be used to solve the EELD problem [19], but implementations need to be made to solve EELD problem. On this chapter six metaheuristic algorithm methods that already have the implementation are going to be presented.

### 3.1. Simulated annealing (SA)

SA is a powerful optimization technique which exploits the resemblance between a minimization process and the cooling of molten metal [1].

The physical annealing process is simulated in the SA technique for the determination of global or near-global optimum solutions for optimization problems. In this algorithm a parameter T, called temperature, is defined [27].

On Figure 1, the simulated annealing algorithm is shown.

Starting from a high temperature, a molten metal is cooled slowly until it is solidified at a low temperature. The iteration number in the SA technique is analogous to the temperature level. In each iteration, a candidate solution is generated. If this solution is a better solution, it will be accepted and used to generate yet another candidate solution. If it is a deteriorated solution, the solution will be accepted when its probability of acceptance Pr (Δ) as given in Eq. (10) is greater than a randomly generated number between 0 and 1 [27]:

E10

where Δ is the amount of deterioration between the new and the current solutions and T is the temperature at which the new solution is generated. Accepting deteriorated solutions in the above manner enables the SA solution to “jump” out of the local optimum solution points and to seek the global optimum solution [1]. The last accepted candidate solution is then taken as the starting solution for the generation of candidate solutions in the next iteration. The reduction of the temperature in successive iterations is governed by the following geometric function:

E11

where v is the iteration number and r is temperature reduction factor. T0 is the initial temperature; its value can be set arbitrarily or estimated [29]. A multi-objective simulated annealing optimization to fix a generation dispatch problem is analyzed in [30], and used too in [1], but with the implementation of turning off the most inefficient generators, and this information will be used to compare with the new methods presented in this chapter.

### 3.2. NSGA II

The Non-dominated Sorting Genetic Algorithm II (NSGA II) has been one of the algorithms most used to solve the multi-objective optimization (MOO) problems in general and EELD problem in particular [31, 32, 33]. It was proposed by Deb et al. in 2002 [34]. NSGA II uses a faster selection, an elitist preservation approach, and a less segmented operating parameter [35]. The operation of NSGA II is shown in Figure 2.

#### 3.2.1. Initial population

Initially, a parent random population is created. The population is ordered based on non-domination. For each solution, an aptitude equal to its level of non-domination is assigned (1 is the best level). Thus, a minimization of aptitude or fitness is assumed. Tournament selection, recombination, and mutation operators are used to create N size descendant populations [36].

The population is initialized based on the problem range and constraints, if any. This population is initialized and ordered based on non-domination.

#### 3.2.2. Selection

In the original NSGA II, the selection by binary tournament (BTS) is used, where tournament is disputed between two solutions and the best one is selected and placed in the crossing tank. Two other solutions are taken again, and another gap in the cross junction is filled. It is performed in such a way that each solution can participate in two tournaments exactly [35].

#### 3.2.3. Crossover

The NSGA II uses the simulated binary crossover (SBX), which works with two parent solutions and generates two descendants. The step-by-step procedure is described in [35].

#### 3.2.4. Mutation

NSGA II uses polynomial mutation, which transforms each solution separately, that is, a stock solution gives offspring after being mutated. The mathematical formulation can be described as in [37].

#### 3.2.5. Crowded tournament selection

An estimate obtained of the density of solutions close to a particular solution i in the population, the average between the two solutions on both sides of solution i together with each one of the targets is taken. This amount is the Crowding distance [35].

The NSGA II presents an improved technique for the maintenance of the diversity of solutions, proposing a method of crowding classification distance. However, in its case the classification technique remained unchanged from its previous version.

In NSGA II, all individuals are classified into combined populations (parents and descendants) based on the Pareto dominance ratio and then classified into several layers based on the front to which an individual is located. At each front, the individuals are arranged in descending order of magnitude of the crowding distance value. In the binary tournament selection process, the algorithm first selects an individual positioned on a better non-dominated front. In cases where the individuals with an identical front are compared, the tournament selection operator chooses the winner based on the crowding distance value [38].

The elitist strategy of NSGA II considers all combinations of non-dominated population solutions as the candidates for solutions to the next generation. If the number of non-dominated solutions is less than the size of the population, they are all maintained as the next-generation solutions. Otherwise, candidates for the next generation are selected by the criterion of crowding distance. This criterion has the advantage of maintaining the diversity of solutions in the population in order to avoid premature convergence [38, 39].

### 3.3. Dragonfly

Recently new stochastic optimization technique was developed by Mirjaili in 2016 [40] named dragonfly optimizer. Dragonflies (Odonata) are fancy insects. There are nearly 3000 different species of this insect around the world. A dragonfly’s lifecycle includes two main milestones: nymph and adult. They spend the major portion of their lifespan in nymph, and they undergo metamorphism to become adult [41].

On Table 1 and Figure 3, the characteristics of dragonfly algorithm (DA) are shown.

General algorithmDragonfly algorithm
Decision variableDragonfly’s position in each dimension
SolutionDragonfly’s position
Old solutionOld position of a dragonfly
New solutionNew position of a dragonfly
Best solutionAny dragonfly with the best fitness
Fitness functionDistance from the food source, the predator, center of the swarm, velocity matching, and collision avoidance
Initial solutionRandomly selected position of a dragonfly
Selection
Process of generating new solutionFlying with a specific velocity and direction

### Table 1.

Characteristics of the DA.

Source: [42].

### Table 5.

Comparative table with all costs and emissions.

The comparison of all the results of the different metaheuristic algorithms is provided in Table 6.

### Table 6.

Comparison between all results of the different programmed algorithms.

## 5. Conclusion

The model was implemented in MATLAB computing. Usually all algorithms have presented good results, because in all cases it is possible to switch off at least two motors, but some differences among results of the different algorithms were evidenced. It was possible to notice vantages in relation to cost and emission of two methods among all of them. The SA achieves the lowest emission of pollutants, while DE obtained the lowest cost for the power plant of ten generating units.

## Acknowledgments

The Institute of Technology and Education “Galileo” from Amazonia (ITEGAM), the Federal University of Para (UFPA), the Research Support Foundation State of Amazonas (FAPEAM), and the National Council of Research (CNPq) Productivity of Research Funds Process 301105/2016-2 for the financial support to this research.

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Jorge de Almeida Brito Júnior, Marcus Vinicius Alves Nunes, Manoel Henrique Reis Nascimento, Jandecy Cabral Leite, Jorge Laureano Moya Rodriguez, Carlos Alberto Oliveira de Freitas, Milton Fonseca Júnior, Edson Farias de Oliveira, David Barbosa de Alencar, Nadime Mustafa Moraes, Tirso Lorenzo Reyes Carvajal and Haroldo Melo de Oliveira (July 18th 2018). Multi-Objective Optimization Techniques to Solve the Economic Emission Load Dispatch Problem Using Various Heuristic and Metaheuristic Algorithms, Optimization and Control of Electrical Machines, Abdel Ghani Aissaoui, Ahmed Tahour and Ilhami Colak, IntechOpen, DOI: 10.5772/intechopen.76666. Available from:

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