Artificial intelligence-based optimization techniques for optimal reactive power dispatch problem: a contemporary survey, experiments, and analysis

The electric power system has become a hot topic over the last few decades due to the exponentially increasing need for electricity in industries, houses, hospitals, transportation, etc. (Mei et al. 2017; Saddique et al. 2020). The optimal reactive power dispatch (ORPD) problem is considered a crucial optimization challenge in this system due to its significant influence on its stability, economic operations, and security (Mei et al. 2017). The ORPD problem is classified as a mixed-integer, nonlinear optimization problem because it involves both continuous and discrete decision variables. Also, it is considered constrained because it contains some equality and inequality constraints that have to be satisfied while searching for the near-optimal values for the decision variables (Mei et al. 2017; Saddique et al. 2020; Abd-El Wahab et al. 2022). Due to its high complexity, traditional techniques, like Newton’s method, quadratic programming, linear programming, non-linear programming, and quadratic programming, could not find an acceptable solution in a reasonable amount of time (Mei et al. 2017; Deeb and Shahidehpour 1988; Lo and Zhu 1991; Granville 1994). Therefore, metaheuristic algorithms have been recently applied to address this problem in an attempt to find an acceptable solution in an acceptable time. The researchers have paid attention to those algorithms due to their ability to solve several complicated problems.

Mei et al. (2017) adapted the moth-flame optimization (MFO) to minimize both the total power loss and voltage deviation for solving the ORPD problem. It is assessed using the IEEE 57-bus system, IEEE 30-bus system, and IEEE 118-bus system, and compared to several optimization techniques to highlight its effectiveness. The experimental findings show that this algorithm could achieve competitive outcomes in comparison to the rival optimizers. Saddique et al. (2020) investigated the performance of four metaheuristic algorithms, such as the whale optimization algorithm, sine cosine algorithm (SCA), particle swarm optimization (PSO), and differential evolution, for solving this problem in terms of four different objective functions. According to the experimental findings, the sine–cosine optimizer is the best alternative metaheuristic algorithm for solving this problem. Li et al. Li et al. (2019) improved the antlion optimization algorithm for presenting a new variant called IALO with better search abilities to accurately solve the ORPD problem. This algorithm was assessed using three IEEE power systems and compared to several optimization algorithms to reveal its effectiveness and efficiency.

Shaheen et al. (2021) combined PSO with the grey wolf optimizer (GWO) to present a better algorithm, referred to as GWO-PSO, for tackling this problem under two different objective functions. This hybrid algorithm was first assessed using four different IEEE bus systems and contrasted with two rival optimizers to observe its efficacy. Abaci and Yamaçli (2017) solved this problem using the differential search algorithm (DSA) to optimize three different objectives. DSA was evaluated using two different power systems (IEEE 57-bus and IEEE 30-bus) and compared to seven optimizers to disclose its effectiveness. Saddique et al. (2022) attempted to minimize the power losses in the transmission system by solving the ORPD problem using the SCA. This algorithm was validated using 57, 14, and 30-bus electric power systems. Also, it was compared to five algorithms to show its effectiveness and efficiency. Results of this comparison show that SCA is better than all the compared algorithms in terms of the final results and computational cost.

The turbulent flow of water-based optimization (TFWO) was improved using the chaotic maps to propose a better variant known as CTFWO that could optimize the real power loss and voltage deviation for solving the ORPD problem more accurately (Abd-El Wahab et al. 2022). This algorithm was tested using two different electric power systems and compared to four algorithms such as artificial ecosystem-based optimization, gradient-based optimizer, classical TFWO, and equilibrium optimizer. The experimental results show that CTFWO is the best-performing algorithm. Shaheen et al. (2023a) enhanced the transient search optimization (TSO) to strengthen its performance for solving the ORPD problem. This enhanced variant was called ITSO and assessed using three different power systems to validate its performance. In addition, it was compared to some rival optimizers to observe its effectiveness. The barnacles mating optimizer was improved and applied to find the near-optimal values for the decision variables of the ORPD problem (Sulaiman, et al. xxxx). Gupta (2022) proposed a simplified variant of the SCA, namely SSCA, to optimize both power loss and net savings for solving the ORPD problem. This algorithm was validated using three electric power systems and compared to some metaheuristic algorithms to show its effectiveness in terms of the final accuracy and convergence speed.

The honey badger algorithm (HBA) was improved using the opposition-based theory to keep its population diversity and increase its convergence speed; this improved variant was called MHBA (Düzenli̇ et al. 2022). The ORPD problem based on five different objective functions, namely transmission cost, voltage deviation power loss, and voltage stability, was solved using MHBA. The IEEE 30-bus power system was used to evaluate the performance of this algorithm. Dora et al. (2023) fused the butterfly optimization algorithm (BOA) with the non-uniform mutation and crossover operators of the exchange market algorithm (EMA) to propose a new variant known as an enhanced BOA (EBOA), which has a high ability to balance between the exploration and exploitation operators. This variant was applied to solve the ORPD problem for optimizing several objectives such as power loss, voltage deviation, and voltage stability. Three electric power systems were used to test and validate the performance of EBOA, and its obtained outcomes for those systems were compared to those of several rival optimizers to disclose their effectiveness.

Nguyen (2019) modified the fractal search algorithm (SFS) to present a modified variant, namely MSFS, with strong search abilities to solve the ORPD problem. In (ElSayed and Elattar 2021), the slime mould algorithm (SMA) was applied to optimize five different objectives, such as power cost, power loss, voltage deviation, system overload, and voltage stability, for solving the ORPD problem. Mugemanyi et al. (2020) used the chaotic maps to improve the exploration operator of the bat algorithm in a new variant termed CBA. This variant has a high ability to avoid stagnation into local optima within the optimization process, thereby achieving better outcomes for the tackled problem. This algorithm was assessed using five power systems and compared with several rival metaheuristic algorithms to reveal its effectiveness. The experimental results show the effectiveness of CBA, compared to the rival optimizers, for solving this problem. Barkavi et al. (xxxx) employed the dragonfly optimization algorithm to search for the near-optimal solution to the ORPD problem, which could minimize the power loss of the electric power systems. This algorithm was tested and verified using the IEEE 30 bus system and compared with some rival optimizers to show its effectiveness.

Table 1 summarizes some studies proposed for tackling the ORPD problem in terms of the used metaheuristic algorithms, publication year, objective functions, benchmarks, and main contributions. In addition, there are several other optimization algorithms proposed recently for tackling this problem, such as the modified ant lion optimizer (Chaitanya et al. 2023), enhanced dwarf mongoose optimization algorithm (Dora, et al. xxxx), coronavirus herd immunity optimizer (Rani and Malakar 2022), artificial hummingbird algorithm (Waleed et al. 2022), hybrid salp swarm algorithm (Kumar and Kumar 2022), hybrid dragonfly algorithm (Emambocus, et al. xxxx), hybrid PSO and pathfinder algorithm (Adegoke and Sun 2022), modified jellyfish optimizer (Gami et al. 2022), enhanced artificial gorilla troops optimizer (Ebeed 2023), improved Pathfinder algorithm (Adegoke 2023), generalized normal distribution optimizer (García-Pineda and Montoya 2023), enhanced PSO (Adegoke et al. 2023), improved Marine predator algorithm (Khan et al. 2022), and several others (Adegoke 2024).

In literature, several algorithms have been proposed to solve the ORPD problem, but they all have at least one of the following problems: stagnation into local minima, low convergence speed, or high computational cost. Those shortcomings prevent finding the near-optimal values for the control variables in the ORPD problem that minimize both transmission power losses and voltage deviation. For example, in Abd-El Wahab 2024a, the artificial rabbit optimizer was integrated with the gradient-based optimizer to propose a hybrid variant, namely AROGBO, to accurately handle the ORPD problem. This variant was assessed using three electric power systems, such as IEEE 30-bus, IEEE 118-bus, and IEEE 57-bus, and compared to several rival optimizers to reveal its effectiveness. Although this algorithm could be better than all the others, its performance still suffers from low convergence speed because it consumes a huge number of function evaluations, up to 25,000. Karmakar (Karmakar and Bhattacharyya 2023) investigated the effectiveness of several metaheuristic algorithms, such as the sine–cosine algorithm, crow search algorithm, particle swarm optimizer, differential evolution, and JAYA algorithm, in optimizing the ORPD problem. The maximum number of iterations and population size in this study were set to 1000 and 20, respectively; this equals 20,000 function evaluations. This means that those algorithms converge slowly, making them not the optimal choice for solving the ORPD problem. Abd-El Wahab (2024b) hybridized the jellyfish search optimizer with the local escape operator of the gradient-based optimizer to propose a robust variant, known as GJSO. This variant was assessed using the IEEE 30-bus and IEEE 57-bus test systems and compared to several competing techniques to reveal its effectiveness. According to the experimental findings, GJSO suffers from low convergence speed due to consuming 25,000 function evaluations to reach the desired outcomes. In addition, its performance for large-scale systems such as IEEE 118-bus and 300-bus test systems had not been validated, so it could not be considered a strong alternative for optimizing the ORPD problem.

Therefore, in this study, four recently published metaheuristic algorithms, namely the mantis search algorithm (MSA), spider wasp optimizer (SWO), nutcracker optimization algorithm (NOA), and artificial gorilla optimizer (GTO), are applied to solve this problem in the hope of fulfilling better outcomes in a reasonable amount of time. These algorithms have been selected because they have robust local optima avoidance and convergence speed acceleration strategies, which help them solve several complicated optimization problems, like parameter estimation of PV models and fuel cells. These algorithms are used to come up with possible solutions to the ORPD problem. Each of those solutions is then mapped onto the load flow data of different electric power systems. Finally, MATPOWER and the Newton–Raphson method are used to figure out the transmission power losses from the load flow data. The summation of those power losses is computed and considered the objective value for the mapped solution. To better solve this problem, a modified variant of NOA, known as MNOA, is proposed. This modified variant does not merge the information from the newly generated solution with the current solution in order to prevent local minima and accelerate convergence. However, MNOA requires more modification to increase its performance for large-scale challenges, so it is combined with a newly proposed improvement mechanism to encourage exploration and exploitation operators. This hybrid variant was dubbed HNOA. Five electric power systems, namely the IEEE 14-bus, IEEE 39-bus, IEEE 57-bus, IEEE 118-bus, and IEEE 300-bus, are used to test and validate the proposed algorithms for the ORPD problem in small-scale, medium-scale, and large-scale. In comparison to several rival optimization algorithms, the experimental results show that MSA could achieve better power losses and voltage deviation for the small-scale system (IEEE 14-bus and 39-bus); MNOA excels for medium-scale systems (IEEE 57-bus); and HNOA is better for medium-scale and large-scale systems (IEEE 118-bus and 300-bus systems). The main contributions of this study are as follows:

• Investigating the performance of four recently published metaheuristic algorithms, named GTO, MSA, SWO, and NOA, to solve the ORPD problem with the purpose of minimizing power losses and voltage deviation.

• Proposing a modified variant of NOA, namely MNOA, with better exploration and exploitation operators to solve this problem more accurately.

• Integrating MNOA with a novel acceleration improvement strategy to further improve its exploitation and exploration operators for better solving large-scale instances, this hybrid variant was called HNOA.

• Validating GTO, MSA, SWO, and NOA using fifteen systems of nonlinear equations and comparing them to some rival optimizers in terms of some statistical metrics, convergence curve, and computational cost to show their effectiveness and efficiency.

• Testing and validating the proposed algorithms for the ORPD problem using Five electric power systems, namely the IEEE 14-bus, IEEE 39-bus, IEEE 57-bus, IEEE 118-bus, and IEEE 300-bus.

• The experimental results show that MSA could achieve better power losses and voltage deviation for the IEEE 14-bus and 39-bus systems; HNOA is better for large-scale systems (IEEE 118-bus and 300-bus systems); and MNOA excels for medium-scale systems (IEEE 57-bus).

The remainder of this paper is organized as follows: Sect. 2 gives an overview of the ORPD problem formulation; Sect. 3 gives a brief overview of the investigated metaheuristic algorithms (the Mantis search algorithm, the nutcracker optimization algorithm, the spider wasp optimizer, and the gorilla troops optimizer); Sect. 4 explains the proposed algorithms; Sect. 5 presents the results and discussions; and Sect. 6 gives a conclusion and future perspectives.

In this paper, the ORPD problem is solved by finding the near-optimal values of three control variables such as generator voltages, reactive shunt compensator, and transformer taps for minimizing the transmission power losses as the first objective \(({F}_{1})\) and voltage deviation as the second objective \(({F}_{2})\) while satisfying several equality and inequality constraints (Saddique et al. 2020). The mathematical models of those objective functions and constraints are described next.

2.1 Transmission power losses \(({{\varvec{F}}}_{1})\)

To solve this problem, several studies in the literature tried to minimize the transmission power losses according to the following formula (Saddique et al. 2020):

where \({V}_{i}\) stands for the \(ith\) bus voltage, \({V}_{j}\) is the \(jth\) bus voltage, \({\theta }_{ij}\) represents the angle between the \(ith\) bus and \(jth\) bus, \(Nb\) stands for the number of buses, and \({g}_{ij}\) represents the conductance between the \(ith\) and \(jth\) buses.

2.2 Voltage deviation \(({{\varvec{F}}}_{2})\)

The voltage deviation could be used as an objective function for solving the ORPD problem. The mathematical model of this objective function is described in the following (Mei et al. 2017; Jamal et al. 2020):

where \(Nl\) represents the number of load buses, \({V}_{Li}\) is the actual voltage at the load bus \(i\), \({V}_{Li}^{sp}\) is the provided voltage (set to 1 p.u.).

2.3 Constraints

2.3.1 Equality constraints

The optimization techniques have been applied to search for the near-optimal solution that could minimize the previous objective functions while satisfying the following equality constraints:

where \({G}_{ij}\) and \({B}_{ij}\) represent, respectively, the conductance and susceptance between the \(ith\) bus and \(jth\) bus; \({P}_{Gi}\) and \({P}_{Di}\) represents active power generation and active load demand, respectively; \({Q}_{Gi}\) and \({Q}_{Di}\) represents reactive power generation and reactive load demand, respectively.

2.3.2 Inequality constraints

Also, the solutions obtained by the optimization algorithms for the ORPD problem must hold true to the following inequality constraints:

where \({N}_{t}\) represents the number of transformers; \({N}_{tl}\) represents the number of transmission lines; \({N}_{g}\) represents the number of generators; \({N}_{c}\) represents the number of reactive compensators; \({P}_{Gi}^{lb}\) and \({P}_{Gi}^{ub}\) represent the lower bound and upper bound active power of the \(ith\) generator, respectively; \({Q}_{Gi}^{lb}\) and \({Q}_{Gi}^{ub}\) represent the lower bound and upper bound reactive power of the \(ith\) generator, respectively; \({V}_{Gi}^{lb}\) and \({V}_{Gi}^{ub}\) represent the lower bound and upper bound power of the \(ith\) generator, respectively; \({T}_{i}^{lb}\) and \({T}_{i}^{ub}\) represent the lower bound and upper bound of the \(ith\) transformer, respectively; \({Q}_{Ci}^{lb}\) and \({Q}_{Ci}^{ub}\) represent the lower bound and upper bound of the \(ith\) reactive compensators, respectively; \({V}_{Li}^{lb}\) and \({V}_{Li}^{ub}\) represent the lower and upper load voltage at the \(ith\) load bus, respectively; \({S}_{li}\) represents the apparent power at the transmission line \(li\); and \({S}_{li}^{ub}\) represents the maximum power at the transmission line \(li\).

Metaheuristic algorithms are considered an indispensable core in soft computing technology due to their effectiveness in solving several complicated optimization problems in a reasonable amount of time. Those algorithms are primarily divided into four categories: the first category is called swarm-based algorithms because it includes the algorithms inspired by the behaviors of birds, animals, and insects in nature; the second category is physics-based algorithms and is based on simulating the physical theories in nature; the third category includes the human-based algorithms that simulate the human behaviors in real life; and the last category is evolution-based algorithms. Over the last few decades, several algorithms belonging to those categories have been proposed as a new attempt to find outstanding algorithms that are able to find better solutions for a wide range of optimization problems. In this paper, four recently published swarm-based optimization algorithms, such as the spider wasp optimizer (SWO), mantis search algorithm (MSA), nutcracker optimization algorithm (NOA), and artificial gorilla troops optimizer (GTO), are applied to solve the systems of nonlinear equations in addition to the optimal reactive dispatch problem as a case study. In the rest of this section, the behaviors of those four algorithms are briefly described.

3.1 Spider wasp optimizer (SWO)

In (Abdel-Basset 2023), the spider wasp optimizer (SWO) was proposed for solving continuous optimization problems, like the parameter estimation problem of photovoltaic models. This algorithm was based on simulating the female spider’s behaviors while searching for the prey, nesting the paralyzed prey, and finally laying the egg over this prey. This algorithm was validated using three different benchmarks: CEC2014, CEC2017, and CEC2020, to observe its exploration and exploitation operators. In addition, it was applied to the parameter estimation problems of double-diode, single-diode, and triple-diode models. Several compared algorithms were used to highlight its effectiveness in terms of several performance metrics and the statistical Wilcoxon rank-sum test. The experimental findings revealed its effectiveness. The mathematical model of this algorithm, in addition to more explanations for its inspiration, is presented in Abdel-Basset (2023).

3.2 Nutcracker optimization algorithm (NOA)

Recently, a new optimization method, dubbed Nutcracker Optimization Algorithm (NOA), has been presented for handling global optimization and engineering design optimization problems (Abdel-Basset, et al. 2023). NOA mimics the nutcracker’s actions while searching for and storing pine seeds and later retrieving them. There are two different nutcracker behaviors that happen at distinct times. The first behavior, more common in the spring and summer, is based on collecting and storing the pine seeds for the coming cold season. During the winter and spring, an alternative behavior is to search for hidden caches that have been marked at various angles with different objects or markers. If the nutcrackers have not found the stored seeds, they will scavenge for them by randomly probing the search area. The mathematical model of NOA is based on two strategies: the forage and storage strategy and the cache-search and recovery method, which are extensively described in Abdel-Basset, et al. (2023).

3.3 Artificial gorilla troops optimizer (GTO)

This algorithm was recently proposed for simulating the gorilla troops’ social behaviors. Similar to the majority of the metaheuristic algorithms, the mathematical model of this algorithm was based on simulating two operators: exploration and exploitation (Abdollahzadeh et al. 2021). The former aims at exploring the search space for finding the region that might involve the global solution, and the latter focuses on those regions for accelerating the convergence speed. This algorithm was assessed using two different mathematical benchmarks: the first benchmark includes the standard test functions that were widely used in the literature; the second includes the test functions available in the challenging CEC2017 competition. It was also applied to seven engineering optimization problems to further verify its performance. Several algorithms were used to observe the effectiveness of this algorithm in terms of several performance metrics. This algorithm has lately been applied to several optimization problems and could achieve outstanding outcomes in comparison to several optimization problems (Shaheen et al. 2023b). The mathematical model and the pseudocode of this algorithm are clearly presented in Abdollahzadeh et al. (2021).

3.4 Mantis search algorithm (MSA)

In (Abdel-Basset et al. 2023), a new nature-inspired optimization algorithm known as the Mantis Search Algorithm (MSA) has recently been presented. This algorithm was inspired by the sexual cannibalism and hunting behaviors of praying mantises in nature. In a nutshell, MSA is comprised of three stages of optimization, namely the search for prey (also known as exploration), the attack of prey (also known as exploitation), and sexual cannibalism. In order to properly address optimization difficulties across a variety of search spaces, those operators are simulated via a variety of mathematical models. The performance of MSA is subjected to stringent testing on fifty-two optimization issues and three real-world applications (five engineering design difficulties, and the parameter estimation problem of solar modules and fuel cells) in order to verify its stability and efficiency for several optimization problems. To highlight its effectiveness, it was compared to several algorithms, such as the artificial gorilla troops optimizer, whale optimization algorithm, differential evolution, gradient-based optimizer, grey wolf optimizer, equilibrium optimizer, African vultures optimization algorithm, weighted mean oF vectOrs, and Runge Kutta method beyond the metaphor. The mathematical model and the pseudocode of this algorithm are presented in Abdel-Basset et al. (2023).

In this study, the performance of four metaheuristic algorithms, termed MSA, SWO, NOA, and GTO, is investigated to reveal their abilities to solve the ORPD problem. Additionally, herein, we present an improved variant of NOA based on hybridizing it with a novel convergence improvement strategy to strengthen its exploration and exploitation operators for solving this problem, especially on a large scale. The main steps to adapt those algorithms to the ORPD problem are extensively described in the next subsections.

4.1 Initialization

This problem involves three main control variables, namely generator voltages, reactive shunt compensator, and transformer taps, where each control variable involves a number of dimensions (\({d}_{j}, j\in \left\{1, 2, 3\right\}\)). The total number of dimensions in this problem is symbolized as \(D\) which must be estimated accurately to minimize \({F}_{1}\) and \({F}_{2}\) while fulfilling several equality and inequality constraints. Similar to the other optimization problems, the proposed algorithms start solving this problem by creating a two-dimensional array of \(N\times D\), where \(N\) represents the number of potential solutions. Then, this array is randomly initialized within the lower bound and upper bound of each control variable as defined in the following formula:

where \(\overrightarrow{r}\) represents a vector including numerical values generated at random according to the uniform distribution, \(\overrightarrow{lb}\) represents the lower bound and \(\overrightarrow{ub}\) represents the upper bound. For example, Fig. 1 is presented to illustrate how to represent the potential solutions to the ORPD problem.

Representation of solutions to the ORPD problem

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4.2 Evaluation

The fitness values of those initial solutions are then evaluated by mapping each solution into the load flow data. Then, the MATPOWER package, with the assistance of the Newton–Raphson method, is used to compute the transmission power loss from the load flow data. This loss value is considered the fitness value for the mapped solution. The fitness values of all solutions are compared with each other to find the solution with the lowest power loss and consider it the best-so-far solution \(\overrightarrow{{x}^{*}}\).

4.3 Adapting SWO, MSA, NOA, and GTO to ORPD

After initializing and evaluating the population, it is updated using those standard metaheuristic algorithms to investigate their performance for finding better load flow data that could minimize the total transmission power losses. Broadly speaking, each algorithm from the proposed algorithms (SWO, MSA, NOA, and GTO) is adapted to the ORPD according to the following steps:

• Step 1: Initialize randomly a population \(P\) of \(N\) solutions according to (12)

• Step 2: Evaluate the initial population according to the MATPOWER platform, and identify the best-so-far solution that has the lowest transmission power loss.

• Step 3: For each \(\overrightarrow{{x}_{i}}\) in \(P\)

  1. o

     Update \(\overrightarrow{{x}_{i}}\) according to the updating mechanism of one of SWO, MSA, NOA, and GTO

  2. o

     Evaluate the new \(\overrightarrow{{x}_{i}}\) and update the local best and best-so-far solution if the new \(\overrightarrow{{x}_{i}}\) is better

• Step 4: Repeat Step 3 until the maximum number of function evaluations (\({T}_{m})\) is achieved.

• Step 5: Return \(\overrightarrow{{x}^{*}}\) as the best solution obtained by the used algorithm. In Fig. 2, the flowchart diagram for adapting those studied algorithms for the ORPD problem is presented.

Flowchart of the proposed algorithms for the ORPD problem

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4.4 The modified NOA (MNOA)

The standard NOA includes two exploration and exploitation operators based on simulating two different behaviors of nutcrackers, which occur at two different seasons. The first pattern represents the foraging and storage behaviors, which are employed to extensively explore the search space to find regions with an abundance of seeds. Those behaviors in the standard algorithm are mathematically simulated based on two operators: the first operator is exploration responsible for exploring the search space, and the second represents exploitation to focus on the best-so-far solution to accelerate the convergence speed. The mathematical model of the exploration operator in the standard algorithm is defined as follows (Abdel-Basset, et al. 2023):

where \({\overrightarrow{x}}_{i}^{t+1}\) represents the updated solution at the function evaluation \(t\); \(j\) represents the \(jth\) dimension in a solution; \(\gamma\) is a number obtained randomly according to the levy-flight; C, A, and B represents the indices of three individuals picked at random from the current population; \({\tau }_{1}\), \({\tau }_{2}\), \(r\), and \({r}_{1}\) are variables assigned random numbers between \(0\) and \(1\); \({x}_{m,j}^{t}\) stands for the mean of the current population; and \(\mu\) is generated as that:

where \({r}_{2}\), \({r}_{3}\), and \({\tau }_{3}\) are variables assigned random numbers between 0 and 1. \({\tau }_{4}\) represents a normal distribution-based random number, and \(\tau_{5}\) is a levy-flight-based random number. In Eq. (13), the classical NOA merges the characteristics of the current and newly-generated solutions to create trial solutions for the optimization problems. This equation might reduce the chance of reaching good outcomes for some optimization problems because merging the old and new solutions might lead to worse trial solutions for two reasons: The first reason is that the old solution might be local optima, thereby causing NOA to permanently stagnate into local optima; the newly generated solution might be the desired solution, and integrating it with the old solution might degrade its performance, making the convergence speed slow. Therefore, this equation is herein modified to make the solutions to ORPD completely depend on the newly generated positions, thereby aiding in avoiding falling into local optima and accelerating the convergence speed. According to that, Eq. (13) is reformulated as follows:

The mathematical model of the exploitation operator in foraging and storage behaviors is mathematically defined as follows:

where \(\uplambda\) is a lèvy flight-based random number, and \(l\) is a linearly reduced factor from 1 to 0. The second and third states in Eq. (16) make the algorithm focus extensively on the best-so-far solution, reducing substantially the population diversity and causing premature convergence. Therefore, this equation is also modified to get rid of premature convergence in an attempt to reach better outcomes, as defined below:

Equations (17) and (15) replace Eqs. (16) and (13), respectively, in the original NOA to present a modified variant, namely MNOA, for better handling ORPD.

4.5 The hybrid MNOA (HNOA)

The proposed MNOA still needs further improvements to promote its exploration and exploitation operators to be able to avoid falling into local minima and accelerate the convergence speed. Therefore, it is integrated with a novel acceleration improvement mechanism based on two folds: the first fold borrows the exploitation operator of generalized normal distribution optimization (GNDO) to observe the solutions around the generalized mean of the current solution as an attempt to accelerate the convergence speed while taking into consideration avoiding premature acceleration, and the second fold employs two different updating schemes to further promote the exploitation and exploration operators. the exploitation operator of GNDO is based on extensively exploiting the regions around the \(ith\) individual’s generalized mean solution (\({\overrightarrow{\mu }}_{i})\), as defined in the following formula (Abdel-Basset et al. 2020):

where \({\overrightarrow{G}}_{0}\)\({{T}_{i}}^{t}\) represents a vector to include the new position of the \(ith\) solution under the first fold, and,\({\gimel }_{1}\), r₂ and \({\gimel }_{2}\) are three numbers randomly generated between 0 and 1. The second fold employs two updating schemes to aid in better exploiting and exploring the search space. The first updating scheme is borrowed from SWO to extensively explore the search space, alleviating falling into local minima during the optimization process. The second scheme is employed to exploit the regions around the best-so-far solution, accelerating the convergence speed. The first updating scheme is mathematically defined as follows (Abdel-Basset 2023):

where \(r\) is a variable to contain a number generated randomly between 1 and -1, \(f\left(\cdot \right)\) stands for the objective function, and \(rn\) and \(r{n}_{1}\) are random numbers generated according to the normal distribution. The second updating scheme is mathematically defined using the following mathematical model to further improve the exploitation operator of MNOA.

where \(r\) is a random number between 0 and 1, \(\overrightarrow{\mathcal{L}}\) is a vector including numerical values generated according to the levy flight, and \(\overrightarrow{r{n}_{2}}\) is a vector including numerical values generated according to the normal distribution. In the proposed mechanism, the tradeoff is achieved between Eqs. (18), (22), (25), and the current solution (\({\overrightarrow{x}}_{i}^{t})\) according to the following formula:

where \({\varvec{r}}\), \({{\varvec{r}}}_{1}\), \({{\varvec{r}}}_{2}\), \({{\varvec{r}}}_{3}\), and \({{\varvec{r}}}_{4}\) are numerical values generated randomly between 0 and 1 according to the normal distribution, and \(\boldsymbol{\rm B}\) and \({\boldsymbol{\rm B}}_{1}\) are predefined controlling parameter to determine the percentage of combination between the current solution and newly generated solutions. This mechanism is effectively hybridized with the proposed MNNOA to propose an additional variant called HNOA, which has outstanding performance for tackling the ORPD problem on a large scale due to its strong exploration and exploitation operators. This hybridization is achieved by dividing the optimization process into four slices: the odd slices apply the acceleration improvement mechanism, and the other slices apply the proposed MNOA. This gives each algorithm a chance to search for a better solution within the search space, significantly exploiting the advantages of each method. The time complexity of the majority of population-based metaheuristic algorithms is based on two factors: the maximum number of function evaluations \(\left({{\varvec{T}}}_{{\varvec{m}}}\right)\) and the number of dimensions \(\left({\varvec{D}}\right)\). According to that, the time complexity of the proposed algorithms in Big-O is \({\varvec{O}}\left({{\varvec{T}}}_{{\varvec{m}}}{\varvec{D}}\right)\); this complexity neglects some other factors in the proposed algorithms, such as the time consumed by the MATPOWER package, due to being the same for all algorithms.

This study is presented to verify and test the performance of four recently published algorithms, such as MSA, SWO, GTO, and NOA, for solving systems of nonlinear equations in general and the ORPD problem in particular. In brief, the experiment section in this study is divided into two parts: the first includes investigating the performance of those algorithms to observe their performance for a set of nonlinear equations, and the second applies those algorithms to the ORPD problem. All experiments are conducted on the same device and implemented in MATLAB R2019a to ensure a fair comparison among the proposed and compared algorithms. To further achieve a fair comparison, the maximum number of function evaluations for all algorithms when applied to the ORPD problem is set to 8000.

5.1 Experiment 1: performance evaluation over nonlinear equation systems

The proposed algorithms (MSA, SWO, NOA, and GTO) are assessed in this section using fifteen systems of nonlinear equations, which are extensively described in Table 2 in terms of the mathematical formulas, search boundary, the number of dimensions (D), and references (Refs). For each system, the fitness values of all nonlinear equations within this system are summed and considered as the overall fitness value of the estimated solution. The following formula is used to compute the summation of the fitness values of all equations in the solved system:

where \({\varvec{n}}\) represents the number of equations in the solved system, and \({{\varvec{f}}}_{{\varvec{i}}}\) represents the \({\varvec{i}}{\varvec{t}}{\varvec{h}}\) equation.

The proposed algorithms are compared to three well-known competitors, such as the golf optimization algorithm (GOA) (Montazeri et al. 2023), Runge Kutta method (RUN) (Ahmadianfar et al. 2021), and grey wolf optimizer (GWO) (Mirjalili et al. 2014) to highlight their effectiveness in terms of the best fitness (BFI), average fitness (AFI), standard deviation (SD), and Friedman mean rank test (FRK). Those algorithms are executed 30 independent runs under the same parameters as those in the cited references, with the exception of the maximum number of function evaluations, which is set to 15,000 to ensure a fair comparison. The obtained outcomes are analyzed in terms of the previously stated performance indicators and reported in Table 3. Inspecting this table shows that MSA is better than all other algorithms in terms of AFI, SD, and FRK for seven nonlinear equation systems; GTO is the best for five systems in terms of the same metrics; GOA ranks first for only one instance; and some algorithms are on par for solving the other systems (F4 and F12). In addition, the convergence curves of each algorithm over some nonlinear equations are depicted in Fig. 3 to further show their effectiveness. This figure indicates that MSA converges faster for F3, F9, F5, F10, and F15; GTO is better for F2 and F5; and GOA converges better for F1, and that confirms the information in the last-mentioned table.

Comparison among algorithms in terms of convergence curve

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5.2 Experiment 2: performance evaluation of the ORPD problem

In this section, the proposed algorithms are tested and validated using six different electric power systems, namely IEEE 14-bus, IEEE 30-bus, IEEE 39-bus, IEEE 57-bus, IEEE 118-bus, and IEEE 300-bus. For these systems, the base power value in megavolt-amperes (MVA) is set to 100 and The branch, bus, and generator data are set as found in the MATPOWER 7.1 package, which is publicly available at (https:, , matpower.org, matpower-7–1-launch, . xxxx). Due to its widespread use in the literature, this package is herein used for simulating those electric power systems that are used to show the performance of the proposed algorithms on small-scale, medium-scale, and large-scale (Zimmerman et al. 2010). In addition, those algorithms are compared to eight rival optimizers, namely the modified pathfinder algorithm (MPFA) (Yapici 2021), crow search algorithm (CSA) (Karmakar and Bhattacharyya 2023), generalized normal distribution optimizer (GNDO) (García-Pineda and Montoya 2023), hybrid artificial rabbits and gradient-based optimization (AROGBO) (Abd-El Wahab 2024a), JAYA algorithm (Roy et al. 2021), improved pathfinder algorithm (IPFA) (Adegoke and Y. 2023), bat algorithm (BAT) (Adegoke et al. xxxx), and jellyfish search optimizer (JSO) (Abd-El Wahab 2024b), under a number of function evaluations up to 8000 to guarantee a fair comparison.

5.2.1 Effect of the population size

The population size has a significant effect on the performance of the metaheuristic algorithms because increasing this parameter unreasonably might maximize the population diversity, minimizing the convergence speed; and reducing it unreasonably might decrease the population diversity, thereby falling the algorithm into local optima. To estimate the best population size for each proposed algorithm in this study, extensive experiments have been conducted under different values, including 5, 10, 15, 20, 25, and 30. Figure 4 presents the average performance of each proposed algorithm for each investigated population size. This figure shows that the performance of NOA, SWO, GTO, and MSA is better when N is equal to 5, 10, 30, and 25, respectively.

Tuning the population size for the proposed algorithms

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5.2.2 Sensitivity analysis

The proposed algorithms include some controlling parameters that need to be accurately estimated to maximize their performance. Therefore, we conduct several experiments in this section to estimate the best value for each controlling parameter in the used algorithms.

The proposed NOA has three controlling parameters that are defined below:

• \({{\varvec{P}}}_{{{\varvec{a}}}_{1}}\)_: Responsible for determining the exploration operator probability.

• \({{\varvec{P}}}_{{{\varvec{a}}}_{2}}\): Responsible for determining the exchange probability between the cache-search stage and the recovery stage.

• \({\varvec{\delta}}\): Responsible for avoiding local optima problems.

Several experiments are conducted to estimate the best value for each parameter, and the results of those experiments in terms of FRK are shown in Fig. 5. This figure shows that NOA performs better when \({{\varvec{P}}}_{{{\varvec{a}}}_{1}}=0.3\), \({{\varvec{P}}}_{{{\varvec{a}}}_{2}}=0.7\), and \({\varvec{\delta}}=0.3\).

Tuning the controlling parameters of NOA

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The outcomes of multiple experiments performed to determine the optimal value for each SWO parameter are illustrated in Fig. 6 in terms of FRK. SWO performs better when TR = 0.3, and CR = 0.7, as illustrated in this figure.

Tuning the controlling parameters of SWO

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MSA has six control parameters, namely A, \({\varvec{a}}\), p, P, \({\varvec{\rho}}\), and\({{\varvec{P}}}_{{\varvec{c}}}\), which significantly affects its performance. Therefore, in this section, extensive experiments are conducted to estimate the best value for each parameter. The results of those experiments in terms of the average fitness are presented in Fig. 7. Inspecting this figure shows that MSA performs well when A = 5, \({\varvec{a}}=0.05\), p = 0.1, P = 10,\({\varvec{\rho}}=10\), and \({{\varvec{P}}}_{{\varvec{c}}}=0.01\).

Tuning the controlling parameters of MSA

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Two control parameters have a substantial impact on the efficacy of GTO. As a result, extensive experiments are conducted in this section to determine the optimal value for each parameter. The average fitness values representing the outcomes of those experiments are illustrated in Fig. 8. This figure reveals that GTO exhibits favorable performance under the following parameters: W = 0.7 and p = 0.01.

Tuning the controlling parameters of GTO

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The proposed HNOA includes two control parameters: \({\varvec{B}}\) and \({{\varvec{B}}}_{1}\) that need to be accurately estimated to maximize its performance for optimizing the ORPD problem. Therefore, several experiments are conducted under various values for each parameter and their findings are reported in Fig. 9. This figure reveals that HNOA exhibits better performance under the following values: \({\varvec{B}}=0.2\) and \({{\varvec{B}}}_{1}=0.7\).

Tuning the controlling parameters of HNOA

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10.1 Comparison between HNOA, MNOA, and NOA over six electric power systems

In this section, we investigate the performance of the proposed variants of NOA for solving ORPD using five studied electric power systems. After executing the proposed variants 20 runs independently, their obtained outcomes are analyzed in terms of BFI, AFI, worst fitness (WFI), SD, FRK, p-value of the Wilcoxon rank-sum test, and computational cost (time) and reported in Table 4. From this table, it is obvious that HNOA has outstanding performance for large-scale systems (IEEE 300-bus and IEEE 118-bus); MNOA has slightly better performance than HNOA for IEEE 57-bus and substantially better than NOA; and three algorithms are on par for IEEE 14-bus. In brief, HNOA is superior in large-scale systems and somewhat competitive with MNOA for medium- and small-scale systems. To further show the effectiveness of both HNOA and MNOA against the classical NOA, the convergence curve obtained by each algorithm on four investigated electric power systems is shown in Fig. 10. MNOA, according to this figure, converges faster for the IEEE 57-bus, while HNOA converges better for the other systems. Consequently, both MNOA and HNOA have better performance than the classical NOA, thus showing that our improvements have a significant positive effect on their performance.

The convergence curves obtained by the proposed variants of NOA for some studied systems

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Performance analytics of algorithms for IEEE 14-bus system over the first objective

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10.2 Performance evaluation over IEEE 14-bus system

In this section, the proposed algorithms are compared to six rival optimizers using the IEEE 14-bus system (Chen et al. 2017). This system contains 10 decision variables, which are divided into five generators at buses 8, 6, 1, 2, and 3; three transformers at branches 4–7, 4–9, and 5–6; and two reactive shunt compensators at buses 9 and 14 (Ghasemi et al. 2015; Shareef and Rao 2018). The search boundaries of those variables, as given in Saddique et al. (2022), are set in Table 5. To estimate the near-optimal values for the controlling variables in this system, all algorithms under two different scenarios are independently executed 20 times. The first scenario, denoted as B, includes initializing the first solution in the population using the base case with a total power loss of 13.6822, while the other solutions are randomly initialized; and the second scenario, denoted as R, involves initializing all solutions randomly within the lower and upper bounds of each decision variable. Under the R and B scenarios, the outcomes obtained by each algorithm for the first objective within 20 independent times are analyzed in terms of BFI, AFI, WFI, SD, FRK, and p-value and reported in Tables 6 and 7. Inspecting these tables shows that the performance of all algorithms except GNDO under both R and B is somewhat similar, as also affirmed in Fig. 12. Among those algorithms, MSA under the B scenario has a slightly better performance in terms of all performance metrics. To further show the effectiveness of MSA for the first objective, Fig. 11 is presented to depict the convergence curves and the multiple comparison test for all algorithms under two considered scenarios. Figure 11(a) confirms that MSA has a convergence speed that is competitive with both HNOA and MNOA and better than all the other compared algorithms. Furthermore, Figure 11(b) affirms that MSA has a mean rank a little better than the proposed MNOA and HNOA and significantly better than all the other algorithms. For the voltage deviation, the same performance metrics in addition to the execution time are used, and their outcomes are reported in Table 8, which shows the effectiveness of MSA for finding the near-optimal values for the control variables in the IEEE 14-bus system. The experiments conducted in this section to observe the performance of the proposed MNOA, HNOA, MSA, GTO, and SWO on the IEEE 14-bus system give the following conclusions:

• Employing the base case in the initial population aids in somewhat improving the convergence speed.

• The performance of MSA is a little better than some algorithms like HNOA and MNOA and significantly better than the others.

• MSA, HNOA, and MNOA almost have the same convergence speed.

• The computational cost is somewhat similar among all algorithms.

Comparison between the performance of algorithms for the IEEE 14-bus under R and B scenarios

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10.3 Performance evaluation over IEEE 39-bus system

The IEEE-39 bus system is employed in this section to further assess the performance of the proposed algorithms against several rival algorithms when applied to solve ORPD. In brief, this system consists of 21 decision variables, which need to be accurately estimated to minimize total power losses (Mugemanyi et al. 2020). Those decision variables encompass the voltages of ten generators at the buses 31, 33, 32, 38, 37, 35, 36, 33, 46, and 39; five transformers at the branches 12–11, 10–32, 22–35, 2–30, and 19–20; and the buses 1, 5, 11, 14, 22, and 27 are used to represent the reactive shunt compensators. The lower and upper bounds of those variables taken from Mugemanyi et al. (2020) are shown in Table 9. Under the R scenario, the proposed and rival optimizers are executed 20 independent times, and within each run, each algorithm creates new solutions for estimating 21 control variables. Those solutions are used to update the generators, transformers, and compensators in the MATPOWER package, which is then fired to compute the total power losses and voltage deviation. The total power losses obtained through 20 independent trials by each algorithm are analyzed in terms of BFI, AFI, worst fitness (WFI), SD, FRK, and p-value and reported in Table 10. Inspecting this table shows that HNOA, under the R scenario, is better than all algorithms in terms of AFI, WFI, and SD, and its BFI and FRK values are a little worse than MPFA. To further show the effectiveness of HNOA for the first objective, Figs. 13(a) and (b) are presented to depict the convergence curves and the multiple comparison test. Those figures show that HNOA is almost on par with MNOA, SWO, and MPFA. Under the B scenario, the proposed and competing optimizers are also run 20 times independently, and their findings are displayed in Table 11. Going over this table reveals that MSA outperforms all other algorithms in terms of all considered performance metrics, followed by HNOA and MNOA as the second and third best, respectively. The convergence curves and the multiple comparison test are displayed in Figs. 13(c) and (d) to further demonstrate the efficacy of the proposed algorithms for the first objective. Those figures illustrate that HNOA, MSA, and MNOA are somewhat competitive for the convergence curve and mean column ranks when applied to solve this system. For the voltage deviation as the second objective, the same performance metrics in addition to the execution time are used, and their outcomes are reported in Table 12, which shows the effectiveness of MNOA for all performance metrics except for the execution time, where MNOA is a little higher than some of the other algorithms. Finally, Finally, Fig. 14 is presented to compare the performance of each algorithm under both R and B scenarios; this figure shows that the performance of some algorithms, such as HNOA, MNOA, JSO, CSA, and MSA, under the B scenario, is better than its performance under the other scenario, while the remaining algorithms perform better under the R scenario. In brief, we conclude that the proposed algorithms (HNOA, MNOA, and MSA) perform better under the B scenario than under the R scenario, and their performance in comparison to the other algorithms is better.

Performance analytics of algorithms for IEEE 39-bus system over the first objective

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Comparison between the performance of algorithms for the IEEE 39-bus under R and B scenarios

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Performance analytics of algorithms for IEEE 57-bus system over the first objective

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10.4 Performance evaluation over IEEE 57-bus system

This section employs another electric power system, namely IEEE 57-bus, with 25 control variables to further observe the performance of the proposed algorithms, especially with increasing the number of dimensions. Those control variables include the voltages of 7 generators at buses 1, 2, 3, 6, 8, 9, and 12; fifteen transformers at the lines 4–18, 4–18, 21–20, 24–25, 7–29, 34–32, 11–41, 15–45, 14–46, 10–51, 13–49, 11–43, 40–56, 39–57, and 9–55; and 3 compensators at the buses 18, 25, and 53 (Chen et al. 2017). In addition, the upper and lower bounds of those control parameters are shown in Table 13, as described in Mei et al. (2017). Under the R scenario, the proposed algorithms, as well as competing optimizers, are independently executed 20 times, and their total power losses as the fitness values are summarized in Table 14. An examination of this table reveals that, under this scenario, HNOA outperforms all other algorithms for this system, and MSA is considered the second-best algorithm with an average performance of 2.5098E + 01. This table also illustrates that all algorithms, except HNOA, MNOA, and MSA, could not satisfy all constraints in all independent runs, leading to high fitness values due to adding a penalty value of \({10}^{5}\) in a case where a constraint has not been satisfied. This implies that those algorithms have poor exploration and exploitation operators, which make them unable to explore the search space as thoroughly as feasible to satisfy all constraints and achieve the lowest power losses. On the other side, the proposed MNOA, HNOA, and MSA could achieve all constraints in all independent runs. It is worth mentioning that we use the R scenario to test the ability of the algorithms to detect the desired solutions without depending on any guide, such as the base case, during the optimization process.

Furthermore, this section tests the performance of the algorithms with the B scenario to witness their exploration and exploitation operators when inserting the base case into the population. The results of all algorithms for this system under the B scenario are shown in Table 15. This table shows that MNOA is the best, HNOA is the second best, and JAYA is the worst-performing algorithm. To compare the performance of algorithms under two scenarios, Fig. 16 is presented. This figure illustrates that the performance of all algorithms except HNOA, MNOA, and MSA under the R scenario is significantly worse than that under the other scenario. As an additional demonstration of HNOA’s efficacy with respect to the total power losses, Fig. 15 illustrates the convergence curves and results of a multiple comparison test under both R and B scenarios. Figure 15(a) demonstrates that HNOA under the R scenario has a better convergence speed than all compared algorithms, followed by MNOA as the second best. Under the B scenario, Fig. 15(c) illustrates the high convergence of MNOA against all compared algorithms.

Comparison between the performance of algorithms for the IEEE 57-bus under R and B scenarios

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The results of the algorithms for voltage deviation are presented in Table 16. Under the R scenario, this table demonstrates the efficacy of HNOA in identifying the control variables that are better than those estimated by the other algorithms. Also, from this table, it is obvious that the BFI for some algorithms is smaller than or a little higher than 1, but the WFI and AFI are so high. This occurs because those algorithms could not satisfy all constraints at all independent times, so high values are added to the fitness values of those solutions to show that they are infeasible for the ORPD problem. Our conclusions from the experiments conducted herein are summarized in the following list:

• Under the R scenario, HNOA performs better than all rival algorithms.

• Under the B scenario, MNOA is the best, followed by HNOA as the second-best, and JAYA is the worst algorithm.

• The performance of all algorithms except MNOA, HNOA, and MSA under the R scenario is significantly worse than their performance under the B scenario due to their inability to satisfy all constraints at all independent times.

• In summary, MNOA under the B scenario is considered the best for finding the control variables of this system.

10.5 Performance evaluation over IEEE 118-bus system

The IEEE 118-bus electric system is herein used to further observe the effectiveness of those algorithms on a large scale. This system involves 77 control variables and is herein investigated under two different cases, where each case is based on a different search space for its control variables as used in Yapici (2021). Those control variables are divided as follows: 54 generators; 9 transformers at the branches 8–5, 26–25, 30–17, 38–37, 63–59, 64–61, 65–66, 68–69, and 81–80; and 14 compensators at the buses 5, 34, 37, 44, 45, 46, 48, 74, 79, 82, 83, 105, 107, and 110 (Mei et al. 2017). The upper and lower bounds of these variables under two cases are depicted in Table 17 and taken from Yapici (2021).

10.5.1 IEEE 118-bus system: Case 1

For this system, under the lower and upper bounds of decision variables in case 1, we run the proposed algorithms alongside competing optimizers 20 times, with each run yielding new solutions for predicting 77 control variables. Tables 18 and 19 summarize the results of an analysis of the total power losses obtained by each algorithm under the R and B scenarios after 20 independent runs, taking into account the algorithm’s BFI, AFI, WFI, SD, FRK, and p-value. Inspecting those tables reveals that HNOA outperforms all other algorithms in terms of all performance metrics. As an additional illustration of HNOA’s efficacy, Fig. 17 displays convergence curves and the multiple comparison test. Inspecting these figures shows that HNOA under both R and B scenarios is better than all compared algorithms for minimizing the total power losses in this system. In addition, Fig. 18 illustrates that the performance of all algorithms, except HNOA, under the B scenario is significantly better than their performance under the R scenario. Table 20 displays the results of algorithms for the voltage deviation; these results demonstrate HNOA’s ability to locate the values of the control variables that could optimize the voltage deviation better than all algorithms, despite having a somewhat greater computational cost than some algorithms.

Performance analytics of algorithms for the IEEE 118-bus system (Case 1) over the first objective

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Comparison between the performance of algorithms for the IEEE 118-bus (Case 1) under R and B scenarios

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Comparison between the performance of algorithms for the IEEE 118-bus (Case 2) under R and B scenarios

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10.5.2 IEEE 118-bus system: Case 2

This section is presented to report the performance of the proposed and compared algorithms under the R and B scenarios for accurately estimating the controlling variables of the IEEE-118-bus system. The lower and upper bounds of those variables are based on Case 2 in Table 17. We run the proposed algorithms 20 times alongside competing optimizers, with each run producing new solutions for estimating 77 control variables in this system. Tables 21 and 22 summarize the findings of an examination of each algorithm’s total power losses under these two scenarios after 20 separate runs, accounting for the algorithm’s BFI, AFI, WFI, SD, FRK, and p-value. Inspecting those tables reveals that HNOA surpasses all other algorithms on all performance metrics. Figure 20 shows convergence curves and the multiple comparison test to further demonstrate HNOA’s efficacy. Inspecting these figures reveals that HNOA, in both R and B situations, outperforms all other algorithms in terms of minimizing total power losses in the system. Furthermore, Fig. 19 shows that, with the exception of HNOA, MSA, and CSA, all algorithms perform much better under the B scenario than under the R scenario. Results of algorithms for the voltage deviation are shown in Table 23, which show that HNOA can find control variable values that can optimize the voltage deviation more effectively than any other algorithm while having a somewhat higher computational cost than all algorithms.

Performance analytics of algorithms for IEEE 118-bus system (Case 2) over the first objective

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10.6 Performance evaluation over IEEE 300-bus system

Finally, to further show the ability of the proposed algorithms to solve the ORPD problem on a large scale, this section employs another electric power system, namely IEEE 300-bus, with 190 control variables. Those variables are divided as follows: Voltages of 69 generators, 107 transformers, and 14 compensators. The details of those variables are extensively described in Mugemanyi et al. (2020). In addition, the upper and lower bounds of those control parameters are reported in Table 24, as described in Mugemanyi et al. (2020). The results of an analysis of the total power losses incurred by each algorithm under the R scenario across 20 independent trials are presented in Table 25. The analysis considers various metrics, including BFI, AFI, WFI, SD, FRK, and p-value. According to this table, HNOA outperforms all other algorithms across all performance metrics, with an average fitness value of 3.81E + 03 and an FRK of 1.15, followed by MNOA in the second rank with an FRK of 1.85. To further show the performance of HNOA under the R scenario, Figs. 21(a) and (b) present convergence curves and the multiple comparison test, respectively. From those figures, HNOA outperforms all other compared algorithms in terms of minimizing overall power losses, followed by MNOA as the second-best. Under the B scenario, all algorithms are executed in 20 independent runs, and their results are reported in Table 26. According to this table, HNOA is more effective than all compared algorithms, having an average performance of 3.77E + 03, which is relatively better than its performance under the R scenario. The second-best algorithm under this scenario is MNOA, and the worst algorithm is the JAYA algorithm. Convergence curves and the multiple comparison test are presented in Figures 21(c) and (d), respectively, to further illustrate how HNOA performs in the B scenario. According to those figures, HNOA performs better than any other compared algorithm in terms of minimizing total power losses, with MNOA coming in second. Moreover, Fig. 22 demonstrates that all algorithms perform significantly better under the B scenario than under the R scenario.

Performance analytics of algorithms for IEEE 300-bus system over the first objective

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Comparison between the performance of algorithms for the IEEE 300-bus under R and B scenarios

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This research paper presents the application of four recently published metaheuristic algorithms, namely MSA, NOA, SWO, and GTO, to address the ORPD problem. These algorithms were chosen herein due to their mechanisms for preventing local optimality and accelerating convergence speed. In addition, a modified variant of NOA, known as MNOA, is herein presented to further improve its performance when solving this problem. This improved version avoids local minima and speeds up convergence by not combining the information from the newly generated solution with the existing solution. However, the performance of MNOA still needs strong improvements to better tackle the ORPD problem on a large scale. Therefore, it is integrated with a newly proposed improvement mechanism to promote its exploration and exploitation operators; this hybrid variant is called HNOA. The objectives of these proposed algorithms are to minimize voltage deviation and power losses in electric power systems. Five electric power systems, including IEEE 14-bus, IEEE 39-bus, IEEE 57-bus, IEEE 118-bus, and IEEE 300-bus, are used to observe the effectiveness of the proposed algorithms against several rival optimizers in optimizing the ORPD problem at small-scale, medium-scale, and large-scale. According to the experimental findings, HNOA outperforms the competing optimizers in terms of power losses and voltage deviation for large-scale systems (the IEEE 118-bus and 300-bus systems), MNOA could achieve better performance for the IEEE 57-bus systems as a medium-scale instance, and MSA demonstrates exceptional performance for the small-scale instances (IEEE 14-bus and 39-bus systems). Quantitatively, MSA could achieve an improvement percentage over the rival optimizers ranging between \(0.001\) and \(46.464\boldsymbol{\%}\) for the IEEE 14-bus system and between \(0.00062\) and \(43.042\boldsymbol{\%}\) for the IEEE 39-bus system; MNOA could achieve an improvement percentage ranging between \(0.5\) and \(27\boldsymbol{\%}\) for the IEEE 57-bus system; and HNOA could reach an improvement percentage between \(0.4\) and \(81\boldsymbol{\%}\) for the IEEE 300-bus system and between \(1.01\) and \(34.81\boldsymbol{\%}\) for the IEEE 115-bus system. Our future work includes further improving the performance of HNOA, either using some optimization techniques such as opposition-based learning and chaotic maps or integrating it with some other metaheuristic algorithms, to strengthen its search abilities for small- and medium-scale systems. In addition, these metaheuristic algorithms will be applied to tackle several other optimization problems, such as:

• 3-D Routing Planning for UAV.

• Lost Target Search with UAV.

• UAV-Assisted IoT Data Collection System.

• Task Scheduling in Cloud Computing.

• Energy efficiency in the IoT networks.

• Energy-Efficient Trajectory Planning for Multi-UAV-Assisted MEC System.