Investigation on New Metaheuristic Algorithms for Solving Dynamic Combined Economic Environmental Dispatch Problems

Abstract

In this paper, the dynamic combined economic environmental dispatch problems (DCEED) with variable real transmission losses are tackled using four metaheuristics techniques. Due to the consideration of the valve-point loading effects (VPE), DCEED have become a non-smooth and more complex optimization problem. The seagull optimization algorithm (SOA), crow search algorithm (CSA), tunicate swarm algorithm (TSA), and firefly algorithm (FFA), as both nature and biologic phenomena-based algorithms, are investigated to solve DCEED problems. Our proposed algorithms, SOA, TSA, and FFA, were evaluated and applied on the IEEE five-unit test system, and the effectiveness of the proposed CSA approach was applied on two-unit, five-unit, and ten-unit systems by considering VPE. We defined CSA for different objective functions, such as cost of production, emission, and CEED, by considering VPE. The obtained results reveal the efficiency and robustness of the CSA compared to SOA, TSA, FFA, and to other optimization algorithms reported recently in the literature. In addition, Matlab simulation results show the advantages of the proposed approaches for solving DCEED problems.

1. Introduction

A competitive electricity generation market necessitates high concentration to solve the economic load dispatch (ELD) problem of power generation. The essential objective of the ELD is to distribute the production of the available production units in such a way that the load demand is covered with a minimum total cost of fuel, taking into account the physical and operational limitations of the electrical system [1].

Due to these physical and operational limitations, the ELD problem is a non-linear optimization problem [1]. As a result, the conventional mathematical methods fall somewhat short in solving such a problem. On the other hand, with the growing concern about polluting emissions and the total cost of fuel, finding a solution to emissions has become a critical issue for the power generation systems around the world. However, the functions of pollutant emission and operating cost are incompatible; reducing emissions increases operating costs and vice versa [2]. Therefore, a grid operator allocates generation units to the tradeoff between the total operation cost reduction and/or reducing emissions under the newest generations, such as prohibited operating zones [3] and ramp rate limits [4], which have a high order of nonlinearities, discontinuities, dimensions, and valve loading points that determine the characteristics of the ELD problem [5]. Therefore, considering more non-convex and multiple local minima increases the complexity of obtaining the optimal fuel cost function. In contrast, metaheuristic optimization techniques have become common to find the best solution to the EED problem. Such algorithms include a colony optimizer [6], multi-verse optimizer [7], particle swarm optimization [8], gray wolf optimizer [9], biogeography-based optimization [10], enhanced exploratory whale optimization algorithm (EEWOA) [7], and hybrid bat–crow search algorithm (HBACSA); they are used exceptionally, in a unique, improved, or hybrid form with others approaches.

Thermal power plants release toxic gases into the atmosphere, endangering human health. Environmental pollution caused by these chemicals has the potential to harm not only humans but also animals and birds. It also has a negative impact on visibility, material quality, and contributes to global warming. DEED provides a solution to these problems by scheduling a renewable energy source and backup power generation based on the predicted load demand to decrease the operational generator’s cost and emissions [2].

2. Literature Review

According to the importance of DEED, researchers have adapted and proposed these new algorithms to address the DEED problem. In [11], the authors proposed a method based on improved sailfish algorithm to solve the hybrid dynamic economic emission dispatch (HDEED). In addition, the paper in [12] presented a novel multi-objective optimization based on enhanced moth-flame optimization approach to locate the optimal solution of the hybrid DEED, including renewable energy generation. Ref. [13] proposed an improved tunicate swarm method to explore the search space for DEED. The proposed ITSA in [13] was applied to the 5, 10, and 15-unit systems, respectively. Ref. [14] proposed a novel approach, namely a multi-objective differential evolution algorithm, to deal with the constraints in DEED problems by considering the difference in the power generation range of the units. Authors in [15] suggested a multi-objective virus colony search algorithm (MOVCS) for solving the DEED problem in the power system integrated electric vehicle and wind units over a 24 h period. Ref. [16] formulated DEED as a third-order polynomial fitting curve method to balance between emissions gas and fuel cost by utilizing renewable energy and PEVs. In [17], the authors proposed an enhanced exploratory whale optimization algorithm (EEWOA) to solve the dynamic economic dispatch (DED), which coordinates the behavior of whales, random exploration, and local random search. The feasibility of EEWOA is validated on 5, 10, and 15 units considering VPE and power loss constraints.

In the last two years, various research works have focused on solving the DEED problems by using novels metaheuristic techniques. The research in [18] developed a new model of the multi-objective DCEED using a hybridized flower pollination algorithm (FPA) with sequential quadratic programming (SQP). The FPA–SQP is tested on 5 and 10-unit systems for a 24 h period. Ref. [19] suggested a novel multi-objective hybrid optimization-algorithm-based equilibrium optimizer (EO) and differential evolution (DE) to solve the DEED. The proposed algorithm is validated and verified on the test system containing ten thermal power generators and one wind farm. The study showed that the hybrid EO–DE method with a constraint management system is able to counter balance the tasks of exploitation and exploration. The results obtained in [20] show that the new chaotic artificial bee colony (IABC) rides the local trap and improves the convergence of the solution in terms of solving the CEED with different constraints. A 10-unit system without and with a wind farm and a 40-unit system were investigated to prove the effectiveness and accuracy of the suggested IABC for tackling the CEED problem. In [21], the improved slime mold algorithm (ISMA) is developed and applied to optimize the single-and bi-objective economic emission dispatch (EED) problems considering VPE. Five test systems (6-unit, 10-unit, 11-unit, 40-unit, and 110-unit) are used to validate the proposed ISMA. Additionally, ref. [22] presented an optimal allocation of the power system for the combined economic emission dispatch problem using a moth swarm algorithm (MSA). The method was tested on two different systems. The first system is a combination of six thermal generators and thirteen solar plants, and the second test system comprises three thermal units combined with thirteen photovoltaic plants.

This paper proposes, presents, and applies more effective methods, such as, TSA, SOA, CSA, and FFA, for solving the DCEED problem, including VPE. The most important contributions of this article can be summarized as follows:

• Improvement of some optimization methods, such as TSA, SOA, CSA, and FFA.
• Application of the proposed algorithms to solve single-and bi-objective DEED problems.
• The four techniques are validated and tested by applying them on the IEEE standard five-unit test system to demonstrate their robustness and accuracy.

The rest of the article is organized as follows. The DEED problem with the mathematical model is presented in Section 3. Section 4 demonstrates the four metaheuristic approaches: TSA, SOA, CSA, and FA. The investigation of the simulation results and discussion are shown in Section 5. Finally, we conclude this paper in Section 6.

3. DCEED Problem Formulation Including VPE

Due to the dynamic behavior of the electrical network and the prodigious variations in load demand on the consumer side, the DCEED problem can be described as a multi-objective mathematical optimization problem, which is non-linear and dynamic. DCEED is a constraint optimization problem that minimizes simultaneously the fuel cost and emission effects in order to meet a power system’s load demand over some appropriate periods while meeting certain equality and inequality constraints [23].

3.1. Objective Function

DCEELDP’s objective is to minimize total fuel costs while also reducing the level of emissions emitted by generating units. Thus, the objective function is mathematically defined as the weighted summation of the production cost of generating units and emissions caused by fossil fuel thermal plants, which is shown below [16]:

$$\min F={\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{i=1}^{Ng}{C}_{i,t}\left(P_{i,t}\right)+pp{f}_i{\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{i=1}^{Ng}{E}_{i,t}\left(P_{i,t}\right)$$

(1)

where F indicates the single objective to be minimized; $C_{i,t}\left(P_{i,t}\right)$ denotes the fuel cost of $Ng$ generators in the $t^{th}$ (t =1, 2, …, T) time interval in USD/h; $E_{i,t}\left(P_{i,t}\right)$ stands for the emissions generated by the generation stations over T dispatch intervals in kg/h; $P_{i,t}$ denotes the dynamic dispatch power in MW. $pp{f}_i$ is the price penalty factor determined by the ratio of $C_i\left(P^{max}\right)$ and $E_i\left(P^{min}\right)$ in USD/kg.

3.1.1. Dynamic Economic Load Dispatch Model (DED)

The objective of the DED problem is to minimize the overall economic cost of fossil fuel during a 24-h period. In some large generators, their cost functions are also non-linear, due to the effect of the valve opening [24]. Consequently, the valve dynamics increase several local minimum points in the cost function, hence complicating the problem. The DED problem involved with VPE is expressed as minimization of the production cost of power dispatch in the following way [25]:

$$C_{i,t}\left(P_{i,t}\right)=a_i+b_i P_{i,t}+c_i P_{i,t}^2+\left|e_i\times sin\left(f_i\times \left( P_i^{min}-P_i\right)\right)\right|$$

(2)

where $a_i, b_i, \mathrm{and}{c}_i$ are the coefficients of the fuel cost corresponding to the generator i; $e_i$, and $f_i$ stand for the fuel cost coefficients of the $i^{th}$ generator due to VPE; and $P_i^{min}$ denotes the minimum real power of the $i^{th}$(i = 1, 2, …, Ng) generating unit.

3.1.2. Dynamic Environmental Dispatch Model (DEnD)

Global warming and increased movements to protect the environment have forced producers to reduce gas emissions caused by the combustion of fossil fuels in various power plants mainly due to sulfur dioxide (SO₂) and nitrate oxide (NO_x) [26,27].

Each thermal power plant will produce its power according to a dynamic non-smooth emission function given by the following quadratic form [28]:

$$E_{i,t}\left(P_{i,t}\right)={\alpha }_i+{\beta }_i P_{i,t}+{\gamma }_i P_{i,t}^2+{\eta }_i\times exp\left({\delta }_i{P}_{i,t}\right)$$

(3)

where ${\alpha }_i, {\beta }_i, {\gamma }_i, {\eta }_i,\mathrm{and} {\delta }_i$ are the emission curve coefficients.

3.2. Constraints Functions

The minimization of the DCEED problem is subject to the following constraints and limits:

3.2.1. Power Balance Constraint

The sum of total power generated by all generators at each time interval t should be matched with the load demand PD and the total transmission losses $P_L$ in the corresponding time period, which is given as follows [29]:

$${\displaystyle \sum }_{i=1}^{Ng}{P}_{i,t}=P_{D,t}+P_{L,t}$$

(4)

The power losses incurred in the transmission lines can be computed by using Kron’s loss coefficients formula given below [30]:

$$P_{L,t}={\displaystyle \sum }_{i=1}^{Ng}{\displaystyle \sum }_{j=1}^{Ng}{P}_{i,t} B_{i,j}{P}_{j,t}$$

(5)

where $B_{i,j}$ denotes the transmission loss coefficients. Wood et al.in [31] provide detailed procedures for calculating the B coefficients.

3.2.2. Power Output Limits

The dispatch active power outputs of each generator must be between the capacities of each specific generating unit at each time interval t [32]:

$$P_i^{min}\le P_{i,t}\ge P_i^{max}$$

(6)

where $P_i^{min}$ and $P_i^{max}$ indicate, respectively, the minimum and the maximum power limits of $P_{i,t}$.

4. Metaheuristic Approaches Applied to DEED

4.1. Seagull Optimization Algorithm (SOA)

The seagull optimization algorithm (SOA) [33] is a recently proposed metaheuristic technique inspired by the natural behaviors of seagulls. The seagull optimization algorithm is a metaheuristic algorithm that mimics gull behavior. Seagulls live in colonies. Seagulls are omnivorous, eating reptiles, earthworms, insects, fish, and other animals.

Seagulls are highly intelligent birds. This aids the seagull in its hunt for prey. The migratory and hunting habits of seagulls are well known. Seagulls migrate in search of a new plentiful food source. After arriving at a new home, seagulls attack their prey (Figure 1 [33]).

The most important thing about seagulls is their migrating and attacking behavior [34]. Thus, the SOA focuses on these two natural behaviors. This behavior is clarified as follows:

4.1.1. Migration (Exploration)

During migration, the members of a seagull swarm should avoid colliding with each other (see Figure 2a). To achieve this purpose, an additional variable A is employed [35]:

$$\overrightarrow{C_s}=A\times \overrightarrow{P_s}\left(j\right)$$

(7)

where $\overrightarrow{C_s}$ indicates the position of each agent, $\overrightarrow{P_s}\left(j\right)$ denotes the current position of seagulls in the jth iteration (j = 1, 2,…, Max_iteration), and A depicts the movement behavior of seagulls.

$$A=a-\left(j\times \left(\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$Ma{x}_{iteration}$}\right.\right)\right)$$

(8)

where a is a constant and responsible for controlling the frequency of employing variable A, which linearly decreases from a to 0.

To find the richest food resources, seagulls move toward the best search agent as shown in Figure 2b.

$$\overrightarrow{M_s}=B\times \left(\overrightarrow{P_{bs}}\left(j\right)-\overrightarrow{P_s}\left(j\right)\right)$$

(9)

where $\overrightarrow{M_s}$ indicates seagull position $\overrightarrow{P_s}\left(j\right)$ toward the best search agent $\overrightarrow{P_{bs}}\left(j\right)$. The coefficient B is a random value responsible for making a tradeoff between exploitation and exploration and is expressed as follows:

$$B=2\times A^2\times rd$$

(10)

where rd is a random number that lies in the interval (0,1).

As seagulls move toward the fittest search agent, they might remain close to each other (see Figure 2c). Thus, seagulls can update their position according to the following rule [33]:

$$\overrightarrow{D_s}=\left|\overrightarrow{C_s}+\overrightarrow{M_s}\right|$$

(11)

where $\overrightarrow{D_s}$ stands for the distance between seagulls and the best search agent.

4.1.2. Attacking (Exploitation)

Seagulls attack their prey in a spiral shape after arriving at a new place, as presented in Figure 2d. Their attacking behavior in the x, y, and z planes is explained as follows [33]:

$$\left\{\begin{array}{c}{x}^{\prime }=r\times cos\left(k\right)\\ y^{\prime }=r\times sin\left(k\right)\\ z^{\prime }=r\times k \end{array}\right.$$

(12)

$$r=u\times exp\left(kv\right)$$

(13)

where r denotes the radius of each turn of the spiral; k is a random number with [0 ≤ k ≤ 2π]; u and v are constants to define the spiral shape [33,36].

The updated position of the search agent $\overrightarrow{P_s}\left(j\right)$ is determined by using Equations (11)–(13) [36].

$$\overrightarrow{P_s}\left(j\right)=\left(\overrightarrow{D_s}\times x^{\prime }\times y^{\prime }\times z^{\prime }\right)+\overrightarrow{P_{bs}}\left(j\right)$$

(14)

4.2. Crow Search Algorithm (CSA)

Crow search algorithm (CSA) is a nature-inspired algorithm suggested by A. Askarzadeh in 2016 [37]. This evolutionary computation technique based-population algorithm imitates crow birds’ conduct and social interaction [38].

One of the best indications of their cleverness is hiding food and remembering its location. Moreover, the mechanism of exploration and exploitation of CSA can be learned from Figure 3 [39]. As shown in Figure 3a, small values of fl (<1) result in a solution to an optimum local; otherwise, if fl of more than 1is selected, the optimal solution leads in the global search, as shown in Figure 3b. Overall, the pseudo code of CSA can be explained as shown in Algorithm 1.

The CSA process is discussed in this subsection [39]:

Step 1: Initializing crow swarm in the d-dimension randomly.

Step 2: A fitness function is used to evaluate each crow, and its value is put as an initial memory value. Each crow stores its hiding place in its memory variable P_i.

Step 3: The crow updates its position by selecting another random crow, (i.e, Mj) and generating a random value. If this value is greater than the awareness probability “AP”, then crow Pi will follow Pj to know Mj.

Step 4: The crow updates its position by selecting another random crow (i.e., Pj) and following it to know Mj. Then new Pj is calculated as follows:

$$P_i^{t+1}=\left\{\begin{array}{c}{P}_i^t+r_i\times f{l}_i^t\times \left(M_j^t-P_i^t\right)r_j\ge A{P}_i^t\\ a random position otherwise \end{array}\right.$$

(15)

where t refers to iteration number; $r_i$ and $r_j$ refer to random numbers; $f{l}_i^t$ is the crow i flight length to denote crow j memory.

Step 5: Updating memory

The crows’ memory is updated as follows:

$$M_i^{t+1}=\left\{\begin{array}{c}{P}_i^{t+1} f\left(P_i^{t+1}\right)<f\left(M_i^t\right)\\ M_i^t otherwise \end{array}\right.$$

(16)

Step 6: Check termination criterion. Steps 3–5 are repeated until tmax is reached.

Algorithm 1. Crow Search Algorithm

1: Input: N number of crows in the population, and Maximum number of iteration (tmax).

2: Output: Optimal crow position

3: Initialize position of crows, and crows’ memory

4: while t < tmax do

5:   for i= 1:N (all N crows of the flock)

6:  Choose a random crow (i.e. Mj), and determine a value of an awareness probability AP

7:     if  $r_j\ge A{P}_i^t$

8:      $P_i^{t+1}=P_i^t+r_i\times f{l}_i^t\times \left(M_j^t-P_i^t\right)$

9:      else

10:       $P_i^{t+1}=a$ random position of search space

11:      end if

12:   end for

13:   Check solution boundaries.

14:   Calculate the fitness of each crow

15:  Update Crows’ memory

16:   end while

4.3. Tunicate Swarm Algorithm (TSA)

The standard tunicate swarm algorithm is a very simple bio-inspired metaheuristic optimization technique, which was first proposed by S. Kaur et al. in 2020 [40]. Its inspiration and performance were proven over the seventy-four benchmark problems compared to several other optimization approaches. Its efficacy and unpretentious structure draw the attention to employ and improve this algorithm for the considered problem. The swarm behavior of TSA is given in Figure 4 [40]. TSA main limitates the swarming behaviors of the marine tunicates and their jet propulsions during its navigation and foraging procedure [41].

In TSA, a population of tunicates (PT) is swarming to search for the best source of food (SF), representing the fitness function. In this swarming, the tunicates update their positions related to the first best tunicates stored and upgraded in each iteration. The TSA begins where the tunicate population is initialized randomly, considering the permissible bounds of the control variables. The dimension of the control variables composes each tunicate (T), which can be initially created as [40]

$$T_n\left(m\right)=T_{min}^n+r\times \left(T_{max}^n-T_{min}^n\right) \forall m\in P{T}_{size }\& n\in Dim$$

(17)

where T(m) stands for the position of each tunicate (m); n refers to each control variable in dimension Dim; r is a random number within the range (0:1); and $P{T}_{size }$ indicates the number of tunicates in the population.

The update process of the tunicates position is executed by the following formula [41]:

$$T_n\left(m\right)=\frac{T_n^{*}\left(m\right)-T_n^{*}\left(m-1\right)}{2+c_1}$$

(18)

where $T^{*}$ denotes the updated position of the $m^{th}$ tunicate based on Equation (18); T(m−1) refers to the neighbor tunicate; $c_1$ is a random number, uniformly distributed between 0 and 1.

$$T_n^{*}\left(m\right)=\left\{\begin{array}{c}SF+A\times \left|SF-rand\times T_n\left(n\right)\right| if rand\ge 0.5\\ SF-A\times \left|SF-rand\times T_n\left(n\right)\right| if rand<0.5\end{array}\right.$$

(19)

where SF is the source of food, which is represented by the best tunicate position in the whole population; A is a randomized vector to avoid any conflicts between tunicates and each other, which is expressed as: [40]

$$A=\frac{c_2+c_3-2c_1}{V{T}_{min}+c_1\left(V{T}_{max}-V{T}_{min}\right)}$$

(20)

where $c_2$ and $c_3$ are random numbers within the range (0:1); $V{T}_{min}$ and $V{T}_{max}$ represent the initial and subordinate speeds to produce social interaction.

The TSA method’s key steps can be described as [42]:

• Step 1: Create the initial tunicate population.
• Step 2: Determine the control units of TSA and stopping criteria.
• Step 3: Compute the fitness values of the initial population.
• Step 4: Select the position of the tunicate with the best fitness value.
• Step 5: Create the new position for each tunicate by using Equation (18).
• Step 6: Update the position of the tunicates that are out of the search space.
• Step 7: Compute the fitness values for the new positions of tunicates.
• Step 8: Until stopping criteria is satisfied, repeat steps 5–8.
• Step 9: After stopping criteria is satisfied, save the best tunicate position.

4.4. Overview of the Firefly Algorithm (FFA)

The firefly algorithm developed by X-S. Yang in 2007 [43] is one of the swarm-based optimization algorithms, which works based on the sparkling performance of all the fireflies. Fireflies use their flashing property to communicate. There are three rules in the firefly algorithm, which are based on the idealized behavior of the flashing characteristics of fireflies [44].

4.4.1. Attractiveness

All fireflies are unisex, and they will move toward more attractive and brighter ones regardless of their sex. In the firefly algorithm, the form of attractiveness function of a firefly is the following monotonically decreasing function [44]:

$$\beta \left(r\right)={\beta }_0\times exp\left(-\gamma r^m\right), m\ge 1$$

(21)

where r is the distance between any two fireflies ${\beta }_0$ is the initial attractiveness at r = 0, and γ is an absorption coefficient which controls the decrease in light intensity.

4.4.2. Distance

The degree of attractiveness of a firefly is proportional to its brightness, which decreases as the distance from the other firefly increases because the air absorbs light. If there is no brighter or more attractive firefly than a particular one, it will move randomly [45].

The distance between any two fireflies i and j, at positions xi and xj, respectively, can be defined as a Cartesian or Euclidean distance as follows [46]:

$$r_{ij}=\Vert x_i-x_j\Vert =\sqrt{{\displaystyle \sum }_{k=1}^d{\left(x_{i,k}-x_{j,k}\right)}^2}$$

(22)

where $x_{i,k}$ is the $k^{th}$ component of the spatial coordinate $x_i$ of the $i^{th}$ firefly.

In the dimensions 2-d, we have:

$$r_{ij}=\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}$$

(23)

4.4.3. Movement

The movement of a firefly i, which is attracted by a more attractive (brighter) firefly j, is calculated by [46]

$$x_i=x_i+{\beta }_0\times exp\left(-\gamma r_{ij}^2\right)\left(x_j-x_i\right)+\alpha \left(rand-\frac{1}{2}\right)$$

(24)

where the second term is due to the attraction, while the third term is randomization with α being the randomization parameter, and rand is a random number within the range (0:1).

5. Simulation Results and Discussion

In order to solve the dynamic combined economic environmental dispatch problem, we developed and executed the CSA, SOA, TSA, and FFA algorithms in MATLAB 2017, and they were run on a personal computer with an Intel Core(TM) i5 with a processor of 2.60 GHz and a Ram of 8.0 GB under MS Windows 8.1. In the first part, the four proposed techniques were tested on the five-unit test system by considering VPE for three case studies, and in the second part, the CSA method was applied on the IEEE ten-unit system, including VPE, for three cases. The constraints involved in all cases were power balance limit with consideration of transmission losses and generator operating limits constraints.

The obtained results were compared with the optimization approaches recently published in the literature.

Table 1. Parameters of CSA, SOA, TSA, and FFA algorithms for DEED problem.

Algorithm	Parameters
SOA	N = 50, tmax = 1000, u = 1, v = 0.011
CSA	N = 50, itermax= 1000, fl = 2, AP = 0.1
TSA	m = 50, itermax = 1000, VTmin= 1, VTmax= 4
FFA	N = 100, itermax= 1000, ${\beta }_0$= 1, γ = 1, α = 0.1,

stands for the parameter values of CSA, SOA, TSA, and FFA algorithms, for all cases studies.

5.1. IEEE Five-Unit Test System

The five-unit test system was derived from [47,48].

Table 2. Cost and emission coefficients of generators for IEEE five-unit power system.

Unit	${\mathit{a}}_{\mathit{i}}$ $\mathbf{\$}/\mathbf{h}$	${\mathit{b}}_{\mathit{i}}$ $\$/\mathbf{M}\mathbf{W}\mathbf{h}$	${\mathit{c}}_{\mathit{i}}$ $\$/\mathbf{M}{\mathbf{W}}^2\mathbf{h}$	${\mathit{e}}_{\mathit{i}}$ $\mathbf{U}\mathbf{S}\mathbf{D}/\mathbf{h}$	${\mathit{f}}_{\mathit{i}}$ $\mathbf{r}\mathbf{a}\mathbf{d}/\mathbf{M}\mathbf{W}$	${\mathit{\alpha }}_{\mathit{i}}$ $\mathit{I}\mathit{b}/\mathbf{h}{ }^{*}$	${\mathit{\beta }}_{\mathit{i}}$ $\mathbf{I}\mathbf{b}/\mathbf{M}\mathbf{W}\mathbf{h}$	${\mathit{\gamma }}_{\mathit{i}}$ $\mathbf{I}\mathbf{b}/\mathbf{M}{\mathit{W}}^2\mathbf{h}$	${\mathit{\eta }}_{\mathit{i}}$ $\mathbf{I}\mathbf{b}/\mathbf{h}$	${\mathit{\delta }}_{\mathit{i}}$ $1/\mathbf{M}\mathbf{W}$	${\mathit{P}}_{\mathit{i}}^{\mathbf{min}}$ $\mathbf{M}\mathbf{W}$	${\mathit{P}}_{\mathit{i}}^{\mathbf{max}}$ $\mathbf{M}\mathbf{W}$	$\mathit{p}\mathit{p}{\mathit{f}}_{\mathit{i}}$ $\$/\mathbf{I}\mathbf{b}$
$P_1$	25	2.0	0.0080	100	0.042	80	−0.805	0.0180	0.6550	0.02846	10	75	1.8201
$P_2$	60	1.8	0.0030	140	0.040	50	−0.555	0.0150	0.5773	0.02446	20	125	1.5436
$P_3$	100	2.1	0.0012	160	0.038	60	−1.355	0.0105	0.4968	0.02270	30	175	3.4911
$P_4$	120	2.0	0.0010	180	0.037	45	−0.600	0.0080	0.4860	0.01948	40	250	1.7278
$P_5$	40	1.8	0.0015	200	0.035	30	−0.555	0.0120	0.5035	0.02075	50	300	0.7578

* Pound per hour (lb/h) = 0.453 59237 kilogram per hour (kg/h).

and

Table 3. Load demands.

Time (h)	Load (MW)	Time (h)	Load (MW)	Time (h)	Load (MW)
1	410	9	690	17	558
2	435	10	704	18	608
3	475	11	720	19	654
4	530	12	740	20	704
5	558	13	704	21	680
6	608	14	690	22	605
7	626	15	654	23	527
8	654	16	580	24	463

include the generation data. In this subsection, the dynamic load dispatch (DLD) cases through the application of the four metaheuristics techniques include:

• Case 1: Dynamic economic dispatch DED;
• Case 2: Dynamic environmental dispatch DEnD;
• Case 3: Dynamic economic emission dispatch DEED.

The B matrix of the test system we use is given by Equation (25) [48]:

$$B={10}^{-4}{}_i\times \left[\begin{array}{ccccc}0.49& 0.14& 0.15& 0.15& 0.20\\ 0.14& 0.45& 0.16& 0.20& 0.18\\ 0.15& 0.16& 0.39& 0.10& 0.12\\ 0.15& 0.20& 0.10& 0.40& 0.14\\ 0.20& 0.18& 0.12& 0.14& 0.35\end{array}\right]$$

(25)

5.1.1. Case 1

The dynamic economic dispatch (DED) considering VPE with variables power losses is studied in this subsection (for total load = 14,577 MW).

The optimal schedule for a five-thermal-generator test system considering VPE and transmission losses, using the crow search algorithm (CSA), is presented in

Table 4. Simulation results of best solutions of CSA for Case 1.

Time h	${\mathit{P}}_1$ MW	${\mathit{P}}_2$ MW	${\mathit{P}}_3$ MW	${\mathit{P}}_4$ MW	${\mathit{P}}_5$ MW	${\mathit{P}}_{\mathit{D}}$ MW	${\mathit{P}}_{\mathit{L}}$ MW	Fuel Cost USD/h	Emission $\mathbf{I}\mathbf{b}/\mathbf{h}$
1	84.804	98.539	50.652	40.000	139.759	410	3.756	1363.640	546.604
2	48.075	98.539	112.673	40.000	139.759	435	4.048	1380.410	510.742
3	88.876	98.539	112.673	40.000	139.759	475	4.849	1423.790	584.122
4	59.954	98.539	112.673	124.908	139.759	530	5.835	1584.710	590.865
5	84.139	98.539	112.673	40.000	229.519	558	6.872	1604.790	969.866
6	55.079	98.539	112.673	209.816	139.759	608	7.868	1777.140	784.062
7	73.529	98.539	112.673	209.816	139.759	626	8.317	1783.770	814.087
8	97.449	98.539	112.673	124.908	229.519	654	9.090	1882.450	1071.515
9	49.619	98.539	112.673	209.816	229.519	690	10.168	1977.660	1175.437
10	64.011	98.539	112.673	209.816	229.519	704	10.559	1996.590	1194.648
11	80.483	98.539	112.673	209.815	229.519	720	11.032	1989.970	1226.653
12	101.112	98.539	112.673	209.816	229.519	740	11.662	2106.450	1282.648
13	64.0110	98.539	112.673	209.816	229.519	704	10.559	1996.590	1194.648
14	49.619	98.539	112.673	209.816	229.519	690	10.168	1977.660	1175.437
15	97.449	98.539	112.673	124.908	229.519	654	9.090	1882.450	1071.515
16	84.800	40.058	112.673	209.816	139.759	580	7.108	1746.070	745.131
17	84.139	98.539	112.673	40.000	229.519	558	6.872	1604.790	969.866
18	55.079	98.539	112.673	209.816	139.759	608	7.868	1777.140	784.062
19	97.449	98.539	112.673	124.908	229.519	654	9.090	1882.450	1071.515
20	64.011	98.539	112.673	209.816	229.519	704	10.559	1996.590	1194.648
21	39.353	98.539	112.673	209.816	229.519	680	9.902	1944.590	1166.578
22	52.007	98.539	112.673	209.816	139.759	605	7.796	1771.650	780.351
23	56.889	98.539	112.673	124.908	139.759	527	5.771	1581.460	586.584
24	81.432	98.539	112.673	124.908	50.000	463	4.553	1392.530	468.968

. The best results are highlighted in bold font.

The comparison results of the optimization techniques (CSA, SOA, TSA, and FFA) and the statistical analysis of the optimal results of Case 1 are given in Table 5. The best values, the robustness and effectiveness of the proposed CSA in finding optimal solutions to the DED problem with a reasonable number of iterations are compared and the constraints verified.

The results shown in Table 4 present the total cost found by the CSA algorithm, which is equal to 42,425.455 USD/h, is lower compared to that found by the SOA, TSA, and FFA algorithms, which is estimated at 48,609.77 USD/h, 46,672.4787 USD/h and 45,474.198 USD/h respectively.

As seen in Table 5, we infer that the CSA obtained the best fuel cost compared to PSO [47], PSOGSA [48], NEHS [49], HS-NPSA [49], and DE-SQP [50].

The CSA algorithm, which offers the best production cost, on the other hand, offers a relatively higher amount of transmission losses compared to FFA, i.e., 193.393 MW for the CSA algorithm, 204.66 MW for SOA, 198.278 MW for TSA, and 191.298 MW for FFA.

Table 5. Comparison of CSA with previous algorithms for DED.

Methods	Total Loss MW	Total Fuel Cost USD/h	Total Emission $\mathbf{I}\mathbf{b}/\mathbf{h}$
CSA	193.393	42,425.455	21,960.553
SOA	204.660	48,609.770	32,652.860
TSA	198.278	46,672.479	27,641.230
FFA	191.298	45,474.198	24,862.338
PSOGSA [48]	NA	42,853.339	22,087.887
NEHS [49]	NA	43,066.073	NA
MHS [49]	NA	45,497.740	NA
HS-NPSA [49]	NA	43,927.305	NA
DE-SQP [50]	NA	45,590.000	23,567.000
PSO [47]	NA	47,852.000	22,405.000

NA means not available in the corresponding literature.

Table 5. Comparison of CSA with previous algorithms for DED.

Methods	Total Loss MW	Total Fuel Cost USD/h	Total Emission $\mathbf{I}\mathbf{b}/\mathbf{h}$
CSA	193.393	42,425.455	21,960.553
SOA	204.660	48,609.770	32,652.860
TSA	198.278	46,672.479	27,641.230
FFA	191.298	45,474.198	24,862.338
PSOGSA [48]	NA	42,853.339	22,087.887
NEHS [49]	NA	43,066.073	NA
MHS [49]	NA	45,497.740	NA
HS-NPSA [49]	NA	43,927.305	NA
DE-SQP [50]	NA	45,590.000	23,567.000
PSO [47]	NA	47,852.000	22,405.000

NA means not available in the corresponding literature.

It is also worth mentioning that in Table 5, the total emissions found by the CSA algorithm, which amount to 21,960.553 lb/h are more reduced compared to those found by the SOA, TSA, and FFA algorithms, which are estimated at 32,652.86 lb/h, 27,641.23 lb/h, and 24,862.338 lb/h respectively.

Figure 5, shows the graphical depiction of the convergence of the DED problem with time (iterations) obtained by applying our proposed methods corresponding to the peak hour (P_D = 740 MW).

In comparison to SOA, TSA, and FFA, we can see that CSA has an incredibly fast convergence ability. In addition, CSA was trapped into the local optimum at around 230 iterations.

5.1.2. Case 2

The dynamic environmental dispatch (DEnD) considering the valve-point loading effect and transmission losses as given in Equation (3) are discussed in this case.

Table 6. Simulation results of best solutions of CSA for Case 2.

Time h	${\mathit{P}}_1$ MW	${\mathit{P}}_2$ MW	${\mathit{P}}_3$ MW	${\mathit{P}}_4$ MW	${\mathit{P}}_5$ MW	${\mathit{P}}_{\mathit{D}}$ MW	${\mathit{P}}_{\mathit{L}}$ MW	Fuel Cost USD/h	Emission $\mathbf{I}\mathbf{b}/\mathbf{h}$
1	54.679	58.236	116.571	110.598	73.364	410	3.448	352.453	1723.627
2	58.067	62.383	121.851	117.982	78.601	435	3.885	385.960	1784.169
3	63.526	69.080	130.221	129.751	87.063	475	4.641	446.642	1912.114
4	71.120	78.439	141.552	145.801	98.890	530	5.797	544.648	2135.506
5	75.032	83.262	147.232	153.895	105.008	558	6.431	601.208	2203.616
6	82.107	92.034	157.218	168.176	116.119	608	7.654	713.801	2241.188
7	84.684	95.241	160.760	173.252	120.183	626	8.121	758.078	2229.487
8	88.729	100.286	166.212	181.072	126.577	654	8.876	831.019	2240.807
9	93.996	106.880	173.119	190.970	134.934	690	9.898	932.292	2270.804
10	96.065	109.478	175.772	194.768	138.227	704	10.311	974.021	2274.187
11	98.446	112.472	178.783	199.072	142.020	720	10.794	1023.365	2305.246
12	101.444	116.253	182.514	204.393	146.809	740	11.414	1087577	2365.552
13	96.065	109.478	175.772	194.768	138.227	704	10.311	974.021	2274.190
14	93.996	106.879	173.119	190.970	134.934	690	9.898	932.292	2270.804
15	88.729	100.286	166.212	181.072	126.577	654	8.876	831.019	2240.804
16	78.130	87.098	151.652	160.208	109.866	580	6.955	648.892	2233.356
17	75.032	83.262	147.233	153.895	105.008	558	6.431	601.208	2203.615
18	82.107	92.034	157.218	168.176	116.119	608	7.654	713.801	2241.189
19	88.729	100.286	166.212	181.072	126.577	654	8.876	831.019	2240.807
20	96.066	109.478	175.772	194.768	138.227	704	10.311	974.021	2274.187
21	92.525	105.036	171.212	188.239	132.596	680	9.609	903.298	2265.782
22	81.678	91.502	156.625	167.326	115.445	605	7.577	706.617	2241.906
23	70.703	77.915	140.940	144.931	98.239	527	5.727	538.858	2126.253
24	61.883	67.063	127.721	126.227	84.514	463	4.407	427.514	1850.281

shows the results of the optimum power dispatch for the five-unit system over 24 h obtained by CSA.

The optimal results of the pollutant power generation, fuel cost, and active power loss obtained by the CSA are depicted in Table 6.

The total losses, total fuel cost, and total emissions computed by our proposed approaches are illustrated in

Table 7. Comparison of CSA with previous algorithms for DEnD.

Methods	Total Loss MW	Total Fuel Cost USD/h	Total Emission $\mathbf{I}\mathbf{b}/\mathbf{h}$
CSA	187.901	51,149.500	17,733.600
SOA	189.766	51,385.300	18,743.100
TSA	188.825	51,878.700	18,602.800
FFA	187.070	51,263.500	19,840.200
NEHS [49]	NA	NA	17,853.003
MHS [49]	NA	NA	17,937.408
HS-NPSA [49]	NA	NA	17,872.348
PSOGSA [48]	NA	51,953.905	17,852.979
DE-SQP [50]	NA	52,611.000	18,955.000
PSO [47]	NA	53,086.000	19,094.000

. We compared them with those determined by other algorithms recently applied in the literature.

By analyzing the results given in Table 7, we infer that the total emissions found by the CSA algorithm, which amount to 17,733.6 lb/h, are lower, compared to those found by the SOA TSA, and FFA algorithms, which are estimated at 18,743.1 lb/h, 18,602.8 lb/h, and 19,840.2 lb/h, respectively.

From the results shown in Table 7, we notice that the total cost determined by the CSA algorithm, which amounts to 51,149.5 USD/h, is lower compared to that found by the SOA, TSA, and FFA algorithms, which is estimated at 51,385.3 USD/h, 51,878.7 USD/h, and 51,263.5 USD/h, respectively.

In addition, when compared to CSA, SOA, and TSA, FFA has a relatively low amount of active power losses. The CS algorithm, which offers the minimum pollutant power generation, in return offers relatively higher transmission losses than FFA, and lower than SOA and TSA, i.e., 187.901 MW, 204.66MW, 188.825 MW, and, 187.07 MW for CSA, SOA, TSA, and FFA, respectively.

Figure 6 represents the convergence curves of our proposed approaches for the DEnD problem. It describes the stable convergence of the objective function of the problem given in Equation (3). The convergence characteristics shown in Figure 6 prove that CSA has better qualities than SOA, TSA, and FFA. We conclude that CSA is favorable for large-scale power systems.

5.1.3. Case 3

In this case, we deal with the dynamic combined economic environmental dispatching (DCEED) with the variable losses by applying the price penalty factor under the impact of the valve-point. Electrical losses are variable depending on the power generated.

Table 8. The best solutions obtained by CSA for the DCEED of the IEEE five-unit test system.

Time h	${\mathit{P}}_1$ MW	${\mathit{P}}_2$ MW	${\mathit{P}}_3$ MW	${\mathit{P}}_4$ MW	${\mathit{P}}_5$ MW	${\mathit{P}}_{\mathit{D}}$ MW	${\mathit{P}}_{\mathit{L}}$ MW	Fuel Cost USD/h	Emission $\mathbf{I}\mathbf{b}/\mathbf{h}$	CEED USD/h
1	23.221	95.066	118.179	125.507	51.646	410.00	3.619	1343.493	400.897	1814.930
2	42.464	84.193	112.348	124.800	75.142	435.00	3.946	1607.846	399.951	2078.248
3	46.363	94.690	111.484	87.447	139.713	475.00	4.697	1670.469	519.509	2254.467
4	26.838	99.530	114.280	126.507	168.745	530.00	5.899	1745.194	669.218	2475.641
5	33.564	92.891	123.185	175.142	139.753	558.00	6.536	1902.752	665.000	2649.852
6	73.336	98.983	157.530	146.016	139.760	608.00	7.625	2031.153	728.731	2893.762
7	57.750	99.767	123.604	210.658	142.515	626.00	8.294	1914.203	816.010	2838.928
8	51.884	88.674	103.432	204.515	214.625	654.00	9.131	2131.606	1040.647	3248.435
9	53.142	97.776	114.207	209.557	225.462	690.00	10.144	2019.883	1153.740	3260.318
10	54.142	109.352	112.662	209.810	228.632	704.00	10.598	2075.861	1204.781	3370.372
11	73.636	100.102	117.640	210.105	229.527	720.00	11.010	2052.002	1223.114	3381.405
12	72.069	116.167	123.841	210.056	229.502	740.00	11.635	2224.386	1274.535	3613.856
13	66.908	98.680	173.744	145.451	229.517	704.00	10.300	2236.929	1154.978	3523.693
14	61.029	98.536	102.648	209.815	228.176	690.00	10.204	2036.995	1171.356	3293.254
15	66.581	117.798	115.698	213.160	149.886	654.00	9.123	2078.304	903.466	3098.415
16	22.005	101.970	112.492	124.576	226.155	580.00	7.200	1701.781	948.532	2700.991
17	24.788	95.120	109.230	195.916	139.614	558.00	6.667	1740.035	706.614	2521.975
18	66.807	83.120	156.707	124.906	184.080	608.00	7.620	2199.492	803.606	3116.141
19	61.025	110.762	116.453	209.863	164.984	654.00	9.087	2122.958	915.351	3145.552
20	61.190	96.517	118.054	209.294	229.473	704.00	10.528	2049.371	1189.705	3333.294
21	61.279	77.640	112.693	208.694	229.519	680.00	9.825	2055.372	1140.486	3280.867
22	66.280	38.471	112.702	209.818	185.459	605.00	7.729	2051.944	873.120	3013.830
23	47.683	93.663	126.697	124.929	139.771	527.00	5.741	1693.775	582.538	2357.800
24	41.562	70.450	112.609	124.880	117.903	463.00	4.404	1696.097	452.880	2212.678

summarizes the best solutions achieved for DCEED for a given load. We can deduce from the same table that CSA and our three competitive algorithms adjust the value of the output power ($P_1-P_5$) such that load demand and the constraint limits are satisfied, and the best fuel cost and pollutants emission results are achieved.

Table 9. The DCEED comparison results of different algorithms of the five-unit test system.

Methods	Total Loss MW	Total Fuel Cost $\mathrm{USD}/\mathbf{h}$	Total Emission $\mathbf{I}\mathbf{b}/\mathbf{h}$	Total Cost $(0.5 \mathbf{Weight}) \mathrm{USD}/\mathbf{h}$	Change % w.r.t CSA
CSA	192.386	46,381.900	20,938.800	33,660.350	//
SOA	196.961	48,500.800	21,130.100	34,815.450	3.43
TSA	194.204	45,816.300	22,424.800	33,942.950	0.84
FFA	192.451	47,030.700	22,069.600	34,550.150	2.64
NEHS [49]	NA	45,398.016	18,392.337	31,895.177	−5.24
MHS [49]	NA	47,390.956	18,423.776	32,907.366	−2.24
MOHDESAT [51]	NA	48,214.000	18,011.000	33,112.500	−1.63
WOA [52]	NA	46,475.090	18,827.980	32,651.530	−2.99
PSOGSA [48]	NA	45,702.6001	18,267.179	31,984.985	−5.24
DE-SQP [50]	NA	46,625.000	20,527.000	33,576.000	−0.002
PSO [47]	NA	50,893.000	20,163.000	35,528.000	5.55
MONNDE [53]	NA	49,135.000	18,233.000	33,684.240	0.06
EP [54]	NA	48,628.000	21,154.000	34,891.000	3.66
SA [51]	NA	48,621.000	21,188.000	33,904.500	0.73
PS [55]	NA	47,911.000	18,927.000	33,419.000	−0.72
MODE [51]	NA	47,330.000	18,116.000	32,723.000	−2.78

shows the values of the active power losses, the best cost, the best emission, and the total cost objectives against 0.5 weight for the test system. The results displayed in the mentioned table are obtained from NEHS, MHS, HS-NPSA, PSO-GSA, DE-SQO, PSO, and our proposed approaches.

Column five of Table 9 depicts the value of the single-objective function of the problem, as specified in Equation (1). The effectiveness of the suggested CSA is demonstrated by the percentage change of some algorithms listed in the same table.

As seen in Table 9, the results reveal that CSA has a good performance for solving DCEED at less than SOA (3.43%), than TSA (0.84%), than FFA (2.64%), than PSO (5.55%), and then EP (3.66%).

According to Table 9, we infer that minimizing the cost of generation and minimizing emissions are contradictory goals. Emissions are highest when the cost of production is minimized.

Figure 7 shows the convergence curves of the total cost with iterations, obtained by applying CSA, SOA, TSA, and FFA methods corresponding to the power demand equal to 740 MW. We infer that CSA and FFA reach the optimum solution more quickly than SOA and TSA, which is shown in Figure 7.

Figure 8 presents a graphical comparison of the total fuel cost, the total emissions, and the total cost of our proposed methods and other algorithms for a five-unit test system. This clearly demonstrates that CSA has the lowest total cost when compared to other algorithms.

Percentage enhancement of the suggested CSA over other algorithms is shown in Figure 9. From this figure, the positive bars show the largest values of the total cost compared to those obtained by CSA, and negative bars represent smaller values of the total cost compared to those obtained by CSA.

The statistical summary of 50 runs for each study case performed (i.e., Case 1 to Case 3) using CSA, SOA, TSA, and FFA approaches is depicted in

Table 10. Statistical summary of case studies using CSA, SOA, TSA, and FFA algorithms for load demand of 740 MW for a IEEE five-unit system.

Case 1	Economic Load Dispatch ELD (USD/h)
Algorithm	Min	Average	Worst	Std Dev
CSA	2142.3181	2164.8135	2593.5470	55.4145
SOA	2456.3468	2537.9394	5577.4323	290.6773
TSA	2269.2981	2293.2777	3635.3608	113.8379
FFA	2265.5019	2310.7733	2926.1582	100.0485
Case 2	Environmental Dispatch EnD (lb/h)
CSA	1090.3380	1111.2373	1421.8938	58.9938
SOA	1160.6740	1757.2506	3536.7399	499.4472
TSA	1295.4652	1506.9932	1728.0597	83.5214
FFA	1241.5852	1323.0107	5121.1719	331.6223
Case 3	Combined Economic Emission Dispatch CEED (USD/h)
CSA	1646.9098	1652.9305	1784.2619	22.1540
SOA	1955.3266	2153.2196	3553.8243	418.7256
TSA	1879.5231	1992.3606	2203.5179	99.1567
FFA	1848.8820	1878.6942	2216.4237	67.0644

.

The columns display the min, average, worst, and standard deviation values for the objective function in each case. According to Table 10, it is clear the CSA method delivers the best fitness, best mean, and best standard deviation values in all the cases.

Figure 10 obtained values (min, mean, and worst values) for 50 runs of the CSA approach for the case of reducing the cost of production (Case 1).

5.2. IEEE 10-Unit Test System

In this case study, the then-unit generating system is utilized to evaluate the performance of the proposed CSA by considering the valve-point loading effect. In this experiment, a 24-h scheduling period with 1-h intervals is used. The original data of the 10-unit system are shown in

Table 11. IEEE 10-unit generator coefficients.

Unit	${\mathit{a}}_{\mathit{i}}$ $\mathbf{\$}/\mathbf{h}$	${\mathit{b}}_{\mathit{i}}$ $\mathbf{\$}/\mathbf{M}\mathbf{W}\mathbf{h}$	${\mathit{c}}_{\mathit{i}}$ $\mathbf{\$}/\mathbf{M}{\mathbf{W}}^2\mathbf{h}$	${\mathit{e}}_{\mathit{i}}$ $\mathbf{\$}/\mathbf{h}$	${\mathit{f}}_{\mathit{i}}$ $\mathbf{r}\mathbf{a}\mathbf{d}/\mathbf{M}\mathbf{W}$	${\mathit{\alpha }}_{\mathit{i}}$ $\mathbf{I}\mathbf{b}/{\mathbf{h}}^{*}$	${\mathit{\beta }}_{\mathit{i}}$ $\mathbf{I}\mathbf{b}/\mathbf{M}\mathbf{W}\mathbf{h}$	${\mathit{\gamma }}_{\mathit{i}}$ $\mathbf{I}\mathbf{b}/\mathbf{M}{\mathbf{W}}^2\mathbf{h}$	${\mathit{\eta }}_{\mathit{i}}$ $\mathbf{I}\mathbf{b}/\mathbf{h}$	${\mathit{\delta }}_{\mathit{i}}$ $1/\mathbf{M}\mathbf{W}$	${\mathit{P}}_{\mathit{i}}^{\mathbf{min}}$ $\mathbf{M}\mathbf{W}$	${\mathit{P}}_{\mathit{i}}^{\mathbf{max}}$ $\mathbf{M}\mathbf{W}$
$P_1$	786.7988	38.5397	0.1524	450	0.0410	103.3908	−2.4444	0.0312	0.5035	0.0207	150	470
$P_2$	451.3251	46.1591	0.1058	600	0.0360	103.3908	−2.4444	0.0312	0.5035	0.0207	135	470
$P_3$	1049.9977	40.3965	0.0280	320	0.0280	300.3910	−4.0695	0.0509	0.4968	0.0202	73	340
$P_4$	1243.5311	38.3055	0.0354	260	0.0520	300.3910	−4.0695	0.0509	0.4968	0.0202	60	300
$P_5$	1658.5696	36.3278	0.0211	280	0.0630	320.0006	−3.8132	0.0344	0.4972	0.0200	73	243
$P_6$	1356.6592	38.2704	0.0179	310	0.0480	320.0006	−3.8132	0.0344	0.4972	0.0200	57	160
$P_7$	1450.7045	36.5104	0.0121	300	0.0860	330.0056	−3.9023	0.0465	0.5163	0.0214	20	130
$P_8$	1450.7045	36.5104	0.0121	340	0.0820	330.0056	−3.9023	0.0465	0.5163	0.0214	47	120
$P_9$	1455.6056	39.5804	0.1090	270	0.0980	350.0056	−3.9524	0.0465	0.5475	0.0234	20	80
$P_{10}$	1469.4026	40.5407	0.1295	380	0.0940	360.0012	−3.9864	0.0470	0.5475	0.0234	10	55

, which are available in the literature [53].

Table 12. IEEE 10-unit B matrix ($\times {10}^{-4}$).

0.49	0.14	0.15	0.15	0.16	0.17	0.17	0.18	0.19	0.20
0.14	0.45	0.16	0.16	0.17	0.15	0.15	0.16	0.18	0.18
0.15	0.16	0.39	0.10	0.12	0.14	0.14	0.16	0.16	0.16
0.15	0.16	0.10	0.40	0.14	0.10	0.11	0.12	0.14	0.15
0.16	0.17	0.12	0.14	0.35	0.11	0.13	0.13	0.15	0.16
0.17	0.15	0.12	0.10	0.11	0.36	0.12	0.12	0.14	0.15
0.17	0.15	0.14	0.11	0.13	0.12	0.38	0.16	0.16	0.18
0.18	0.16	0.14	0.12	0.13	0.12	0.16	0.40	0.15	0.16
0.19	0.18	0.16	0.14	0.15	0.14	0.16	0.15	0.42	0.19
0.20	0.18	0.16	0.15	0.16	0.15	0.18	0.16	0.19	0.44

and

Table 13. Load demands.

Time (h)	Load (MW)	Time (h)	Load (MW)	Time (h)	Load (MW)
1	1036	9	1924	17	1480
2	1110	10	2022	18	1628
3	1258	11	2106	19	1776
4	1406	12	2150	20	1972
5	1480	13	2072	21	1924
6	1628	14	1924	22	1628
7	1702	15	1776	23	1332
8	1776	16	1554	24	1184

are the 10-unit generator loads demand and the B matrix, respectively, which are derived from [53]. In this research, the weighting function approach (w.f.a) is used to converts multi-objective functions into a single problem [56]. Hence, by the usage of the w. f. a, Equation (1) can be reformulated as:

$$\min F=w_1{\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{i=1}^{Ng}{C}_{i,t}\left(P_{i,t}\right)+w_2{\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{i=1}^{Ng}{E}_{i,t}\left(P_{i,t}\right)$$

(26)

where ${\displaystyle \sum }_{i=1}^n{w}_i=1$ (n: number of problems).

Dynamic load dispatch (DLD) cases in this subsection include:

• Case 4: Dynamic economic dispatch DED $\left(w_1=1 , w_2=0\right)$;
• Case 5: Dynamic environmental dispatch DEnD $\left(w_1=0 , w_2=1\right)$;
• Case 6: Dynamic economic emission dispatch DEED $\left(w_1=0.5, w_2=0.5\right)$;

5.2.1. Case 4

The dynamic economic dispatch (DED) considering VPE with variables power losses is studied in this subsection (for total load = 39.848 MW).

Hourly generation (MW) schedule obtained from DED using CSA for a 10-unit system by considering VPE and transmission losses are presented in

Table 14. Best solutions obtained from CSA for test Case 4.

Time h	$$\begin{gathered} {\mathit{P}}_1 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_2 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_3 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_4 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_5 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_6 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_7 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_8 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_9 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_{10} \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_{\mathit{L}} \\ \mathbf{MW} \end{gathered}$$	Fuel Cost $\mathbf{U}\mathbf{S}\mathbf{D}/\mathbf{h}$	Emission $\mathbf{I}\mathbf{b}/\mathbf{h}$
1	150.510	135.104	80.705	62.302	169.338	122.675	125.946	119.882	54.877	34.491	19.832	61,600.519	3803.894
2	150.006	135.034	146.567	120.465	175.863	122.983	128.759	118.303	20.993	13.373	22.336	64,906.264	4689.769
3	150.279	135.165	181.085	182.838	221.095	138.049	129.652	89.063	49.058	10.227	28.514	72,304.859	6228.359
4	150.009	135.007	224.237	191.995	242.419	159.944	129.963	119.998	44.613	43.418	35.606	80,271.190	7718.284
5	150.034	135.001	290.188	242.097	222.606	124.793	129.457	117.073	61.531	47.082	39.862	84,408.221	9763.984
6	150.032	144.110	339.873	298.516	242.647	159.853	98.936	118.058	79.997	44.509	48.532	94,297.297	13,307.920
7	222.445	135.032	330.607	299.754	239.465	159.924	129.960	116.990	70.059	51.333	53.563	100,946.056	13,761.749
8	224.611	213.686	339.936	291.807	242.827	139.143	129.115	119.997	79.959	54.035	59.116	107,884.313	14,564.055
9	272.168	323.499	339.776	299.530	242.880	159.038	129.950	120.000	65.817	42.489	71.142	124,932.384	17,534.742
10	299.063	395.517	339.995	293.040	242.928	159.991	122.587	114.212	79.889	55.000	80.219	137,626.028	20,656.966
11	341.347	449.088	339.950	296.452	243.000	159.982	129.965	110.813	79.997	44.020	88.609	150,876.695	26,836.868
12	386.873	450.590	339.134	299.972	242.996	156.337	129.852	119.998	79.960	37.402	93.114	157,522.576	29,006.003
13	312.707	441.906	332.353	299.267	242.973	144.923	128.829	119.947	79.461	54.994	85.359	145,630.761	24,855.409
14	272.168	323.499	339.776	299.530	242.880	159.038	129.950	120.000	65.817	42.489	71.142	124,932.384	17,534.742
15	224.611	213.686	339.936	291.807	242.827	139.143	129.115	119.997	79.959	54.035	59.116	107,884.313	14,564.055
16	152.831	135.356	293.140	299.998	237.653	158.625	129.971	93.751	52.366	44.230	43.915	89,095.065	11,635.439
17	150.034	135.001	290.188	242.097	222.606	124.793	129.457	117.073	61.531	47.082	39.862	84,408.221	9763.984
18	150.032	144.110	339.873	298.516	242.647	159.853	98.936	118.058	79.997	44.509	48.532	94,297.297	13,307.920
19	224.611	213.686	339.936	291.807	242.827	139.143	129.115	119.997	79.959	54.035	59.116	107,884.313	14,564.055
20	247.097	405.611	338.935	300.000	241.615	159.957	130.000	116.665	64.768	43.158	75.807	131,752.041	20,566.632
21	272.168	323.499	339.776	299.530	242.880	159.038	129.950	120.000	65.817	42.489	71.142	124,932.384	17,534.742
22	150.032	144.110	339.873	298.516	242.647	159.853	98.936	118.058	79.997	44.509	48.532	94,297.297	13,307.920
23	150.015	135.032	215.063	180.844	222.867	143.145	129.793	119.964	56.899	10.332	31.955	76,082.212	7006.619
24	150.001	135.024	184.626	131.224	222.612	123.171	129.588	86.119	24.598	22.486	25.447	68,742.483	5589.037

. The best results are highlighted in bold font.

Total fuel cost and total emission results obtained from the CSA algorithm are compared with other methods in Table 15. As can be seen from Table 15, the proposed CSA algorithm produces fewer fuel cost value by 2.487512 × 10⁶ USD than other algorithms. CSA was able to achieve an enhanced in total fuel cost in rang 10⁶ of 0.01659 USD, 0.013315 USD and 0.029288 USD, respectively as compared to MHS [49], SA [57], and NSGAII [58]. IBFA [59], GCABC [60], and TLBO [61] gives a best solutions (less than this obtained by our proposed approach of 5812 USD, 13,039 USD, and 15,396 USD, respectively).

To elaborate the transition of CSA for different phases of the search convergence characteristics for the 10-unit test system under a fixed hour (P_D = 2150 MW) for 1000 iterations is shown in Figure 11. From the figure, we can be seen that CSA has an incredibly fast convergence ability it has been quickly trapped the optimal values.

Table 15. Comparison of the 10-unit test system with previous algorithms for best minimum cost solution (Case 4).

Methods	Total Fuel Cost $\left(\times {10}^6\right) \mathbf{U}\mathbf{S}\mathbf{D}$	$$\begin{gathered} \mathbf{Total} \mathbf{Emission} \\ \left(\times {10}^5\right)\mathbf{I}\mathbf{b} \end{gathered}$$
CSA	2.487512	3.38103
MHS [49]	2.504106	NA
SA [57]	2.537200	NA
NSGAII [58]	2.516800	3.17400
IBFA [59]	2.481700	3.27500
GCABC [60]	2.474473	NA
TLBO [61]	2.472116	3.30411

Table 15. Comparison of the 10-unit test system with previous algorithms for best minimum cost solution (Case 4).

Methods	Total Fuel Cost $\left(\times {10}^6\right) \mathbf{U}\mathbf{S}\mathbf{D}$	$$\begin{gathered} \mathbf{Total} \mathbf{Emission} \\ \left(\times {10}^5\right)\mathbf{I}\mathbf{b} \end{gathered}$$
CSA	2.487512	3.38103
MHS [49]	2.504106	NA
SA [57]	2.537200	NA
NSGAII [58]	2.516800	3.17400
IBFA [59]	2.481700	3.27500
GCABC [60]	2.474473	NA
TLBO [61]	2.472116	3.30411

5.2.2. Case 5

The dynamic environmental dispatch (DEnD) considering the valve-point loading effect and transmission losses as given in Equation (3) is discussed in this case.

Table 16. Best solutions obtained from CSA for test Case 5.

Time h	$$\begin{gathered} {\mathit{P}}_1 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_2 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_3 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_4 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_5 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_6 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_7 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_8 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_9 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_{10} \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_{\mathit{L}} \\ \mathbf{MW} \end{gathered}$$	Fuel Cost $\mathbf{U}\mathbf{S}\mathbf{D}/\mathbf{h}$	Emission $\mathbf{I}\mathbf{b}/\mathbf{h}$
1	150.002	135.718	90.566	91.469	118.016	136.066	109.538	94.773	74.663	54.913	19.725	63,282.134	3479.107
2	157.856	153.661	110.735	93.173	142.569	139.932	110.889	89.120	79.790	54.989	22.712	68,271.277	3945.886
3	160.271	167.053	120.413	132.252	178.038	155.820	122.997	119.962	75.131	54.997	28.936	75,457.550	5060.796
4	204.728	221.786	134.213	140.423	197.409	159.938	129.528	119.976	79.999	55.000	37.000	86,669.943	6532.429
5	188.357	229.602	151.075	171.912	239.342	158.088	129.982	119.990	77.798	54.576	40.723	89,962.573	7507.212
6	277.420	270.540	161.914	183.738	242.939	159.996	130.000	117.692	79.999	54.583	50.822	105,456.159	9689.460
7	303.679	253.234	215.080	207.820	234.626	159.789	129.996	118.557	80.000	54.731	55.512	110,376.988	11,046.974
8	284.213	276.631	242.805	257.079	232.163	159.480	129.906	119.925	79.154	54.847	60.204	114,622.220	12,591.579
9	329.264	351.175	263.618	277.043	237.413	153.536	129.956	119.969	79.838	54.610	72.424	131,422.312	16,492.225
10	391.022	365.874	298.133	265.150	242.871	159.900	129.864	119.824	75.641	54.994	81.275	143,558.138	20,215.772
11	403.610	398.763	323.390	299.975	233.988	159.970	129.997	112.664	79.995	52.485	88.838	152,258.869	24,291.715
12	415.007	433.955	340.000	295.831	242.986	159.987	129.954	98.248	79.986	47.515	93.474	159,891.196	28,568.055
13	403.752	376.588	323.914	265.781	242.984	159.996	129.998	119.990	79.991	54.733	85.728	148,855.883	22,349.281
14	329.264	351.175	263.618	277.043	237.413	153.536	129.956	119.969	79.838	54.610	72.424	131,422.312	16,492.225
15	284.213	276.631	242.805	257.079	232.163	159.480	129.906	119.925	79.154	54.847	60.204	114,622.220	12,591.579
16	223.433	240.864	174.031	189.920	227.097	159.983	129.867	119.965	79.171	54.950	45.282	95,730.049	8464.578
17	188.357	229.602	151.075	171.912	239.342	158.088	129.982	119.990	77.798	54.576	40.723	89,962.573	7507.212
18	277.420	270.540	161.914	183.738	242.939	159.996	130.000	117.692	79.999	54.583	50.822	105,456.159	9689.460
19	284.213	276.631	242.805	257.079	232.163	159.480	129.906	119.925	79.154	54.847	60.204	114,622.220	12,591.579
20	349.961	338.292	287.234	299.742	242.953	157.864	129.890	119.878	68.079	54.236	76.132	134,994.987	18,062.886
21	329.264	351.175	263.618	277.043	237.413	153.536	129.956	119.969	79.838	54.610	72.424	131,422.312	16,492.225
22	277.420	270.540	161.914	183.738	242.939	159.996	130.000	117.692	79.999	54.583	50.822	105,456.159	9689.460
23	167.768	180.865	146.464	142.876	189.040	159.016	129.864	113.622	79.999	54.997	32.511	80,208.210	5752.844
24	166.459	153.534	105.409	106.892	152.447	159.990	110.254	119.831	79.968	54.989	25.774	71,963.076	4438.329

shows the results of the optimum power dispatch for the 10-unit system over 24 h obtained by CSA. The optimal results of the pollutant power generation and fuel cost obtained by the CSA are depicted in Table 16. The total fuel cost and total emissions computed by the CSA method are illustrated in Table 17. We compared the obtained solutions with those determined by other algorithms recently published in the literature.

By analyzing the results given in Table 17, we infer that the total emissions found by the CSA algorithm, which amount to 2.9354 × 10⁵lb, are lower, compared to that found by IHS [49], MHS in [49], NSGA-II [58], TLBO [61], CRO [62], and HCRO [62], which are estimated at 296,044 lb, 302,093 lb, 304,120 lb, 294,153 lb, 317,400 lb, and 327,500 lb, respectively.

Method GCABC in [60] obtains the less total emission values of 293,416 lb compared with those values obtained by the methods cited in Table 17.

According to Table 17, we notice that the total cost determined by the CSA algorithm which is equal to 2.62594 × 10⁶ USD is lower compared to that found by the NSGA-II [58] algorithm, which is estimated at 2.6563 × 10⁶ USD.

Table 17. Comparison of the 10-unit test system with previous algorithms for best minimum cost solution (Case 5).

Methods	Total Fuel Cost $\left(\times {10}^6\right) \mathbf{U}\mathbf{S}\mathbf{D}$	Total Emission $\left(\times {10}^5\right)\mathbf{I}\mathbf{b}$
CSA	2.625940	2.93540
GCABC [60]	NA	2.93416
TLBO [61]	2.594148	2.94153
IHS [49]	NA	2.96044
MHS [49]	NA	3.02093
NSGA-II [58]	2.656300	3.04120
CRO [62]	NA	3.17400
HCRO [62]	NA	3.27500

Table 17. Comparison of the 10-unit test system with previous algorithms for best minimum cost solution (Case 5).

Methods	Total Fuel Cost $\left(\times {10}^6\right) \mathbf{U}\mathbf{S}\mathbf{D}$	Total Emission $\left(\times {10}^5\right)\mathbf{I}\mathbf{b}$
CSA	2.625940	2.93540
GCABC [60]	NA	2.93416
TLBO [61]	2.594148	2.94153
IHS [49]	NA	2.96044
MHS [49]	NA	3.02093
NSGA-II [58]	2.656300	3.04120
CRO [62]	NA	3.17400
HCRO [62]	NA	3.27500

Figure 12 represents the convergence curves of our proposed approach for the DEnD problem. It describes the stable convergence of the objective function of the problem given in (3) (Case 5). The convergence characteristics shown in Figure 12 prove that CSA has good qualities. We conclude that CSA is favorable for large-scale power systems.

5.2.3. Case 6

In this case, we deal with the dynamic combined economic environmental dispatching (DCEED) by considering the power losses and the valve point-effect loading.

Table 18. The best solutions obtained by CSA for the DCEED of the IEEE 10-unit test system.

Time h	$$\begin{gathered} {\mathit{P}}_1 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_2 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_3 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_4 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_5 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_6 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_7 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_8 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_9 \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_{10} \\ \mathbf{MW} \end{gathered}$$	$$\begin{gathered} {\mathit{P}}_{\mathit{L}} \\ \mathbf{MW} \end{gathered}$$	FC $\mathbf{U}\mathbf{S}\mathbf{D}/\mathbf{h}$	E $\mathbf{I}\mathbf{b}/\mathbf{h}$	TC $\mathbf{USD}/\mathbf{h}$
1	150.406	135.246	85.444	94.209	181.135	122.376	129.280	94.366	20.605	42.622	19.691	61,700.233	3934.478	32,955.958
2	150.059	135.192	148.162	120.431	172.562	122.461	93.065	119.995	22.348	48.148	22.423	65,198.802	4454.610	34,966.155
3	150.093	135.208	101.472	182.259	221.659	149.384	129.872	119.939	55.572	41.141	28.607	72,288.444	5650.903	39,113.119
4	150.060	135.019	190.752	204.190	224.454	159.847	124.653	119.999	79.982	52.633	35.589	80,461.980	7112.270	43,928.263
5	150.231	135.171	241.644	242.567	241.338	159.980	129.383	119.991	52.892	46.297	39.493	84,292.688	9023.037	46,796.853
6	152.020	164.386	320.567	277.526	222.190	159.921	129.914	119.986	80.000	49.903	48.411	94,530.118	12,104.35	53,457.173
7	161.851	223.048	310.663	299.756	235.796	141.568	129.795	119.940	78.244	55.000	53.662	100,886.75	13,127.88	57,150.064
8	150.069	269.756	339.843	299.405	242.988	158.986	124.745	119.937	75.404	53.775	58.903	107,645.16	14,941.13	61,436.135
9	289.083	313.982	339.999	280.201	234.134	156.640	129.595	120.000	78.241	53.458	71.328	125,468.68	16,952.94	71,353.584
10	364.296	316.930	339.330	299.170	242.913	159.675	129.700	119.025	76.855	54.122	80.023	137,757.86	19,776.86	78,914.408
11	391.758	400.840	336.936	293.570	242.999	159.970	129.998	119.999	63.537	54.997	88.603	151,135.50	24,136.97	87,778.694
12	442.035	396.653	337.897	299.108	242.982	157.061	127.818	119.996	79.999	39.772	93.323	158,774.96	28,337.51	93,695.444
13	359.815	391.269	339.912	299.467	242.964	151.758	126.070	118.443	79.987	47.449	85.138	145,326.69	22,318.79	83,964.399
14	289.083	313.982	339.999	280.201	234.134	156.640	129.595	120.000	78.241	53.458	71.328	125,468.68	16,952.94	71,353.584
15	150.069	269.756	339.843	299.405	242.988	158.986	124.745	119.937	75.404	53.775	58.903	107,645.16	14,941.13	61,436.135
16	150.156	135.036	276.528	283.184	240.900	159.073	129.657	120.000	51.168	52.036	43.739	88,890.958	10,861.13	50,016.892
17	150.231	135.171	241.644	242.567	241.338	159.980	129.383	119.991	52.892	46.297	39.493	84,292.688	9023.037	46,796.853
18	152.020	164.386	320.567	277.526	222.190	159.921	129.914	119.986	80.000	49.903	48.411	94,530.118	12,104.35	53,457.173
19	150.069	269.756	339.843	299.405	242.988	158.986	124.745	119.937	75.404	53.775	58.903	107,645.16	14,941.13	61,436.135
20	326.074	347.521	309.290	299.994	242.997	160.000	129.970	119.999	67.957	44.017	75.820	133,249.35	18,331.48	75,929.053
21	289.083	313.982	339.999	280.201	234.134	156.640	129.595	120.000	78.241	53.458	71.328	125,468.68	16,952.94	71,353.584
22	152.020	164.386	320.567	277.526	222.190	159.921	129.914	119.986	80.000	49.903	48.411	94,530.118	12,104.35	53,457.173
23	150.173	137.534	180.818	163.649	238.569	149.328	129.739	117.812	51.515	44.869	32.009	76,243.211	6439.102	41,480.912
24	150.015	135.021	126.085	104.167	220.872	159.474	129.251	119.979	21.132	43.459	25.457	68,625.467	5068.830	36,985.679

Note: FC: Fuel cost, E: Emission and TC: Total cost (FC+E).

summarizes the best compromise solutions achieved for DCEED for a given load.

According to Table 18, we can deduce that CSA adjust the value of the output power ($P_1-P_{10}$) such that load demand and the constraint limits are satisfied, and the best fuel cost and pollutants emission results are achieved. However, the statistical comparative results of CSA method along with various other methods for both cost and emission objective functions are provided in

Table 19. The DCEED comparison results of different algorithms of the 10-unit test system.

Methods	Total Loss MW	Total Fuel Cost $\left(\times {10}^6\right)\mathbf{USD}$	Total Emission $\left(\times {10}^5\right)\mathbf{I}\mathbf{b}$	Total Cost $\left(\times {10}^6\right)\mathbf{USD}$
CSA	1298.9956	2.492050	3.19592	2.843985
MONNDE [53]	1307.8000	2.557900	2.95220	2.769100
NSGA-II [58]	NA	2.521000	3.12460	2.823600
IBFA [59]	1299.8760	2.517117	2.99037	2.753743
TLBO [61]	1301.1900	2.472116	2.94153	2.776644
HCRO [62]	1299.8723	2.517076	2.99065	2.753863
CRO [62]	1298.4666	2.517821	3.01941	2.768615
NSPSO [63]	NA	2.474472	2.93416	2.704316
PSO-CSC [64]	1303.1000	2.524700	3.05240	2.788550

. The minimum cost obtained by the proposed CSA approach is 2,492,050 USD among 50 trials in the economic dispatch problem and considered as the ever best solution. It is also clear from the simulation results that the proposed approach provides better statistical results as compared with many other available methods given in Table 19.

Figure 13 shows the convergence curves of the CEED with time (iterations) obtained by applying the CSA method corresponding to a fixed hour with a power demand of 2150 MW. It is observed from the convergence curve that a better reduction in combined cost and emission can be obtained with less number of iterations using the proposed CSA.

Figure 14 displays the obtained values (best, average, and max values) among 50 runs of the CSA technique for the case of minimizing the fuel cost of production (Case 4).

6. Conclusions and Future Work

In this paper, new optimization approaches were proposed, presented and applied to solve the problem of the dynamic economic emissions dispatch of generator units considering the valve-point loading effects. The main inspiration of these optimization techniques are the fact that metaheuristic algorithms are easy to implement and can be used for a variety of other problems.

The proposed strategies are validated by simulating MATLAB and testing on the two standard IEEE power systems, 5-unit and 10-unit systems. The numerical results of this system are presented to show the capabilities of the proposal algorithms to establish an optimal solution of the dynamics problem of combined economic emissions dispatch in several passages. From Table 4, Table 6, Table 8, Table 14, Table 16, and Table 18, it is obviously clear that the optimal generation schedule of a 5-units and 10-unit system obtained by CSA satisfy power balance constraint with considering power losses and generator operating limits constraint. The proposed CSA gives better performance compared to methods cited in the literature.

In all cases, our proposed algorithms can reach the optimum solution more quickly which. In future works, we intend to combine CSA with TSA, and introduce it to other kinds of optimization issues, such as large-scale economic load dispatch problems, integrated renewable energy sources, multi-objective ED problems with many complex constraints.