A Novel Hybrid Crow Search Arithmetic Optimization Algorithm for Solving Weighted Combined Economic Emission Dispatch with Load-Shifting Practice

Abstract

The crow search arithmetic optimization algorithm (CSAOA) method is introduced in this article as a novel hybrid optimization technique. This proposed strategy is a population-based metaheuristic method inspired by crows’ food-hiding techniques and merged with a recently created simple yet robust arithmetic optimization algorithm (AOA). The proposed method’s performance and superiority over other existing methods is evaluated using six benchmark functions that are unimodal and multimodal in nature, and real-time optimization problems related to power systems, such as the weighted dynamic economic emission dispatch (DEED) problem. A load-shifting mechanism is also implemented, which reduces the system’s generation cost even further. An extensive technical study is carried out to compare the weighted DEED to the penalty factor-based DEED and arrive at a superior compromise option. The effects of CO₂, SO₂, and NOx are studied independently to determine their impact on system emissions. In addition, the weights are modified from 0.1 to 0.9, and the effects on generating cost and emission are investigated. Nonparametric statistical analysis asserts that the proposed CSAOA is superior and robust.

1. Introduction

The generation and use of electrical energy is crucial to modern society. The term economic cost dispatch (ECD) that is associated with the power system, which may minimize the cost of power generating while still satisfying the operating requirements, is a highly essential problem given that thermal power generation is the predominant power generation at present. But, as environmental issues worsen, more people are realizing that we need to consider emissions of dangerous gases such as NO_x, SO₂, and CO₂ in addition to just the generation cost. The cost of producing electricity and the need to reduce emissions are both taken into account by CEED. In common ECD problems, generator cost functions are approximated by quadratic functions. The most frequent form of expressing the CEED issue is as a quadratic function, reflecting the fact that it is a multi-objective optimization problem. Higher order polynomials have been shown to enhance solution methodologies, and studies have indicated that advanced functions can provide an accurate representation of the power generating system’s response; however, this will only serve to further exacerbate the issue at hand. To address CEED issues, some scientists use optimization strategies grounded on conventional mathematical modeling. The traditional approach may be used in several power-generating test environments. Benefits include optimality shown mathematically and the absence of problem-specific factors. The conventional method that relies on a coordination equation to tackle the issue of economic emission load dispatch takes into account the limitation of line flow. Using the Min-Max price penalty factor brings about a decrease in the total fuel expense of CEED. Nevertheless, standard mathematical approaches face significant difficulties in resolving these nonlinear issues because of intrinsic nonconvexity and nonlinearity of the existing power generating system and other constraints in the actual production process. Many evolutionary programming and AI-based strategies have been proposed to address CEED issues, including but not limited to the bat algorithm [1], artificial bee colony [2], teaching-and-learning-based optimization [3], cuckoo search [4], flower pollination algorithm [5], firefly algorithm [6], bacterial foraging optimization [7], genetic algorithm [8], differential evolution [9], and particle swarm optimization [10].

1.1. The Literature Review

Authors of [11] exhibit a mathematical model of the mud ring feeding methodology, and its advantages over other current optimization methods are tested using two case studies. A data-driven surrogate-assisted technique is presented in [12] for dealing with high-dimensional large-scale MACEED challenges. This work, outlined in reference [13], presents a hybrid dynamic economic environmental dispatch model that combines an energy storage device, wind turbines, and solar systems, along with thermal power units. The aim of this model is to stabilize the production of renewable energy sources. The authors in [14] propose an advanced algorithm (the knee-guided algorithm (KGA))to simply addresses the EED problem. In this algorithm, the solution is determined using the lowest Manhattan distance technique, which defines the optimal solution. The suggested algorithm aims on finding a solution near the knee point, improving convergence, and delivering the knee solution rather than the entire POF. This approach increases the accessibility of the algorithm’s results to policymakers in thermal power plants. Study [15] suggests a hybrid optimization solution for the electrical power systems’ multi-objective economic emission dispatch (MOEED) issue. In study [16], researchers provide and execute an adapted iteration of the modified marine predators algorithm (MMPA) to tackle both single- and multi-objective CEED problems. It is recommended that MMPA be used to enhance the efficiency of regular MPA. To forestall the untimely aggregation of knowledge, it incorporates a complete learning strategy in which the best practices of all participants are shared. For reliability-based dynamic economic emission dispatch (DEEDR), the authors of [17] have developed an enhanced NSGA-III (I-NSGA-III). I-NSGA-III incorporates a distinctive crossover operator by using an angle-based connection and normal distribution approach. The authors of article [18] propose a distributed optimization strategy on a hybrid microgrid system to decrease the expenses associated with power generation. The authors of study [19] propose a nondominated sorting genetic algorithm III with three crossover strategies (NSGA-III-TCS) to address the challenges related to combined heat and power dynamic economic emission dispatch (CHPDEED), both with and without forbidden operation zones. In study [20], to handle DEED-PEV, the authors suggest a new NSGA-II (NNSGA-II). In NNSGA-II, the simulated binary crossover operator is swapped out for a Gaussian-based one, and the crossover frequency of each individual is adaptively adjusted depending on its position in the population. In paper [21], the researcher combines demand side management (DSM) and multi-objective dynamic economic and emission dispatch (MODEED) to examine the advantages of DSM on the generation side to minimize economic and emission dispatch problems separately and simultaneously with and without DSM. In paper [22], the competitive swarm optimization (CSO) algorithm, a newly discovered evolutionary method, is used to find the optimal solution for several objectives, including minimizing production costs, carbon emissions, voltage fluctuations, and power losses. The CSO algorithm’s efficiency is measured against that of numerous cutting-edge metaheuristics, including GA, PSO, CSA, ABC, and SHADE-SF. The economic emission dispatch problems for combined heat and power production are tackled in article [23] by proposing an improved version of the bare-bone multi-objective particle swarm optimization (IBBMOPSO). In research paper [24], a primal-dual technique is used to replace the bi-level profit-maximizing model with linear single-level optimization. To address an environmental economic dispatch challenge, the authors of article [25] suggest an ordered optimization outline for the MMEG arrangement, which would allow for the energy coordination strategy to be produced in a decentralized fashion. In research article [26], the researchers suggest a cooperative optimization approach for demand side management (DSM)-assisted grid-tied residential microgrid (MG) planning and operation. A combined model of dynamic economic and emission dispatch (DEED) and DSM with the inclusion of renewable energy resources (RERs) is presented by the researcher in study [27]. Research [28] introduces a novel multi-objective evolutionary algorithm dubbed the improved multi-objective exchange market algorithm to solve the MDEEDP when electric vehicles (EVs) are present, when EV drivers exhibit random behavior, and when WP uncertainty exists. A MOEED model that takes solar power variability into account is presented in [29]. Both underestimating and overestimating solar power are modeled, with the latter case incurring additional costs in operation. The authors of article [30] propose a multi-objective optimization technique to address the problem related to economic emission dispatch in integrated energy and heating systems (IEHS). This strategy considers system uncertainties and incorporates multi-energy demand response (MEDR) and carbon capture power plants (CCPPs). The influence of errors in the investigational source of the input/output characteristics of conventional power plants is studied by the authors of [31]. Taking into account LCA and risk cost, research [32] suggests a low-carbon economic schedule for IES. In article [33], the authors develop a crow search optimization algorithm (CSA) based on swarm intelligence to deal with the difficult restricted MEEDP with the modified predictive element of RES. The author of article [34] conducts an analysis to examine how the inclusion of renewable energy sources, an electric vehicle parking lot, and an integrated demand response program affects the economic emission dispatch of a multi-county combined heat and power system. Using a real-world complex three-county test system, researchers employ the strength pareto evolutionary algorithm 2 and the nondominated sorting genetic algorithm-III. In study [35], the authors present an ECSO method, which stands for extended crisscross search optimization. In order to address the combined heat and power economic emission dispatch (CHPEED) issue in a nonlinear and nonconvex area, the authors of research [36] offer a multi-objective multi-criterion decision-making (MCDM) technique with constraint-handling rules tailored to this specific situation. Study [37] introduces a novel method for incorporating loss prediction using artificial neural networks into the dynamic economic emission dispatch model. Economic power dispatch over a whole 24 h period is posed as an optimization issue in [38]. The load dispatch simulation incorporates the modeling of thermal, water resources, and renewable for demand-side management, aiming to create a realistic situation. To tackle the MO-CHPEED problem in a fuzzy environment, a new developed algorithm is developed by [39]—dynamically controlled whale optimization algorithm (DCWOA)—through which problems can be solved like multi-objective nonconvex optimization. The benchmark functions in line with CEC-BC-2017 is outlined in work [40]. The authors’ first step involves comparing the original AOA with five enhanced versions of AOA in order to choose the most optimal improved strategy. Subsequently, the authors use simulations to compare the improved method with other intelligent optimization algorithms, therefore confirming its effectiveness. In [41], the three-layer optimization issue is solved using the column-and-constraint generation technique (CCG) and, with this, the best scheduling plan and worst case operating domain are achieved. Paper [42] examines the economic emission dispatch issue for many zones of combined heat and power generation and uses a simulated system to test the efficacy of the method. In [43], researchers optimize the energy costs and then compare the outcomes to choose the best hybrid system. After generating mathematical models for typical benchmark functions, TCSC is deployed at weak locations across many echelons in the proposed work of [44,45,46] to improve the system voltage profile. The authors’ proposal in [47,48,49] aims to evaluate the impact of photovoltaic (PV) penetration on active power loss, reactive power loss, and enhancement of voltage profile. This is achieved by assessing the location for PV deployment using the voltage collapse proximity index technique. Therefore, an effort is made to analyze the loading characteristic of the IEEE 14 bus system in conditions of potential voltage collapse, such as when the network is exposed to the use of photovoltaic solar energy sources.

With column headings such as “objective”, “optimization algorithm”, “system description”, and “RES type”,

Table 1. Analysis of the literature survey.

Objective	Optimization Models Used	System Description	Incorporation of RES	Year	Reference
Dynamic economic load dispatch	Bottlenose dolphin optimizer (BDO)	Twenty-nine functions including seven uni-modal functions	PV, WT	2022	[11]
Minimize operation cost and pollution	Improved mayfly optimization algorithm	Thermal power plant along with PV, WT, and BESS	PV, WT	2022	[13]
Minimize operation cost and pollution	Knee-guided algorithm (KGA)	6, 10, and 11 generating units	NA	2022	[14]
Decreasing the emission of greenhouse gases and the fuel cost	Marine predators algorithm (MMPA)	3, 5, 6, and 26 generating units	NA	2022	[16]
Fuel cost, pollutant emission, and system reliability	NSGA-III (I-NSGA-III)	5, 10, and 30 units	NA	2022	[17]
Minimize the total generation cost in a dynamic economic dispatch problem	Distributed management algorithm for DEDP	6 generating units	NA	2023	[18]
Multi-objective dynamic economic and emission dispatch	Multi-objective particle swarm optimization (MOPSO)	6 generating units with DSM implementation	NA	2018	[21]
Reduction of carbon emissions, in addition to low cost and high efficiency	Competitive swarm optimization (CSO) algorithm	Thermal–solar (TS), thermal–wind (TW), and thermal–wind–solar (TWS)	PV, WT	2022	[22]
Environmental economic dispatch problem	Distributed augmented Lagrangian (ADAL) method	4 multi-micro-energy grid systems	Battery	2022	[25]
Minimization of MG total annual cost and total annual emission	Multi-objective optimization	Residential MG consisting of 1000 smart homes with different DSM participation levels	PV, WT, BESS	2019	[26]
Generation side operational benefits and reduction in environmental pollution level	(NSGA-II) and Monte Carlo simulation (MCS)	DSM-based six thermal generating units, one solar-powered generator, and one wind-powered generator	6 MT, along with 1 PV and WT	PV, WT	[27]
Reducing carbon emissions and improving wind power consumption	Multi-objective optimization	IEEE-30 bus power system and a 6 bus district heating system	WT	2022	[30]
Energy sustainability and climatic benefits	Crow search optimization algorithm (CSA)	Six benchmark test systems with multi-dimensional constraints	PV, WT	2022	[33]
Reduction in cost of economic operation and the pollutant emission	Crisscross search optimization (CSO) algorithm	IEEE-30 bus system, 40 generators system, and hydrothermal generation system	NA	2022	[35]
Reduce the overall generation cost of the system.	CSAJAYA	DSM-implemented two microgrid distribution systems	WT	2023	[50]

displays the papers covered in the literature.

1.2. Research Gap and Objective of the Paper

The extensive literature review suggests that combined economic emission dispatch problems have always been an emerging topic for the power system engineers. Researchers have developed various optimization algorithms to solve this problem for decades and some algorithms have proven better than others. The implementation of an economic strategy called demand side management (DSM) is rarely used, but the literature has proven that shifting the loads to lesser priced hours results in a much lower operational cost of a distribution system. This article implements DSM on two dynamic systems and thereafter studies the impact of it on obtaining a balanced and distributed minimal cost and emission on the test structures. An exclusive hybrid CSAOA is provided as the optimization tool of the research study.

1.3. Methodology and Contributions

The aim of the paper is to minimize both cost and emission simultaneously. Cost refers to the fuel cost utilized by the generators in supplying power, and emission refers to the pollutants emitted by the combustion of fossil fuels during the process of producing power. Simultaneous minimization of both cost and emission with proper balance is called combined economic emission dispatch (CEED) and there are 3 ways to evaluate CEED. All generators have maximum and minimum limits within which they operate, and this is called the inequality constraints. The sum of the power outputs of the generators should be exactly equal to the load demand every hour and this is called equality constraint. A proposed hybrid CSAOA is utilized to minimize the cost (ECD) and all the three types of CEED, which are the fitness functions of the work. An economic strategy called DSM is also implemented as a step before CEED to restructure the load demand, which helps in reducing the cost component furthermore. All the mathematical modeling of fitness functions and formation of CSAOA are explained in further sections of the manuscript.

The novel contributions of the work conducted in this article can be listed as follows:

• A unique hybrid CSAOA is suggested as the optimization technique for this study. Prior to using it to address the DEED issue, the suggested methodology is additionally tested on six multi-dimensional and varied modal benchmark functions.
• The test system leverages a combination of conventional generating plants and combined heat and power (CHP) plants. This allows for the evaluation of various dispatch strategies, including dynamic economic dispatch, emission dispatch, and weighted economic emission dispatch, contributing to the optimization of power generation based on cost, emissions, or a balance of both.
• A load-shifting policy called demand side management (DSM) is also considered and all the above-mentioned studies are also conducted in the presence of DSM to highlight its benefits.

The remainder of this paper is set out as follows: in Section 2, we outline the formalization of the issue; in Section 3, we focus on how the suggested hybrid algorithm is used here; the simulation findings are described in Section 4, and this paper is wrapped up in Section 5.

2. Objective Function Formulation

The objective of economic cost dispatch (ECD) is to minimize the cost of electricity generation while ensuring compliance with all relevant fairness and equity requirements. Dynamic economic load dispatch has been implemented to handle changing demand and hourly power schedules.

2.1. Cost Function for DG Units

Fuel costs money. The cost of generating one unit of electricity from a fossil-fuel generator is known as its generation cost. The generating cost function of generating units is represented by a linear quadratic equation. Equation (1) denotes a quadratic equation [50,51].

$$ECD={\displaystyle \sum _t^{24}{\displaystyle \sum _{j=1}^n{\displaystyle ({\alpha }_j{P_{j,t}}^3+b_j{P_{j,t}}^2+c_j{P}_{j,t}+d_j}}})$$

(1)

Multiplying the power output (P_j) of the jth DG unit by the associated costs (a_j, b_j, c_j, and d_j) yields the total system cost. Considering that n is the total number of DG units, the overall price is ECD. The total expenditure for a 24 h period is calculated based on an economic dynamic load, where the variable “t” represents the specific number of hours.

2.2. Emission Dispatch for DG Units

While producing power, alternative fossil-fuel generators release harmful emissions into the environment. Toxic gases, such as nitrogen oxides, sulfur, and carbon, are often emitted into the air in the form of thick black smoke. Scheduling power plants to reduce emissions of hazardous gases is known as “emission dispatch” (EMD). By utilizing Equation (2) and having knowledge of the emission coefficients, one can determine the objective function of the emission dispatch. The total emission can be represented as EMD, while emission coefficients can be represented as α_j, β_j, and γ_j [52].

$$EMD={\displaystyle \sum _{t=1}^{24}{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{C{O}_{2}N{O}_{X}S{O}_2}({\alpha }_j{P_{j,t}}^3+{\beta }_j{P_{j,t}}^2+{\gamma }_j{P}_{j,t}+{\delta }_j}}})$$

(2)

2.3. PPF-Based Combined Economic Emission Dispatch

Fuel cost reduction is the focus of ECD, whereas pollution control is the focus of EMD, with regards to traditional fossil-fuel generators and their impact on the environment. Hence, it is essential to find a mutually acceptable resolution that may effectively achieve the objectives of reducing fuel expenses and mitigating pollution emissions. To combine Equations (1) and (2) into a CEED, we use the price penalty factor (PPF), which is a parameter that incorporates both ECD and EMD in a mixed objective function. This is shown in Equation (3) [50].

$$CEED={\displaystyle \sum _t^{24}{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{C{O}_{2}N{O}_{X}S{O}_2}\begin{array}{l}[(a_j{P_{j,t}}^3+b_j{P_{j,t}}^2+c_j{P}_{j,t}+d_j)+\\ ppf*({\alpha }_j{P_{j,t}}^3+{\beta }_j{P_{j,t}}^2+{\gamma }_j{P}_{j,t}+\delta )]\end{array}}}}$$

(3)

Equation (4) provides the form of price penalty factors (ppf). The jth generator’s maximum values are denoted by $P_j^{\max }$.

$$pp{f}_{j,\max ,\max }=\frac{ECD\left(P_j^{\max }\right)}{EMD\left(P_j^{\max }\right)}$$

(4)

2.4. FP-Based Combined Economic Emission Dispatch

Through the given approach, two objective functions that are incompatible are analyzed here that share common choice and control variables, and resolve them by computing their ratio. For instance, (1) represents the economic dispatch equation, while (2) represents the emission function; together, they are referred to as ECD and EMD. Then, by minimizing the ratio of EMD to ECD, a compromise solution may be reached using the FP technique. Equation (5) provides the numerical expression for this [50].

$$FP=\frac{EMD}{ECD}$$

(5)

2.5. The Proposed Environment-Constrained Economic Dispatch (ECED)

Both of the aforementioned CEED strategies have as their primary goal the lessening of atmospheric pollution. The system’s generating cost increases substantially, surpassing the highest value attained by economic dispatch. Equation (6) shows how to combine two objective functions with distinct aims in order to achieve a better quality compromise solution [50,51]. Whether a system is unimodal or multimodal is determined by (1) and (2). The characteristics of the economic dispatch equation and the emission dispatch equation are given in accordance with this equation.

$$ECED=\mu \left[\frac{ECD-EC{D}_{\min }}{EC{D}_{\max }-EC{D}_{\min }}\right]+\left(1-\mu \right)\left[\frac{EMD-EM{D}_{\min }}{EM{D}_{\max }-EM{D}_{\min }}\right]$$

(6)

where μ is between 0 and 1, ECD_min is the minimum generation cost, EMD_min is the minimum amount of pollutants emitted, ECD_max is the maximum generation cost, and EMD_max is the maximum volume of pollutants released attained by replacing the optimal constraints of EMD_min in Equations (1) and (2). The data acquired [51] also suggest the following three procedures and presumptions:

• Setting μ = 0.5, or providing equal weight to both goal functions, is known to provide the fastest and most effective steps towards obtaining the best compromised solution.
• The choice with the largest quality compromise will have a smallest CPI-EPI difference, where the CPI is cost performance index and EPI is emission performance index. The formulae for CPI and EPI are denoted by Formulas (7) and (8).

  $$CPI=\left[\frac{ECD-EC{D}_{\min }}{EC{D}_{\max }-EC{D}_{\min }}\right]\times 100\%$$

  (7)

  $$EPI=\left[\frac{EMD-EM{D}_{\min }}{EM{D}_{\max }-EM{D}_{\min }}\right]\times 100\%$$

  (8)
• The generating cost of the higher quality compromise will be closer to ECD_min, while the pollution output will be lower than EMD_min.

2.6. Equality and Inequality Constraints

The constraints of equality are given in Equation (9), whereas the constraints of inequality are shown in Equation (10) to ensure the values of the DERs are confined.

$${\displaystyle \sum _{j=1}^n{P}_{j,t}}=D_t$$

(9)

$$P_{j,\min }\le P_j\le P_{j,\max }$$

(10)

where D_t is the power demand at the same tth time and P_j,t represents the power of the generator.

2.7. Utilization Percentage

Equation (11) may be used to obtain the utilization rate.

$$UP=\frac{{\displaystyle \sum _T{P}_j^t}}{24\times P_j^{\max }}$$

(11)

The term “UP” is often used to convey the hourly outputs of test systems with a large number of distributed energy resources (DERs) in a manner that is both clear and comprehensible, where G_j^t represents hourly output with respect to time.

2.8. Demand Side Management

The study of microgrid energy management, particularly focusing on economic operation, has been a significant area of research in recent years and is expected to be a major issue in the future. Nevertheless, the economic operation of a microgrid system remains incomplete until the demand side management (DSM) approach is taken into account. The adoption of DSM would reduce costs across the board for the articles discussed in the review of the relevant literature. DSM priorities shift elastic loads on the studied network to cheaper times of day for the grid. Despite the fact that the overall load demand remains the same at the conclusion of the forecast period, which is characteristically a day, there is a substantial decrease in peak demand, resulting in an increase in the load factor. Load shifting, peak clipping, strategic expansion, strategic conversion, variable load shape, and valley filling are just a few of DSM’s load-shaping tactics.

Listed below are the specific measures that should be taken while putting DSM into practice:

Step 1: provide the dynamic duration of the time T of the load data in hours.

Step 2: enter the TOU price of energy consumption on the market for T hours.

Step 3: enter the total DSM involvement in % (if elastic loads are not specified).

Step 4: categorize the loads as either elastic or inelastic according to the percentage of engagement in DSM.

For example, when we say x% DSM, it means that x% of the hourly load demand may be adjusted based on elasticity, but the remaining (100 − x) % is not adjustable. Optimal planning is required for elastic load requirements.

Step 5: Calculate the minimum and maximum values, and the total, for the inelastic load. It is important to note that the control variables that need to be optimized have a flexible load prerequisite.

Step 6: apply the optimization approach [52],

$$Minimize [{\mathrm{cost}}_{grid}^t\times (load\_elasti{c}^t+load\_inelasti{c}^t)]$$

(12)

where

$$0\le load\_elasti{c}^t\le load\_inelasti{c}^{max}$$

(13)

$$\mathrm{Total} \mathrm{load} \mathrm{demand}={\displaystyle \sum _{t=1}^T({load\_elastic}^t+load\_inelasti{c}^t)}$$

(14)

Step 7: the total load demand model that has been redesigned incorporating the approach of demand side management (DSM) is determined by summing the load demand in terms of inelastic for each hour with the elastic loads with better optimized values.

3. Optimization Techniques

3.1. Crow Search Algorithm

Crows have a clever strategy for discovering hidden food sources. They observe the movements of their fellow crows and closely monitor other bird populations in the area. Once they are alone, they revisit these locations to conduct further investigations. In addition, when a crow’s food is taken by another crow, it becomes more concerned and actively seeks out different locations to avoid having its food stolen again. Furthermore, it utilizes its own specialized knowledge to deter potential intruders. Based on the above, the CSA has been developed by the authors of [53].

The purpose of this metaheuristic is to enable a specific crow to go in the same direction as another crow in order to locate the location of its concealed food supply. This operation should be carried out in such a way that the location of the crow is progressively updated. In addition, the crow is required to migrate to a new area if food is appropriated. According to the theory, there is a d-dimensional environment that contains a great number of crows. Every crow has a memory that recalls the precise location of its hiding place. Every time the iteration is performed, the position of the crow’s hiding area is given. When it comes to positions, this is the best one that a crow has ever held. It is said that every crow recalls the place where it had its most memorable experience. It is common for crows to move around the region in search of better food sources. Consider the possibility that in the subsequent iteration, crow “b” wants to go to the site that was identified by crow “a” that came before it. At this point in the process, the first crow makes the decision to follow another crow to the spot where the second crow is hidden. Because of this circumstance, there are two conceivable outcomes.

Case 1: Crow “b” is unaware of the fact that crow “a” is following. Consequently, the first crow “a” will go towards the location where the food of second crow “b” is hidden. In this scenario, the updated location of the first crow “a” is determined by generating a random integer that follows a uniform distribution between 0 and 1, and multiplying it by the flight length of the first crow “a” during the current iteration.

Case 2: Crow “b” is aware that crow “a” is following. In order to safeguard its cache from theft, crow “b” will deceive crow “a” by relocating to a different spot inside the search zone.

In both instances, the following mathematical configurations are possible [53]:

$$M^{a,iter+1}=\{\begin{array}{l}{M}^{a,iter}+ran{d}_a\times f{l}^a\times (m^{b,iter}-M^{a,iter}), ifran{d}_b\ge A{P}^b\\ a random position,otherwise\end{array}$$

(15)

where M is the crow’s food position. The distance “fl^a” in the above equation represents the direct distance traveled by the ath crow, whereas the random values “rand_a” and “rand_b” follow a stable distribution between 0 and 1. In both Case 1 and Case 2, there is the possibility of automatic updates being applied to the memory, m.

$$m^{a,iter+1}=\{\begin{array}{l}{M}^{a,iter+1}, if f(M^{a,iter+1}) < f(m^{a,iter})\\ m^{a,iter},otherwise\end{array}$$

(16)

where f(.) is the notation used to refer to the function that is responsible for determining fitness.

“fl” is a value that shows how near the search space is to being reached. AP may discuss crows in terms of their awareness probability. The value of an AP might fluctuate between 0 and 1 when it is used as a probability factor. For the purpose of gaining a better understanding of the current situation, AP may make use of the crow’s search approach.

3.2. Arithmetic Optimization Algorithm (AOA)

When used for mathematical calculations, the arithmetic operators [54] imply that the multiplication and divisibility operators are the ones that are most often utilized, yielding diverse values or assessments across different domains, which is important for the investigative search process. However, unlike subtractive and additive operators, the divisive and multiplicative prerequisite aid by swiftly accomplishing the objective owing to their significant dispersion. With the help of these operations, we demonstrate a new function to show how the distributions of different operators are related to one another. Because of this, it is feasible that the optimal solution may be originated by an experimental search, which can be performed through repeated experimentation. Additionally, search controllers based on divisive and multiplicative operators are used to enhance the exploitation phase of the search process by means of improved communication during the optimization phase. The mathematical model for updating the location in AOA is provided below as Equation (17):

$$\begin{array}{l}if r_1>MOA\\ x_{i,j}(P_{\_iter}+1)=\{\begin{array}{l}best(x_j)÷(MOP+\epsilon )\times \left(\left(U{L}_j-L{L}_j\right)\times \lambda +L{L}_j\right),r_2<0.5\\ best(x_j)\times MOP\times \left(\left(U{L}_j-L{L}_j\right)\times \lambda +L{L}_j\right),Otherwise\end{array}\\ else\\ x_{i,j}(P_{\_iter}+1)=\{\begin{array}{l}best(x_j)-MOP\times \left(\left(U{L}_j-L{L}_j\right)\times \lambda +L{L}_j\right),r_3<0.5\\ best(x_j)+MOP\times \left(\left(U{L}_j-L{L}_j\right)\times \lambda +L{L}_j\right),Otherwise\end{array}\end{array}$$

(17)

The control variables’ upper and lower limits are denoted by UL and LL, respectively. As shown in Equations (18) and (19), the MOA and MOP are adjusted with each iteration of the mathematics optimizer (MO). P_iter represents present iteration. x_i,j denotes the jth position of the ith solution at the current iteration, and best (x_j) is the jth position in the best obtained solution so far. The tuning parameters z and λ are assigned the specific values of 5 and 0.5, correspondingly.

$$MOA\left(P_{-}iter\right)=Min+P_{-}iter\times \left(\frac{Max-Min}{M_{-}Iter}\right)$$

(18)

$$MOP\left(P_{-}iter\right)=1-\frac{P_{-}ite{r}^{1/z}}{M_{-}Ite{r}^{1/z}}$$

(19)

Max and Min are the upper and lower limits of the allowable values for MOA. “M_iter” represents the maximum number of iterations. Figure 1 shows the flowchart of AOA.

3.3. Hybrid CSAOA

A prominent algorithm that uses greedy search to continuously increase the fitness function in each iteration has a major impact on the proposed hybrid CSAOA [52,55]. Through the process of substituting the disparity between the upper and lower limits of variables with the optimum solution that is generated from the current iteration of the solution set, Equation (17) of AOA is changed. A representation of the alteration may be seen in Equation (20) below.

$$\begin{array}{l}if r_1>MOA\\ x_{i,j}(P_{\_iter}+1)=\{\begin{array}{l}best(x_j)÷(MOP+\epsilon )\times \left(\left(best(x_j)\right)\times rand\times fl+L{L}_j\right),r_2<0.5\\ best(x_j)\times MOP\times \left(\left(best(x_j)\right)\times rand\times fl+L{L}_j\right),Otherwise\end{array}\\ else\\ x_{i,j}(P_{\_iter}+1)=\{\begin{array}{l}best(x_j)-MOP\times \left(\left(best(x_j)\right)\times rand\times fl+L{L}_j\right),r_3<0.5\\ best(x_j)+MOP\times \left(\left(best(x_j)\right)\times rand\times fl+L{L}_j\right),Otherwise\end{array}\end{array}$$

(20)

The outcome of implementing the projected hybrid CSAOA on several benchmark functions and the subsequent part provides a comprehensive analysis of the findings obtained from this implementation. After that, microgrid systems use the algorithm to execute bilevel DSM for energy management.

3.4. Realization on Benchmark Functions

Metaheuristic algorithms are naturally stochastic, resulting in varying performances across different runs as they strive to find the optimal solution for a given problem. In order to assess the suitability and efficiency of the projected CSAOA algorithm, it is developed and subjected to testing using a predefined set of benchmark functions. In this study, the researchers implement a collection of six benchmark functions utilized by [55] to evaluate the suggested CSAOA approach. The functions F1–F2 are referred to as unimodal functions since they do have just one distinct global optimum instead of a local optimum. These operations determine the efficacy of any multimodal technique in exploiting opportunities. The functions F3-F4 exhibit single global optimum and many local optimal. In order to evaluate the exploratory capabilities of the metaheuristic approach, these functions are of the utmost importance. Additionally, they serve as benchmarks for multimodal optimization. Functions F5 and F6 are the benchmark functions for the fixed dimension multimodal system.

Figure 2 illustrates how well the recommended techniques perform for each of the six functions that serve as benchmarks, which are F1 through F6. A representation of the convergence characteristics of the proposed techniques is shown in Figure 2 for a range of benchmark functions in proportion to the total number of iterations and shows how these characteristics change over time. The convergence characteristics are graphed by calculating the average values of the best solutions from 30 distinct runs in each iteration. The solution that has been proposed demonstrates a mixture of behaviors from a large number of integrated algorithms, which ultimately results in a system performance that is very efficient.

The suggested approach is used to conduct a statistical analysis on the benchmark functions, and the results are provided in

Table 2. Specifications of the benchmark functions [54].

Function	Dim	Range	fmin
$f_1\left(x\right)={\displaystyle \sum _{i=1}^n{x}_i^2}$	30	[−100, 100]	0	Unimodal Benchmark Functions
$f_2\left(x\right)={\displaystyle \sum _{i=1}^n\left|x_i\right|+{\displaystyle {\prod }_{i=1}^n\left|x_i\right|}}$	30	[−10, 10]	0
$f_3\left(x\right)={\displaystyle \sum _{i=1}^n-x_i\sin \left(\sqrt{\left|x_i\right|}\right)}$	30	[−500, 500]	−418.9829 × 5	Multimodal Benchmark Functions
$\begin{array}{l}{f}_4\left(x\right)=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{x}_i^2}}\right)-\\ \exp \left(\frac{1}{n}{\displaystyle {\sum }_{i=1}^n\cos \left(2\Pi x_i\right)}\right)+20+e\end{array}$	30	[−32, 32]	0
$f_5\left(x\right)=-{{\displaystyle \sum _{i=1}^5\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}}^{-1}$	30	[0, 10]	−10.1532	Fixed Dimension Multimodal Benchmark Functions
$f_6\left(x\right)=-{{\displaystyle \sum _{i=1}^{10}\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}}^{-1}$	30	[0, 10]	−10.5363

. By virtue of the same algorithm boundaries, the new method has been recommended with better comparison with other techniques, where the total population has been considered as 100 and 300 number of iterations.

Table 3. Statistical study of benchmark function results using “CSA”, “AOA”, and “CSAOA”.

Function	Unimodal benchmark functions	F1			F2
Algorithm		CSA	AOA	CSAOA	CSA	AOA	CSAOA
Best		82.18	117.14	0.0096	4.19	3.58	0.01
Worst		332.1716	1061.23	0.14	9.00	14.25	0.07
Mean		150.3337	409.08	0.03	6.02	6.60	0.04
SD		48.6507	246.52	0.02	1.13	2.50	0.01
Function	Multimodal benchmark functions	F3			F4
Algorithm		CSA	AOA	CSAOA	CSA	AOA	CSAOA
Best		−7320.13	−5886.74	−7915.45	5.42	4.17	0.01
Worst		−1967.51	−3502.98	−2871.64	9.53	20.55	0.12
Mean		−4414.72	−4454.94	−5513.72	6.76	8.88	0.04
SD		1691.85	536.98	1076.63	0.81	5.46	0.01
Function	Fixed-dimension multimodal benchmark functions	F5			F6
Algorithm		CSA	AOA	CSAOA	CSA	AOA	CSAOA
Best		−10.14	−9.81	−10.15	−10.53	−10.40	−10.53
Worst		−2.62	−2.37	−2.68	−2.41	−2.32	−2.42
Mean		−7.29	−6.03	−6.41	−8.52	−8.04	−6.23
SD		3.55	2.95	3.79	3.36	2.57	3.83

consists of the mean value (F_mean), standard deviation (FSD) as given in (21) and (22), and best (F_best) and worst (F_worst) optimal values obtained over 25 individual runs.

$$F_{mean}=\frac{{\displaystyle \sum _{t=1}^N{f}_x}}{N}$$

(21)

$$F_{SD}=\sqrt{{\frac{{\displaystyle \sum _{t=1}^N\left(f_x-F_{mean}\right)}}{N}}^2}$$

(22)

Let N represent the total number of distinct runs, which in this case is 30.

Based on the statistical data shown in Table 3, it is evident that the suggested CSAOA algorithm demonstrates good performance across various kinds of functions. The uniformity of the various methods may be assessed by generating a boxplot over several iterations, which visually displays the range of ideal values achieved in each iteration. A boxplot displays the highest and lowest values as cross symbols at the top and bottom, respectively. The rectangular box represents the range where 50% of the data are located. It is obvious how the suggested methods prevent users from becoming stuck in local optimums and how the distribution of optimal values acquired over runs may be viewed.

4. Case Study and Proof of Concept

Two distribution test systems are considered for the evaluation of ECD, EMD, and ECED with and without incorporating DSM strategy. Fossil-fuel generators consider the valve point loading effect, resulting in cost and emission fitness functions that are both nonconvex and nonlinear. The whole of the work conducted in this study is divided into three distinct parts. The first step involves incorporating demand side management (DSM) into the projected load demand model, taking into account the willingness of 40% of consumers to engage in the DSM approach. ECD and EMD are evaluated in the second stage regarding both load demand models, both with and without the DSM strategy being implemented. ECED is evaluated in the third stage to obtain a balanced trade-off result among minimum generation cost and pollutants emitted. An algorithm that was developed recently (AOA) and its variations serve as the optimization tool for this research. The following section provides a detailed discussion of the findings achieved, which are directly related to the issues outlined in Section 2.

4.1. Test System 1

Table 4. Conventional and 3 CHP generator cost and emission coefficients.

Power-Only Generators i	1	CHP-Based Generators	1	2	3
a (USD/MW3)	2.55 × 102	a (USD/MW3)	1.25 × 103	2.65 × 103	1.57 × 103
b (USD/MW2)	7.70 × 100	b (USD/MW2)	3.60 × 101	3.45 × 101	2.00 × 101
c (USD/MW)	1.72 × 10−3	c(USD/MW)	4.35 × 10−2	1.04 × 10−1	7.20 × 10−2
d (USD)	1.15 × 10−4	d (USD)	6.00 × 10−1	2.20 × 100	2.30 × 100
α (tons/MW2)	4.09 × 10−4	α (tons/MW2)	2.70 × 10−2	2.50 × 10−2	2.00 × 10−2
β (tons/MW2)	−0.0005554	β (tons/MW2)	1.10 × 10−2	5.10 × 10−2	4.00 × 10−2
γ (tons)	6.49 × 10−4	γ (tons)	1.65 × 10−3	2.20 × 10−3	1.10 × 10−3
Pj,max (MW)	1.35 × 102
Pj,min (MW)	3.50 × 101

shows the operating limits, cost, and emission coefficients of the CHP and fossil fueled units that deliver power to the load demand of distribution test system 1. The hourly load demand data for this test system are gathered from article [19].

The load model for the first stage has been changed to account for 40% of the loads participating in the DSM approach. During the restructuring, the elastic loads are optimally transferred using the proposed technique. It is determined that a DSM participation level of 40% would result in the total demand, mean demand, peak demand, and load factor computed. Peak demand is lowered by up to 16% when the DSM technique is implemented, according to

Table 5. Advantages of DSM implementation.

	Without DSM	With DSM
Total Demand (kW)	7171	7171
Mean Demand (kW)	298.7	298.7
Peak Demand (kW)	399	334.62
Reduction in Peak (%)	-	16.14%
Load Factor	0.7489	0.8929

. Similarly, a change in the distribution of elastic loads lead to an increase in the load factor, which increases from 0.7489 to 0.8929. Including a DSM approach in the operation of a distribution system can bring various advantages. The total and average demand of the distribution system remains the same both with and without the presence of DSM, which is another important point to take into consideration. Figure 3 depicts the adjusted load model for different levels of DSM participation.

Each fitness function is minimized seriatim using the proposed CSAOA for both the load profiles, and their minimal values obtained are recorded and displayed in Table 3. The results obtained in

Table 6. Fitness function values with proposed CSAOA.

	Without DSM		With DSM
Fitness Function	Cost (Thousands of USD)	Emission (tons)	Cost (Thousands of USD)	Emission (tons)
ECD	296.744	286	293.098	287
EMD	359.919	54	367.956	31
ECED (µ = 0.5)	324.066	134	322.365	133

point toward the following inferences:

• The minimum generation cost is found to be USD 296,744 using CSAOA, which is further reduced to USD 293,098 when DSM integrated load demand is accounted for.
• The minimum emission is 54 tons and 31 tons without and with DSM, respectively.
• When ECED is assessed to find a trade-off solution between least cost and pollutants (using µ = 0.5), the solution set is found to be (USD 32,4067, 134 tons) which is further improved to (USD 322,365.5, 133 tons) when DSM incorporated load demand model is considered.

Thereafter, the hourly contributions of the four DERs are analyzed during the minimization of ECD, EMD, and ECED (with µ = 0.5). Figure 4 above shows the hourly output of CHP1, CHP2, CHP3, and G1. The DERs (G1) with less values of cost coefficients are utilized more during ECD minimization and the DERs (CHP1) with less values of emission coefficients are utilized more during EMD minimization. When ECED is minimized giving equal weightage to both cost and emission, a balanced amount of all the DERs is utilized to deliver the power during every hour. Figure 5 represents the percentage utilization of DERs during different fitness functions, as discussed in Figure 4. Only 33% of the total capacity of CHP1 is utilized when ECD is minimized, as the cost coefficients of CHP1 are high, whereas almost 100% of the total capacity of G1 is utilized for the same, as G1 has low-cost coefficients. On the contrary, CHP1 does not emit any harmful toxic pollutants during its operation, whereas fossil-fueled G1 has high emission coefficients. Hence, when EMD is minimized, 90% of CHP1 and only 27% of G1 is utilized. CHP3 is utilized to the maximum extent for all the fitness functions, as it has both low cost and emission coefficients compared with the rest.

Table 7. Measure of central tendencies for algorithms when ECED is minimized.

	Best	Worst	Mean	Hits	STD	Time (s)
CSA [S]	0.37458	0.39503	0.380715	21	0.009532	8.0068
AOA [S]	0.37299	0.38851	0.378163	20	0.007441	7.8993
CSAOA [P]	0.37282	0.37662	0.373707	23	0.001635	7.0934

shows the measures of central tendencies when ECED was evaluated for 30 individual trials using CSA, AOA and CSAOA. Hits refer to the number of times the minimum value of ECED was obtained among 30 trials. Lower values of standard deviation indicate the robustness of the algorithm.

4.2. Test System 2

Test system 2, which is used to assess the effectiveness described in Section 2, comprises six fossil-fueled generators (FFGs).

Table 8. Generator power limit and fuel cost factor.

Unit	ai	bi	ci	di	Pi,min	Pi,max
	USD/MW	USD/MW	USD/MW	USD/MW	(MW)	(MW)
P1	1.00 × 10−1	9.20 × 10−2	1.45 × 10−1	−1.36 × 10−1	5.00 × 10−1	2.00 × 100
P2	4.00 × 10−1	2.50 × 10−2	2.20 × 10−1	−3.50 × 10−3	2.00 × 10−1	8.00 × 10−1
P3	6.00 × 10−1	7.50 × 10−2	2.30 × 10−1	−8.10 × 10−2	1.50 × 10−1	5.00 × 10−1
P4	2.00 × 10−1	1.00 × 10−1	1.35 × 10−1	−1.45 × 10−2	1.00 × 10−1	5.00 × 10−1
P5	1.30 × 10−1	1.20 × 10−1	1.15 × 10−1	−9.80 × 10−3	1.00 × 10−1	5.00 × 10−1
P6	4.00 × 10−1	8.40 × 10−2	1.25 × 10−1	−7.56 × 10−2	1.20 × 10−1	4.00 × 10−1

,

Table 9. Maximum SO₂ penalty factor and emission coefficient in 6 generator sets.

Unit	Emission Coefficients of SO2				Penalty Factor of SO2
	αSO2	βSO2	γSO2	δSO2	hs
	tons/kW	tons/kW	tons/kW	tons/kW	tons/kW
1	5.0000 × 10−4	1.5000 × 10−1	1.7000 × 101	−9.0000 × 101	1.0852 × 100
2	1.4000 × 10−3	5.5000 × 10−2	1.2000 × 101	−3.0500 × 101	1.0616 × 100
3	1.0000 × 10−3	3.5000 × 10−2	1.0000 × 101	−8.0000 × 101	2.1051 × 100
4	2.0000 × 10−3	7.0000 × 10−2	2.3500 × 101	−3.4500 × 101	5.9760 × 10−1
5	1.3000 × 10−3	1.2000 × 10−1	2.1500 × 101	−1.9750 × 101	6.7720 × 10−1
6	2.1000 × 10−3	8.0000 × 10−2	2.2500 × 101	−2.5600 × 101	6.1920 × 10−1

,

Table 10. Maximum NO_x penalty factor and emission coefficient in 6 generator sets.

Unit	Emission Coefficients of NOx				Penalty Factor of NOx
	αNOX	βNOX	γNOX	δNOX	hn
	tons/kW	tons/kW	tons/kW	tons/kW	tons/kW
1	1.2000 × 10−3	5.2000 × 10−2	1.8500 × 101	−2.6000 × 101	9.4070 × 10−1
2	4.0000 × 10−4	4.5000 × 10−2	1.2000 × 101	−3.5000 × 101	1.4962 × 100
3	1.6000 × 10−3	5.0000 × 10−2	1.3000 × 101	−1.5000 × 101	1.3870 × 100
4	1.2000 × 10−3	7.0000 × 10−2	1.7500 × 101	−7.4000 × 101	8.3080 × 10−1
5	3.0000 × 10−4	4.0000 × 10−2	8.5000 × 100	−8.9000 × 101	2.1705 × 100
6	1.4000 × 10−3	2.4000 × 10−2	1.5500 × 101	−7.5000 × 101	1.0930 × 100

and

Table 11. Maximum CO₂ penalty factor and emission coefficient in 6 generator sets.

Unit	Emission Coefficients of CO2				Penalty Factor of CO2
	αCO2	βCO2	γCO2	δCO2	hc
	tons/kW	tons/kW	tons/kW	tons/kW	tons/kW
1	1.5000 × 10−3	9.2000 × 10−2	1.4000 × 101	− 16.00	7.8230 × 10−1
2	1.4000 × 10−3	2.5000 × 10−2	1.2500 × 101	− 93.50	1.1895 × 100
3	1.6000 × 10−3	5.5000 × 10−2	1.3500 × 101	− 85.00	1.4356 × 100
4	1.2000 × 10−3	1.0000 × 10−2	1.3500 × 101	− 24.50	1.1333 × 100
5	2.3000 × 10−3	4.0000 × 10−2	2.1000 × 101	− 59.00	7.4560 × 10−1
6	1.4000 × 10−3	8.0000 × 10−2	2.2000 × 101	− 70.00	7.1580 × 10−1

display the DER limitations, associated costs, SO₂, NO_X, and CO₂ emission coefficients, and their corresponding penalty factors, as obtained from reference [40]. The peak load demand is 225 MW. This study aims to evaluate the list of fitness functions described in earlier sections. The arithmetic optimization algorithm (AOA) and the crow search algorithm (CSA) in conjunction with the planned CSAOA are used in order to test the fitness functions. For each of the algorithms, the population size is set at 100, and the fitness function is set at 1000. The codes have been evaluated in MATLAB 2019a environments with a laptop configuration of Intel Core i5 8th generation, 8 GB RAM.

Along with the integration of the demand side management (DSM) technique, as outlined in Section 2, utilizing AOA, the test system’s expected load demand is adjusted. Figure 6 displays the recently reorganized load requirement of the system. DSM offers a significant advantage, resulting in a 12.46% reduction in peak load, from 225 MW to 207 MW, and an increase in the load factor from 0.8291 to 0.8996. The statistical data are additionally documented in

Table 12. Beneficial effects of DSM applications.

	Without DSM	With 40% DSM
Peak Load Demand (MW)	225	207.385
Ave. Load Demand (MW)	186.56	186.56
Total Load Demand (MW)	4477.5	4477.4990
Load Factor	0.8291	0.8996
Reduction in Peak (%)	Reference	7.829%

. It is significant to note that in both the pre- and post-DSM scenarios, at the end of the day, total and average load are the same.

The description of results as displayed in

Table 13. Outcomes of various fitness functions when evaluated using CSAOA.

	Cost and Emission Profiles	Without DSM	With DSM
ECD	Cost minimization (thousands of USD)	76.085	74.774
EMD	Emission minimization (tons)	240	236
CEED (PPF)	Cost (thousands of USD)	314.678	310.264
	Emission (tons)	256	254
CEED (FP)	Cost (thousands of USD)	90.775	92.192
	Emission (tons)	249	244
ECED (µ = 0.5)	Cost (thousands of USD)	78.304	78.064
	Emission (tons)	246	243

( the outcome of implementing CSAOA on all the equations mentioned in Section 2) is as follows:

(a)

Cost (ECD) minimization performed on test system 2. The minimum generation cost of the system is found to be USD 76,085 without DSM, which further reduces to USD 74,774 with DSM, respectively. Figure 7 and Figure 8 illustrate, respectively, the hourly production of distributed energy resources (DERs) in the graphical representation, when the cost of generating is USD 76,085 and USD 74,774. On comparing Figure 6, Figure 7 and Figure 8, the hourly load pattern can be easily traced, meaning that the total load demands every hour are fulfilled by the DERs.

(b)

Emission (EMD) minimization is performed on test system 2. The minimum emissions of the plant could be reduced to 236 tons ultimately when DSM is implemented, and the load demand is restructured.

(c)

CEED is performed as per the PPF method and FP method. With the PPF method, the cost and emission combination is USD 314,678, 256 tons and, with the FP method, the cost and emission combination is found to be USD 90,775, 249 tons. These values produce much more improved outcomes when the load demand is reorganized utilizing the DSM approach. It can be clearly seen that the FP method of CEED is a better option for obtaining a minimum cost and emission combination compared with PPF-based CEED.

(d)

Weighted combined economic emissions dispatch with equal weightage to both cost and emissions function is found to be the best measure of CEED, with a minimum value of USD 78,304 and 246 tons for cost and emissions, respectively. These values, as mentioned above, reduce further for the DSM-based load demand model to USD 78,064 and 243 tons. The hourly output of DERs is shown in Figure 9 and Figure 10, where the weighted ECED is used as the fitness function. Both the cost and emission functions are given equal weights. Similarly, Figure 11 and Figure 12 display the hourly output of DERs when PPF-based CEED is considered as the fitness function.

(e)

Thereafter, the minimization of CO₂, SO₂, and NO_x are individually considered as EMD fitness functions and are minimized one at a time. When CO₂ is individually minimized to 75 tons, the values of SO₂ and NO_x are 93 tons and 77 tons, which add to an amount of 246 tons of total emission value. Similarly, SO₂ and NO_x are minimized individually, and all the results are mentioned in

Table 14. Emission minimization (tons).

	CO2 Minimum	SO2 Minimum	NOx Minimum	Minimum Emission
CO2	75	80	88	78
SO2	93	84	97	86
NOx	77	75	66	71
Overall	246	240	252	236

. The values of CO₂, SO₂, and NO_x when the total emission is minimized is also shown in Table 14. Figure 13a–d are the pie-chart representations of Table 14, which highlights the shares of individual emission components as a part of the total emission. The average share of SO₂, CO₂, and NO_x are 37%, 33%, and 30%, respectively.

Both 2D and 3D curves are plotted for cost and emission values of test system 2, with different weightage values ranging from 0.1 and 0.9, and the same is displayed in Figure 14 and Figure 15. The 3D graph shows the coordinates of a balanced compromised solution between cost (USD 78,060) and emission (243 tons), which is obtained when µ = 0.5. µ is represented using the word “mu” in the figure.

Measures of central tendencies are evaluated and displayed in

Table 15. Statistical analysis of generation costs for test system 2 after 30 trials.

	Minimum Attained Cost (USD)	Maximum Attained Cost (USD)	Average Attained Cost (USD)	Standard Deviations	Hits to Minimum Cost	Execution Time (s)
CSA	74,792	74,829	74,806.80	18.4361	18	5.06
AOA	74,775	74,822	74,789.10	21.9063	21	4.05
CSAOA	74,774	74,782	74,775.60	3.2547	24	3.20

to statistically analyze the performance of the proposed CSAOA with CSA and AOA. The economic load dispatch equation is minimized in 30 individual trials using all the three algorithms and thereafter the minimum cost, maximum cost, and total number of times the minimum cost is yielded by each algorithm is recorded.

The suggested CSAOA has a greater level of robustness, as shown by the smallest value of the standard deviation and the maximum hits. Figure 16 shows the boxplot that is drawn using the data mentioned in Table 15. The cost convergence characteristics in Figure 17 show the change in generation cost with the change in iterations for various algorithms. A sensitivity assessment is also operated to comment on the effects of change in tuning parameters fl, and z. fl is changed from 1.5, 2 and 2.5 for z ranging between 1, 5, and 9, as shown in

Table 16. Sensitivity analysis for tuning parameters of CSAOA.

Scenario	fl	z	Cost with DSM (USD)	Execution Time (s)
1	1.5	1	74,774.7027	4.8003
2	2	5	74,774.6995	4.8003
3	2.5	9	74,774.6984	4.8003
4	1.5	1	74,774.7054	3.2005
5	2	5	74,774.6986	3.2005
6	2.5	9	74,774.6987	3.2005
7	1.5	1	74,774.7075	9.6006
8	2	5	74,774.9023	9.6006
9	2.5	9	74,774.6983	9.6006

. The generation cost remains unchanged, whereas the effect of change in tuning parameters is only shown in the execution time.

5. Conclusions

The purpose of this article was to build a unique hybrid CSAOA as an optimization tool, which was then applied to address CEED issues after the same was realized on six benchmark functions. CSAOA exhibited good exploration and exploitation capabilities and maintained an adaptive balance between both. Proposed CSAOA also showcased a better convergence rate compared with other existing well-known optimization techniques. Sensitivity analysis performed on CSAOA claimed that it was least affected by change in tuning parameters. The generating cost of the system was reduced from USD 76,085 to USD 74,774 because of DSM initiatives. In addition to this, it improved the load factor of the system and reduced the peak demand from 223 MW to 207 MW. Weighted DEED proved to be a better and economic compromised solution compared with penalty factor-based DEED. Among the limitations used in this paper, voltage profile improvement and transmission loss minimization could have been incorporated, along with the reduction of cost and emissions for both the test systems. Also, DSM implementation requires a set of hourly electricity market prices that is usually available while solving microgrid energy management problem. In this case, since it was not available for the subject test systems, the same had to be collected from the literature.

As an opportunity of forthcoming research, the robustness of the recommended algorithm can be implemented on complex distribution systems such as a microgrid, wherein the effect of battery energy storage systems, renewable energy sources, etc., can be analyzed.