Novel Heuristic Optimization Technique to Solve Economic Load Dispatch and Economic Emission Load Dispatch Problems

Abstract

The fundamental objective of economic load dispatch is to operate the available generating units such that the needed load demand satisfies the lowest generation cost and also complies with the various constraints. With proper power system operation planning using optimized generation limits, it is possible to reduce the cost of power generation. To fulfill the needs of such objectives, proper planning and economic load dispatch can help to plan the operation of the electrical power system. To optimize the economic load dispatch problems, various classical and new evolutionary optimization approaches have been used in research articles. Classical optimization techniques are outdated due to many limitations and are also unable to provide a global solution to the ELD problem. This work uses a new variant of particle swarm optimization techniques called modified particle swarm optimization, which is effective and efficient at finding optimum solutions for single as well as multi-objective economic load dispatch problems. The proposed MPSO is used to solve single and multi-objective problems. This work considers constraints like power balance and power generation limits. The proposed techniques are tested for three different case studies of ELD and EELD problems. (1) The first case is tested using the data of 13 generating unit systems along with the valve point loading effect; (2) the second case is tested using 15 generating unit systems along with the ramp rate limits; and (3) the third case is tested using the economic emission dispatch (EELD) as a multi-objective problem for 6 generating unit systems. The outcomes of the suggested procedures are contrasted with those of alternative optimization methods. The results show that the suggested strategy is efficient and produces superior optimization outcomes than existing optimization techniques.

1. Introduction

Since fuel resources (coal) are limited but the demand for electrical power is increasing, part of this research is to find the best solution to reduce the cost of power generation and reduce emissions so that the global warming problem also reduces. The above-discussed issues are the main motivation for considering this topic in our work [1].

The primary objective of the proposed study is to identify the ideal producing point where the power production system will function and fuel costs may be reduced. If the generating stations operate at the optimum generation point, the generation cost should be minimal for any particular load demand. The second motive is to reduce emissions; if the burning of fuel definitely reduces, the emission of toxic gases also decreases, reducing the pollution level [2,3].

Various classical approaches are listed for finding the optimum solution to the ELD problem, such as linear programming (LP) [3], pattern search PS [4], quadratic programming (QP) [5], etc. Classical methods have many drawbacks, like the LP and QP methods, which need linear mathematical objectives. Pattern search methods are sensitive to changes in the parameters, and the DS (direct search) method’s performance for large data is not good.

The objective of the ELD problem is to maintain the generating output power according to the load demand and reduce the cost of power generation [1]. ELD problems are also associated with constraints like power demand, generation limits, line losses, etc. Due to the presence of such constraints, economic load dispatch is a nonlinear computation problem. So, the optimization of such a problem requires efficient evolutionary techniques [2].

Some classical and heuristic methods are not effective for solving nonlinear optimization problems because they have many limitations and take a long time to compute. Most of the modern heuristic techniques are more effective and provide the best solution to the problem [6]. Heuristic optimizations techniques are the newest optimization techniques, as explained in various articles, and are very efficient, simple, and fast at computation. Such techniques effectively optimize the ELD data within seconds and cannot deviate in the direction of a local solution [7].

The latest optimization techniques have many features; they can optimize big data and provide a global solution without deviation from the constraints. These methods are unaffected by initial conditions or changes in the variables [8]. Optimization is the task of finding the most suitable solution to a problem. The optimization process starts randomly to find the solution of the objective function (constraints are also defined) in the search space within the defined limits and with a number of agents, or swarms, and finally finds the solution to the problem, which is the global optimum solution, by exploring as small a set of solutions as possible [9].

The proposed work considered a modified PSO (MPSO), which is the latest variant of particle swarm optimization and is very effective for single objective (ELD) and multi-objective (EELD) problems. This is a new variant of the PSO that has an attraction factor that helps the particle move in the direction of a solution; it controls the movement of particles in such a way that they cannot deviate from the search area.

Other variants of the PSO need to update the velocity and position of the particle continuously until they can find a solution to the problem. This work proposes a new modified PSO, where the velocity of the particles is controlled by the attraction factor in such a way that they cannot stop during the iteration process; hence, there is no need to update the velocity of the particles; only the position of the particles is required to be updated. That is the big advantage of the proposed MPSO. Hence, the speed of computation increases and the particles are also not moved to the local solution. The MPSO always provides a global solution to the proposed ELD and EELD problems.

This paper’s remaining sections are structured as follows: the second part demonstrates the works published by the researchers. Here, we considered the last 20 years’ papers for the study. The third part of the article presents the mathematical formulation of the economic load dispatch and economic emission dispatch. The fourth section of the article presents the mathematical model and algorithm of the new PSO. The fifth section shows the different case studies and the results of the new PSO compared with other optimization techniques. In the last section, the conclusion of the work is provided.

2. Literature Review

The economic load dispatch is formulated and optimized with various constraints using different optimization techniques to find out the minimum generation cost and fulfill the load demand. This section discusses the various optimization techniques used for finding the optimum solution to the ELD problem.

To address the ELD issues, a biogeography-based optimization technique was proposed. The recommended approach primarily uses the two processes of migration and mutation to obtain the global optimum [1]. A new evolutionary technique is suggested for the optimization of multi-objective ELD problems [2].

The linear programming method is used to obtain the real and reactive power of the electrical generation system; however, such methods take a long computation time and are sometimes unable to provide a global solution for large data sets [3].

The pattern search method was proposed for finding the optimum solution to the ELD problem and the valve loading effects were also considered. The proposed algorithm was tested on various test data for validation of the results and compared with other optimization techniques [4]. Quadratic programming was used for the solution of the ELD problem; this consists of a DC load flow and network security constraints [5]. The PSO techniques used the ELD problem along with transmission losses, dynamic operation limits, and restricted operating zones [6].

Discrete and continuous ELD problem solutions are obtained by using a new variant of the PSO (EPSO). In this article, the ELD problem is formulated with penalties and constraints [7].

Article [8] solved the ELD problem, including the ramp rate limits and the transient stability, by using a novel particle swarm optimization called the chaotic quantum-behaved PSO algorithm.

The authors of [9] considered class-topping optimization techniques that are based on human intelligence. They solve the ELD problem as well as the CEED using four different case studies.

Article [10] proposes QPSO and HQPSO to solve the ELD problem. They solve a case ELD problem consisting of data from 13 generating unit systems along with the valve loading effect, and the results are compared to classical PSO.

To resolve the pollution and economic dispatch issues, a new combined genetic-tabu search technique is provided. The suggested technique is designed such that a basic GA serves as the first search to guide the search towards the ideal region [11].

Article [12] optimized the ELD problem along with the valve loading using the EPSO. The research also includes constraints like the prohibited operating zone.

The QPSO technique was used to find the optimum solution to the ELD problem. The suggested PSO particles move in the search area with good speed and explore the solution quickly [13]. Two test cases of 10 and 15 generating units were taken and optimized for the ELD problem using the PPSO [14]. Article [15] used a combination of differential evolution and a PSO called quantum particle swarm optimization (QPSO) as a solution for economic load dispatch optimization.

An artificial bee colony approach is discussed in the article to resolve the economic dispatch problem [16]. The suggested technique is used to solve the economic dispatch problem along with the power balance and network loss constraints.

The BAT method is demonstrated for the optimization of multi-objective problems. This work considered the economic emission dispatch as a multi-objective problem and found the overall generation cost of the plant [17]. The Chameleon Swarm Algorithm is proposed for the minimization of emissions as well as the generation cost of the power plant [18].

The ELD problem with a valve-point effect is solved using a hybrid method [19]. A novel evolutionary optimization approach is proposed to resolve the economic load dispatch issue. The standard harmony search random selection process is replaced with a mutation process based on wavelet theory to increase the performance of the suggested strategy [20].

An optimization technique that is based on the teaching-learning approach is used to solve ELD problems without taking transmission losses into account. The suggested solution can handle an ELD while taking a nonlinearity, such as valve point loading, into account [21]. A single objective ELD problem along with a prohibited zone and ramp rate and a multi-objective ELD problem are optimized using C andGPSO [22].

An effective evolutionary technique is using the novel adaptive particle swarm optimization for a limited ED issue [23]. The differential evolution algorithm was investigated for the purpose of resolving power system economic load dispatch issues. Five ELD issues with various features were utilized to test the proposed methods [24]. A contemporary search and rescue optimization approach that is motivated by the behavior of humans during search and rescue operations was used to solve the combined emission and economic dispatch [25].

To overcome the ELD difficulties, an enhanced arithmetic optimization is suggested. Two crucial variables in the original AOA are math optimizer acceleration and math optimizer probability [26]. To optimize multi-objective ELDs, data mining technology is used. The proposed techniques provide the best decision for the sample test data [27].

The “Grey Wolf” approach is taken to solve the challenging nonconvex ELD optimization problems effectively and reliably. The grey wolf optimizer algorithm’s search agents are helped by the sine and cosine functions in such a way that the solution cannot move to the local optima [28].

Harris Hawks’ techniques are used to address the issues of economic load dispatch. The Harris Hawks Optimizer is used to find the number of possible solutions in the search space; each of these regions can be thoroughly searched for the best local solution via adaptive hill climbing [29].

An optimization algorithm based on the theory of demand and supply was used to optimize the ELD model. The proposed algorithm efficiently moves the particles in the search space between the local and global searches [30].

To address the complex economic load dispatch issue, a new hybrid grey wolf optimization method with a strong learning mechanism has been developed [31]. The foraging optimization method is based on the Spiral Foraging approach given for the solution of the ELD problem. Spiral foraging strategies improve global search capability and convergence velocity [32]. A multi-objective ELD problem was formulated using a wind energy system and a distributed generation system [33].

A hybrid generation system can help reduce the cost of power generation as well as carbon emissions [34]. An adaptive multi-population-based differential evolutionary technique is proposed that improves ELD global solutions [35].

A diffusion model with heterogeneous coefficients and a generic nonlinear incidence rate for brucellosis was used for the solution of economic load dispatch [36]. For the solution of economic load dispatch with emissions subject to power balance and generation limit constraints, a new particle swarm optimization technique was proposed [37]. The model proposed is a nonlinear distributed delayed periodic AG-ecosystem with competition on the time scales. Our approach integrates and generalizes the discrete and continuous situations in terms of the time scale [38].

The Bumble Bee Mating Optimization algorithm is used for economic load dispatch optimization. The proposed BBMO works in three different modes: the queen, the workers, and the drones (males) [39]. A multi-objective optimization method is used in this article for the optimal planning of the distributed generators in electric distribution networks. Reducing the annual cost and network loss, along with improving the reliability of the network, are the main objectives of a multi-objective algorithm [40].

3. Formulation of ELD Problem

The mathematical model for minimizing the cost of generation is given as

$${\mathrm{F}}_{\mathrm{cT}}=\mathrm{Minimize}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{FC}}_{\mathrm{i}}\left({\mathrm{P}}_{\mathrm{i}}\right)$$

(1)

$${\mathrm{FC}}_{\mathrm{i}}\left({\mathrm{P}}_{\mathrm{i}}\right)={\mathrm{a}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}}^2+{\mathrm{b}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}}+{\mathrm{c}}_{\mathrm{i}}$$

(2)

3.1. Formulation of Economic Dispatch along with Valve Point

When using a multi-valve steam turbine system, the fuel cost function of the generating units may vary [4]. The cost of generation increases with the use of multi-valve steam turbines. The generator heat rate curve is the result of the multi-valve steam turbine valve opening operation. The impact of valve-point loading is shown in Figure 1 [22]. In Figure 1, A, B, C, D, and E are the points where the valve can be opened to control the speed of the generator if the load suddenly increases. Due to the opening of the valve, the normal fuel ignition characteristic of the generating plant changes.

The importance of this impact lies in the fact that the huge steam-producing cost curve’s function is, in reality, nonlinear rather than continuous. In reality, the input–output curves for generating units are not a smooth cost function when considering a multi-valve steam operating system. The representation of the cost functions of the generating units is changed by the inclusion of the valve-point loading. Hence, the economic load dispatch cost function is modified when considering the valve point effects [22]; it is mathematically formulated as given in Equation (3).

$${\mathrm{C}}_{\mathrm{iv}}\left({\mathrm{P}}_{\mathrm{i}}\right)={\mathrm{a}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}}^2+{\mathrm{b}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}}+{\mathrm{c}}_{\mathrm{i}}+\left|{\mathrm{E}}_{\mathrm{i}}\ast \sin ({\mathrm{F}}_{\mathrm{i}}\left({\mathrm{P}}_{\mathrm{i}}^{\min }-{\mathrm{P}}_{\mathrm{i}}\right)\right|$$

(3)

3.2. Formulation of the ELD with a Ramp Rate Limit

When generating units are operated online, the actual operating ranges depend on the ramp rate limits [14]. The operation of the generating units at the different intervals is shown in Figure 2. The steady-state condition of generation is shown in Figure 2a; when generation increases, it is represented in Figure 2b; Figure 2c shown when generation decreases.

When power generation increases, as shown in Figure 2b, it is represented as

$${\mathrm{P}}_{\mathrm{i}}\left(\mathrm{t}\right)+{\mathrm{P}}_{\mathrm{i}}\left(\mathrm{t}-1\right)\le {\mathrm{UR}}_{\mathrm{i}}$$

(4)

and when power generation decreases, as shown in Figure 2c, it is given as

$${\mathrm{P}}_{\mathrm{i}}\left(\mathrm{t}-1\right)-{\mathrm{P}}_{\mathrm{i}}\left(\mathrm{t}\right)\ge {\mathrm{DR}}_{\mathrm{i}}$$

(5)

Now the economic load dispatch, with an up rate and down rate of power, is defined as

$${\mathrm{P}}_{\min }=\mathrm{Max}\left[\right({\mathrm{P}}_{\mathrm{i}}^{\min },{\mathrm{P}}_{\mathrm{i}}\left(\mathrm{t}-1\right)-{\mathrm{DR}}_{\mathrm{i}}\left)\right]$$

(6)

$${\mathrm{P}}_{\max }=\mathrm{Min}\left[\right({\mathrm{P}}_{\mathrm{i}}^{\max },{\mathrm{P}}_{\mathrm{i}}\left(\mathrm{t}-1\right)-{\mathrm{UR}}_{\mathrm{i}}\left)\right]$$

(7)

$${\mathrm{P}}_{\mathrm{i},\min }\le {\mathrm{P}}_{\mathrm{i}}\le {\mathrm{P}}_{\mathrm{i},\max }$$

(8)

where UR_i and DR_i are the upper and down ramp rate limits, respectively.

3.3. Formulation of the EELD Multi-Objective Problem

When generating power in a thermal power plant, coal is used as fuel; due to the burning of coal, toxic gases like COx, NOx, and SOx are emitted. Such toxic gases not only pollute the environment but also affect human health. So, reducing the emission of gases is a great achievement of the ELD solution. So, when considering the minimization of generation costs and environmental emissions, it is formulated as a multi-objective EELD (economic emission load dispatch) problem [2].

The first objective is the minimization of the cost function already discussed, given in Equations (1) and (2); the minimization of the environmental emission considered as the second objective is given as

$${\mathrm{EC}}_{\mathrm{T}}=\mathrm{Minimize}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{FE}}_{\mathrm{i}}\left({\mathrm{P}}_{\mathrm{i}}\right)$$

(9)

$${\mathrm{FE}}_{\mathrm{i}}\left({\mathrm{P}}_{\mathrm{i}}\right)={\mathrm{f}}_{\mathrm{i}}+{\mathrm{e}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}}+{\mathrm{d}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}}^2$$

(10)

To obtain the economic emission dispatch, both objectives are combined using the penalty factor given in Equation (11). The ratio of the power produced by the plant to the actual power requirement of the load being satisfied after line loss is known as the penalty factor (h_i)

$$\frac{{\mathrm{Fc}}_{\mathrm{i}}\left({\mathrm{P}}_{\mathrm{i}.\max }\right)}{{\mathrm{FE}}_{\mathrm{i}}\left({\mathrm{P}}_{\mathrm{i}.\max }\right)}={\mathrm{h}}_{\mathrm{i}}$$

(11)

where

$${\mathrm{FC}}_{\mathrm{i}}\left({\mathrm{P}}_{\mathrm{i}.\max }\right)={\mathrm{a}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}.\max }^2+{\mathrm{b}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}.\max }+{\mathrm{c}}_{\mathrm{i}}\mathrm{and}$$

(12)

$${\mathrm{FE}}_{\mathrm{i}}\left({\mathrm{P}}_{\mathrm{i}.\max }\right)={\mathrm{f}}_{\mathrm{i}}+{\mathrm{e}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}.\max }+{\mathrm{d}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}.\max }^2$$

(13)

$${\mathrm{E}}_{\mathrm{T}}={\mathrm{FC}}_{\mathrm{T}}+{\mathrm{h}}_{\mathrm{i}}{\ast \mathrm{FE}}_{\mathrm{i}}$$

(14)

where E_T is the total economic emission cost.

3.4. Constraints

When optimized, the ELD problem is required to follow some restrictions or limitations, called constraints. During the formulation of the ELD mathematical model, keep in mind that the constraints should be inserted and then optimized. This work included the following constraints.

3.4.1. Power Balance

The power balancing restrictions make sure that the total amount of power generated by all the different kinds of generating unit is sufficient to satisfy the power load in each time period.

$${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{L}}={\mathrm{P}}_{\mathrm{D}}$$

(15)

3.4.2. Power Limits

The generator’s output power must not exceed its nominal value and must not fall below the level required for the boiler to operate steadily. As a result, the production is capped at a certain minimum and maximum range. The following equation may be used to represent each productivity unit with an expected production in the circuit.

$${\mathrm{P}}_{\min }\le {\mathrm{P}}_{\mathrm{i}}\le {\mathrm{P}}_{\max }$$

(16)

4. Heuristic Optimization Technique

The ELD problem is nonlinear in nature because of the presence of different constraints. For the optimization of such nonlinear ELD problems, classical methods were found to be inefficient, and hence, new heuristic optimization techniques were required. The PSO was first introduced in 1995 [1]. After that, many new variants of PSO were investigated and implemented in optimization problems. The PSO is used for the optimization of various nonlinear problems; however, it has some disadvantages, such as difficulties in premature convergence, lagging performance for large test data, diversity in the local solutions, and the appropriate tuning of its parameters.

This work is considered a modified particle swarm optimization technique. A modified PSO is an effective variant of a PSO and can easily optimize single and multi-objective ELD problems. In the MPSO, we are using an attraction factor as shown in Equation (25) that has the following features:

• It controls the movements of the particles in such a way that they cannot move away from the search area;
• It continuously changes the particles’ positions so the particles cannot stop during the iteration process until the final results are obtained; therefore, there is no need to update the velocity of the particles after every iteration;
• It also attracts the particles so they cannot move to the local solution.

4.1. Initialization of the Swarm

To optimize the proposed technique, the swarm is required to be initialized randomly within the effective real power minimum and maximum limits of the power generated (P_min and P_max) using the generating units, as given in Equation (17), and the velocity of the initialized particles, as given in Equation (18).

$${\mathrm{S}}_{\mathrm{i}}^{\mathrm{k}}={\mathrm{P}}_{\min }+\mathrm{rand}\left({\mathrm{P}}_{\max }-{\mathrm{P}}_{\min }\right)$$

(17)

$${\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}}={\mathrm{V}}_{\min }+\mathrm{rand}\left({\mathrm{V}}_{\max }-{\mathrm{V}}_{\min }\right)$$

(18)

where P_initial and V_initial are the initial values of the swarm’s position and its velocities are randomly generated using a random positive number (rand) between zero and one using MATLAB software when the algorithm is executed.

Now, initialize the limits of the velocity of the particles using Equations (19) and (20).

$${\mathrm{V}}_{\max }=({\mathrm{P}}_{\max }-{\mathrm{P}}_{\min })/10$$

(19)

$${\mathrm{V}}_{\min }=-{\mathrm{V}}_{\max }$$

(20)

4.2. Updating the Velocity of the Particle

The velocity of the particle is updated using the personal best and the global best value of the swarm is given as

$${\mathrm{V}}_{\mathrm{inew}}^{\left(\mathrm{K}+1\right)}={\mathrm{WV}}_{\mathrm{i}}^{\mathrm{K}}+{\mathrm{rand}\ast \mathrm{c}}_1\left({\mathrm{P}}_{\mathrm{best},\mathrm{i}}-{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}\right)+{\mathrm{rand}\ast \mathrm{c}}_2\left({\mathrm{g}}_{\mathrm{best},\mathrm{i}}-{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}\right)$$

(21)

Inertia weight helps to accelerate the particles in the search space and is given as

$$\mathrm{W}={\mathrm{W}}_{\max }-\frac{{\mathrm{W}}_{\max }-{\mathrm{W}}_{\min }}{{\mathrm{iter}}_{\max }}\times \mathrm{iteration}$$

(22)

During the considered optimization, there were 100 iterations: W_max and W_min were considered 0.9 and 0.4, respectively, and c₁ and c₂ were 2.05.

4.3. Updating the Particles’ Positions

This PSO used one attraction factor (S_d) that can attract the particles in the search area and find the global solution to the problem. N is the number of generating units. It is represented as follows.

$${\mathrm{S}}_{\mathrm{d}}={\mathrm{P}}_{\mathrm{best}}\ast \mathrm{rand}+{\mathrm{g}}_{\mathrm{i},\mathrm{best}}\ast \left(1-\mathrm{rand}\right)$$

(23)

where rand is a random variable generated using MATLAB between 0 and 1, P_best is the best value of the particle, and g_i,best is the global value.

The positions of the particles are updated by using the best global position of the particles and an attraction factor, as follows.

$${\mathrm{n}}_{\mathrm{i},\mathrm{best}}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}\frac{{\mathrm{P}}_{\mathrm{i},\mathrm{best}}}{\mathrm{N}}$$

(24)

$$\mathsf{\sigma }=\frac{{\mathrm{R}}_1-{\mathrm{R}}_2}{{\mathrm{R}}_3}$$

(25)

where R₁ and R₂ are random variables within [0, 1] and R₃ within [−1, 1].

$${\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}+1}={\mathrm{S}}_{\mathrm{d}}+\mathsf{\alpha }\ast \mathsf{\sigma }\left({\mathrm{n}}_{\mathrm{i},\mathrm{best}}-{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}\right)$$

(26)

4.4. Algorithm of the Proposed MPSO

⮚

Consider the number of particles;

⮚

Define the number of iterations;

⮚

Initialize the swarm using the minimum and maximum values of the generated power, as given in Equations (17) and (18);

⮚

Initialize the velocity of the particles, as shown in Equations (19) and (20);

⮚

Define the objective function of the ELD;

⮚

Define the constraints;

⮚

Initialize the pbest;

⮚

Select the best global values;

⮚

Update the velocity of the particles;

⮚

Insert an attraction factor;

⮚

Update the position of the particles;

⮚

After completing the iteration, collect the best global values;

⮚

Terminate the algorithm if the objective is fulfilled; otherwise, repeat the process again.

A pseudo code of the new modified PSO is given in Figure 3. The conversance characteristic of the modified PSO (MPSO) is shown in Figure 4. It was obtained between the objective functions (ELD and EELD) and the number of iterations taken. In the entire test case, 100 iterations were considered.

5. Case Study and Result Analysis

All the case studies were run in MATLAB 2022 using a personal laptop, HP P5 8th generation, for the 100 iterations. For each case, 50 trials were considered because after 50 iterations the results were repeated; therefore, only 100 iterations were considered.

5.1. Case 1

For this case study, the test data from 13 generating systems were considered, as shown in

Table 1. Test data for case 1 of 13 generating units for the load demand of 1800 MW.

Gen. Units	$P_{\mathbf{i}}^{\mathbf{m}\mathbf{i}\mathbf{n}}$	${\mathbf{P}}_{\mathbf{i}}^{\mathbf{M}\mathbf{a}\mathbf{x}}$	${\mathbf{c}}_{\mathbf{i} }$	${\mathbf{b}}_{\mathbf{i}}$	${\mathbf{a}}_{\mathbf{i}}$	${\mathbf{E}}_{\mathbf{i}}$	${\mathbf{F}}_{\mathbf{i} }$
1	0	680	550	8.10	0.00028	300	0.035
2	0	360	309	8.10	0.00056	200	0.042
3	0	360	307	8.10	0.00056	150	0.042
4	60	180	240	7.74	0.00324	150	0.063
5	60	180	240	7.74	0.00324	150	0.063
6	60	180	240	7.74	0.00324	150	0.063
7	60	180	240	7.74	0.00324	150	0.063
8	60	180	240	7.74	0.00324	150	0.063
9	60	180	240	7.74	0.00324	150	0.063
10	40	120	126	8.60	0.00284	100	0.084
11	40	120	126	8.60	0.00284	100	0.084
12	55	120	126	8.60	0.00284	100	0.084
13	55	120	126	8.60	0.00284	100	0.084

. This case was analyzed with the valve loading effect (E_i and F_i), for the load demand of 1800 MW, included [10,19,20,21,22]. Here, a_i, b_i, and c_i are the cost coefficients.

The optimum results obtained with the proposed modified PSO are shown in

Table 2. Comparative results of 13 generating unit systems.

Power Output	HS [20]	DE [19]	HQPSO [10]	TLBO [21]	GPSO-w [22]	QPGPSO [22]	HHO [29]	ESCSDO [30]	PSO [36]	MPSO
P1(MW)	628.318	628.317	628.318	364.99	628.3185	628.3185		538.5587		628.29
P2(MW)	149.59	149.24	149.109	227.95	224.3707	223.3356		75.6427		149.68
P3(MW)	222.74	223.168	223.322	217.46	148.7126	298.6696		224.3995		222.76
P4(MW)	109.86	109.85	109.865	95.225	60	109.8547		109.8666		109.88
P5(MW)	60	109.86	109.862	106.67	109.865	60		109.8666		60
P6(MW)	109.87	109.866	109.866	123.54	109.6557	60		109.8666		109.85
P7(MW)	109.87	109.82	109.791	112.53	60	60		109.8666		109.854
P8(MW)	109.86	109.82	60.000	144.22	159.73	109.866		109.8666		109.785
P9(MW)	109.686	60	109.866	126.07	109.5848	60		109.8666		109.899
P10(MW)	40	40	40	60.236	40	40		40		40
P11(MW)	40	40	40	48.475	40	40		77.39996		40
P12(MW)	55	55	55	91.364	55	55		92.39991		55
P13(MW)	55	55	55	81.239	55	55		92.39991		55
Power output (MW)	1800	1800	1800	1800	1800	1800		1800	1800	1800
Total fuel cost (USD/h)	17,963.83	17,963.94	17,963.95	18,141.02	17,978.62	19,971.85	17,986.03	18,028.19	18,205.78	17,962.72
CPU time (s)	13.7	5.42	7.58	31.6	38.4	40	5.7	3.82	77.37	1.8921

. The obtained results are compared with other techniques for the validation of the results and show the effectiveness of the proposed algorithm. The results obtained using the MPSO are compared with the HQPSO [10], DE [19], harmony search HS [20], TLBO [21], QPGPSO [22], GPSO-w [22], HHO [29], and ESCSDO [30]. The proposed MPSO has a minimum generation cost of USD 17,962.72/h and a very low computation time of 0.8921 s.

5.2. Case 2

In this case study, the test data from 15 generating unit systems are considered. This test data system was analyzed with ramp-rate limits. The test data was tested for a load of 2650 MW [14,22]. This system’s test data are shown in

Table 3. Fuel cost coefficients, ramp rate limits and the generation limits of 15 generating unit systems for a load demand of 2650 MW.

No. Unit	${\mathbf{c}}_{\mathbf{i}}$	${\mathbf{b}}_{\mathbf{i}}$	${\mathbf{a}}_{\mathbf{i} }$	${\mathbf{P}}_{\mathbf{i}}^{\mathbf{m}\mathbf{i}\mathbf{n}}$	${\mathbf{P}}_{\mathbf{i}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$	${\mathbf{P}}_{\mathbf{i}}$	$\mathbf{U}{\mathbf{R}}_{\mathbf{i}}$	$\mathbf{D}{\mathbf{R}}_{\mathbf{i}}$
1	671	10.1	0.000299	150	455	400	80	120
2	574	10.2	0.000183	150	455	300	80	120
3	374	8.8	0.001126	20	130	105	130	130
4	374	8.8	0.001126	20	130	100	130	130
5	461	10.4	0.000205	150	470	90	80	120
6	630	10.1	0.000301	135	460	400	80	120
7	548	9.8	0.000364	135	465	350	80	120
8	227	11.2	0.000338	60	300	95	65	100
9	173	11.2	0.000807	25	162	105	60	100
10	175	10.7	0.001203	25	160	110	60	100
11	186	10.2	0.003586	20	80	60	80	80
12	230	9.9	0.005513	20	80	40	80	80
13	225	13.1	0.000371	25	85	30	80	80
14	309	12.1	0.001929	15	55	20	55	55
15	323	12.4	0.004447	15	55	20	55	55

.

The optimum results obtained with the proposed MPSO are listed in

Table 4. Test results of case 2 for 15 generating unit systems.

Power Output (MW)	PPSO [14]	Fuzzy and APSO [23]	GPSO-w [22]	GA [24]	G- SCNHGWO [28]	ESCSDO [30]	PSO [23]	MPSO
P1	455	455	-	-	455	455	454.9999	421.4
P2	455	455	-	-	380	379.9996	454.9999	455
P3	130	130	-	-	130	129.9999	130	130.6
P4	130	130	-	-	130	130	130	131.6
P5	231.05	271.78	-	-	170	170	234.2005	341
P6	460	460	-	-	460	460	460	460
P7	465	465	-	-	430	430	464.9999	465
P8	60	60	-	-	67.9593	70.257	60	70
P9	25	25	-	-	58.0137	59.332	25	21.6
P10	35.5224	25	-	-	159.99	159.9	30.9939	20
P11	74.29	43.41	-	-	80	80	76.7014	20
P12	80	55	-	-	80	80	79.9999	63.2
P13	25	25	-	-	25	25	25	22
P14	15	15	-	-	17.9118	15	15	13.6
P15	15	15	-	-	15	15	15	15
Total Power (MW)	2650	2650	2650	2650	2650	2650	2650	2650
Fuel Cost (USD/h)	32,543.289	32,548.06	32,548.6	33,063.5	32,687.10	32,692.401	32,858.01	32,465.69
CPU time (s)	3.47	8.7	40	33.5	14.92	2.90	11.3	1.87

. The results obtained with the proposed algorithm are compared for validation with recently developed optimization algorithms like the PPSO [14], Fuzzy and APSO [23], GPSO-w [22], GA [24], G-SCNHGWO [28], and ESCSDO [30]. Results shown in the table clearly show the effectiveness of the modified PSO; it performs better and has a minimum generation cost. The minimum fuel cost given by the MPSO is USD 32,465.69/h and the computation time is 0.824 s.

5.3. Case 3

This case study is considered multi-objective test data. In this case, we evaluated the economic load dispatch and environmental emissions. Then, both objectives are combined to find the overall economic emissions dispatch. The test data for this case study are shown in

Table 5. Test data for case 3 for a load demand of 900 MW.

Gen. Unit	${\mathbf{a}}_{\mathbf{i}}$	${\mathbf{b}}_{\mathbf{i}}$	${\mathbf{c}}_{\mathbf{i}}$	${\mathbf{d}}_{\mathbf{i}}$	${\mathbf{e}}_{\mathbf{i}}$	${\mathbf{f}}_{\mathbf{i}}$	${\mathbf{P}}_{\mathbf{i}}^{\mathbf{m}\mathbf{i}\mathbf{n}}$	${\mathbf{P}}_{\mathbf{i}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$
1	0.1525	38.54	756.8	0.0042	0.33	13.86	10	125
2	0.106	46.2	451.4	0.004	0.33	13.9	10	150
3	0.02083	40.159	1049.99	0.00683	−0.55	40.27	35	225
4	0.0356	38.31	1234.5	0.0068	−0.55	40.27	35	210
5	0.0211	36.33	1658.6	0.0046	−0.52	42.7	130	325
6	0.0179	38.27	1356.7	0.0042	−0.52	42.7	125	315

. The case data are tested for a load demand of 900 MW for six generating units along with the emission and fuel cost coefficients.

Now, the six generating units’ data are analyzed for the load demand of 900 MW. The case data are tested with environmental emissions. The results obtained with the proposed modified PSO technique are shown in

Table 6. Results of the six generating units for the economic emission dispatch for a demand of 900 MW.

Output Power	BAT [17]	ABC [16]	RGA [17]	ACB [18]	MPSO
P1(MW)	92.3288	92.3297		92.315	79.4
P2(MW)	98.3910	98.3912		98.3707	99.98
P3(MW)	150.1132	150.1948		150.1997	154.4
P4(MW)	148.586	148.5588		148.5549	145.84
P5(MW)	220.4007	220.4043		220.4051	223.26
P6(MW)	218.1267	218.1307		218.115	224.14
Losses(MW)	28.008975	28.009673	29.725	28.004	27.26
Power output (MW)	928	928	929.725	928.004	927.26
Fuel cost (USD/h)	48,350.163	48,350.683	48,567.7	48,108	47,889.45
Emission (Ton/h)	693.772	693.788	694.19	693.791	669.3217
Total cost(USD/h)	81,527.739	81,529	81,764	81,527	81,489.12
Computation time(s)	11.47	5.28	2.94	3.81	1.72

. The MSPO results were compared, for validation, with the BAT [17], ABC [16], RGA [17], and ACB [18] and the proposed algorithm was found to gives the best results compared to the other techniques, as shown in Table 6. The MPSO gives a minimum generation cost of USD 47,889.45/h, a minimum emission discharge of 669.3217 T/h, and a total economic emission dispatch cost of 81,489.12 for a load demand of 900 MW.

Before combining the fuel cost and emission required values of the penalty factor shown in Equation (11) using the Equations (12) and (13), the penalty factor (hi) is calculated for six generating units, as given in

Table 7. Penalty factor h values.

Generating Units	hi
1	65.9
2	61.6
3	42.4
4	47.89
5	43.64
6	51.3

.

6. Conclusions

Policies are required for the generation of power in such a way that it fulfils load demands, generates the power at a low cost, and emits fewer toxic gases into the environment. ELD is one of the best optimization approaches that can help generate power with a minimum fuel cost. This work proposes a new PSO for the optimization of economic load dispatch as a single objective for thirteen generating units with valve loading effects and for fifteen generating units with ramp rate limits. The performance of the proposed PSO is important in terms of the minimum generation cost and the computation time taken. According to the results shown for both single objective cases, the performance of the proposed new PSO is better than the other optimization techniques; it also has a minimum generation cost and takes less computation time (the results are tabulated in Table 2 and Table 4). Similarly, the proposed new PSO algorithm was tested for a multi-objective economic emission dispatch problem with line losses. In this case, the performance of the proposed algorithm was the best in comparison with the other techniques (the results are tabulated in Table 6). The proposed algorithm was able to provide the minimum generation cost, the minimum emission, and the minimum total cost, with less computation time (generation +emission). Overall, the proposed technique’s performance was the best and it was effective for single as well as multi-objective problems.