2. Formulation of combined economic emission dispatch (CEED) problem

The combined economic emission dispatch problem has two conflicting objectives. The first objective can be stated as determining the optimal power generation schedule from a set of online generating units to satisfy the load demand subject to several physical and operational constraints to minimize the fuel cost. The second objective can be stated as determining the optimal power generation schedule from a set of online generating units to satisfy the load demand to minimize the pollutant emissions produced by the generating units. Both the conflicting objectives have to be minimized at the same time because operating the system with minimum cost will result in higher emission and considering only the minimum environmental impact is not practical which results in high production cost of the system. This section formulates the objective functions of the CEED problem along with equality and inequality constraints to maintain rigorous standards to meet the practical requirements of the power system. The goal of this chapter is to find the Pareto-optimal solutions of the CEED problem which minimize both these objectives subject to constraints. The mathematical formulation is as follows.

## 2.1. Objective functions of CEED problem

These objectives are desirable to obtain great economic benefit [1] and to reduce the nitrogen oxide (NOx), sulfur oxide (SOx) and carbon dioxide (CO2) pollutants which cause harmful effect on human beings [2]. These conflicting objectives can be formulated as a multi-objective combined economic emission dispatch (CEED) problem. This CEED problem can be solved using traditional mathematical programming techniques such as lambda iteration, gradient search [1] and can also be solved using modern heuristics optimization techniques. The numerous advantages of solving the CEED problem using heuristic optimization methods compared to the traditional mathematical programming techniques are they are population-based, do not require any derivative information, do not use gradient information in search process, use stochastic operators in search process, they are simple to implement and flexible, have inbuilt parallel

A single optimal solution cannot be obtained for a multi-objective CEED problem which simultaneously minimizes the conflicting objectives of fuel cost and emission. Thus the simultaneous minimization of conflicting objectives in a multi-objective optimization problem (MOP) gives rise to a set of tradeoff solution called as Pareto-optimal (PO) solutions [3] which needs further processing to arrive at a single preferred solution. In literature domination based framework using multi-objective evolutionary algorithms (MOEA) which simultaneously minimizes the fuel cost and emission have been employed to solve the CEED problem. These population-based approaches can obtain the multiple non dominated solutions in a single simulation run. These non-dominated solutions portray the tradeoff between fuel cost and emission objectives of CEED problem. Modern meta-heuristic optimization algorithms like Genetic Algorithm [4, 5], Biogeography Based Optimization [6], Particle Swarm Optimization [7], Bacterial Foraging Algorithm [8], Scatter Search [9], Teaching Learning Based Optimization [10], Differential Evolution [11] and Harmony Search Algorithm [12] have been developed and successfully implemented to solve this complex, highly nonlinear, non-convex CEED problem.

The multiple objective CEED problem can also be transformed into a single objective problem using a weighted sum approach and h parameter values. The h parameters are used to overcome the dimensionality problem when combining multi-objectives and the converted single objective problem is then solved using evolutionary algorithms [13–15]. Another technique to solve CEED problem without the h parameter is to normalize the fuel cost and emission components [6] and solve the single objective function using evolutionary algorithms (EA). In these approaches for the chosen value of weights will give one particular PO solution at a time. However, the disadvantage of these methods is that it requires multiple runs to find the set of PO solutions.

Each evolutionary algorithm has its own characteristics and merits; therefore it is natural to think of integrating these different algorithms to handle a complex problem like CEED. In the research field of Evolutionary Algorithms merging of two or more optimization algorithms into a single framework is called hybridization. In [16–21] hybrid multi-objective optimization algorithms have been successfully applied to solve CEED, various complex engineering problems, and standard test functions. The results indicate that the hybrid algorithms are effective, can exchange elite knowledge within the hybrid framework, can do parallel processing, can improve the exploration and exploitation capabilities and can yield more favorable perfor-

mance than any single algorithm.

architecture and they are scalable and are also computationally quick.

82 Particle Swarm Optimization with Applications

The general formulation for a multi-objective optimization problem (MOOP) is to minimize the number of objective functions simultaneously. A general mathematical model is represented as follows [21]:

$$\text{Minimize}\,f(\mathbf{x}) = \left\{f\_1(\mathbf{x}), f\_2(\mathbf{x}), \dotsm f\_m(\mathbf{x})\right\}, \mathbf{x} \in \mathcal{D} \tag{1}$$

where ai, bi, ci represent the cost coefficients of the generator i. ei and f <sup>i</sup> are coefficients to model

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

The second objective function f <sup>2</sup>ð Þx is an emission function which takes into account the major pollutants caused by the fossil fuel fired power plants. The main pollutants from the power plants are the sulfur oxides and nitrogen oxides. The sulfur oxide emissions are proportional to the fuel consumed by the power plants and have the same form as that of the fuel cost function

The nitrogen oxides emissions are difficult to evaluate as the nitrogen is available in air and also in the fuel. The production of nitrogen gas is related to boiler temperature and air content. The modeling of the nitrogen oxides consists of straight lines and exponential terms. The

The total emission function is obtained by adding the coefficients of (8) and (9) which gives the combination of the mixture of sulfur oxides and nitrogen oxides pollutants [7]. The total

The total emission function given by (10) has a quadratic term and an exponential term which

generator i. The modeling of the emission function is very important because according to the Amendments of the Clean Air Act regulatory agencies might decide to limit power plant

In order ensure that the total real power generation exactly match with the total load demand Pd and transmission loss Pl in the system a power balance equality constraint given in (11) should

The transmission losses in the power network are function of Pg and can be represented using

BijPgj <sup>þ</sup>X<sup>n</sup>

i¼1

Si,<sup>1</sup> <sup>þ</sup> Si,2Pgi <sup>þ</sup> Si, <sup>3</sup>Pg<sup>2</sup>

Ni,<sup>1</sup> þ Ni, <sup>2</sup>Pgi þ Ni, <sup>3</sup>e

Pgi þ γ<sup>i</sup>

, γi , ηi

� � <sup>þ</sup> <sup>η</sup><sup>i</sup>

Pg<sup>2</sup> i

e δiPgi � � (10)

Pgi � Pd � Pl ¼ 0 (11)

B0iPgi þ B<sup>00</sup> (12)

<sup>i</sup> (8)

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85

Ni,4Pgi (9)

, δ<sup>i</sup> are the emission coefficients of the

given by (6). The sulfur oxide emission function can be stated as follows [7].

i¼1

i¼1

<sup>10</sup>�<sup>2</sup> <sup>α</sup><sup>i</sup> <sup>þ</sup> <sup>β</sup><sup>i</sup>

emission in the areas where there are high concentrations of harmful contaminants.

h xð Þ¼ <sup>X</sup><sup>n</sup>

i¼1

Xn j¼1 Pgi

<sup>f</sup> <sup>2</sup>,soð Þ¼ Pg <sup>X</sup><sup>n</sup>

nitrogen oxides emission function can be stated as follows

<sup>f</sup> <sup>2</sup>ð Þ¼ Pg <sup>X</sup><sup>n</sup>

2.2. Active power balance equality constraint and bounds

B-matrix coefficients (Kron's loss formula [1]) as follows

Pl Pg ð Þ¼ <sup>X</sup><sup>n</sup>

i¼1

makes the function highly nonlinear. In (10) αi, β<sup>i</sup>

emission function can be stated as follows

be satisfied.

<sup>f</sup> <sup>2</sup>,Noð Þ¼ Pg <sup>X</sup><sup>n</sup>

i¼1

the effect of valve point of the generator i.

where f xð Þ represents the vector of objectives and f <sup>i</sup> ð Þx , i ¼ 1, 2, ⋯, m is a scalar decision variable which maps decision variable <sup>x</sup> into objective space <sup>f</sup> <sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>n</sup> ! <sup>R</sup>: The n-dimensional variable x is restricted to lie in a feasible region D which is constrained by j in-equality constraint and k equality constraint, i.e.

$$\mathcal{D} = \left\{ \mathbf{x} : \mathbf{g}\_j(\mathbf{x}) \le 0, h\_k(\mathbf{x}) = 0, j = 1, 2, \cdots \mathbf{j}; k = 1, 2, \cdots, K \right\} \tag{2}$$

The decision variable x can be written more suitably as

$$\mathbf{x} = [\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \dots, \mathbf{x}\_n]^T \tag{3}$$

where T is the transposition of the column vector to the row vector. The decision variables are restricted to take a value within a lower x ð Þ min <sup>i</sup> and upper x ð Þ max <sup>i</sup> bounds. These bounds are called the decision space [3].

In MO CEED problem the number of objectives m ¼ 2.The mathematical model of CEED is represented as follows:

$$\text{Minimize}\,f(\mathbf{x}) = \{f\_1(\mathbf{x}), f\_2(\mathbf{x})\}, \mathbf{x} \in \mathcal{D} \tag{4}$$

subject to power balance equality constraints h xð Þ and bounds. The function f <sup>1</sup>ð Þx represents the minimization of total fuel cost function and the function f <sup>2</sup>ð Þx represents the minimization of the emissions from the fossil fuel fired plants. The decision variable x consists of the real power generation of the n generating units and can be written as

$$\mathbf{x} = \begin{bmatrix} P\mathbf{g}\_1, P\mathbf{g}\_2, P\mathbf{g}\_3, \dots, P\mathbf{g}\_n \end{bmatrix}^T \tag{5}$$

where Pgi is the real power output of the i th generator.

Power plants commonly have multiple valves that are used to control the power output of the units. In a practical generating unit, when steam admission valves in thermal units are first opened, a sudden increase in losses is registered which results in ripples in the cost function. In order to model these ripples accurately, sinusoidal functions are added to the quadratic cost function [24]. The resulting cost function contains higher order nonlinearity and makes the problem non-differentiable and non-convex. Hence there are two versions of the fuel cost function, the quadratic function represented by f <sup>1</sup>ð Þx and the combination of quadratic and a sinusoidal (valve-point) function represented by f <sup>1</sup>,Vð Þx . The two versions of the fuel cost functions are given below

$$\_1f\_1(\text{Pg}) = \sum\_{i=1}^n a\_i \text{Pg}\_i^2 + b\_i \text{Pg}\_i + c\_i \tag{6}$$

$$f\_{1,V}(\mathbf{Pg}) = \sum\_{i=1}^{n} a\_i \mathbf{Pg}\_i^2 + b\_i \mathbf{Pg}\_i + c\_i + \left| e\_i \sin \left\{ f\_i \left( \mathbf{Pg}\_i^{\min} - \mathbf{Pg}\_i \right) \right\} \right| \tag{7}$$

where ai, bi, ci represent the cost coefficients of the generator i. ei and f <sup>i</sup> are coefficients to model the effect of valve point of the generator i.

Minimize f xð Þ¼ <sup>f</sup> <sup>1</sup>ð Þ<sup>x</sup> ; <sup>f</sup> <sup>2</sup>ð Þ<sup>x</sup> ; <sup>⋯</sup><sup>f</sup> <sup>m</sup>ð Þ<sup>x</sup> � �, x<sup>∈</sup> <sup>D</sup> (1)

<sup>x</sup> <sup>¼</sup> ½ � <sup>x</sup>1; <sup>x</sup>2; <sup>x</sup>3; <sup>⋯</sup>; xn <sup>T</sup> (3)

ð Þ max

Minimize f xð Þ¼ <sup>f</sup> <sup>1</sup>ð Þ<sup>x</sup> ; <sup>f</sup> <sup>2</sup>ð Þ<sup>x</sup> � �, x <sup>∈</sup> <sup>D</sup> (4)

� �<sup>T</sup> (5)

<sup>i</sup> þ biPgi þ ci (6)

� (7)

<sup>i</sup> � Pgi � � � � � � �

variable which maps decision variable <sup>x</sup> into objective space <sup>f</sup> <sup>i</sup> <sup>¼</sup> <sup>R</sup><sup>n</sup> ! <sup>R</sup>: The n-dimensional variable x is restricted to lie in a feasible region D which is constrained by j in-equality

where T is the transposition of the column vector to the row vector. The decision variables are

In MO CEED problem the number of objectives m ¼ 2.The mathematical model of CEED is

subject to power balance equality constraints h xð Þ and bounds. The function f <sup>1</sup>ð Þx represents the minimization of total fuel cost function and the function f <sup>2</sup>ð Þx represents the minimization of the emissions from the fossil fuel fired plants. The decision variable x consists of the real

x ¼ Pg1; Pg2; Pg3; ⋯; Pgn

th generator.

Power plants commonly have multiple valves that are used to control the power output of the units. In a practical generating unit, when steam admission valves in thermal units are first opened, a sudden increase in losses is registered which results in ripples in the cost function. In order to model these ripples accurately, sinusoidal functions are added to the quadratic cost function [24]. The resulting cost function contains higher order nonlinearity and makes the problem non-differentiable and non-convex. Hence there are two versions of the fuel cost function, the quadratic function represented by f <sup>1</sup>ð Þx and the combination of quadratic and a sinusoidal (valve-point) function represented by f <sup>1</sup>,Vð Þx . The two versions of the fuel cost

<sup>f</sup> <sup>1</sup>ð Þ¼ Pg <sup>X</sup><sup>n</sup>

i¼1

aiPg<sup>2</sup>

<sup>i</sup> <sup>þ</sup> biPgi <sup>þ</sup> ci <sup>þ</sup> ei sin <sup>f</sup> <sup>i</sup> Pgmin

ð Þ min

ð Þx ≤ 0; hkð Þ¼ x 0; j ¼ 1; 2; ⋯J; k ¼ 1; 2; ⋯;K n o

<sup>i</sup> and upper x

ð Þx , i ¼ 1, 2, ⋯, m is a scalar decision

<sup>i</sup> bounds. These bounds are

(2)

where f xð Þ represents the vector of objectives and f <sup>i</sup>

D ¼ x : gj

restricted to take a value within a lower x

where Pgi is the real power output of the i

<sup>f</sup> <sup>1</sup>,Vð Þ¼ Pg <sup>X</sup><sup>n</sup>

i¼1

aiPg<sup>2</sup>

functions are given below

called the decision space [3].

represented as follows:

The decision variable x can be written more suitably as

power generation of the n generating units and can be written as

constraint and k equality constraint, i.e.

84 Particle Swarm Optimization with Applications

The second objective function f <sup>2</sup>ð Þx is an emission function which takes into account the major pollutants caused by the fossil fuel fired power plants. The main pollutants from the power plants are the sulfur oxides and nitrogen oxides. The sulfur oxide emissions are proportional to the fuel consumed by the power plants and have the same form as that of the fuel cost function given by (6). The sulfur oxide emission function can be stated as follows [7].

$$f\_{2,so}(\text{Pg}) = \sum\_{i=1}^{n} \text{S}\_{i,1} + \text{S}\_{i,2}\text{Pg}\_i + \text{S}\_{i,3}\text{Pg}\_i^2 \tag{8}$$

The nitrogen oxides emissions are difficult to evaluate as the nitrogen is available in air and also in the fuel. The production of nitrogen gas is related to boiler temperature and air content. The modeling of the nitrogen oxides consists of straight lines and exponential terms. The nitrogen oxides emission function can be stated as follows

$$f\_{\mathbf{2},\mathbf{No}}(\mathbf{Pg}) = \sum\_{i=1}^{n} N\_{i,1} + N\_{i,2} P \mathbf{g}\_i + N\_{i,3} e^{N\_{i,4} P \mathbf{g}\_i} \tag{9}$$

The total emission function is obtained by adding the coefficients of (8) and (9) which gives the combination of the mixture of sulfur oxides and nitrogen oxides pollutants [7]. The total emission function can be stated as follows

$$f\_2(Pg) = \sum\_{i=1}^{n} \left[ 10^{-2} \left( \alpha\_i + \beta\_i Pg\_i + \gamma\_i Pg\_i^2 \right) + \eta\_i e^{\delta\_i Pg\_i} \right] \tag{10}$$

The total emission function given by (10) has a quadratic term and an exponential term which makes the function highly nonlinear. In (10) αi, β<sup>i</sup> , γi , ηi , δ<sup>i</sup> are the emission coefficients of the generator i. The modeling of the emission function is very important because according to the Amendments of the Clean Air Act regulatory agencies might decide to limit power plant emission in the areas where there are high concentrations of harmful contaminants.

#### 2.2. Active power balance equality constraint and bounds

In order ensure that the total real power generation exactly match with the total load demand Pd and transmission loss Pl in the system a power balance equality constraint given in (11) should be satisfied.

$$h(\mathbf{x}) = \sum\_{i=1}^{n} P \mathbf{g}\_i - P d - P l = 0 \tag{11}$$

The transmission losses in the power network are function of Pg and can be represented using B-matrix coefficients (Kron's loss formula [1]) as follows

$$\text{Pl}(\text{Pg}) = \sum\_{i=1}^{n} \sum\_{j=1}^{n} \text{Pg}\_{i} \text{B}\_{\text{i}} \text{Pg}\_{j} + \sum\_{i=1}^{n} \text{B}\_{0i} \text{Pg}\_{i} + \text{B}\_{00} \tag{12}$$

where Bij, B0i, B<sup>00</sup> are transmission loss coefficients. There are instances in literature where the power losses in the system is neglected and the power balance equation given by (11) is curtailed as follows

$$h(\mathbf{x}) = \sum\_{i=1}^{n} P \mathbf{g}\_i - P d = 0 \tag{13}$$

k

th variable) <sup>n</sup>

the following steps.

P�1 i¼1 Pgi

evolutionary algorithm.

Pgi <sup>¼</sup> Pgmin

BkkPg<sup>2</sup>

X j∈ α<sup>n</sup> Pgi

Pd <sup>þ</sup> <sup>X</sup> i∈ α<sup>n</sup>

0 @

usually set as 0:001 p:u:

Step 5. Recalculate Pgi using Eq. (16).

Step 7. Stop the constraint handling procedure.

<sup>k</sup> þ 2

outside the bounds the value of Pgk is set equal to Pgmin

candidate solution will always lie within the decision space.

X i∈ α<sup>n</sup>

BijPgj <sup>þ</sup> <sup>X</sup>

i∈ α<sup>n</sup>

integer which lies in the range of 1½ � ; n

from the power demand Pd. If the power transmission losses are consid-

<sup>i</sup> � Pgmax i

� �; i∈α<sup>n</sup> (16)

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<sup>i</sup> ; i∈ αn. Regulate the

(17)

87

i¼1 Pgi .

<sup>i</sup> ; i∈ αn.

ered, to determine Pgk and to maintain the equality constraint becomes hard. It is done using

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

Step 1. Update the variables belonging to the set α<sup>n</sup> by normal optimization process of an

Here rand is a uniformly distributed random number in the range of ½ � 0; 1 . The set α<sup>n</sup> contains all the integers in the range 1½ � ; n except k, where k is a randomly generated

Step 2. If updating of the variables is carried out using any other technique then regulate the

Step 3. Obtain the value of the <sup>k</sup>th variable of the candidate solution Pgk by solving the following quadratic equation (17) whose coefficients are associated with the variables belonging to the set α<sup>n</sup> and the transmission loss coefficients [7]. To improve the potential candidate solution and also to improve the flexibility and diversity of the optimization algorithm the value of k is randomly generated integer between 1 and n.

> BkiPgi þ B0<sup>k</sup> � 1 !

Out of the two roots of the quadratic equation (17), one root will be selected as the value of the variable Pgk using the following procedure. If both the roots of the quadratic equation lie within the bounds then the root which has the minimum value is selected. If only one root lies within the bounds, this root is selected as the value of Pgk and the other root which lies outside the bounds is neglected. If both the roots lay

Step 4. Calculate the residue PRD by subtracting the total system demand Pd and the total system transmission loss Pl from the sum of the total power generation <sup>P</sup><sup>n</sup>

Step 6. Repeat step 3, step 4 and step 5 until j j PRD < tol. This step will ensure that the

If j j PRD < tol, then go to step 7; otherwise go to step 5. Here, tol is the demand tolerance

<sup>B</sup>0iPgi � <sup>X</sup>

i ∈α<sup>n</sup>

<sup>k</sup> .

Pgkþ

Pgi þ B<sup>00</sup>

1 A ¼ 0

<sup>i</sup> � rand<sup>∗</sup> Pgmin

updated variables which violate the lower bounds as Pgi <sup>¼</sup> Pgmin

updated variables which violate the upper bounds as Pgi <sup>¼</sup> Pgmax

The above equations given by (11) and (13) are most common form of power balance equation found in the literature.

The power output of each generator i should lie within its minimum limit (Pgmin <sup>i</sup> ) and maximum limit (Pgmax <sup>i</sup> ) given by

$$P\mathbf{g}\_i^{\min} \le P\mathbf{g}\_i \le P\mathbf{g}\_i^{\max}; \mathbf{i} = \mathbf{1}, \mathbf{2}, \mathbf{3} \dots, \mathbf{n} \tag{14}$$

## 2.3. Combined economic emission dispatch

The purpose of the CEED problem is to determine the Pareto-optimal real power generation vector <sup>x</sup><sup>∗</sup> <sup>¼</sup> Pg<sup>∗</sup> <sup>1</sup>; Pg<sup>∗</sup> <sup>2</sup>; Pg<sup>∗</sup> <sup>3</sup>; ⋯; Pg<sup>∗</sup> n � �<sup>T</sup> that minimize the two conflicting objective given by (7) and (10) while satisfying the real power equality constraint given by (11) and the bounds given by (14). The bi-objective CEED problem can be formulated as

$$\text{Minimize}\,f = \begin{bmatrix} f\_{1,V}(P\text{g}), f\_2(P\text{g}) \end{bmatrix} \tag{15}$$

In MO CEED problem, the economic and emission objectives will conflict with each other and is not possible to satisfy them simultaneously. There is no way of improving these objectives without degrading at least one of these objectives and the resulting set of non-dominated solutions thus obtained are called Pareto-optimal set. The objective function values of all elements in the PO set in the objective space constitute the Pareto front. When the sufficient number of PO solutions is available for the CEED problem then it is possible to find a convex curve containing these solutions to produce the Pareto front. The two main goals of MO CEED problem:


## 3. Constraint handling mechanism

At any stage of the algorithm whenever a new population is being generated it is very important to make sure that the population lies within the decision space. While solving the CEED problem this implies that the population should satisfy the equality constraints and bounds. If the transmission losses are neglected than the k th variable of the candidate solution Pgk can be calculated by subtracting the sum of the power generations (excluding the k th variable) <sup>n</sup> P�1 i¼1 Pgi from the power demand Pd. If the power transmission losses are considered, to determine Pgk and to maintain the equality constraint becomes hard. It is done using

Step 1. Update the variables belonging to the set α<sup>n</sup> by normal optimization process of an evolutionary algorithm.

$$P\mathbf{g}\_i = P\mathbf{g}\_i^{\min} - \mathbf{rand} \ast \left( P\mathbf{g}\_i^{\min} - P\mathbf{g}\_i^{\max} \right); i \in \alpha\_n \tag{16}$$

Here rand is a uniformly distributed random number in the range of ½ � 0; 1 . The set α<sup>n</sup> contains all the integers in the range 1½ � ; n except k, where k is a randomly generated integer which lies in the range of 1½ � ; n


$$B\_{kl}P\mathcal{g}\_k^2 + \left(2\sum\_{i\in a\_n} B\_{kl}P\mathcal{g}\_i + B\_{0k} - 1\right)P\mathcal{g}\_k +$$

$$\left(Pd + \sum\_{i\in a\_n} \sum\_{j\in a\_n} P\mathcal{g}\_i B\_{ij} P\mathcal{g}\_j + \sum\_{i\in a\_n} B\_{0i}P\mathcal{g}\_i - \sum\_{i\in a\_n} P\mathcal{g}\_i + B\_{00}\right) = 0$$

Out of the two roots of the quadratic equation (17), one root will be selected as the value of the variable Pgk using the following procedure. If both the roots of the quadratic equation lie within the bounds then the root which has the minimum value is selected. If only one root lies within the bounds, this root is selected as the value of Pgk and the other root which lies outside the bounds is neglected. If both the roots lay outside the bounds the value of Pgk is set equal to Pgmin <sup>k</sup> .


the following steps.

where Bij, B0i, B<sup>00</sup> are transmission loss coefficients. There are instances in literature where the power losses in the system is neglected and the power balance equation given by (11) is

Pgi � Pd ¼ 0 (13)

<sup>i</sup> ; i ¼ 1, 2, 3⋯, n (14)

<sup>i</sup> ) and maxi-

(15)

th variable of the candidate

h xð Þ¼ <sup>X</sup><sup>n</sup>

The power output of each generator i should lie within its minimum limit (Pgmin

<sup>i</sup> <sup>≤</sup> Pgi <sup>≤</sup> Pgmax

Pgmin

<sup>3</sup>; ⋯; Pg<sup>∗</sup> n

by (14). The bi-objective CEED problem can be formulated as

i¼1

The above equations given by (11) and (13) are most common form of power balance equation

The purpose of the CEED problem is to determine the Pareto-optimal real power generation

and (10) while satisfying the real power equality constraint given by (11) and the bounds given

Minimize f ¼ f <sup>1</sup>,Vð Þ Pg ; f <sup>2</sup>ð Þ Pg

In MO CEED problem, the economic and emission objectives will conflict with each other and is not possible to satisfy them simultaneously. There is no way of improving these objectives without degrading at least one of these objectives and the resulting set of non-dominated solutions thus obtained are called Pareto-optimal set. The objective function values of all elements in the PO set in the objective space constitute the Pareto front. When the sufficient number of PO solutions is available for the CEED problem then it is possible to find a convex curve containing these solutions to produce the Pareto front. The two main goals of MO CEED

2. Find a wide spread of non-dominated solutions to represent the entire range of the Pareto-

At any stage of the algorithm whenever a new population is being generated it is very important to make sure that the population lies within the decision space. While solving the CEED problem this implies that the population should satisfy the equality constraints

solution Pgk can be calculated by subtracting the sum of the power generations (excluding the

1. Find a set of non-dominated solutions which lie on the Pareto-optimal front

and bounds. If the transmission losses are neglected than the k

� �<sup>T</sup> that minimize the two conflicting objective given by (7)

h i

curtailed as follows

86 Particle Swarm Optimization with Applications

found in the literature.

<sup>i</sup> ) given by

2.3. Combined economic emission dispatch

<sup>1</sup>; Pg<sup>∗</sup> <sup>2</sup>; Pg<sup>∗</sup>

mum limit (Pgmax

vector <sup>x</sup><sup>∗</sup> <sup>¼</sup> Pg<sup>∗</sup>

problem:

optimal front.

3. Constraint handling mechanism


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 of the algorithm always lies within the decision space.

• Maximum number of iteration MaxIt

• Generate a random nPop size population

posed in Section 3 is carried out.

• Once the random population is initialized the Constraint Handling Mechanism pro-

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

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89

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

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

• Calculate crowding distance and rank based on non-domination fronts

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

• Calculate Crowding Distance and rank the population based on Non Dominated fronts

• Crossover Percentage pCrossover • Mutation Percentage pMutation

• Mutation rate mu

• Initialize Population

• Mutation step size sigma

• Evaluate the objective functions

• Perform Non Domination Sorting

• Create offspring population

• Selection, Crossover and Mutation

• Merge the parent and offspring population

• Each front is filled in ascending order

• 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

• Last front-descending order of crowding distance

• Perform non domination sorting

• Apply Constraint Handling Mechanism

• For each generation do

• Select solutions

• End for
