Exploring Evolutionary Algorithms for Optimal Power Flow: A Comprehensive Review and Analysis

Abstract

It has been more than five decades since optimum power flow (OPF) emerged as one of the most famous and frequently used nonlinear optimization problems in power systems. Despite its long-standing existence, the OPF problem continues to be widely researched due to its critical role in electrical network planning and operations. The general formulation of OPF is complex, representing a large-scale optimization model with nonlinear and nonconvex characteristics, incorporating both discrete and continuous control variables. The inclusion of control factors such as transformer taps and shunt capacitors, and the integration of renewable energy sources like wind power further complicates the system’s design and solution. To address these challenges, a variety of classical, evolutionary, and improved optimization techniques have been developed. These techniques not only provide new solution pathways but also enhance the quality of existing solutions, contributing to reductions in computational cost and operational efficiency. Multi-objective approaches are frequently employed in modern OPF problems to balance trade-offs between competing objectives like cost minimization, loss reduction, and environmental impact. This article presents an in-depth review of various OPF problems and the wide array of algorithms, both traditional and evolutionary, applied to solve these problems, paying special attention to wind power integration and multi-objective optimization strategies.

1. Introduction

When it comes to designing and operating power systems, particular tools are necessary for power engineers. Almost all of these tools may be seen as optimization issues. Since its debut in the early 1960s, the optimum power flow (OPF) has been intensively studied, and it was developed by Dommel and Tinney [1,2].

As a result, the OPF was originally designed as a logical extension of classical economic dispatch (ED) [3]. Despite the fact that both optimization functions may have the same goal, there are differences among them. As a result of ED, the whole power grid is reduced to a single equality constraint. The OPF issue, on the other hand, directly presents all of the system’s key components. It is no longer only the extended ED calculation that is connected with the generic word OPF. An array of optimization issues common to research on power systems are instead presented. Historically, OPF problems primarily concentrated on key objectives within power networks, namely the minimization of economic costs (running costs of thermal generators), the mitigation of active and reactive power losses, and the enhancement of power transfer and control efficiency, reliability, and emissions [4]. In the initial stages, simpler single-objective problems with fewer constraints prevailed due to their operational efficiency. The prevalent methodologies were rooted in mathematical model-based and convex relation-based approaches. Nevertheless, with the enhancing complexity of power networks, the emergence of metaheuristic and evolutionary-based optimization techniques showcased their effectiveness in managing multiple objectives and constraints within larger power networks [5]. Unlike earlier technologies that often overlooked specific objectives, namely emissions and reliability constraints, contemporary OPF considerations adopt a more comprehensive approach. Present-day OPF objectives encompass resiliency, emissions, and reliability as integral components. This inclusivity is achieved through the enhancement of reliability, reduction in reserve and running costs, and the consideration of an overarching system-wide cost function. Over the last two decades, escalating environmental concerns, particularly related to emissions from power plants, mandated the incorporation of such objectives for addressing specific functions. Consequently, the present-day landscape requires the accurate and timely resolution of multiple objectives to adeptly navigate the evolving challenges in power system optimization [6].

With the ever-changing nature of power systems, OPF research has evolved from its original form. The deregulation of the electric power business, advancements in power electronics, and environmental rules placed on power plants have all played a vital role in the OPF’s continual reformulation, and some of the OPF-related problems are given in Figure 1. Since its inception, the OPF has been intimately linked to developments in numerical optimization methods [7]. To solve the OPF, researchers have used a wide variety of optimization strategies.

An OPF is a method for determining a power system network’s optimum setting so that a specific objective function may be best served, while fulfilling all of the system’s power flows, safety requirements, and operating restrictions. Due to this, several control parameters are changed to create the best network configuration. The most often utilized OPF optimization variables are as follows [7]:

• Generator real power results;
• Generator bus voltages;
• Shunt reactors;
• Transformers taps;
• FACTS devices and phase shifters.

The dependent control variables are given as follows:

• Load bus voltages;
• MVA line flows;
• Generator reactive power limits;
• Slack bus active powers.

The contributions of the paper are given below:

• This comprehensive review paper addresses several challenges associated with OPF issues. The discussion encompasses a range of methodologies, including convex, linear, and nonlinear approaches, and explores the integration of modern techniques such as Artificial Intelligence. The examination also extends to various constraints within AC optimal power flow (ACOPF), aiming to provide a deeper understanding of how these constraints impact and influence the optimal power flow.
• OPF challenge involves incorporating the anticipated active power production from renewable energy sources (RESs) and evaluating system’s performance across multiple indices. This analysis encompasses operating costs, voltage profile, and power losses, aiming to comprehend and optimize the intricate interplay between these factors.

The remaining article is arranged as shown below: Section 2 exhibits the formulation of OPF problems, Section 3 explains various techniques used for OPF issues so far, and the conclusion is given in Section 4.

2. Mathematical Formulation

Generally, the OPF problem includes the highly constrained nonconvex, nonlinear large-scale static optimization issue, which is given as [8]

$$min\hspace{0.17em}\hspace{0.17em}f(x,u)$$

(1)

$$\left\{\begin{array}{l}g(x,u)=0\\ h(x,u)\le 0\end{array}\right.$$

(2)

where $f$ indicates an objective function, $x$ and u express dependent and control parameters vectors together, and $g(x,u)\&h(x,u)$ denote equality and inequality limitations.

The OPF problem has made use of a variety of objectives. The primary function of achieving OPF is to keep generator production costs as low as possible. Having voltages close to rated values at different bus points is essential for the power system to function properly. With increased load, the voltage drop on the high-pressured TLs results in an unstable voltage. Reactive power sources that can be controlled are expected to be available in the grid and provide the necessary power to enhance voltage profile. Furthermore, in light of growing environmental awareness, reducing pollutant emissions has been identified as a component of the OPF issue. Therefore, the OPF research may include goals such as reducing emissions, reducing power loss, and reducing the L-index, and those objectives are outlined clearly in the following sections [9].

2.1. Objective Functions

2.1.1. Minimization of Fuel Cost

The basic quadratic cost curve used to depict the fuel costs of thermal generators is given as shown below [9]:

$$\mathrm{f}={\displaystyle {\sum }_{m=1}^{NG}\left(a_m+b_m{P}_{Gm}+c_m{P}_{Gm}^2\right)}$$

(3)

where $a_m, b_m, c_m$ indicate cost coefficients of the $m^{th}$ generator, $NG$ denotes the total generators, and $P_{Gm}$ denotes power at the $m^{th}$ generator.

2.1.2. Minimization of Cost with Valve Points

When the steam admission valves open, the large-scale steam turbine may experience wire pulling effects. As a result, the temperature increases rapidly. To model the valve point loading effect, we add sinusoidal components to a quadratic production cost model. Nonconvex production cost functions may be stated in this manner [9]:

$$f={\displaystyle \sum _{m=1}^{NG}\left(a_m+b_m{P}_{Gm}+c_m{P}_{Gm}^2+\left|d_m\ast \sin \left(e_m\ast \left(P_{Gm}^{\min }-P_{Gm}\right)\right)\right|\right)}$$

(4)

where $d_m, e_m$ denote sinusoidal cost coefficients of the $m^{th}$ unit, and $P_{Gm}^{\min }$ indicates the minimum limit of the $m^{th}$ generator.

2.1.3. Minimization of the L-Index

The voltages on all load buses in a power system should be within their permitted limits. Non-optimized control variables can cause considerable and gradual voltage loss, which can lead to a systemic voltage collapse if the system is disturbed in any way. The voltage stability margin is assessed using the L-index, which was developed in [10]. For example, a bus’ voltage collapse status is indicated by its current value. As a rule, the range of values for the L-index extends from 0 to 1. The following is a mathematical expression for the L-goal index’s function [9]:

$$f=\min (\max (L_m))$$

(5)

where $L_m$ indicates the L-index of $m^{th}$ load bus expressed as [10] follows:

$$L_m=\left|1-{\displaystyle \frac{{\displaystyle \sum _{n=1}^{NG}{H}_{mn}{V}_n}}{V_m}}\right| \mathrm{where} m=1,2,\dots ,NL;$$

(6)

$$H_{mn}=-[inv(Y_{mm})]\ast [Y_{mn}]$$

(7)

where $Y_{mm}\&Y_{mn}$ denote the admittance of the $m^{th}$ bus and the $m^{th}\&n^{th}$ buses.

2.1.4. Minimization of Transmission Loss

The objective function for reducing transmission loss is defined mathematically as follows [9]:

$$f={\displaystyle \sum _{n=1}^{nl}{G}_n\left(V_m^2+V_n^2-2V_m{V}_n\cos {\theta }_{mn}\right)}$$

(8)

where $G_n$ indicates the $n^{th}$ line conductance.

2.1.5. Minimization of Emission Pollution

Currently, society expects not just reliable power but also a low amount of pollution from thermal plants. As a result, one of the objectives for the OPF issue is emission pollution (EP), which may be described as below [9]:

$$f={\displaystyle \sum _{k=1}^{NG}\left({\mathsf{\alpha }}_k+{\mathsf{\beta }}_k{P}_{Gk}+{\gamma }_k{P}_{Gk}^2+{\mu }_k\exp \left({\mathsf{\xi }}_k{P}_{Gk}\right)\right)}$$

(9)

where ${\alpha }_k,{\beta }_k,{\gamma }_k,{\mu }_k,{\xi }_k$ denote $k^{th}$ unit emission factors.

2.2. Constraints

The equality and inequality constraints are mathematically explained as shown below:

$$\begin{array}{l}{P}_{Gm}-P_{Dm}-V_m{\displaystyle {\displaystyle \sum }_{\mathrm{n} \in k}}{V}_n\left(G_{mn}\cos {\mathsf{\theta }}_{mn}+B_{mn}\sin {\mathsf{\theta }}_{mn}\right)=0\\ Q_{Gm}-Q_{Dm}-V_m{\displaystyle {\displaystyle \sum }_{\mathrm{n} \in k}}{V}_n\left(G_{mn}\sin {\mathsf{\theta }}_{mn}+B_{mn}\cos {\mathsf{\theta }}_{mn}\right)=0\end{array}$$

(10)

$$\left\{\begin{array}{l}{P}_{Gm}^{\min }\le P_{Gm}\le P_{Gm}^{\max }\\ V_{Gm}^{\min }\le V_{Gm}\le V_{Gm}^{\max }\\ Q_{Gm}^{\min }\le Q_{Gm}\le Q_{Gm}^{\max }\end{array}\right. m=1,2,\dots ,NG$$

(11)

$$t_m^{\min }\le t_m\le t_m^{\max } m=1,2,\dots ,NT$$

(12)

$$b_{Ck}^{\min }\le b_{Ck}\le b_{Ck}^{\max } k=1,2,\dots ,NC$$

(13)

$$V_{Lm}^{\min }\le V_{Lm}\le V_{Lm}^{\max } k=1,2,\dots ,LB$$

(14)

where $NT,NC,LB,NT,NC$ indicate the number of transformer taps, shunt compensators, load buses, transformers and capacitors, respectively; $P_{Gm}^{\min },P_{Gm}^{\max }$ indicate the lower and higher active power limits of the $m^{th}$ unit; $V_{Gm},V_{Gm}^{\min },V_{Gm}^{\max }$ indicate generator voltage and its lower and higher voltage limits for the $m^{th}$ unit, respectively; $Q_{Gm},Q_{Gm}^{\min },Q_G^{\max }$ denote reactive power and its lower and higher limits for the $m^{th}$ unit, respectively; $t_m,t_m^{\min },t_m^{\max }$ denote the transformer taps and their lower and higher limits for the $m^{th}$ transformer tap, respectively; $Q_{Ck},Q_{Cm}^{\min },Q_{Cm}^{\max }$ indicate the shunt compensator and its lower and higher limits for $m^{th}$, respectively; $V_{Lm},V_{Lm}^{\min },V_{Lm}^{\max }$ denote load bus and its lower and higher voltages for the $m^{th}$ load bus, respectively; and $S_{lm},S_{lm}^{\max }$ denote MVA and maximum MVA flows in the $m^{th}$ transmission line, respectively.

3. Numerous Algorithms Applied to Solve OPF Problems

Different optimization techniques have been applied to evaluate the solution for power system optimal planning and operation problems since the starting of the last century [11,12,13,14,15]. In [11,12], the authors discussed the applications and limitations of different classical techniques for solving OPF issues. In [13], the authors elaborated on major challenges associated with a continually changing load and how computation intelligence tools helped in solving OPF with these challenges. The review in [14] explained the various deterministic techniques applied to solve OPF and concluded that these techniques are only local solvers and may not provide a guaranty in nonconvex problems. In [15], the authors identified challenges and provided solutions to convert current power grid into a smart grid and mainly focused on different OPF issues like OPF uncertainty, risk-based OPF, and conventional OPF. The authors mainly focused on various power system planning objectives for conventional and RES systems and techniques applied to solve these objectives [16]. The major contribution of the current review paper is that different researchers discussed several methods that included classical, iterative, evolutionary and hybrid evolutionary approaches used in solving the OPF issue, which are shown in Figure 2.

Table 1. The taxonomies of different algorithms applied to various OPF problems.

Ref.	Year	Objective Functions							Test Systems Considered	Objectives	With RES	Method	Major Findings
		M1	M2	M3	M4	M5	M6	M7
[17]	1985	Y						Y	IEEE 6, 30 bus	S		GM	Constraints are used without penalties
[18]	1968	Y							IEEE 6-bus	S		GM	Constraints are used without penalties
[19]	1984	Y							IEEE 5-bus	S		Newton	Identifying binding inequalities is a challenging task
[20]	1973	Y							IEEE 57, 118 bus	S		QP	Both inequality and equality considered
[21]	1977	Y	Y						IEEE 30-bus	S		IGA	Line contingency is considered
[22]	2002	Y							IEEE RTS	S		EGA	Enhanced GA is introduced
[23]	2007	Y							IEEE 57 bus	S		HGA	MATPOWER is included with the GA
[24]	2014	Y							IEEE 30-bus	S		GA	Necessity of control parameters is explained
[25]	2002	Y		Y	Y		Y		IEEE 30-bus	S		PSO	New algorithm is introduced
[26]	2005	Y							IEEE 5-bus	S		HPSO	Enhanced PSO is introduced
[27]	2005	Y							IEEE 118-bus	S		MPSO	Multi-objective PSO is introduced
[28]	2004	Y							IEEE 9, 30 118 bus	S		IPSO	Improved PSO is introduced
[29]	2010	Y		Y	Y		Y		IEEE 30-bus	S&M		DE	Numerical based DE is introduced
[30]	2007							Y	IEEE 9, 33, 66, 132	S		HDE	Hybrid DE is introduced
[31]	2011	Y		Y					IEEE 30-bus	S&M		MDE	Weighted matrix for MO
[32]	2017	Y			Y	Y		Y	IEEE 30, 57, Algerian 59 bus	S&M		ESDE-MC	Non-dominated sorting MO
[33]	2011			Y				Y	IEEE 30 bus	S		GSA	Reactive power dispatch is considered
[34]	2012	Y							IEEE 30 bus	S		GSA	Included FACTS devices
[35]	2012	Y	Y	Y	Y		Y		IEEE 30, 57 bus	S&M		GSA	Weighted matrix for MO
[36]	2012	Y							IEEE 9, 30 bus	S		ABC	Transient stability constraints are considered
[37]	2012			Y				Y	IEEE 30, 118 bus	S		ABC	Reactive power dispatch is considered
[38]	2013	Y							IEEE 25, 30 bus	S		ABC-mGA	Security constraints are considered
[39]	2013							Y	IEEE 14, 57 bus	S		ABC	Reactive power dispatch is considered
[40]	2003	Y							IEEE 30-bus	S		SA	Quadratic constraint considered
[41]	2004	Y		Y				Y	IEEE 30-bus	S&M		SA	Weighted matrix for MO is considered
[42]	2015	Y		Y				Y	IEEE 9, 26 bus	S&M		CBO	Weighted matrix for MO is considered
[43]	2016	Y		Y	Y			Y	IEEE 3bus	S		CBO	New EA is introduced
[44]	2016	Y	Y	Y	Y		Y	Y	IEEE 30, 57, 118	S&M		ICBO	Weighted matrix for MO is considered
[45]	2014	Y		Y	Y			Y	IEEE 30, Algerian 59	S&M		BHBO	Weighted matrix for MO is considered
[46]	2015		Y						26 bus, IEEE 30, 57	S		IGSO	Improved version of GSO is introduced
[47]	2015	Y	Y			Y		Y	IEEE 9, 30 bus	S		BBO	New EA is introduced
[48]	2015	Y		Y	Y	Y		Y	IEEE 30, 57 bus	S&M		ARBBO	Weighted matrix for MO is considered
[49]	2014	Y		Y	Y		Y		IEEE 30, 118 bus	S&M		TLBO	Weighted matrix for MO is considered
[50]	2015	Y	Y			Y	Y		IEEE 30, 57 bus	S&M		ITLBO	Enhanced version of TLBO is introduced
[51]	2016	Y	Y		Y	Y		Y	IEEE 14, 30 & 57 bus	S		SKH	Stud GA is added to KH algorithm
[52]	2017	Y	Y		Y	Y		Y	IEEE 30, Algerian 59, 118	S		SKH	Stud GA is added to KH algorithm
[53]	2011	Y		Y				Y	IEEE 30-bus	M		HSA	Non-dominated technique is used
[54]	2016	Y						Y	IEEE 30, 118-bus	S		FHSA	Fuzzy technique is introduced with HSA
[55]	2018	Y		Y				Y	IEEE 30, 118-bus	S		MSCA	Modified version of SCA is used
[56]	2015	Y							IEEE 30, 57	S	Y	MCS	WRIG generator is considered
[57]	2016	Y	Y			Y		Y	IEEE 30, 57 bus	S&M		SOS	Weighted matrix for MO
[58]	2016	Y							IEEE 30, 57 bus	S		G-ABC	Temperature of the TL is considered
[59]	2020	Y			Y			Y	IEEE 30, 118 bus	S		Jaya	Weighted matrix for MO is considered
[60]	2014	Y	Y	Y	Y			Y	IEEE 30, 57, 118 bus	S		KH	Weighted matrix for MO is considered
[61]	2012		Y		Y			Y	IEEE 30, 118 bus	S&M		DE	Weighted matrix for MO is considered
[62]	2014	Y			Y			Y	IEEE 30 bus	S&M		ABC	Non-dominated technique is used
[63]	2014	Y		Y				Y	IEEE 14, 30, 118 bus	S&M		TLBO	Non-dominated technique is used
[64]	2015	Y		Y	Y			Y	IEEE 30 bus	S&M		GSA	Non-dominated technique is used
[65]	2017	Y			Y	Y		Y	IEEE 30, 118 Algerian 59,	S&M		ESDE	Non-dominated technique is used
[66]	2022	Y				Y			IEEE 30 bus	S&M		MRF	Non-dominated technique is used
[67]	2024	Y		Y		Y			IEEE 30 bus	S&M		SCA	Non-dominated technique is used
[68]	2024	Y				Y			IEEE 30 bus	S&M		DPA	Non-dominated technique is used
[69]	2015	Y	Y	Y	Y	Y		Y	IEEE 30, 118 bus	S&M		PSO GSA	Weighted matrix for MO is considered
[70]	2021	Y							PEGASE 13, bus	S	Y	PPOPF	Intermittent nature of RES is considered
[71]	2023	Y							IEEE 33, 69, 118 bus	S	Y	PSO	Intermittent nature of RES is considered
[72]	2022	Y		Y				Y	IEEE 30 bus	S	Y	FFA	Penalty and reserve costs are considered
[73]	2010	Y							IEEE 30 bus	S	Y	W&S	Wind forecast is considered
[74]	2011	Y							IEEE 5-bus	S	Y		Wind model is generated
[75]	2012	Y							IEEE 30- bus	S	Y	ABC	Fixed speed WT is considered
[76]	2009	Y							IEEE 30-bus	S	Y	PSO	Dynamic OPF is considered
[77]	2012	Y							IEEE 39-bus	S	Y	EP	The Monte Carlo simulation is used
[78]	2009	Y							IEEE 30-bus	S	Y	QP	Stochastic model of wind power
[79]	2011	Y							IEEE 118-bus	S	Y	IP	Up and down spinning reserves are considered
[80]	2014	Y						Y	IEEE 30-bus	S	Y	BFA	WECS is considered
[81]	2005	Y							IEEE 5-bus	S	Y	GM	New Q-V model of IG is considered
[82]	2017	Y						Y	IEEE 30, 75 bus	S	Y	PSO-APO	Security constrained OPF is considered
[83]	2016	Y						Y	IEEE 30 & 75 bus	S	Y	FAAPO	Security constrained OPF is considered
[84]	2017	Y							England 39-bus	S	Y	SC	Stochastic model of wind power
[85]	2016	Y							IEEE 30-bus	S	Y	PSO	PDF used for wind power generation
[86]	2017	Y				Y			IEEE 5-bus	S	Y	DE	Wind PDF is considered
[87]	2019	Y	Y			Y		Y	IEEE 30, 118-bus	S	Y	MJaya	Solar, wind, and hydro generation are considered
[88]	2020	Y	Y					Y	IEEE 30-bus	S	Y	PSO	Wind PDF is considered
[89]	2021	Y				Y		Y	IEEE 30-bus	S&M	Y	GOA	Stochastic model of wind power
[90]	2019	Y						Y	IEEE 30, 57 & 118 bus	S	Y	MSA_GSA	Analysis with and without wind power
[91]	2020	Y						Y	14 & 124-bus	S	Y	P-ELM	Penalty and reserve costs of wind and solar
[92]	2023	Y		Y		Y			IEEE 30-bus	S	Y	SCO	Modelling of solar and wind is considered
[93]	2023	Y	Y			Y		Y	IEEE 30-bus	S	Y	WHO	Stochastic model of wind and solar is considered
[94]	2024	Y							-	S	Y	Manto Carlo	Micro grid with EVs is considered
[95]	2024	Y							-	S	Y	-	Case Study at Kuwaiti Roundabout
[96]	2024	Y							IEEE 30 bus	S	Y	WGA	Wind uncertainties are considered
[97]	2024	Y							IEEE 30 bus	S	Y	DRL	Deep learning techniques are considered
[98]	2021
[99]	2024

M1: Minimization of total cost; M2: Minimization of total cost with valve point loading effect; M3: Voltage profile improvement M4: Voltage stability enhancement; M5: Emission pollution; M6: Piecewise quadratic cost; M7: Reactive/Real power loss S: Single-objective problem; M: Multi-objective problem; RESs: Renewable energy sources (solar, wind, hydro); MO: Multi-objective; Y: Objective function is considered in the paper.

depicts the taxonomies of different algorithms applied to various OPF problems.

3.1. Single-Objective Optimization

Several traditional methods have been used throughout the years. Specifically, the OPF issue was solved using a gradient method (GM) in [17]. Nevertheless, GM has a sluggish convergence rate with the steepest direction of descent. In [18], the economic dispatch (ED) issue was resolved using a linear programming (LP) method with security limits. The objective functions and limitations were framed in linear models throughout the LP process, and a solution using the simplex algorithm was obtained. However, the limitation of LP is that it can only be applied to functions and constraints that are linear-only [15]. The Newton method [19] is utilized for network sparsity techniques to find a better solution for a reactive power problem. However, a drawback of this technic is that it is sensitive to an initial guess and may differ if starting values are not selected properly. The quadratic programming (QP) technique was applied to solve the ED issue [20]. For a gradient step size assessment, QP does not require any penalty variables and is in a position to handle both equality and inequality limitations. QP, however, has certain limitations associated with the approximation of quadratic objective functions and constraints that should be linear. To find a solution to the Security Constrained ED (SCED) problem, the interior point (IP) method [100] was used, and it used the benefits of Cholesky factorization, which helps to minimize CPU time while seeking an optimal solution [100]. The IP method, however, restricts the inappropriate collection of the initial stage, resulting in an infeasible solution [15]. While classical techniques are capable of offering nearly optimal solutions to convex OPF issues, if the issues are nonlinear, the techniques become difficult to implement. To address the limitations of traditional approaches and provide a better optimal solution, evolutionary methods were documented.

These approaches are population-based and offer optimal solutions through the proper adjustment of their control parameters with a desired and reasonable execution, and they include genetic algorithms (GAs) [21,22,23,24,101], particle swarm optimization (PSO) [25,26,27,28,102], differential evolution (DE) [29,30,31,32,103], gravitational search algorithms (GSAs) [33,34,35,104], artificial bee colony (ABC) [36,37,38,39,105], simulated annealing (SA) [40,41], colliding bodies optimization (CBO) [42,43,44], black-hole-based optimization (BHBO) [45], group search optimization (GSO) [46], biogeography-based optimization (BBO) [47,48], teaching–learning-based optimization (TLBO) [49,50], krill herd (KH) [51,52,106,107], the harmony search (HS) method [53,108], the electromagnetism-like mechanism [EM] method [54], sine-cosine algorithms [55], cuckoo search [56] and symbiotic organisms search (SOS) [57].

3.1.1. Genetic Algorithm

A GA is based on random selection, and natural genetics is operated in a population of individuals; every individual is an effective solution [101]. GA has many advantages in comparison to conventional techniques, which can be listed as follows: (i) GA deals with solution-set coding itself. (ii) A group of solutions, rather a single search solution, is used, which plays an important part in enhancing the robustness of the GAs and helps the solution to escape from suboptimal conditions, which results in more chance of reaching the global optimum. (iii) GAs use probabilistic operations, while traditional approaches use deterministic operations of transition for optimization, i.e., GAs do not use deterministic rules. (iv) GAs do not need prior knowledge about space restrictions or special characteristics such as convexity or existents of derivatives.

A simple GA (SGA) was reported in [21] that used only basic operators like selection, crossover, and mutation to solve OPF problem. Only the bit chromosome was utilized to encode every control variable in order to keep the GA chromosome limited to a small size. Better results are obtained in different operating conditions compared to the gradient-based conventional method. In addition, an enhanced GA (EGA) was introduced in [22]. Here, a number of specialized genetic operators are also included in an SGA to increase the robustness of the solutions’ GA efficiency. The specialized operators used in the implementation of EGA are as follows: (a) Elitism—From the old generation to the current generation, the first strongest chromosome or few chromosomes are never missing. The most appropriate solution for each generation is in an arbitrary selected chromosome in the next generation. (b) Hill climbing—In smooth areas, it is used to improve the search speed. (c) Fitness scaling—To eliminate premature convergence and slow finishing, linear scaling is taken into account here. In terms of cost and measurement time, optimum solutions were obtained for the IEEE 30 bus, with EGA achieving better cost in comparison to an SGA in a shorter execution time. However, the disadvantage of a GA is that the fitness of a population may be constant over a number of iterations until an optimal individual is acquired. In general, a GA is terminated after a specified number of generations and the best one is checked in the last iteration. If the results are not within acceptable limits, the GA may have to start afresh. In order to avoid this, one combination of a GA with MATPOWER was reported in [23]. In this method, first, the GA worked for a specified number of generations; thereafter, MATPOWER was used to find solution.

Ref. [24] reported a new GA based on the selection of tuning parameters for control variables and state variables. The authors proposed three different algorithms based on the selection of various combinations of tuned parameters that could be tested to find the effective values on the convergence of the SGA. The obtained results were better compared to those from the Lagrange multiplier, the Newton method, and other classical methods.

3.1.2. Particle Swarm Optimization

Kenedy and Eberhart [102] developed PSO, and it is based on bird flocking or fish schooling. In PSO, the search procedure is operated on a population, in which individuals known as agents are represented by their positions in multidimensional space. In flight, each agent adjusts its location due to its experience, called ‘pbest’, and based on the neighboring agents’ experience, called ‘gbest’. The authors of [25] reported the application of PSO to find an optimal solution for OPF problems, i.e., to optimize the total fuel cost, and the L-index. Different IEEE standard test schemes were considered to evaluate performance and obtained improved results relative to those recorded in the literature.

Modified PSO (MPSO) was introduced in [26] for OPF issues. In PSO, agents consider information on only self-experience, and then it is easily trapped into local minima, but in MPSO, agents consider information not only from self-experience and the best information sources also from other agents who are in the swarm. This enhances the probability of the research discovering a global solution and reduces the impact of the starting location of particles. In contrast to EP and hybrid EP, the technique suggested was evaluated on the five-bus system, and the results were optimal. The authors developed a hybrid PSO (HPSO) method in [27] to solve OPF problem. In some examples, PSO leads to premature convergence and does not actually rely on the convergence of the algorithm to a local minimum; it only represents the convergence of all the particles in the best place that the swarm has found so far. This phenomenon is known as stagnation. In order to prevent stagnation in HPSO, the GA mutation mechanism is used in PSO. This approach allows the search to be conducted in various search space areas, which lets the particles avoid being trapped in a suboptimal state. The findings show that the HPSO is more effective than alternative approaches, as indicated in the literature.

Improved PSO (IMPSO), introduced in [28], utilizes the ‘constriction factor’ to enhance the effectiveness of PSO. Due to this, in the study, the maximum velocity Vmax was limited to the dynamic range of the parameter Xmax. The results achieved by PSO with constriction factor methodology have more precision than PSO, which does not use a constriction factor, and this greatly helps with converging the optimal values more quickly.

3.1.3. Differential Evolution

Kenneth V. Price and R. Storn [22] suggested differential evolution (DE) in 1995. DE is one among the most useful techniques for optimizing the stochastic real parameters. Contrary to conventional algorithms, the current generation is interrupted by scaling differences between arbitrary and various members of the population. So, for producing the offspring, there is no need to use a different probability distribution. Another advantage is that DE has developed a unique differential operator system to produce new offspring from parent chromosomes rather than a conventional crossover or mutation. Due to this, DE has become self-adaptive, and the selection process for DE is a greedy one, as it selects the best aspects of a new solution and its parents, due to which DE may escape from premature convergence.

DE was allegedly utilized by the authors of [29] to solve single-objective OPF. The comparison of the achieved results of DE with those of other techniques shows that DE is most effective and better in comparison to the classical and heuristic methods. When handling integer variables with DE, there is a possibility of deteriorating the function of the DE mutation strategy, which results in weakening the solution search ability. To avoid such drawback, a robust searching hybrid DE (RSHDE) [30] algorithm was introduced, and it consists of two different schemes, as mentioned in [30]. The results obtained were compared to several other methods, which revealed a significant improvement in RSHDE’s performance. Another drawback of classical DE is that the selection operator depends on the ‘fitness considered objective function’. For two feasible solutions, the one that has the best value is able to determine whether if a person is feasible and another is unfeasible, as well as the one who is feasible and the other who is favored in all restriction functions if both are infeasible.

Modified DE (MDE) was recommended in [31] to prevent such a situation. In MDE, data may be provided and can only be deleted by adding a new mutation operator and selection mechanism to original DE. So, if number of generations increase, the unfeasible solution rate decreases. Further, in [103], chaotic DE and QP (CCDEQP) were combined to solve OPF problems. In traditional DE, selection, crossover, and mutation are key factors affecting its convergence, but these parameters cannot assure the optimizations completely in the search space because these values are constant and thus easily become trapped in local minima.

In CCDEQP, three novel methods such as chaotic mapping, irregularity, and stochastic characteristics of DE may overcome this situation, enhancing convergence. Due to this chaotic sequence, EAs may be more helpful in comparison to conventional EAs by being able to escape the local minima. DE’s performance relies heavily on the control variables’ adjustment. Improper selection of these control variables can contribute to early convergence.

Therefore, [32] developed DE with mixed crossover (ESDE-MC) for different OPF issues. The control parameter values were changed dynamically by considering the fitnesses of target and trial vectors, which helped the solution to escape from premature convergence.

3.1.4. Gravitational Search Algorithm

The gravitational search algorithm (GSA) was developed by Rashedi [104] in 2009 with the goal of solving optimization problems. The GSA is an agent-based system impacted by Newton’s laws. This technique consists of multiple agents that communicate with each other by gravitational force. In GSA, each agent has several parameters: location, gravitational, and inertial masses. The location of the masses indicates the solution of the issue, and the masses are computed by using fitness values. The technique converges on the basis of the alteration of the gravitational and inertial masses, and the heaviest mass is the best solution.

In [33], a GSA was used to solve OPF with various objective functions on various test systems. The findings acquired from the GSA were compared with several optimization approaches published in the literature, which demonstrated that the GSA was effective for finding solutions for OPF problems. A GSA was also reportedly utilized in [34] to find a solution for OPF incorporating FACTS devices. In [35], a GSA was used for OPF, and the obtained results indicated that the GSA improved the solution quality of nonlinear problems when compared to other optimization methods in a shorter execution time.

3.1.5. Artificial Bee Colony

The artificial bee colony (ABC) technique was introduced in 2005 by Karboga [105] to find a solution to optimization problems, and it is an algorithm focused on swarm intelligence influenced by the collective action of social insects’ communities. In [36], the authors applied ABC to solve a transient stability OPF problem. Here, ABC is divided into two groups, which consist of workers and non-worker bees. A local search is carried out by the onlooker bees around the bees after the measurement of the volume of food in that specific supply. If an alternate nutritional food source is available, this food source may be substituted by new one. If the onlooker bees do not find a better food source in a definite period, they generate a random one.

In [37], ABC is introduced for optimizing the power loss. The advantages of ABC is that it does not need any tuning parameters values, which are very difficult to determine in GA, PSO, and other heuristic approaches. The global search capability is implanted by considering the neighborhood food source mechanism, which is equal to the mutation process. The effectiveness of the ABC method has been tested on different test systems, and proven findings have shown that the suggested approach is superior to the other methods [37].

The authors introduced a combination of ABC and micro-GA (MGA) [38]. In MGA, the population size is very small, the crossover probability is equal to one, and no mutation process is used such that it is quickly stuck in a local minimum. To avoid this premature convergence to a non-optimal solution, the ABC technique was applied first: it explored the search space and gave a solution near to the optimal value; next, the MGA was applied to move closer to the optimal solution with great precision. The formulated algorithm was verified with IEEE 25 bus, and optimum results were obtained in comparison to the individual ABC technique and MGA applied to the same bus system.

DE [39] needs a high population to prevent local optima that increases DE’s convergence time. In order to overcome the disadvantage of requiring a large population in DE, a combination of ABC and DE was developed in [39] to find a solution to the optimal reactive problem. The evolving detection and acceleration processes of the ABC technique were coupled with the population development approach of DE. The article reported tests conducted with several bus systems and achieved better results with a shorter computational time.

Simulated annealing (SA) was used to solve OPF in [40], which was based on the molecular distribution and energy equation formation, and it found near-optimal solutions. In [41], a two-layer SA (TLSA) application for multi-objective VAR source planning was presented. In TLSA, the outermost part solves the maximization problem and innermost part solves the minimization problem simultaneously, which results in reducing the computation burden.

In [42,43], the authors presented a colliding bodies optimization (CBO) technique for solving the OPF issue and compared it to other approaches in terms of computing efficiency. However, CBO attained good convergence, which was close to the global optimal, and it also avoided the major drawback of premature convergence. An improved CBO (ICBO) method was introduced for OPF problems in [44]. In ICBO, the addition of a memory concept to the original CBO provided a smooth curve compared to other methods.

In [45], black-hole-based optimization (BHBO) was used; here, the Schwarzschild radius was an essential factor for controlling both exploration and exploitation capabilities. An improved group search optimization (IGSO) approach was implemented in [46] to solve OPF problems including valve point. The authors extended the GSO by presenting a mathematical idea and also making significant improvements to the conventional GSO parameters.

In [47], BBO was used for single-objective OPF issues. BBO comprises migration and mutation operators, which aid in the search for the best solution. The superiority of the BBO was evaluated on IEEE 30 bus using a variety of single-objective scenarios. However, in [48], the performance of BBO was enhanced by adding a Gaussian mutation operator and developing real-coded BBO (RCBBO), which resulted in improving convergence characteristics in comparison to original BBO.

For running both single- and multi-objective OPF [49], teaching–learning-based optimization (TLBO) was used. The concepts of the instructor and learner phases in TLBO efficiently investigated the searching location and aided in the discovery of a global solution while being independent of algorithmic factors. The efficiency of TLBO was further increased using the Levy mutation operator, which lead to Levy TLBO (LTLBO) [50]. When compared to TLBO, the numerical results of the LTLBO published in [47] turned out to be the superior in terms of speed, solution, time, and lowest objective function values.

The krill herd (KH) method was developed in [106] to address ED with the valve point loading effect for different test systems. KH consists of two local and global strategies that make KH more powerful. However, like all various methods, KH also suffers from premature convergence [51]. Further improvements were reported in [51,52]; the stud KH (SKH) method proposed was capable of solving various OPF issues. The effectiveness of the KH process was enhanced by adding a genetic function known as stud selection and crossover [107], which resulted in the creation of the SKH approach. The SKH approach produced superior simulated results in terms of objective function values in a shorter execution time than KH and the other methods presented in the literature.

Ref. [53] proposed a harmony search (HS) method for multiple-objective functions in OPF. The selection of the pitch adjust rate (PAR) and bandwidth (BW) were the most critical step in the HSA. In the early stages and throughout the iteration process, these ideals were kept constant. The unsuitable selection of PAR and BW values in the initial stage may have provided the worst results and needed further iterations to find the right solution. Fuzzy HSA (FHSA) for OPF problems was introduced in [108].

In FHSA, during the evolution process, the fuzzy technique was applied to monitor the PAR and BW values, and simulation results with FHSA showed that the reported method provided the best values in fewer iterations compared to HSA and the other algorithms. The improved version of electromagnetism-like mechanism (IEM) was introduced to solve OPF with a different objective function in [54]. In IEM, the ‘bounds normalization concept’ was introduced in EM, which helped to reduce the search area to find an optimal solution that helped to find best solution in less time compared to the EM method. In [55], the authors defined an improved version of the sine-cosine algorithm, in which Levy flights were embedded into the original sine-cosine algorithm and added to knowledge sources to influence the DE operator.

A OPF model incorporating wind power generation using a wound rotor induction generator (WRIG) and solutions obtained using the modified cuckoo Search (MCS) algorithm were given in [56]. Here, the performance of the cuckoo search (CS) method was improved by adding a Levy flight operator, which replaced the worst solutions with potentially better solutions that resulted in MCS, providing optimal values in comparison to CS. To address the OPF issue, a new biologically inspired symbiotic organisms search (SOS) method was presented in [57]. It was reliant on the symbiotic communication policies that the organisms used to survive in the environment. The key factor in SOS was that it does not have any control parameters, which resulted in the possibility of organisms trapped in local minima being less compared to the other algorithms. Temperature-dependent OPF was solved by using gbest-guided ABC for the IEEE 30 and 57 bus systems [58]. Here, the temperature effect of the branches was also considered one of the major constraints, while solving the problem that helped the authors to find the accurate active power loss of the TLs.

3.2. Multi-Objective Optimization Problem

Many real-world issues have several objective functions, some of which are in conflict with one another. The difficulty of achieving these goals necessitates the use of multi-objective optimization techniques to discover a global solution to multi-objective optimization problems (MOOPs). The following is a common way to express MOOPs, which have multiple goals bound by different equality and inequality constraints [109]:

$$min\hspace{0.17em}f\left(x,u\right)={\left[f_1\left(x,u\right),f_2\left(x,u\right),\dots ,f_m\left(x,u\right),\dots ,f_M\left(x,u\right)\right]}^T$$

(15)

$$s.t\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left\{\begin{array}{l}{g}_o\left(x,u\right)=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}o=1,2,\dots ,N\\ h_o\left(x,u\right)\le 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}o=\hspace{0.17em}\hspace{0.17em}1,2,\dots ,O\end{array}\right.$$

(16)

where $f_m\left(x,u\right)$ indicates the $m^{th}$ objective, and $g_o\left(x,u\right),h_o\left(x,u\right)$ indicates the $o^{th}$ equality and inequality constraints; thus, MOOPs are solved in two different ways.

3.2.1. Weighted Sum Technique

The weighted sum technique is used to transform the MOOPF issue into a single-objective challenge by utilizing a linear combination of diverse objectives. Typically, the chosen objective functions exhibit conflicting relationships. Consequently, it becomes necessary to identify solutions that provide a balance or compromise solution among the chosen objectives. The model objective function is given below [59]:

$$min\hspace{0.17em}f={\lambda }_1{f}_1+{\lambda }_2{f}_2+\cdots \cdots +{\lambda }_m{f}_m+\cdots \cdots {\lambda }_M{f}_M$$

(17)

where $f_1,f_2,f_m\&f_M$ indicate conflict objective functions, and ${\lambda }_1,{\lambda }_2,{\lambda }_m\&{\lambda }_M$ represent weight factors; the value of these weights will be chosen based on the objective functions.

3.2.2. Non-Dominated Sorting Technique

With regard to all of the objective functions, no one solution to the MOOP issue can be deemed better than another solution. If a MOOP has two solutions, X1 and X2, and the solutions X1 and X2 satisfy the following criterion:

$$fm(X_1)\le fm(X_2)$$

(18)

then, as a result, X₁ and X₂ are classified as non-dominated and inferior solutions. If, on the other hand, Equation (18) is not fulfilled all of the time, these solutions are said to not be substandard. Because of the competitive nature and non-commensurable qualities of objective functions, MOOPs consist of several optimum or non-dominated solutions, and this collection of solutions is known as the Pareto optimal front [109].

In [60], a multi-objective issue is treated as a single-objective issue by using a valuable trade-off ratio. Still, it is a quite challenging task for the researchers to achieve a uniformly distributed collection of non-dominated solutions using this procedure. The problem with this method is that many trial runs are necessary to accomplish a set of non-dominated solutions. To address the aforementioned draw backs and make a trade-off among the conflicting functions in multi-objective OPF (MOOPF), a variety of multi-objective techniques were proposed, namely multi-objective DE [61], decomposition-based multi-objective ABC [108], modified TLBO [63], a non-dominated sorting multi-objective GSA [64], enhanced self-adaptive DE [65], a multi-objective manta ray foraging optimizer [66], multi-objective squirrel search algorithm [67], and a non-dominated democratic political algorithm [68]. In all these methods, non-dominated sorting and crowding distance procedures were employed to extract a set of non-dominated points over the population of the conflicting objectives. Finally, a fuzzy decision procedure was employed to find the best compromise solution. Moreover, several hybrid methods were introduced, and these were GA with SA [110], hybrid DE with BBO [111], sequential quadratic program-based DE [112], double differential with modified TLBO [113], and PSO with GSA [69]. In these hybrid methods, additional features based on the searching processes were included that essentially helped us to find the near-optimal solution.

3.3. Stochastic Optimal Flow Problem

In the last few decades, conventional power plants like thermal, gas, and oil have been the primary sources of electricity generation, causing of 35.29% of all pollutant emissions responsible for climate change and global warming. The rapid rise in power use, as well as concerns about the degradation of conventional energy supplies, has led scientists to focus on renewable energy sources (RESs). RESs have shown great potential in terms of reducing fuel consumption and pollutant emissions in the context of national power savings and emission reductions. By the end of 2023, the global installed capacities of renewables such as solar, wind, hydropower, geothermal, marine, biogas, etc., reached about 3372 GW. The global renewable energy market is expected to continue its upward growth over the next years, growing at a rate of 4.22%. This growth in the RE market reflects a global shift towards renewable and sustainable energy technologies. China and the United States lead the global PV market, with 760 and 265 GW of installed RE capacity, respectively. The installed capacity in Africa amounts to approximately 221 GW. Most of this capacity exists in the form of hydropower plants, though PV solar plants and wind energy technologies have made significant progress in this respect during the last two decades because to their cheap cost, pure nature, and accessibility in comparison to other RESs.

Assumptions, Limitations, and Uncertainties related to RESs:

Assumptions:

• It is assumed that renewable energy sources, like solar and wind, have predictable patterns of availability that can be modeled for grid integration;
• The seamless integration of renewable energy into existing grids is assumed to not cause major disruptions, relying on current grid flexibility and capacity;
• The assumption is made that supportive government policies, such as subsidies and tax incentives for renewable energy projects, will remain stable, driving long-term adoption.

Limitations:

• A key limitation is underdeveloped energy storage technology, which restricts the potential to store and use renewable energy during non-generating periods S;
• The geographical dependency of renewable energy sources limits their effectiveness in regions lacking suitable solar or wind conditions F;
• The high initial cost of deploying large-scale renewable energy systems, including infrastructure upgrades and storage, presents a financial limitation;
• Existing grid infrastructure may not be fully capable of handling the intermittent and distributed nature of renewable energy sources, leading to a limitation in integration potential S.

Uncertainties:

• Climate change introduces uncertainty in the long-term availability and consistency of renewable resources like wind and solar energy;
• Fluctuations in energy market prices due to changing demand, supply conditions, and policy shifts introduce significant economic uncertainty for renewable energy investors;
• Economic uncertainties, such as fluctuating electricity prices and policy changes, affect the long-term feasibility of and investment in renewable energy systems.

Nonetheless, the drawbacks of using wind and solar energies are their unpredictability and the discontinuous behavior of wind velocity and solar radiation [70,71].

3.3.1. Wind Generator Cost Modelling

The power generated by wind turbines is given below [70]:

$$w=\left\{\begin{array}{l}0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\hspace{0.17em} for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0<v<v_{in}\\ w_r{\displaystyle \frac{(v-v_{in})}{(v_r-v_{in})}}\hspace{0.17em}\hspace{0.17em} ;\hspace{0.17em}\hspace{0.17em} for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{v}_{in}\le v<v_r\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}v>v_o\end{array}\right.$$

(19)

where $w\&w_r$ indicate generated and rated wind power, respectively, and $v,\hspace{0.17em}{v}_{in},\hspace{0.17em}{v}_o\&v_r$ indicate wind velocity, cut-in, cut-out, and rated speed, respectively [72];

3.3.2. Solar Energy Cost Modelling

The generated solar power capacity is determined by

$$E_t=P_{STC}[1+{\beta }_p(T_{cell}-T_{STC})]H_t/H_{STC}$$

(20)

where E_t indicates output power, T_STC and T_cell indicate the cell’s surface temperature under the Standard Test Condition, and βp represents power temperature coefficient. The cell surface Tcell is determined as follows:

$$T_{cell}=T_{\infty }+7.8\ast 10^-2H_t$$

(21)

The OPF problem addressed here involves optimizing the objective function consisting of thermal generator costs, including wind and solar energy, which is defined as

$$\mathit{min}\hspace{0.17em}f={\displaystyle {\sum }_{m=1}^{NG}\left(a_m+b_m{P}_{Gm}+c_m{P}_{Gm}^2\right)}+F(P_{wp})+F(P_s)$$

(22)

where $F(P_{wp})\&F(P_s)$ indicate wind and solar cost, respectively.

Apart from the constraints mentioned in Equations (10)–(14), the additional constraints are mentioned here;

$${\displaystyle {\sum }_{n=1}^{N_G}{P}_{Gn}+P_s+P_w=P_D}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(23)

where $P_{Gn},P_s,P_w,P_D$ indicate thermal, solar, wind power, and load demand, respectively.

The authors in [114] deal with the unpredictability of wind behavior, and additional cost considerations must be included in optimum power flow (OPF). The authors discussed the anticipated penalty and reserve costs for not utilizing current wind power, as well as the wind power shortfall. In [73], the authors described the impact of wind power’s stochastic nature on overall operational costs. Ref. [74] created a wind generator simulation and integrated it with a conventional power grid. In [75], the authors addressed an extended OPF problem with wind power incorporation but only examined fixed speed wind generators. OPF was designed using combined-variable and fixed-speed wind energy generators, and the results were achieved using the shuffle frog leap method [76].

In [77], the authors utilized the Monte Carlo method to calculate the cost of wind power; additionally, small signal stability limitations were added to the OPF problem that were handled via self-adaptive EP. The authors created a wind-producing power model based on a relative histogram, which would be incorporated into an enhanced conic quadratic OPF [78]. Another study [115] developed a dispatching model that took combined wind and thermal generators into account, and the effect of wind energy was represented using an incomplete gamma function. Ref. [79] presented an OPF model that incorporated wind energy fluctuation, up and down spinning reserves, and thermal electricity production costs. Ref. [80] used a bacterial foraging method, which included wind power generation utilizing a doubly fed induction generator (DFIG).

Ref. [116] described two different kinds of asynchronous wind turbines used in the power flow research, PQ and RX. Nevertheless, since the PQ or RX models require two iterations, the load flow calculations using these methods are slower in the case of larger power systems. As a consequence, the QV concept of asynchronous wind generators was proposed in [81]. In [82], hybrid PSO was utilised to overcome the security constraints in wind power integration related to objective OPF, with the objectives of optimizing the actual power loss and total operating cost of both the baseline and contingency situations. SCOPF was solved using the fuzzy artificial physics optimization technique in [83], with wind power included in the normal and contingency phases. Wind speed was simulated using the Weibull distribution, and the solution was evaluated with penalty and reserve costs taken into account. The probability distribution function (PDF) took into consideration of the randomness of wind power and load and focuses on reducing heat production in [84]. PSO was utilized to assess OPF integration with and without wind power, while Weibull PDF was employed to account for wind speed uncertainty [85]. Ref. [86] used a success–history–variable adaptation approach to address OPF with wind and sun integration of an IEEE 30-bus system.

An improved Jaya algorithm [87] with RESs was recently applied to OPF. Ref. [88] used a triangular FMF-based model to assess the optimal-wind-power-to-total-generation ratio. The stochastic character of wind speed, wind frequency distributions, and Weibull fitting were accomplished using Monte Carlo Scenarios. The researchers solved OPF with wind power in [89,90], and the present research indicates that a wind farm with variable wind turbines capable of supplying proactive energy to the grid would have an interaction with the wind farm. In [91], the authors addressed the OPF problem, which included wind power, solar energy, and thermal power. To forecast wind speed and solar insolation, a p-ELM technique was utilized as a hybrid of persistence and the extreme learning machine method. In [92], the authors considered carbon tax in addition to solar and wind generations for the OPF issue.

The aim of the researchers in [93] was to enhance the penetration level of RESs in the power system to maximize efficiency by integrating RESs with the OPF problem, and their solution was found by using an improved wild horse optimizer (WHO). The authors in [94] explored stochastic power flow behavior influenced by electric vehicles (EVs) participating in vehicle-to-grid (V2G) systems. They employed Monte Carlo simulations to assess the impact of EVs on grid performance under uncertain conditions and enhanced power grid reliability and stability by evaluating potential variations in load and generation due to V2G interactions. In [95], the authors investigated the feasibility of using PV solar energy to power street lighting in the Gaza Strip. The study assessed the technical and economic viability of a solar-based street lighting system, highlighting the potential for cost savings and environmental benefits in a region facing energy shortages. The paper reviewed power management strategies and sizing optimization techniques for hybrid energy systems (HESs), emphasizing the role of feature selection. It highlighted how selecting key variables improves system performance and optimization efficiency. The study focused on enhancing HES design by integrating renewable energy for better reliability and cost-effectiveness. Ref. [117] reviewed power management strategies and sizing optimization techniques for hybrid energy systems (HESs), emphasizing the role of feature selection. It highlighted how selecting key variables improved system performance and optimization efficiency. The study focused on enhancing HES design by integrating renewable energy for better reliability and cost-effectiveness.

The economic and environmental analysis in the context of the optimal power flow (OPF) problem often involves balancing cost reduction and minimizing environmental impacts, particularly with the integration of renewable energy sources like solar and wind. In studies like the one using the Self-Adaptive Wild Geese Algorithm (SAWGA), the economic analysis focused on reducing fuel consumption and operational costs by optimizing the mix of energy from traditional thermal and renewable sources. This led to lower generation costs while maintaining system reliability. From an environmental perspective, the inclusion of stochastic wind and solar power in OPF models helped to reduce CO₂ emissions. By optimizing the energy dispatch with renewable sources, the system reduced its dependence on fossil fuels, thus cutting greenhouse gas emissions. Additionally, the environmental impact was assessed through the minimization of pollutants alongside economic objectives, creating a dual-focus optimization model that prioritized both cost-efficiency and sustainability, in [96].

Ref. [97] presented a novel hybrid model combining Deep Reinforcement Learning (DRL) with Quantum-Inspired GA (QIGA) to address the OPF in HRES. The experimental results concluded that this approach effectively optimized power generation in real time, adapting to changes in operational demands while enhancing the search process for optimal solutions through quantum principles. Ref. [98] proposed a data-driven OPF approach using a Stacked Extreme Learning Machine (SELM) framework, which allowed for fast training without extensive parameter tuning. By decomposing OPF model features into three stages, the method improved learning efficiency and corrected biases. Numerical evaluations of benchmark systems showed that it outperformed existing methods and could be adapted to different systems with minimal hyperparameter adjustments. Ref. [99] evaluated different power flow models for their effectiveness in OPF studies. It compared four unbalanced three-phase models and introduced a linear model for low-voltage transformers that enhanced accuracy in convex power flow studies. The results show the linear approximation model, with transformer linearization and phase coupling, delivers superior accuracy for large-scale distribution networks.

After thoroughly reviewing the literature on optimal power flow (OPF) problems, several critical research gaps have emerged that warrant further investigation, which are mentioned below:

• Need for real-time solutions to manage dynamic uncertainties in renewable generation;
• Limited exploration of integrating large-scale energy storage systems into OPF models;
• Challenges in the scalability of computational algorithms for large power systems with diverse energy sources;
• Insufficient integration of economic market dynamics into OPF solutions;
• Need for expanding environmental metrics in OPF models to include lifecycle analyses;
• Lack of development of multi-objective optimization models that balance technical, economic, and environmental factors.

4. Conclusions

Techniques that address economic dispatch and OPF problems using methods like PSO, hybrid GA, SA, and bio-inspired techniques improve reliability and accuracy. While these methods have strengths and weaknesses, they have shown promising results in large-scale networks, with acceptable accuracy and low installation costs. For multipurpose functions, hybrid metaheuristic systems like ESDE-MC, IGSA, and PSO-DE provide superior solutions, marking a new era in power systems with the integration of RES. This research provides a broad understanding of OPF methods. The research extensively examines and contrasts OPF methods from various perspectives to provide a comprehensive understanding of the issue, highlighting diverse formulations and broad coverage. Despite these advances, several research gaps remain. These include the need for real-time solutions to manage dynamic uncertainties in renewable generation, limited integration of large-scale energy storage systems, challenges in scaling computational algorithms for diverse energy sources, and insufficient inclusion of economic market dynamics in OPF models. Moreover, expanding environmental metrics in OPF to include lifecycle analyses and developing multi-objective optimization models that balance technical, economic, and environmental factors are crucial areas for future exploration. Addressing these gaps will be essential to meet the evolving demands of modern power systems. Future research directions could explore data science methodologies, machine learning, and soft computing to further address OPF challenges and enhance system performance through scalability and parallelism in software–power system implementations.