Optimizing Economic Dispatch with Renewable Energy and Natural Gas Using Fractional-Order Fish Migration Algorithm

Abstract

This work presents a model for solving the Economic-Environmental Dispatch (EED) challenge, which addresses the integration of thermal, renewable energy schemes, and natural gas (NG) units, that consider both toxin emission and fuel costs as its primary objectives. Three cases are examined using the IEEE 30-bus system, where thermal units (TUs) are replaced with NGs to minimize toxin emissions and fuel costs. The system constraints include equality and inequality conditions. A detailed modeling of NGs is performed, which also incorporates the pressure pipelines and the flow velocity of gas as procedure limitations. To obtain Pareto optimal solutions for fuel costs and emissions, three optimization algorithms, namely Fractional-Order Fish Migration Optimization (FOFMO), Coati Optimization Algorithm (COA), and Non-Dominated Sorting Genetic Algorithm (NSGA-II) are employed. Three cases are investigated to validate the effectiveness of the proposed model when applied to the IEEE 30-bus system with the integration of renewable energy sources (RESs) and natural gas units. The results from Case III, where NGs are installed in place of two thermal units (TUs), demonstrate that the economic dispatching approach presented in this study significantly reduces emission levels to 0.4232 t/h and achieves a lower fuel cost of 796.478 USD/MWh. Furthermore, the findings indicate that FOFMO outperforms COA and NSGA-II in effectively addressing the EED problem.

1. Introduction

As a reaction to diverse environmental, social, economic, technical, and political concerns, there has been a notable growth in the capacity of renewable energy systems (RESs) within contemporary power grids [1,2]. Consequently, the environmental economic dispatch (EED) problem has emerged as a crucial optimization challenge in power system operation and planning. This is particularly significant due to the growing apprehension about global climate change and environmental contamination associated with traditional fossil fuel-based electricity production [3,4]. In the realm of power generation, the traditional EED issue revolves around determining the optimal output levels of generating units to meet the load demand while minimizing operational expenses. Simultaneously, the problem must adhere to system constraints and address environmental concerns to minimize pollution from thermal power plants. Mathematically, the EED issue is non-convex [5,6], involving inconsistent objectives and non-linear restrictions derived from power flow limitations, grid compliance requirements, valve point impacts, and prohibited operating zones for units. Efficient algorithms are necessary to find the optimal trade-off between environmental and economic considerations. Moreover, the integration of RESs into electricity systems further adds complexity to the problem [7,8].

In the last decades, there has been a growing trend of integrating cost and emissions considerations in the development and management of electric systems, leading to the emergence of the EED challenge [9]. The objective of EED is to determine the optimal allocation of power generation among different generating units that are both cost-effective and environmentally responsible. However, EED poses significant difficulties as the fuel cost function of generators often exhibits nonlinearity and non-convexity. To tackle this challenge, various optimization methods have been proposed in the literature. These approaches can be grouped into three classifications:

• Conventional techniques that address convex problems using established approaches.
• Unconventional methods that focus on practical and non-convex problems, offering alternative solutions.
• Hybrid techniques overcome the limitations of individual nonconventional algorithms by combining their strengths to solve complex problems. These hybrid methods have demonstrated success in obtaining globally optimal solutions for EED problems under various constraints.

Iterative or conventional techniques are commonly used as conventional approaches in addressing the EED problem. These techniques include the linear programming approach [10], gradient iterative approach [11], quadratic programming approach [12], Lagrange’s method (LM) [13,14], hybrid mixed-integer linear programming and the interior point technique [15], and Newton–Raphson approach [16]. In a study [17], the EED issue has been reformulated into a quadratic programming problem, which can be further expressed as a semidefinite programming formulation. Through the utilization of convex iteration and branch-and-bound techniques, this problem can be solved iteratively. However, the efficiency of these techniques heavily relies on the initial conditions, often requiring a large number of iterations to achieve satisfactory results, and they may struggle to converge to a global solution. Consequently, these methods are quite intricate, involving computational complexities and complex mathematical expressions. As a result, unconventional methods have emerged as effective alternatives for addressing the optimal EED problem.

In contrast, researchers have proposed various metaheuristic methods as alternative approaches to tackle this challenge. The advantages of metaheuristic algorithms have been well-established in addressing complex optimization problems [18]. In a study [19], the authors introduced a varied version of the sine cosine optimization (SCO) algorithm to solve the optimal EED problem. They made improvements in two key aspects. Firstly, an enhanced elite leadership technique was incorporated throughout the particle position update phase, resulting in improved search capabilities of the algorithm. Secondly, a combination of crossover and optimal choices was utilized to block the algorithm from being trapped in local optima. Additionally, a dimension-by-dimension variation technique was implemented to enhance the optimization accuracy of SCO and increase population diversity. The presented algorithm was tested on two systems: the IEEE 30-bus and the 10-unit test schemes [19]. Another approach, inspired by human behavior during search and rescue operations, was employed in addressing the EED problem using the search and rescue optimizer [20]. Furthermore, a squirrel search optimization method with function was projected to solve the EED issue [21]. The weighted sum method was utilized to convert the multi-objective function into a single-objective function, demonstrating the efficiency of the algorithm in resolving the multi-objective power system optimization issue through experimental tests. Additionally, in a different study [22], the Monte Carlo method was applied to a power dispatching strategy for the baseline case. This strategy prioritized wind and solar PV generators to assist hydropower plants in meeting demand while considering the loss of load probability (LOLP) and expected demand not supplied (EDNS).

The genetic algorithm (GA) has been projected as a solution to the optimal EED problem [23]. However, dependent on the size of the scheme under analysis, the computational time required for execution may lead to suboptimal solutions. In a study [24], the Moth Swarm Algorithm was tested on two EED test systems comprising a combination of thermal and PV plants across a 24 h period, while considering spinning reserve allocation. To address the coupled EED problem, both GA and particle swarm optimization (PSO) were proposed [25]. These methods were implemented on a self-sufficient power plant in Pakistan, considering varying load requirements. The results obtained confirmed that PSO outperformed GA. In another study [26], the authors introduced a multi-objective learning backtracking search approach for the EED issue. This method integrated leader selection and guidance as learning methodologies to enhance the uniformity and diversity of the Pareto frontier. Furthermore, for the coupled EED problem, a multi-objective membrane search approach was proposed [27]. The evaluation of this algorithm across multiple test cases demonstrated its high spatial detection capability and ability to generate improved solutions for the EED problem.

In order to address the EED issue, a varied version of the shuffle frog-jump approach (SFLA) was proposed [28]. This modified version incorporates a local search method based on the inertia equation, which enhances the performance of the original SFLA. Furthermore, the integration of crossover and mutation operators from the GA improves the global search process. In a separate study [29], the Shark Optimization Algorithm (SOA) was developed to allocate dispersed generations in distribution systems. The objective was to minimize losses while simultaneously improving bus voltage values and stability. However, this analysis did not consider emissions reduction. To tackle the EED problem, a hybrid approach combining a modified version of the artificial bee colony technique and a Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) was suggested [30]. This hybrid method was employed in three schemes with 10, 20, and 40 producing units, demonstrating its effectiveness. Another proposed hybrid strategy, presented in [31], combines the particle diversity of PSO with the quick convergence of differential evolution (DE) techniques. The crossover operators from GA were also incorporated into this hybrid method. A parameter adaptive control technique was manipulated to modernize the crossover probability and improve optimization findings. Additionally, the issue of multi-objective optimization was addressed by introducing a penalty factor.

In order to predict future load conditions, a functional link artificial neural network (ANN) model based on single-layer black widow optimization was presented by the authors in [32]. Additionally, a nondominated sorting multi-objective instructional learning-based optimizer was presented to address the EED issue in a grid with a PV source, taking into account the anticipated future load. The effectiveness of the recommended method was assessed under two distinct circumstances with varying sun radiation. In a study [33], an ANN controller was utilized to install a photovoltaic distributed generator in an IEEE 33-bus distribution feeder. However, it should be noted that the ANN-based technology provided was only suitable for a single unit. Furthermore, various optimization approaches have been suggested to tackle the EED issue. These methods include the flower pollination algorithm [34], pigeon-inspired optimizer [35], improved bare-bone multi-objective PSO [36], lightning flash algorithm [37], chameleon swarm algorithm [38], niching penalized chimp algorithm [39], multi-verse optimization algorithm [40], bat algorithm [41], teaching–learning-based algorithm [42], hybrid crow search–JAYA algorithm [43], simulated annealing algorithm [44], improved marine predators algorithm [45], osprey optimization algorithm [9], quantum-behaved bat algorithm [46], hybrid differential evaluation and crow search approach [47], adaptive Hooke and Jeeves algorithm [48], mantis search algorithm [49], and hybrid heap-based and jellyfish search approach [50].

In this research, the conventional IEEE 30-bus system was modified to incorporate a restricted set of thermal units (TUs), allowing for the integration of solar PV, natural gas (NG), and wind generation units. The study extensively examines the stochastic characteristics of RESs such as wind and solar. Probability density functions (PDFs) including Weibull and lognormal distributions are utilized to capture their uncertainty. To account for the variability and intermittency of these RESs, the proposed cost model includes penalty costs for underestimation and reserve costs for overestimation. Subsequently, a restricted optimization issue is developed to minimize the EED issue while satisfying various constraints. These constraints encompass power flow equality and inequality limits, NG limitations, and prohibited operating zones (POZs) bounds. To solve this EED issue, three strategies are employed: Fractional-Order Fish Migration Optimization (FOFMO) [51], Coati Optimization Algorithm (COA) [52], and NSGA-II [53]. It is worth noting that most multi-objective optimization techniques are typically tailored for unconstrained optimization problems. The key contributions of this study can be summed up as follows:

• Formulation of the EED issue considers a combination of thermal, solar, wind, and NG units.
• Detailed stochastic study of RESs using appropriate PDFs.
• Inclusion of all system constraints, including security, power flow conditions (equality and inequality), POZs, and NG constraints, are formulated in the EED issue.
• Utilization of optimization techniques, specifically FOFMO, COA, and NSGA-II are employed.
• Comparative analysis of the solutions attained from the utilized optimization techniques.

The rest of the paper is organized as follows:

• Section 2 introduces the system configuration being studied.
• Section 3 includes a mathematical analysis of the EED problem, taking into account the incorporation of RESs and NG. The interpretation of the EED issue, system constraints, and the optimization methods employed in this work are also presented in this section.
• Section 4 depicts and elucidates the obtained illustrations and results.
• Finally, in Section 5, conclusions drawn from the study and suggestions for future research are provided.

2. System Investigated

The primary objective of power grids is to optimize generator output to meet demand requirements while minimizing fuel costs, with emissions not being a factor. Presently, every nation is implementing various strategies and innovative approaches to reduce pollutants and safeguard the atmosphere in alignment with national and global environmental conservation standards. Furthermore, the minimization of fuel costs for generation units is a critical factor in addressing the EED problem and meeting load requirements. Simultaneously, reducing emissions (referred to as the environmental objective) holds great importance in mitigating climate change and environmental pollution. Traditionally, the EED problem solely focused on the economic objective concern. However, due to the urgency of combating climate change, greater attention has been given to adjusting greenhouse gas productions. Hence, it is highly significant to address the EED challenge, which involves balancing both economic and environmental goals. To realize this, the integration of clean and cost-effective RESs into electrical networks becomes a crucial enabler for minimizing environmental emissions.

This study examines a modified version of the IEEE standard 30-bus scheme, which incorporates both TUs and RESs. The system model, illustrated in Figure 1, includes three TUs located at buses 1, 2, and 8, as well as three distinct RESs, namely wind and two PV systems situated at buses 5, 11, and 13, respectively. The specifications of the scheme under investigation can be found in

Table 1. The model descriptions [54].

Item	Quantity	Specifications
Units	6	3 TUs and 3 RESs
TUs/NGs	3	Bus 1 (slack), bus 2, and bus 8
Wind turbine (WT)	25	Bus 5, 75 MW
PV	2	Bus 11 and 13, 50 MW
Active loads	-	283.4 MW
Reactive loads	-	126.2 MVAr
Number of load buses	24	-

.

To achieve the simultaneous reduction of emissions and fuel cost, three distinct cases are examined. In each case, three RESs comprising wind and two PV units are implemented at buses 5, 11, and 13, respectively. Three optimization methods of FOFMO, COA, and NSGA-II are applied to the problem under investigation. Figure 2 illustrates the methodology for the presented techniques. The breakdown of these cases is as follows:

• Case I: Utilizing three TUs and three RESs [55].
• Case II: Changing the fuel of TU at bus 1 into NG.
• Case III: Changing the fuel of TUs at buses 2 and 8 into NGs.

3. Formulation of the Optimization Problem

3.1. Objective Functions

The EED issue to be addressed is one in which the first objective function reflects the economic elements ($C_{tot}$) and the second represents the environmental implications ($E_{tot})$, as described in Equation (1).

$$\mathrm{Minimize}\left(EED\right)=\mathrm{m}\mathrm{i}\mathrm{n}(C_{tot},E_{tot})$$

(1)

3.1.1. Fuel Costs

The overall cost of produced electricity is the sum of the costs of TUs, RESs, and NGs (if NGs are included), as given in Equation (2).

$$C_{tot}=C_{tot}\left(P_{TU}\right)+C_{tot}\left(P_{RESs}\right)+\alpha \times C_{tot}\left(P_{NG}\right)$$

(2)

where $C_{tot}\left(P_{TU}\right)$ denotes the total TUs cost, $C_{tot}\left(P_{RES}\right)$ denotes the total RESs cost, and $C_{tot}\left(P_{NG}\right)$ denotes the total NGs cost, where $\alpha$=0 with no NGs involved and $\alpha$ = 1 if NGs are involved.

Fuel Cost Study of TUs

The TU cost should take into account the quantity of steam to the turbine’s blades, as well as any rapid changes in the valve’s condition. Steam in these facilities is regulated by valves that direct the turbine via an independent set of nozzles. To attain maximum efficiency, this set of nozzles has to be operated at full power [56]. For obligatory production, these valves open progressively. Equation (3) represents the cost function of TUs [57].

$$C_{tot}\left(P_{TU}\right)=\sum _{i=1}^{N_{TU}}{a}_{{TU}_i}+b_{{TU}_i}{P}_{{TU}_i}+c_{{TU}_i}{P}_{T{U}_i}^2+\left|d_{{TU}_i}\mathit{sin}\left(e_{{TU}_i}\times \left(P_{{TU}_i}^{min}-P_{{TU}_i}\right)\right)\right|$$

(3)

where $a_{{TU}_i}$, $b_{{TU}_i}$, and $c_{{TU}_i}$ denote the cost factors of the ith thermal generator unit ($P_{{TU}_i})$. $d_{{TU}_i}$ and $e_{{TU}_i}$ represent the valve point impacts.$P_{{TU}_i}^{min}$ denotes the minimum power of $P_{{TU}_i}$ during the generator operation. These factors are described in

Table 2. Emission and cost factors of the TUs [57].

Emission Factors
Generator	Bus	${\mathit{\phi }}_{\mathit{T}\mathit{U}}$	${\mathit{\psi }}_{\mathit{T}\mathit{U}}$	${\mathit{\omega }}_{\mathit{T}\mathit{U}}$	${\mathit{\tau }}_{\mathit{T}\mathit{U}}$	${\mathit{\xi }}_{\mathit{T}\mathit{U}}$
		(t/h)	(t/pu. MWh)	(t/pu. MW2h)	(t/h)	(pu. MW−1)
TU1	1	0.04091	−0.05554	0.0649	0.0002	6.667
TU2	2	0.02543	−0.06047	0.05638	0.0005	3.333
TU3	8	0.05326	−0.0355	0.0338	0.002	2
Cost Factors
Generator	Bus	${\mathit{a}}_{\mathit{T}\mathit{U}}$	${\mathit{b}}_{\mathit{T}\mathit{U}}$	${\mathit{c}}_{\mathit{T}\mathit{U}}$	${\mathit{d}}_{\mathit{T}\mathit{U}}$	${\mathit{e}}_{\mathit{T}\mathit{U}}$
		(USD/h)	(USD/MWh)	(USD/MW2h)	(USD/h)	(MW−1)
TU1	1	30	2	0.00375	18	0.037
TU2	2	25	1.75	0.0175	16	0.038
TU3	8	20	3.25	0.00834	12	0.045

.

Fuel Cost Study of RESs

The cost of the RESs, in USD/MWh, represents the total costs of WPs $\left(C_{tot}\left(P_{WP}\right)\right)$ and PVs $\left(C_{tot}\left(P_{PV}\right)\right)$ which can be expressed as in Equation (4):

$$C_{tot}\left(P_{RES}\right)=C_{tot}\left(P_{WP}\right)+C_{tot}\left(P_{PV}\right)$$

(4)

However, each RES possesses a distinct cost function. In order to address the intermittent nature of RESs, two approaches can be employed. First, standby power units (SPUs) can be installed to compensate for the deficit in generated power compared to the scheduled amount. Second, energy storage (ES) can be fitted to store any surplus power generated beyond the planned capacity.

To establish cost expressions, casual data for wind speed and solar irradiance can be fitted using the Weibull and lognormal distributions respectively. These distributions are utilized in the subsequent subsections to formulate the cost expressions.

A.

Cost estimation of wind plants

The cost calculation for WPs incorporates various factors, including the initial investment costs and the expenses associated with standby and energy storage (ES) units. The direct cost, denoted as $C_{d_{WP}}$, accounts for the initial investment, operational, and maintenance costs. Equation (5) defines this cost relationship as follows:

$$C_{d_{WP}}=K_{d_{WP}}{P_{WP}}_{sch}$$

(5)

where $K_{d_{WP}}$ and ${P_W}_{sch}$ denote the direct cost factor and the scheduled power of the WPs, respectively. When the actual power provided by the turbines drops below the scheduled power, the system uses reserve units to satisfy demand. The reserve capacity $\left(C_{r_{WP}}\right)$ cost is expressed in Equation (6).

$$C_{r_{WP}}=K_{r_{WP}}\left({{P_{WP}}_{sch}-P}_{{WP}_{act}}\right)=K_{r_{WP}}{\int }_0^{P_{W_{act}}}\left({P_{WP}}_{sch}-p_{WP}\right)f_w\left(p_{WP}\right)d{p}_w$$

(6)

The cost factor for the standby elements is denoted as $K_{r_{WP}}$, while $P_{{WP}_{act}}$ represents the actual power delivered by the wind turbines. Similarly, when the actual power, $P_{{WP}_{act}}$, exceeds the ${P_{WP}}_{sch}$, the cost related to the storage units can be expressed by Equation (7) as follows:

$$C_{s_{WP}}=K_{s_{WP}}\left(P_{{WP}_{act}}-{P_{WP}}_{sch}\right)=K_{s_{WP}}{\int }_{{P_{WP}}_{sch}}^{{P_{WP}}_r}\left(p_{WP}-{P_{WP}}_{sch}\right)f_w\left(p_{WP}\right)d{p}_w$$

(7)

where the variable ${P_{WP}}_r$, $f_w\left(p_{WP}\right)$ signifies the rated wind power and the PDF of the wind speed. Typically, the PDF of the wind power is modeled using the widely recognized Weibull distribution, which effectively captures the arbitrary occurrence of different wind velocity (v) levels [58,59]. In Figure 3a, the Weibull PDF of the wind velocity statistics is depicted, generated through 8000 Monte Carlo situations that consider the scale and shape factors (represented as α and β). In this case, α is set to 9, and β is set to 2. The probability $\left(f_v\left(v\right)\right)$ of a specific (v) can be described using Equation (8) [58]:

$$f_v\left(v\right)=\left(\frac{\beta }{\alpha }\right){\left(\frac{v}{\alpha }\right)}^{\left(\beta -1\right)}{e}^{-{\left(\raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.\right)}^{\beta }}$$

(8)

The cost factors associated with wind power are presented in Appendix A. The equation describing the power output of wind turbines, which is contingent upon the wind speed, is provided as Equation (9) [59]:

$$p_{WP}=\left\{\begin{array}{cc}0& v_{out}\le v\le v_{in}\\ {P_{WP}}_r\left(\frac{v-v_{in}}{v_r-v_{in}}\right)& v_{in}\le v\le v_r\\ {P_{WP}}_r& v_r\le v\le v_{out}\end{array}\right.$$

(9)

where $v_{in},$ $v_r,$ and $v_{out}$ indicate the cut-in speed, nominal speed, and cutout speed of the WPs, respectively. The wind power prospect can be stated as in Equation (10):

$$f_w\left(p_{WP}\right)=\frac{\beta \left(v_r-v_{in}\right)}{{\alpha }^{\beta }\times {P_W}_r}{\left[v_{in}+\frac{p_{WP}}{{P_{WP}}_r}\left(v_r-v_{in}\right)\right]}^{\beta -1}\times \mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left(\frac{v_{in}+\frac{p_{WP}}{{P_{WP}}_r}\left(v_r-v_{in}\right)}{\alpha }\right)}^{\beta }\right]$$

(10)

Lastly, the total cost of the WP is stated in Equation (11):

$$C_{{tot}_{WP}}=C_{d_{WP}}+C_{r_{WP}}+C_{s_{WP}}$$

(11)

B.

Cost Estimation of the PV Plant

Similarly, the overall cost equation for the PV plant can be formulated following the same approach employed to estimate the cost equations for the WP.

The direct cost, denoted as $C_{d_{PV}}$, signifies the initial investment, operational, and maintenance expenses associated with PVs. Equation (12) defines this cost relationship as follows:

$$C_{d_{PV}}=K_{d_{PV}}{P_{PV}}_{sch}$$

(12)

In this equation, $K_{d_{PV}}$ and ${P_{PV}}_{sch}$ correspond to the direct cost factor and the scheduled power of the PV, respectively. When the actual power ($P_{{PV}_{act}})$ drops below the scheduled power ${{(P}_{PV}}_{sch})$, the system uses reserve units to satisfy demand [60]. The reserve capacity $\left(C_{r_{PV}}\right)$ cost is expressed in Equation (13).

$$C_{r_{PV}}=K_{r_{PV}}\left({{P_{PV}}_{sch}-P}_{{PV}_{act}}\right)=K_{r_{PV}}\left({{P_{PV}}_{sch}-p}_{PV}\right)\times f_{PV}\left(p_{PV}\right)$$

(13)

$K_{r_{PV}}$ represents the cost factor of the standby elements. If $P_{{PV}_{act}}$ exceeds ${P_{PV}}_{sch},$ the cost related with the storage units can be expressed by Equation (14) as follows

$$C_{s_{PV}}=K_{s_{PV}}\left(P_{{PV}_{act}}-{P_{PV}}_{sch}\right)=K_{s_{PV}}\left(p_{PV}-{P_{PV}}_{sch}\right)\times f_{PV}\left(p_{PV}\right)$$

(14)

The power provided by the backup and storage elements is proportional to the solar irradiance PDF $G$. The PDF of $G$ is fitted using a lognormal model [60,61,62], as shown in Figure 3b. Equation (15) gives the probability of $G$ with lognormal fit values of $\mu =5.61$ and $\sigma =0.61$; therefore:

$$f_{PV}\left(G\right)=\frac{1}{G\sigma \sqrt{2\pi }}\mathrm{e}\mathrm{x}\mathrm{p}\left\{\frac{-\left(lnG-{\mu }^2\right)}{2{\sigma }^2}\right\}, \forall G>0$$

(15)

The accessible power from the PV scheme can be revealed as follows:

$$p_{PV}\left(G\right)=\left\{\begin{array}{c}{P}_{PVr}\left(\frac{G^2}{G_{std}}\right), 0<G<R_c\\ P_{PVr}\left(\frac{G}{G_{std}}\right), G\ge R_c\end{array}\right.$$

(16)

where the standard solar irradiation $G_{std}$ is equivalent to 1000 W/m². Through the operation irradiance $R_c$ is adjust as 120 W/m². $P_{PVr}$ denotes the nominal output power of the PV units. The cost factors of PV can be explained in Appendix A.

Ultimately, the total cost of the PV is stated as follows:

$$C_{{tot}_{PV}}=C_{d_{PV}}+C_{r_{PV}}+C_{s_{PV}}$$

(17)

Fuel Cost Study of NGs

In this study, NGs are used rather than TUs to reduce fuel costs and pollution emissions as much as feasible. The overall cost of NGs is comprised of many spending components: original cost, operation and maintenance cost, and fuel cost. Therefore, the overall cost of the NGs $\left(C_{tot}\left(P_{NG}\right)\right)$ is given in Equation (18) [63,64], thus:

$${ C}_{tot}\left(P_{NG}\right)=\sum _{i=1}^{N_{NG}}{\mathrm{g}}_{{NG}_i}\times P_{{NG}_i}+\sum _{i=1}^{N_{NG}}{a}_{{NG}_i}+b_{{NG}_i}{P}_{{NG}_i}+c_{{NG}_i}{P}_{{NG}_i}^2$$

(18)

where $a_{{NG}_i}$, $b_{{NG}_i}$, and $c_{{NG}_i}$ signify the cost factors of the ith NG ($P_{{NG}_i})$, and ${\mathrm{g}}_{{NG}_i}$ indicates the factor of the initial and operating costs of NGs. These factors are detailed in

Table 3. Emission and cost factors of the NGs [65].

Emission Factors
Generator	Bus	${\mathit{\phi }}_{\mathit{N}\mathit{G}}$	${\mathit{\psi }}_{\mathit{N}\mathit{G}}$	${\mathit{\omega }}_{\mathit{N}\mathit{G}}$
		(t/h)	(t/pu. MWh)	(t/pu. MW2h)
NG1	1	0.02091	−0.07554	0.04490
NG2	2	0.02543	−0.05047	0.03638
NG3	8	0.03326	−0.05550	0.01380
Cost Factors
Generator	Bus	${\mathit{a}}_{\mathit{N}\mathit{G}}$	${\mathit{b}}_{\mathit{N}\mathit{G}}$	${\mathit{c}}_{\mathit{N}\mathit{G}}$
		(USD/h)	(USD/MWh)	(USD/MW2h)
NG1	1	14	1.06	0.00175
NG2	2	15	1.05	0.0105
NG3	8	17	1.25	0.02434

.

To summarize, $C_{tot}$ of the complete scheme can be written as in Equation (19).

$$\begin{gathered} C_{tot}=\sum _{i=1}^{N_{TU}}{a}_{{TU}_i}+b_{{TU}_i}{P}_{{TU}_i}+c_{{TU}_i}{P}_{T{U}_i}^2+\left|d_{{TU}_i}\times \sin \left(e_{{TU}_i}\times \left(P_{{TU}_i}^{min}-P_{{TU}_i}\right)\right)\right| +C_{d_{WP}}+C_{r_{WP}}+C_{s_{WP}}+C_{d_{PV}}+C_{r_{PV}}+C_{s_{PV}}+\sum _{i=1}^{N_{NG}}{\mathrm{g}}_{{NG}_i}{P}_{{NG}_i}+ \\ \sum _{i=1}^{N_{NG}}{a}_{{NG}_i}+b_{{NG}_i}{P}_{{NG}_i}+c_{{NG}_i}{P}_{{NG}_i}^2 \end{gathered}$$

(19)

3.1.2. Emission Levels

Equation (20) describes the total emissions $\left(E_{tot}\right)$ resulting from thermal and natural gas units, as the RESs do not emit any emissions. Only the emissions of TUs and NGs units are accounted for in this presentation.

$$E_{tot}=\sum _{i=1}^{N_{TU}}{E_{TU}}_i+\sum _{i=1}^{N_{NG}}{E_{NG}}_i$$

(20)

where ${E_{TU}}_i$ and ${E_{NG}}_i$ denote the overall emissions of TU and NG, respectively.

Emission Study of TUs

Equation (21) provides a method to calculate the total emission $E_{tot}\left(TU\right)$ in tons per hour (t/h) of environmentally harmful gases such as SOx and NOx, which are directly linked to the generated output power.

$$E_{tot}\left(TU\right)=\sum _{i=1}^{N_{TU}}\left[{\phi }_{{TU}_i}+\left({\psi }_{\begin{array}{c}{TU}_i\\ \end{array}}{\times P}_{{TU}_i}\right)+({\omega }_{{TU}_i}\times {P_{{TU}_i}}^2)+{\tau }_{{TU}_i}\times e^{{{\xi }_{TU}}_i{P}_{{TU}_i}}\right]$$

(21)

where ${\phi }_{{TU}_i}$, ${\psi }_{{TU}_i}$, ${\omega }_{{TU}_i}$, ${\tau }_{{TU}_i}$, and ${\xi }_{{TU}_i}$ denote the factors of emission levels correlated with the ith $TU$. These factors can be identified in Table 2.

Emission Study of NGs

Equation (22) [66] describes the total emissions resulting from NGs, which is equivalent to the total emissions resulting from the TUs.

$$E_{tot}\left(NGs\right)=\sum _{i=1}^{N_{NG}}\left[{\phi }_{{NG}_i}+\left({\psi }_{{NG}_i}{\times P}_{{NG}_i}\right)+({\omega }_{{NG}_i}\times P_{{NG}_i}^2)\right]$$

(22)

where ${\phi }_{{NG}_i}$, ${\psi }_{{NG}_i}$, and ${\omega }_{{NG}_i}$ denote the factors of emission levels correlated with the ith $NG$. These factors can be identified in Table 3.

3.2. Limitations

When solving the objective function for any configuration, it is crucial to take into account the following key restrictions.

3.2.1. Power Balance Constraints

The power balance constraints pertain to both active and reactive powers which can be described as follows:

$$P_{GU}=P_{L_i}+P_{{Loss}_i}$$

(23)

$$Q_{GU}=Q_{L_i}+Q_{{Loss}_i}$$

(24)

3.2.2. Active and Reactive Powers Limits

The operational restrictions of all utilized elements of the active and reactive powers, involving thermal, wind, solar, and NGs, are expressed by Equations (25) to (32) respectively.

$$P_{{TU}_i}^{min}\le P_{T{U}_i}\le P_{{TU}_i}^{max} \forall i\in N_{TU}$$

(25)

$$P_{WP}^{min}\le P_{WP}\le P_{WP}^{max}$$

(26)

$$P_{PV}^{min}\le P_{PV}\le P_{PV}^{max}$$

(27)

$$P_{NG}^{min}\le P_{NG}\le P_{NG}^{max}$$

(28)

$$Q_{{TU}_i}^{min}\le Q_{T{U}_i}\le Q_{{TU}_i}^{max} \forall i\in N_{TU}$$

(29)

$$Q_{WP}^{min}\le Q_{WP}\le Q_{WP}^{max}$$

(30)

$$Q_{PV}^{min}\le Q_{PV}\le Q_{PV}^{max}$$

(31)

$$Q_{NG}^{min}\le Q_{NG}\le Q_{NG}^{max}$$

(32)

3.2.3. Prohibited Operating Zones Boundaries

In order to account for physical limitations experienced by TUs, such as stillness in shaft bearings or failures in pumps and boilers, Power Operating Zones (POZs) can be permitted within specific operating districts. These POZs result in a non-continuous operation of TUs. Equation (33) provides a description of the POZs.

$$P_{{TU}_i}^{minPOZ,j}\le {POZ}_{{TU}_i}^j\le P_{{TU}_i}^{maxPOZ,j}$$

(33)

where $P_{{TU}_i}^{minPOZ,j}$ and $P_{{TU}_i}^{maxPOZ,j}$ denote the min and max restrictions of the jth POZ of the ith TUs, respectively.

3.2.4. Security Limitations

Equations (34) and (35) depict the voltage security constrictions for the alternators’ bus and the load bus voltage, respectively. Additionally, Equation (36) outlines the limitations on the capacity of the branches.

$$V_{G_i}^{min}\le V_{G_i}\le V_{G_i}^{max} \forall i\in N_G$$

(34)

$$V_{L_i}^{min}\le V_{L_i}\le V_{L_i}^{max} \forall i\in N_L$$

(35)

$$S_{L_i}\le S_{L_i}^{max} \forall i\in nl$$

(36)

where $V_{G_i}$ denotes the voltage of the ith generator bus, while $V_{L_i}$ represents the voltage of the ith load bus. $N_G$, $N_L$, and $nl$ signify the count of alternator, load, and branches buses in the utility, respectively.

Furthermore, two additional parameters, network power loss ($P_{loss}$) and voltage deviation (VD), are considered as network constraints. They can be computed using Equations (37) and (38), respectively.

$$P_{loss}=\sum _{q=1}^{nl}{G}_{q\left(ij\right)}\left[{V_i}^2+{V_j}^2-2V_i{V}_j\cos \left({\delta }_{ij}\right)\right]$$

(37)

$$VD=\left(\sum _{p=1}^{N_L}\left|V_{L_p}-1\right|\right)$$

(38)

where $G_{q\left(ij\right)}$ and ${\delta }_{ij}$ indicate the transconductance of branch q connected to bus i and bus j and the phase difference between ${\delta }_i$ and ${\delta }_j$, respectively.

3.2.5. Natural Gas Boundaries

To be more precise, it is necessary to examine the impact of the required volume of natural gas (${\mathcal{V}}_{NG}$) on the remaining loads within the natural gas supply network [67]. The specific volume of natural gas needed to generate a particular amount of electrical power can be determined using Equation (39).

$${\mathcal{V}}_{{NG}_i}=\frac{0.278{\eta }_{{NG}_i}HHV}{p_{{NG}_i}}$$

(39)

The efficiency of the gas turbine is denoted as ${\eta }_{{NG}_i}$, and HHV signifies the high heat value of NG. The characteristics of the NG, including its distance, diameter, material kind, pressure, and mean flow rate (${\tilde{q}}_{xy,t}$), play a crucial role prior to bringing the NG to a gas turbine. This relationship can be expressed mathematically through Equation (40).

$${\tilde{q}}_{xy,t}=\pm C_{NG}\frac{T_O}{p_o}{\eta }_p\sqrt{\frac{\left(p_x^2-p_y^2\right)D_p^5}{S_{NG}Z{T}_{av}{L}_{p}f}}$$

(40)

$${\tilde{q}}_{xy,t}=\frac{q_{xy,t}^{in}+q_{xy,t}^{out}}{2}$$

(41)

where $C_{NG}$ signifies the constant factor, $T_o$ is the typical temperature, $p_o$ indicates the absolute pressure under atmospheric environment, and $p_x$ and $p_y$ signify the absolute upstream pressure and absolute downstream pressure, respectively. $D_P$ signifies the inner diameter of the pipe in (mm), $S_{NG}$ is the specific gravity of NG, and Z is the factor that captures the contrast between the actual state of the gas and its ideal state. $T_{av}$ represents the average temperature. $L_P$ refers to the span of the pipeline in meters, while f represents the hydraulic friction factor.

When the flow speed surpasses 20 m/s, the movement of dust atoms can have detrimental effects on cooking purposes and may even cause erosion on the internal surface of the pipeline. Therefore, the flow velocity $\left(U_{NG}\right)$ in (m/s) can be determined using Equation (42).

$$U_{NG}=\frac{353\times q_{NG}\times p_o}{D_p^2\sqrt{p_p^2-\frac{3730 f L_p{{\tilde{q}}_{xy,t}}^2}{D_p^5}}}$$

(42)

The pipeline’s flow velocity and pressure are interrelated and act as constraints within the system during the transition from thermal to gas generation units. Both the $U_{NG}$ and pipeline pressure ($p_p$) need to be taken into account [66,67].

Equations (43) and (44) define the equality restrictions for the line pack of pipeline x-y at hour t ($L_{x-y,t}$). Equation (45) specifies that the initial and final estimates of the line pack are equal. Additionally, Equations (46) to (49) describe the inequality restrictions pertaining to the flow speed, pressure, flow rate, and air compressor limitations, respectively.

$$L_{x-y, t}=G_{x-y}\left(\frac{p_{x, t}+p_{y, t}}{2}\right)$$

(43)

$$L_{x-y, t}=L_{x-y, t-1}+q_{xy, t}-q_{y, t}$$

(44)

$$\sum _t{q}_{xy,t}^{in}=\sum _t{q}_{xy,t}^{out}$$

(45)

$$U_{NG}\le 20 \mathrm{m}/\sec$$

(46)

$${p_{pipe}^{min}\le p}_{pipe}\le p_{pipe}^{max}$$

(47)

$${q_{pipe}^{min}\le q}_{pipe}\le q_{pipe}^{max}$$

(48)

$$p_{x, t}\le {\mathsf{\Gamma }}_C p_{y, t}$$

(49)

3.3. Optimization Techniques

3.3.1. Fish Migration Optimization (FMO) Algorithm

In 2010, the FMO approach was introduced as a swarm intelligence technique [68,69]. It emulates the growth, migration procedures, and predation approach observed in fish biology. A notable distinction between FMO and other meta-heuristic approaches is that its optimization formularies were derived from insights provided by biologists. In comparison to PSO, FMO demonstrates superior accuracy and reasonable time consumption. However, when it comes to low-dimensional complex functions, FMO’s optimization performance is not particularly outstanding. This is primarily because the integer order speed update in FMO tends to overlook the optimal solution, resulting in suboptimal optimization outcomes. On the other hand, the fractional order speed update in FMO employs fractions to adjust the step size and incorporates historical speed information, enabling the algorithm to yield more precise results.

Each species in nature possesses unique survival strategies, methods of predation, and reproductive mechanisms. Furthermore, they must adapt to challenging environments, as they constantly face the risk of being captured by natural predators. Not every fish can successfully reach adulthood. Biologists have observed that fish swim in water for various reasons. Drawing inspiration from the grayling fish, the FMO algorithm is introduced, leveraging its characteristics. The life cycle of the grayling fish is divided into five stages, which serve as the basis for the procedure.

• Stage 0+: Newborn and young (age ranging from 0 to 1 year).
• Stage 1+: Juvenile (age ranging from 1 to 2 years).
• Stage 2+: Sub-adult (age ranging from 2 to 3 years).
• Stage 3+: Adult (age ranging from 3 to 4 years).
• Stage 4+: Adult (age ranging from 4 to 5 years).

Each stage of the fish’s life cycle is associated with distinct fecundity rates. The life cycle graph of the grayling, depicted in Figure 4, illustrates this. Within the graph, F₂, F₃, and F₄ represent the fecundity rates corresponding to stages 2+, 3+, and 4+ respectively.

The FMO algorithm is specifically developed to optimize processes through two main components: the swim and the migration processes. The swim procedure simulates the behavior of graylings as they navigate and grow in the water while seeking food sources. During this procedure, energy consumption aligns with the fish’s change, as defined by Equation (50).

$$E_{r, d}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}.\mathrm{E}$$

(50)

$E_{r, d}$ represents the energy expended in the d dimension, while rand denotes a randomly generated number. E stands as a constant that states the maximum energy consumption in a single dimension. For this particular procedure, we assign a value of 2 to E. The operative correlation among the remoteness traveled and energy consumption can be expressed as:

$${dis}_{offset, d}=\frac{E_{r, d}.U_{s,d}}{a+b.({U_{s,d})}^x}$$

(51)

The moving distance is represented by disoffset; d, while $U_{s,d}$ denotes the swimming velocity in the d dimension. Constants a, b, and x can be involved, where a represents the standard metabolic rate, b is a scaling constant, and x is the velocity exponent. The values for a, b, and x are documented as 2.25, 36.2, and 2.23, respectively. To calculate the new position, the value of every dimension is updated using Equation (52), where offset represents the moving space of the particle.

$$p_{new}=p_{old}+p_{offset}$$

(52)

If the fitness value of $p_{new}$ is superior to that of $p_{old}$, $p_{old}$ will be revised using $p_{new}$ according to Equation (52). At the same time, the speed will be adjusted using Equation (53).

$$U_s=2.U_s$$

(53)

As the fish reach maturity, a portion of them return to their birthplace for the purpose of reproducing offspring, which is known as the migration process. This migration process occurs exclusively in stages 2+, 3+, and 4+, as fish in stages 0+ and 1+ are unable to reproduce. The fecundity rates for these three stages are 5%, 10%, and 100% respectively. When fish discover a new potential location, the coordinates are renewed using Equation (54).

$$p_{new}={(d}_{max}-d_{min}).rand+d_{min}$$

(54)

$d_{max}$ and $d_{min}$ represent the upper and lower limits, respectively.

To determine the fitness value of another candidate, the velocity will be adjusted as follows:

$$\mathrm{U}=\left\{\begin{array}{c}\pi .U_s, F\left(P\right)<F\left(P_{best}\right)\\ U_s, otherwise\end{array}\right.$$

(55)

where $U_s$ represents the initial velocity.

Fractional Calculus (FC)

FC is an extension of the traditional perception of calculus, encompassing fractional derivatives and fractional integrals, much like classical calculus [70,71]. However, the key distinction lies in the order of derivatives and integrals: while classical calculus deals exclusively with integer orders, FC allows for fractional orders. This fundamental difference enables FC to precisely capture the memory and inherited traits of diverse constituents and their evolutionary procedures. Since the inception of FC, its associated philosophy has found successful applications across numerous fields.

The FC, originating from classical calculus with a history spanning over 300 years, had limited advancements until modern times caused by the emergence of employed disciplines like fluid mechanics, cybernetics, and biology. However, its practical significance gradually became apparent, leading to increased scholarly interest in FC. FC is a branch of mathematical analysis that offers several advantages. Firstly, it provides a mathematical perspective on the inevitability of historical development. Additionally, FC is known for its memorability. Moreover, compared to nonlinear models, the concise expression of the FC model allows for a better fit with the real world. There are three definitions of fractional derivatives (FDs), each associated with an order α.

The first definition is the Riemann–Liouville FD, and it is described as follows [70]:

$$RL D_{\alpha ,t}^{\alpha }\mathrm{f}(\mathrm{x})=\left\{\begin{array}{c}\frac{d^{n}f}{{dt}^n}, \alpha =n,\\ \frac{1}{\mathsf{\Gamma }(n-\alpha )} \frac{d^n}{{dx}^n}{\int }_{\alpha }^x{\left(x-t\right)}^{n-\alpha -1}{f}^n\left(t\right), n-1<\alpha <n\end{array}\right.$$

(56)

The second definition is the Caputo FD, and it is described as follows [70]:

$$C D_{\alpha ,t}^{\alpha }\mathrm{f}(\mathrm{x})=\left\{\begin{array}{c}\frac{d^{n}f}{{dt}^n}, \alpha =n,\\ \frac{1}{\mathsf{\Gamma }(n-\alpha )} {\int }_{\alpha }^x{\left(x-t\right)}^{n-\alpha -1}{f}^n\left(t\right), n-1<\alpha <n\end{array}\right.$$

(57)

The third definition is the Grünwald–Letnikov FD, and it is described as follows [70]:

$$GL D^{\alpha }\mathrm{f}(\mathrm{x})=\underset{\mathrm{h}\to 0}{\lim }\left(\frac{1}{{\mathrm{h}}^{\alpha }}\sum _{k=0}^{+\infty }\frac{{\left(-1\right)}^k\mathsf{\Gamma }(\alpha +1)\mathrm{f}(x-kh)}{\mathsf{\Gamma }(k+1)\mathsf{\Gamma }(\alpha -k+1)}\right)$$

(58)

To approximate the discrete time situation using Grünwald–Letnikov FD, the following expression can be used (referred to as Equation (59)).

$$GL D^{\alpha }\mathrm{f}\left(\mathrm{t}\right)=\frac{1}{{\mathrm{T}}^{\alpha }}\sum _{k=0}^r\frac{{\left(-1\right)}^k\mathsf{\Gamma }(\alpha +1)\mathrm{f}(t-kT)}{\mathsf{\Gamma }(k+1)\mathsf{\Gamma }(\alpha -k+1)}$$

(59)

The time increment is denoted as T, and the truncation order as r in the approximation of the discrete time situation using the Grünwald–Letnikov FD (referring to Equation (59)). The Gamma function, denoted as $\mathsf{\Gamma }$(x), is present in all three FD definitions mentioned above. The three FC definitions mentioned above exhibit the following properties:

(a)

Linearity $D^{\alpha }\left[\alpha \mathrm{f}\left(\mathrm{x}\right)+\mathrm{b}\mathrm{g}\left(\mathrm{x}\right)\right]=\alpha D^{\alpha }\mathrm{f}\left(\mathrm{x}\right)+\mathrm{b}{D}^{\alpha }\mathrm{g}\left(\mathrm{x}\right)$

(b)

The index theory $D^{\alpha +\beta }\mathrm{f}\left(\mathrm{x}\right)=D^{\alpha }{D}^{\beta }\mathrm{f}\left(\mathrm{x}\right)$

(c)

Generalized Leibniz rule $D^{\alpha }\left[\mathrm{f}\left(\mathrm{x}\right).\mathrm{g}\left(\mathrm{x}\right)\right]=\sum _{i=0}^{\infty }{D}^{i}f\left(x\right).D^{\alpha -i}\mathrm{g}\left(x\right)$

where a and b are constants.

Proposed Fractional-Order Fish Migration Optimization Algorithm

This section describes in detail a novel algorithm known as Fractional-Order FMO (FOFMO). The FOFMO algorithm combines the FMO method with the idea of the Grünwald–Letnikov FD. There are two major differences between the FMO technique and the FOFMO technique. First, the FOFMO employs a new technique for fractional-order velocity. Second, the FOFMO produces new offspring placements using the global optimal particles.

A.

Fractional-Order Velocity

The velocity update equations in the FMO algorithm are given by (53) and (55). While the FMO procedure excels in penetrating for global optimum solutions, it exhibits a limited exploitation ability as it requires considerable time for exploration. To enhance the exploitation capability, the FC is employed for velocity updates. Assuming a time interval of 1, the following expression is obtained:

$$d_{offset}=\frac{E_r.U_s^t}{a+b.({U_s^t)}^x}$$

(60)

$$U_s^t=P^t-P^{t-1}$$

Based on Equation (60), with $a=2.25$; $b=36.2$; $x=2.23$, it is evident that the denominator is significantly larger than the numerator in the overall fraction. As the iteration progresses, the fish’s speed gradually decreases, leading to a potential algorithm stagnation.

From (58), let $\alpha =1$, we get

$$GL D^{\alpha }\left[f_{t+1}\right]=f_{t+1}-f_t$$

(61)

Equation (61) denotes the derivative of order 1 in the discrete case. For $\mathsf{\Gamma }$(x), we get

$$\begin{gathered} \mathsf{\Gamma }\left(\alpha +1\right)=\alpha \mathsf{\Gamma }\left(\alpha \right), \\ \mathsf{\Gamma }\left(\alpha +1\right)=\alpha (\alpha -1)\mathsf{\Gamma }\left(\alpha -1\right), \\ \mathsf{\Gamma }\left(\alpha +1\right)=\alpha (\alpha -1)(\alpha -2)\mathsf{\Gamma }\left(\alpha -2\right), \\ \mathsf{\Gamma }\left(\alpha +1\right)=\alpha (\alpha -1)(\alpha -2)(\alpha -3)\mathsf{\Gamma }\left(\alpha -3\right) \end{gathered}$$

In order to generalize, let T = 1; r = 4 [72,73,74], Equation (62) can be attained.

$$\begin{gathered} GL D^{\alpha }\mathrm{f}\left(\mathrm{t}\right)=\frac{1}{{\mathrm{T}}^{\alpha }}\sum _{k=0}^r\frac{{\left(-1\right)}^k\mathsf{\Gamma }(\alpha +1){\mathrm{f}}_{t+1-kT}}{\mathsf{\Gamma }(k+1)\mathsf{\Gamma }(\alpha -k+1)} \\ ={\mathrm{f}}_{t+1}-\alpha f_t-\frac{1}{0.5}\alpha \left(1-\alpha \right)f_{t-1}-\frac{1}{6}\alpha \left(1-\alpha \right)\left(2-\alpha \right)f_{t-2}-\frac{1}{24}\alpha \left(1-\alpha \right)\left(2-\alpha \right)\left(3-\alpha \right)f_{t-3} \end{gathered}$$

(62)

In the proposed algorithm, with a fish population size of ps and dimension Dim, the position matrix P is defined as P(ps; Dim) = [p₁; p₂; · · ·; p_ps]^T, where p_i = [p_i;1; p_i;2; · · ·; p_i;Dim] represents the position of particle i. Similarly, P^pre = [P^pre1; P^pre2; P^pre3; P^pre4] denotes the past positions of the particles utilized for fractional-order speed calculation. Precisely, $p_1^{pre\left\{h\right\}}(ps; Dim)={\left[p_1^{pre\left\{h\right\}},p_2^{pre\left\{h\right\}},\dots .,p_{ps}^{pre\left\{h\right\}} \right]}^T$, $p_i^{pre\left\{h\right\}}=\left[p_{i,1}^{pre\left\{h\right\}},p_{i,2}^{pre\left\{h\right\}},\dots .,p_{i,Dim}^{pre\left\{h\right\}} \right]$, where h = 1, 2, 3, 4; i = 1, 2,…, ps. Thus, the velocity of the particle is updated by (64), where

$$U_{s,d}=P_{i,d}-\alpha p_{i,d}^{pre1}-\frac{1}{2} \left(1-\alpha \right)p_{i,d}^{pre2}-\frac{1}{6} \left(1-\alpha \right)\left(2-\alpha \right)p_{i,d}^{pre3}-\frac{1}{24} \left(1-\alpha \right)\left(2-\alpha \right)\left(3-\alpha \right)p_{i,d}^{pre4}$$

(63)

$$\begin{gathered} p_{i,d}^{new}=\frac{E_{r, d}.U_{s,d}}{a+b.({U_{s,d})}^x}, \\ {p}_{i,d}^{pre4}=p_{i,d}^{pre3}=p_{i,d}^{pre2}{=p}_{i,d}^{pre1}=p_{i,d}^{new} \end{gathered}$$

(64)

B.

New Positions of the Offspring

Furthermore, graylings migrate back to replicate new offspring once they reach maturity as a group. Graylings need to breed in a location that enhances their chances of survival.

Hence, the positions of the new offspring should be near the global finest particle. Equation (54) is substituted with Equation (65).

$$p_i^{new}=p_{gbest}+rand. (p_i^{old}-p_{gbest})$$

(65)

This section provides a detailed description of the innovative algorithm utilizing fractional-order speed and new positions. The flowchart of the FOFMO can be found in Figure 5. Algorithm 1 presents the pseudo code of the FOFMO procedure, with Rate_fecundity representing the fecundity rates at specific stages, E_min as the minimum energy threshold for elimination, x_g,i indicating the growth status, and x_eng,i representing the energy of particle i.

Algorithm 1 FOFMO

The dimension Dim, population size ps, a, b, x, Maximum iteration Maxitn, Initialize the searching space V, itn = 1, position matrix Pitn Past position matrix Ppre,itn, the energy $X_{eng}^{itn}$ Compute fitness function values ${f(P}_i^{itn})$, set grow status $X_g^{itn}=0+$ While itn < MaxitnΙΙ!stopCriterion do for t = 1 to ps do       for d = 1 to Dim do       Calculate consuming energy by Equation (50)       Calculate fractional-order position $P_{i,d}^{itn}$, update the history by (64)       Calculate the fitness value of new position $P_{i,d}^{itn}$, if ${f(P}_i^{itn})< {f(P}_{gbest}^{itn})$ then             ${f(P}_{gbest}^{itn})={f(P}_i^{itn})$             $P_{gbest}^{itn}=P_i^{itn}$             Increased energy $x_{eng,i}=x_{eng,i}+rand.E_{max}$       End if             Consuming energy by $x_{eng,i}=x_{eng,i}-{f(P}_i^{itn})/{\sum }_i{f(P}_i^{itn})$         $\mathbf{If} x_{eng,i}<E_{min}$, then             $x_{g,i}$ = 5, the grayling died       End if       $\mathbf{If} x_{g,i}=0+‖x_{g,i}=1+$ then             $x_{g,i}=\left|x_{g,i}+1\right|+$       Else if $x_{g,i}=2+‖x_{g,i}=3+$ then       If rand()<Ratefecundity then             Immigrate and produce offspring by Equation (65)             $x_{g,i}=0$       Else             $x_{g,i}=\left|x_{g,i}+1\right|+$       End if       Else if $x_{g,i}=4+$ then                 Immigrate and produce offspring by Equation (65)                 $x_{g,i}=0$       End if End for                 Itn = itn + 1 End while Output: The global optimal solution: $p_{gbest}, f(p_{gbest})$

3.3.2. Coati Optimization Algorithm (COA) and Non-Dominated Sorting Genetic Algorithm (NSGA-II) Techniques

The COA and NSGA-II are optimization algorithms employed within the study to deal with the EED problem [52,53].

The COA is an unconventional optimization approach that specializes in realistic and non-convex issues [52,75]. It offers an alternative approach to dealing with complex optimization-demanding situations. In the context of the EED problem, COA is utilized as one of the optimization algorithms to acquire Pareto premiere solutions for fuel costs and emissions. It contributes to finding the superior trade-off between monetary and environmental concerns.

On the other hand, NSGA-II is a hybrid optimization technique that combines the strengths of more than one nonconventional algorithm to resolve complex troubles [53]. It is a well-known set of rules for multi-goal optimization. In the study, NSGA-II is employed as one of the optimization algorithms to attain Pareto optimal solutions for the EED issue. It helps to concurrently optimize a couple of goals, which include minimizing toxin emissions and fuel costs, even considering system constraints.

Both COA and NSGA-II play significant roles in the proposed model for optimizing economic dispatch with RESs and NGs. They contribute to locating efficient alternatives that address the mixing of thermal, renewable schemes, and natural gas units while considering environmental and financial objectives.

4. Simulation Results and Comparative Analysis

Three cases are evaluated in order to minimize both emissions and fuel costs. In general, each case includes three RESs: wind and two PVs at buses 5, 11, and 13. This can be described as follows:

• Case I: involves three TUs and three RESs [55].
• Case II: Changing the fuel of TU at bus 1 to NG.
• Case III: Changing the fuel of TUs at buses 2 and 8 to NGs.

4.1. First Case

In this reported literature case, a comparison was made between the Pareto fronts (PFs) generated by the SMODE and MOEA/D techniques, as documented in [55] and presented in

Table 4. Complete statistical findings of the optimization approaches for Case I.

Variables	Units	Min.	Max.	MOEA/D	SMODE	FOFMO	COA	NSGA-II
PTU1	MW	50	140	117.117	111.912	122.054	111.247	120.045
PTU2		20	80	65	65	42.574	42.849	43.490
PTU3		10	35	18.404	23.556	19.02	17.089	16.025
PWP		0	75	55.448	54.059	56.448	57.059	58.448
Ppv (Bus 11)		0	50	17.650	18.437	20.650	19.437	21.650
Ppv (Bus 13)		0	50	15.327	15.756	17.327	18.756	18.327
Ploss				5.5430	5.3149	5.1254	5.4861	5.5258
Q1	MVAr	−50	140	2.129	2.789	−1.035	−0.944	−4.775
Q2		−20	60	21.420	34.505	21.402	14.351	14.801
Q5		−15	70	37.728	36.170	25.870	35.001	35.002
Q8		−30	60	27.103	20.377	40.001	40.001	40.002
Q11		−20	30	24.912	23.420	21.005	20.1907	19.183
Q13		−20	25	20.329	15.630	22.278	21.334	25.002
V1	pu	0.96	1.10	1.077	1.077	1.077	1.078	1.079
V2		0.96	1.10	1.0649	1.067	1.070	1.072	1.074
V5		0.96	1.10	1.0445	1.0363	1.064	1.065	1.067
V8		0.96	1.10	1.0403	1.0363	1.0428	1.043	1.045
V11		0.96	1.10	1.0879	1.0779	1.042	1.041	1.045
V13		0.96	1.10	1.0603	1.0433	1.027	1.027	1.031
VD				0.4531	0.422	0.412	0.4684	0.4861
Total cost (USD/MWh)				919.041	927.050	911.854	912.554	914.247
Emission (t/h)				0.6222	0.4722	0.5473	0.4248	0.5487

The boldface value denotes the optimal outcome obtained through the proposed algorithms.

. The study reveals that SMODE exhibits superior diversity compared to MOEA/D, particularly regarding the cost objective. Notably, SMODE achieves fewer emission levels, specifically 0.4722 t/h, while MOEA/D attains a slightly lower fuel cost of 919.041 USD/MWh.

The variables attained utilizing the proposed FOFMO method are compared with four other considered optimizers, as shown in Table 4. Notably, the proposed FOFMO achieves a minimum cost of 911.854 USD/MWh, a minimum power loss of 5.1254 MW, and a minimum VD of 0.412 pu. While the COA technique achieves a minimum emissions level of 0.4248 t/h.

Furthermore, the FOFMO demonstrates fast solution convergence, with a computational time of approximately 120.798 s, outperforming the other algorithms, followed by the COA algorithm with a computational time of around 133.063 s. (See Figure 6).

4.2. Second Case

In this case, it is replaced the largest TU at bus 1 to NG incorporating stochastic RESs. Figure 7 exhibits the PFs of FOFMO, COA, and NSGA-II methods. Additionally,

Table 5. Complete statistical findings of the optimization approaches for Case II.

Variables		Min.	Max.	FOFMO	COA	NSGA-II
PNG1	MW	50	140	72.841	64.878	75.541
PTU2		20	80	80.002	74.221	42.574
PTU3		10	35	35.001	34.982	10.578
PWP		0	75	33.744	31.351	32.744
Ppv (Bus 11)		0	50	39.737	39.605	39.247
Ppv (Bus 13)		0	50	28.431	37.834	32.481
Ploss				4.501	4.892	5.682
Q1	MVAr	−50	140	27.312	−29.199	−11.36590
Q2		−20	60	16.332	67.744	21.86743
Q5		−15	70	30.881	64.072	25.95090
Q8		−30	60	51.183	28.012	40.00000
Q11		−20	30	2.452	11.821	20.85680
Q13		−20	25	3.457	−8.321	22.28300
V1	pu	0.96	1.10	1.091	1.062	1.07869
V2		0.96	1.10	1.085	1.102	1.07306
V5		0.96	1.10	1.055	1.084	1.06652
V8		0.96	1.10	1.054	1.033	1.04903
V11		0.96	1.10	1.011	1.032	1.04444
V13		0.96	1.10	1.012	0.971	1.03090
VD				0.567	0.794	0.94580
Wgencost				102.394	124.889	126.9591
Sgencost (Bus 11)				147.248	150.245	114.9636
Sgencost (Bus 13)				97.385	143.721	39.9283
mass flow (m3)				63.552	71.365	78.345
P1_cost				9.534	10.705	10.975
Total cost (USD/h)				798.182	804.584	812.4878
Emission (t/h)				0.5222	0.4523	0.6823
Fuelvlvcost				340.821	327.402	353.139

The boldface value denotes the optimal outcome obtained through the proposed algorithms.

shows the statistical simulation findings for solutions from three optimization strategies used in this event using stochastic RESs.

The variables obtained using the proposed FOFMO method are compared with two other considered optimizers, as illustrated in Table 5. Notably, the proposed FOFMO achieves a minimum cost of 798.182 USD/MWh, a minimum power loss of 4.501 MW, and minimum VD of 0.567 pu. Furthermore, the FOFMO demonstrates fast solution convergence, with a computational time of approximately 128.248 s, outperforming the other algorithms, followed by the COA algorithm with a computational time of around 145.735 s.

In terms of emission minimization, COA obtains lower emission levels of 0.4523 t/h, whilst FOFMO reaches emission levels of 0.5222 t/h. As a result, COA has a higher variety of PF than FOFMO in the direction of the cost function. In comparison to overall fuel expenses, FOFMO obtains a lower fuel cost value of 798.182 USD/MWh, whereas COA achieves a fuel cost of 804.584 USD/MWh.

4.3. Third Case

In this case, the largest thermal unit at bus 1 is retained, while the TUs at buses 2 and 8 are replaced with NGs incorporating stochastic RESs. Figure 8 showcases the PFs achieved through the utilization of the FOFMO, COA, and NSGA-II techniques. Furthermore,

Table 6. Complete statistical findings of the optimization approach for Case III.

Variables	Units	Min	Max	FOFMO	COA	NSGA-II
PTU1	MW	50	140	121.482	52.811	112.811
PNG2		20	80	80.002	80.001	80.001
PNG3		10	45	35.005	43.198	41.178
Pw		0	75	69.564	52.296	62.246
Ppv (Bus 11)		0	50	23.937	32.959	42.159
Ppv (Bus 13)		0	50	18.556	28.588	34.588
Ploss				3.214	4.044	5.374
Q1	MVAr	−50	140	30.127	−5.729	50.127
Q2		−20	60	−5.867	6.156	−9.847
Q5		−15	40	17.857	36.589	21.357
Q8		−30	50	48.558	49.179	43.518
Q11		−20	25	5.523	17.464	8.573
Q13		−20	25	15.687	26.712	21.648
V1	pu	0.96	1.10	1.058	1.006	1.01
V2		0.96	1.10	1.059	1.007	1.004
V5		0.96	1.10	1.008	0.991	1.07
V8		0.96	1.10	1.013	1.014	1.019
V11		0.96	1.10	1.027	1.069	1.078
V13		0.96	1.10	1.087	1.058	1.096
VD				0.509	0.644	0.601
Wgencost				99.534	193.203	99.534
Sgencost (Bus 11)				32.902	51.606	32.902
Sgencost (Bus 13)				27.892	39.781	27.892
mass flow_02				172.832	172.837	172.832
mass flow_08				75.621	93.317	75.621
Total cost (USD/MWh)				796.478	807.782	816.451
Emission (t/h)				0.5591	0.4232	0.4181
Fuelvlvcost				336.871	147.932	325.171

The boldface value denotes the optimal outcome obtained through the proposed algorithms.

presents the statistical simulation findings of solutions obtained from these three optimization methods employed in this case, considering the incorporation of stochastic RESs.

The variables obtained using the proposed FOFMO method are compared with two other considered optimizers, as illustrated in Table 6. Notably, the proposed FOFMO achieves a minimum cost of 796.478 USD/MWh, a minimum power loss of 3.214 MW, and a minimum VD of 0.509 pu. Furthermore, the FOFMO demonstrates fast solution convergence, with a computational time of approximately 136.116 s, outperforming the other algorithms, followed by the COA algorithm with a computational time of around 151.245 s. In terms of convergence, FOFMO slightly outperforms COA and NSGA-II, as shown in Figure 8.

Similarly, when comparing the minimization of emission levels, NSGA-II attains lower emission levels at a value of 0.4181 t/h, while FOFMO and COA obtain emission levels valued at 0.5591 t/h and 0.4232 t/h, respectively. Hence, NSGA-II reveals better diversity in the PFs compared to FOFMO in terms of the cost function. In contrast, regarding overall fuel costs, FOFMO attains a lower fuel cost valued at 796.478 USD/MWh, while COA and NSGA-II obtain a fuel cost of 807.782 USD/MWh and 816.451 USD/MWh, respectively.

4.4. Comparative Study

A comparative study is presented in two cases, Cases II and III, where TUs are replaced with NGs. These cases offer a framework for comparison, encompassing two perspectives: comparing the algorithms within each case and comparing the cases themselves.

The results indicate that the FOFMO technique exhibits high emission levels, making it environmentally unfavorable. However, its cost objective may encourage investment. In contrast, COA and NSGA-II achieve the lowest emission levels but have a higher cost objective in Case II and Case III, respectively.

The complexity of these findings arises from the multidisciplinary nature of the objectives. To resolve this complexity, a holistic approach is suggested, where the two cases are compared to determine which one aligns with governmental regulations and provides benefits to the community.

The exclusion of Case I from the analysis is justified due to its significantly higher emission levels and fuel costs compared to the proposed cases (Cases II and III), as illustrated in

Table 7. Comparative study between optimization methods for three cases.

Comparison	Case I		Case II		Case III
Approaches	FOFMO	COA	FOFMO	COA	FOFMO	COA
No. of iterations	300	300	300	300	300	300
No. of population	100	100	100	100	100	100
Computation time (s)	120.798	133.063	128.248	145.735	136.116	151.245
Total cost (USD/MWh)	911.854	912.554	798.182	804.584	796.478	807.782
Emission (t/h)	0.5473	0.4248	0.5222	0.4523	0.5591	0.4232

The boldface value denotes the optimal outcome obtained through the proposed algorithms.

.

Upon examining the outcomes of Case I, it becomes apparent that FOFMO yields the highest emission levels of 0.5473 t/h. Conversely, in Case II, COA achieves the lowest emission levels of 0.4523 t/h, while FOFMO attains the lowest fuel costs of 798.182 USD/MWh. In Case III, COA achieves the lowest emission levels of 0.4232 t/h, while FOFMO attains the lowest fuel costs of 796.478 USD/MWh.

The analysis of the three cases reveals that Case III stands out for achieving the lowest fuel costs and emission levels. This outcome is attributed to the combination of the COA technique, which minimizes emissions, and the FOFMO technique, which reduces fuel costs. As a result, Case III successfully meets the system requirements by striking a balance between environmental and economic considerations.

The question remains as to which optimization technique in Case III provides the best compromise solution. Armed with the data provided, the power system operator can make an informed decision regarding the optimal optimization technique.

In summary, Case III offers the most favorable solution by retaining the largest thermal unit at bus 1 and replacing thermal units at buses 2 and 8 with NG units, incorporating stochastic renewable energy sources. This approach not only minimizes emissions but also reduces fuel costs, making it the most effective solution among the three cases.

Figure 9 and Figure 10 illustrate the load bus voltage profiles for two proposed cases, integrating RESs and NGs, with the worst VD values using three optimization methods. In Case II, the worst VD values obtained using FOFMO and COA are 0.56 pu and 0.79 pu, respectively. Similarly, in Case III, the worst VD values are 0.5 pu and 0.64 pu, respectively, using FOFMO and COA. The variation in VD values is attributed to the higher diversity of COA, which results in smaller emission levels or larger cost objectives.

5. Conclusions

This study presents an economic-environmental dispatch model that aims to determine the optimal solutions for an integrated IEEE 30-bus system comprising conventional thermal units, natural gas units, and renewable energy sources such as wind and two photovoltaic systems. The model takes into account both the emission and total cost functions to evaluate and identify the optimal operating points that simultaneously minimize emissions and total costs.

To achieve the simultaneous minimization of emissions and fuel or generation costs, three distinct scenarios were tested. Each scenario consisted of three RESs, specifically wind and two PV units located at buses 5, 11, and 13, respectively. The first scenario involved three TUs and three RESs. In Case II, the fuel of the TU at bus 1 was replaced with NG. Case III replaced the fuels of TUs at buses 2 and 8 with NGs. Through the obtained results, it was determined that the third scenario provided the optimal compromise solution, achieving the lowest emissions and total costs.

To obtain Pareto optimal solutions simultaneously, three optimization techniques were employed: FOFMO, COA, and NSGA-II. All system constraints, including equality, inequality, and natural gas limits, were successfully met. A comparative analysis was conducted between the recent optimization methods to determine the optimal values for the EED pollutant emissions and fuel costs. The results demonstrated that FOFMO outperformed COA and NSGA-II.

Further investigation can be conducted on the EED formulation by exploring alternative optimization techniques such as reinforcement learning (RL), and others. It would be valuable to combine these techniques with suitable constraint-handling methods. Additionally, addressing the dynamic EED issue, which considers variations in load demands over time, generator ramping rates, and uncertainties associated with Renewable Energy Sources (RESs) and system limitations, presents an important area for future research.