A hybrid capuchin search algorithm with gradient search algorithm for economic dispatch problem

This paper presents an effective approach for solving economic load dispatch problems contemplating the scheduling a set of thermal generating units to produce a specific power at low consumption costs. These problems can be thought of as nonlinear, non-convex, and highly constrained optimization problems with a large number of local minima. To cope with the above issues in solving such problems, a new meta-heuristic named capuchin search algorithm was adopted. To boost the search performance of this algorithm as well as to mitigate its early convergence and regression to the local optimum, it was hybridized with another algorithm and improved using several positive amendments. First, a memory element was added to this algorithm to ameliorate its position and velocity update mechanisms in order to exploit the most encouraging candidate solutions. Second, two adaptive parametric functions were used to manage the exploration and exploitation features of this algorithm and balance them appropriately. Finally, the hybridization was made using the gradient-based optimizer to strengthen the intensification ability of this algorithm and balance its searching ability to fulfill sensible search performance. The proficiency of the proposed algorithm was divulged by assessing it on computationally difficult economic load dispatch problems under 6 different tests with a generator of 3, 13, 40, 80, and 140 units, each with different constraints and load conditions. The proposed algorithm provided the best performance among many other competitors. Its superiority and practicality were revealed by obtaining optimal solutions for large-scale test cases such as 40-unit and 140-unit test systems.


Introduction
In the digital age, the processes of the power system can be deemed the main driver of revolutions in the industrial, economic, social, and other related fields.Reduction of the cost of power consumption attracts the attention of power systems' engineers to achieve efficient use of operations.Economic load dispatch (ELD) can be utilized in power systems to efficiently handle the cost of energy consumption.In generators, scheduling the thermal generating units in an economical way to gain the target power output in relation to the specified operating constraints is the ultimate goal.These constraints are related to power demand, power output, valve-point loading effect, transmission damage, restricted operating areas, ramp-rate limits, etc.Specifically, the ELD problem can be designed as an optimization problem with nonlinear, non-convex, highly constrained, and with a large number of local minima when the valve point effect is taken into account.
In the initial stage of addressing ELD problems, many researchers presented several classical mathematical methods to accomplish this stage, such as nonlinear programming (Reid and Hasdorff 1973), dynamic programming (Travers and Kaye 1998), Lagrange relaxation (Boyd et al. 2004), and mixed integer programming (Wang et al. 2014).Due to the mature theoretical basis and rapid convergence of these classical methods, they can quickly tackle small-dimensional ELD problems, but they may fail to deal with large-scale ELD problems with a large number of constraints.The main difficulties with these mathematical methods are their sensitivity to oscillations, discontinuities, and nonlinearity.Currently, complex ELD problems are typically handled by meta-heuristic (MH) algorithms to get a reasonable level of performance.MH optimization algorithms have attracted the attention of a lot of academic researchers and research communities working in energy scheduling systems to overcome the shortcomings of classical mathematical methods (Zhang et al. 2021).These algorithms provide an optimization framework that is able to explore and exploit the search space of ELD problems through well-tuned learning factors to efficiently search for optimal solutions (Zhang et al. 2021).
There are a lot of merits of MH algorithms, which can be easily channeled to solve a broad range of optimization problems in a variety of domains.Essentially, these algorithms strive to produce a suitably useful solution to any real-world problem that is complex and difficult to address to an optimum level (Braik et al. 2023).These types of algorithms encapsulate problem-specific knowledge in the form of mathematical objective functions and formulation of solutions.They are derivative-free optimization methods with no specific parameters in most cases.Moreover, they are quite flexible and do not depend on the nature of the problems of interest.Lastly, they adopt effective stochastic mechanisms to escape from local optima (Braik et al. 2021).Eminently, many MH algorithms of various classes and inspirations have been used promisingly to address complex real-world problems such as image enhancement (Braik 2022), feature selection problems (Braik 2023), and many other applications in a diversity of fields (Yigit et al. 2023).This effective and widespread use of MH algorithms in addressing realworld problems is ascribed to their potential in controlling their exploration and exploitation aspects.
As can be inferred above, the ELD problem is formulated and solved as an optimization problem (D' Angelo et al. 2022).Regarding the use of MH algorithms in solving ELD problems, remarkably, large numbers of MHs were used to solve ELD problems in the literature, each problem with different search spaces and different complexities.There are many examples found in the literature, including, but not limited to, particle swarm optimization (PSO) (Zhang et al. 2021), flower pollination algorithm (FPA) (Ramli et al. 2020), and chameleon swarm algorithm (Braik et al. 2023).
The standard MH algorithms may not often perform well in solving ELD problems, as their initial results might often not be satisfactory.This is typically attributed to the intricacy and cambered shape of the search spaces of these problems.Therefore, most of the original MH algorithms used to tackle these problems have either been improved (i.e., modified) or hybridized with other algorithms to strengthen their exploration, exploitation, and convergence behaviors in solving ELD problems.Examples of modified MH algorithms used to tackle ELD problems include the island-based harmony search algorithm (Al-Betar 2021), the multi-group marine predator algorithm (Pan et al. 2021), the modified krill herd algorithm (KHA) (Kaur et al. 2021), and many more.There are also a lot of hybrid MH algorithms used for handling ELD problems such as hybrid salp swarm algorithm (Alkoffash et al. 2021), hybrid gray wolf optimizer (GWO) (Al-Betar et al. 2020), and many others.
As ELD problems typically have many constraints in their nature, each oriented approach used to solve such problems must be prepared with a constraint handling approach.In general, there are two main strategies for dealing with constraints in ELD solutions: penalty or feasibility (Ramli et al. 2020).In the penalty strategy, the violations of the constraints may occur in the solution and the significant penalty weight for each constraint violated will be embedded in the evaluation function (Pan et al. 2021).Also, an independent objective function can be drafted to handle the violated constraints (Alkoffash et al. 2021).The feasibility strategy precludes any constraint violation in the solution.As a result, any impractical solution regardless of the value of its objective function is discarded (Al-Betar et al. 2020).To turn an infeasible solution into a feasible one, a heuristic repair function was tailored in agreement with the constraints of ELD problems.Examples include handling inequality constraints using a barrier function and handling equality constraints using Lagrangian multipliers (Boyd et al. 2004).
In fact, much of the previous works reported in the literature for solving ELD problems in power systems behaved well in all small-scale cases of ELD problems, but gave poor performance in some large-scale cases of ELD problems with generators of 80 or 140 units.This involves that every MH algorithm has limited functionality in handling a large number of units in generators and various constraints while solving such problems.This points out that there is no general approach reported in the literature that is able to solve all small and large scale cases of ELD problems with the best solution or provide a high level of performance for all cases.This deterioration in the accuracy of the achievements in reliably solving ELD problems is associated with several issues such as problem size and intricate search space especially when considering the valve point effect of these problems.Thus, the search for other effective solutions to ELD problems is still in high demand.This is also centered in the no-free-lunch (NFL) theory (Wolpert and Macready 1997), whereby a particular MH cannot act very well for all sets of optimization problems or for all different cases of the same problem.Each problem has its own search space, and the MH requires a proper synergy between exploration and exploration to fine-tune convergence behavior.Therefore, there is room for improvements to enhance the robustness in finding more accurate results for problems of interest.
Indeed, this motivated us to search for a new and reliable MH that can better deal with ELD problems to deliver a better level of efficiency than those currently found in the literature.In line with this, a recently developed MH algorithm named capuchin search algorithm (CSA) (Braik et al. 2021) was adopted in this paper to solve a variety of small-and largescale cases of ELD problems.The CSA algorithm mimics the natural conduct of capuchin monkeys during the foraging process.Superbly, this algorithm has received a lot of attention from researchers due to its ease of implementation and simple structure with few control parameters that require adjustment.As widely reported in the literature, this algorithm has been improved, modified, or hybridized with other algorithms for specific applications.Such updated versions of CSA were applied to solve various optimization problems in different fields such as feature selection (Abd Elaziz et al. 2023), wind power prediction (Al-qaness et al. 2022), and modeling of industrial systems (Braik 2021), and many other fields.
Although the basic CSA algorithm has divulged reasonable levels of performance in addressing complex problems when dealing with various functions of nonlinear continuous search spaces, it did not ensure general cost-optimal in all of these addressed problems (Braik et al. 2021).This is because this algorithm faces few drawbacks in its basic mathematical model, such as deviating from the global optimal or falling into local optimal values.Other deficiencies of this algorithm might be related to the difficulty of appropriately balancing between exploration and exploitation features.Quite simply, these issues can be dealt with by enhancing these features of CSA within the search space of the problems of interest.A convincing balance between these two aspects will help lessen computational cost and obtain an efficient optimization.This, in turn, will help address real-world problems with convoluted search spaces.
Based on the points discussed above, it is fundamentally important to improve the main functions of CSA to keep pace with those challenges in tackling these problems.In this, after a deep study of the optimization framework of CSA, we propose a hybrid improved CSA (HICSA).In this hybrid algorithm, the exploration and exploitation conducts of the basic CSA are ameliorated to be applicable to the challenging nature of ELD problems.In sum, the key contributions of the proposed HICSA algorithm for solving ELD problems can be briefly described as follows: • First, a memory element is added to the search agents of CSA in order to promote the follow-up of the best former solutions got to date and to find optimal food places.• Second, two adaptive functions were applied to replace the fixed social and cognitive parameters of CSA, which were implemented during the iterative loop steps of HICSA.• Third, the best solutions of CSA are iteratively fed to the gradient-based optimizer (GBO) to instruct HICSA to search for supportive candidate solutions.• Finally, a repair method was utilized at each iterative loop of the proposed HICSA to handle any constraint violations that occurred throughout the search process in order to maintain the feasibility of the solutions.
The above updates are expected to mature a reasonable diversity of the population of HICSA during optimization.As per this, this proposed HICSA is anticipated to have promising accuracy in achieving the global optimal solutions associated with efficient convergence rate in solving the ELD problems considered in this study.
The overall performance of the proposed HICSA algorithm was assessed on ELD problems with 6 different test cases with a generator of 3, 13, 40, 80, and 140 units.To judge the efficacy and robustness of HICSA among other competitors, a comprehensive comparative study was conducted between it, the basic CSA, and other state-of-the-art methods.
In Sect.2, several methods to solving ELD problems are reviewed.The definition and mathematical formulation of ELD problems are presented in Sect.3. A description of the basic CSA and GBO is provided in Sect. 4. The proposed method is discussed in detail in Sect. 5.The computational results are presented, discussed, and analyzed in Sect.6 with closing comments and possible future works in Sect.7.

Related works
A large number of MH algorithms have been modified, improved, or hybridized with others and then applied to address a large variety of ELD problems.This section reviews the most recent studies that have addressed ELD problems using MH algorithms that were modified, improved, or hybridized with other ones.
A hybrid salp swarm algorithm (HSSA) was combined with β-hill climbing algorithm to tackle different cases of ELD problems with valve point effect (Alkoffash et al. 2021).
The key aim of HSSA is to lessen the cost of fuel in relation to equality and inequality constraints.The HSSA algorithm has an improved balance between exploitation and exploration aspects which is reflected in improved overall performance while solving ELD problems.Pan et al. (2021) have evolved an improved variant of the marine predator algorithm (MPA) to address ELD problems in power systems.This improved variant exploited the multigroup mechanism which encourages faster convergence and faster production of better solutions.In Kaur et al. (2021), the ELD problem was addressed by implementing a modified version of the KHA algorithm (MKHA), which uses crossover and mutation operators to strengthen local search capabilities.The MKHA algorithm used a repair operation to handle the equality constraints in ELD problems, thus altering the total power generation upon user power consumption.The MKHA algorithm was evaluated on three power systems to verify the efficacy of the solutions obtained.
A reasonable approach for solving ELD problems in power systems is presented in Kapelinski et al. (2021).This approach proposed a new variation of the firefly optimization algorithm with a non-homogeneous population.In this approach, the population size was increased four times, that is, from 4 to 4 × n, where n represents the population size, and this approach also used two methods to initialize the population.The results yielded from the experimental results confirmed the competitiveness and effectiveness of the proposed approach compared to other best reported methods in the relevant fields.In another study, the idea of clustering techniques was embedded with cuckoo search optimization to address six real cases of the ELD problem in a power system (Yu et al. 2020).The initialization step generates clustered population.The evolution step updated the positions based on the relative average fitness between each cluster and the entire population.This idea revealed the viability of the proposed method through comparative results.
In Spea (2020), crow search algorithm was modified to cope with complex nature of the ELD problem.This method only considered the feasible areas of the search space.Three cases of ELD problem were used for the purpose of evaluation.The performance of this algorithm was demonstrated by comparative evaluation and statistical analysis.Another study reported in Al-Betar et al. (2020), addressed the ELD problem efficiently using a hybridization of β-hill climbing (βHC) with the iterative loops of GWO.That method showed its applicability when tested on five real-world ELD cases.It was able to achieve very competitive results compared to other state-of-the-art methods.The ELD problem was also addressed by a modified version of the dragonfly algorithm (DA) Das et al. (2020).Thermal energy, plant wind, and solar renewable systems were deemed in the objective function.Four ELD systems were used to assess the performance.The evaluation results against well-established methods showed the feasibility of the modified variants of DA in terms of computational time and solution cost.Braik et al. (2023) evolved an enhanced chameleon swarm algorithm (ECSA) by amalgamating Lé vy flight and roulette wheel selection techniques to solve five different cases of ELD problems in a power system with a generator having different number of units, constrains, and under different conditions.The results of the ECSA algorithm showed that it is better than the basic CSA and other competing methods designed to solve the same cases of ELD problems.This method showed its superiority in obtaining hot best solutions for ELD cases with 140-unit and 40-unit power systems.Examples of other recent studies reported in the literature applied to solve ELD problems in various power systems are modified versions of the Jaya optimization algorithm (Balamurugan et al. 2020), and FPA (Ramli et al. 2020).In a general sense, the modified and hybrid versions of these algorithms performed better than the original ones.
Although there has been tremendous progress reported in recent studies to deal with ELD problems, the state of the art in handling these types of problems is still evolving.Each of the approaches discussed above or reported in the literature acted effectively in some cases of ELD, but some of them fell short in other cases, such as those ELD problems involving generators with large-scale cases such as 80-or 140-unit systems.Furthermore, no general approach presented in the literature is capable of providing high performance for all cases of ELD or dealing with all ELD problems with optimal solutions for all situations.To find more convincing and effective solutions to ELD problems than those presented in the literature, the capuchin search algorithm as a newly developed MH algorithm was adopted and improved to investigate its performance in handling ELD problems.The following sections describe the ELD problem and the algorithms adopted to solve this problem.

Economic load dispatch problem
ELD is one of the important problems in power systems due to the fact that the natural sources of oil are decreasing year by year.The main objective of ELD problems is to reduce the total fuel cost consumption while satisfying the pre-determined objective function and various practical constraints as described below.

Objective function
The objective function of the ELD problem has been modeled as a quadratic polynomial as given by Eq. (1). (1) where F i (x i ) is the total fuel cost of the generation unit at the ith position, x i is the real power output of the generating unit at the ith position, and a i , b i , and c i are the fuel cost coefficients for generating units at the ith position.
As shown in Eq. ( 1), the objective value is calculated by adding the cost of fuel for all generating units in each solution.The ELD solution is represented as a one-dimensional array x = (x 1 , x 2 , . . ., x d ), where d represents the number of generating units.
As a matter of fact, the generating units have multiple fuel input valves.This leads to changes in the fuel cost of the generating unit, so the result is a non-convex behavior of the generating units.Therefore, the rectified sinusoidal components are added to Eq. ( 2) as shown below: where e i and f i are the fuel cost coefficients for the generating unit at the ith position with valve-point effects, and x min i is the minimum limits for the generating units at the ith position.
It should be noticed that valve-point effects increase the number of local optima and thus increase the complexity of the search space of the ELD problem (Al-Betar et al. 2020).

ELD constraints
The ELD problem is a multi-constrained optimization problem, which includes power balance, generation limits, prohibited operating zones, and ramp rate limits constraints.It should be noted that the power balance and power output limits constraints are only the ones that were taken into account when solving the ELD problems under study as presented in the results section.This fits with other relevant comparative methods mentioned in the literature.A description of these used constraints is presented below:

Power balance constraint
This constraint is classed as an equality constraint, where the total generation of all generating units must be equal to the sum of the expected total demand and the total transmission loss as shown below: where x D is the total anticipated demand and x L is the total transmission loss computed using B coefficient methods as shown below: where B 00 is the value of the loss coefficient, B 0i is the value of the loss coefficient of the generating unit at position i in a vector space, and B i j is the loss coefficient matrix.
For easiness, the overall transmission loss was overlooked in this work as did by other researchers in the literature (Braik et al. 2023).

Power output limits
This constraint is classed as an inequality constraint.The power output for each generating unit in the system shall be within the feasible range as follows: where x min i and x max i are the minimum and maximum power output limits of the generating unit at the ith position.

Capuchin search algorithm
CSA is a new meta-heuristic evolved by the natural conduct and life activities of capuchins while foraging in real life.The population of capuchins in CSA can be split into two main collections: the leader (i.e., the alpha) and the followers.The leader is accompanied by other capuchins that are liable for locating food sources for the other capuchins in the CSA algorithm.The rest of the capuchins are the followers that update their position by going behind the leaders inwards the group.The capuchins in CSA, whether as leaders or followers, move through the forest in search for food sources with a velocity defined as given by Eq. ( 7) (Braik et al. 2021).
where v k+1 i, j and v k i, j denote the next and current velocity of capuchin i at dimension j at the respective iterations k+1 and k, x k i, j is the current position of capuchin i at dimension j and iteration k, lbest k i, j is the best position got so far for capuchin i at dimension j and iteration k, gbest k j represents the best position for food at dimension j and iteration k, c 1 = 1.50 and c 2 = 1.50 are two acceleration factors that manage the impact of lbest k i, j and gbest k j on the velocity, r 1 and r 2 are stochastic values that can be configured independently in the range between 0 and 1, and χ represents the weight of inertia that manages the impact of the former velocity on the capuchin motion which is defined in Eq. (8).
where k and K represent the present and maximum iteration values, respectively, and w l = 0.1 and w u = 0.9 refer to the maximum and minimum coefficient values of the inertia weight, respectively.As reported in Braik et al. (2021), the following five movement tactics can be used by the leaders and followers in CSA while foraging.These movement strategies form the bases and rudiments of the standard CSA, which can be mathematically drafted as follows: • The position of the leaders while jumping on trees can be settled as follows: where x k i, j identifies the current position of the leaders at dimension j, gbest k j identifies the position of the food at dimension j, η is a random number created in the range [0, 1], P b f = 0.75 identifies the balance probability granted by the capuchins' tails, g is the gravitational force with a value of 9.81, and θ is the hopping angle of the leaders, which is given in Eq. (10).
where rand is a uniformly generated random number in the range [0, 1].• The leaders' position while foraging on riversides using leaping mechanism can be settled as follows: x k+1 i, j = gbest k j + P e f P b f (v k+1 i, j ) 2 sin(2θ) g i < n/2 0.15 < η ≤ 0.20 (11) where P e f = 19 is the probability of elasticity of movement of the capuchins on the ground.• The leaders' position while foraging on ground using normal walking can be settled as follows: • The leaders' position while foraging on trees using swinging strategy can be settled as follows: x k+1 i, j = gbest k j + P b f × sin(θ ) i < n/2; 0.75 < η ≤ 0.90 (13) • The leaders' position while foraging on trees using climbing strategy can be settled as follows: In the CSA algorithm, the random path of leaders can be used to forage at different locations, which can be identified as given by Eq. (15).
where Pr indicates the random search probability of the leaders having a value of 0.1, ub j and lb j denote the upper and lower limits of the search domain at dimension j, and τ represents a function of time defined as shown in Eq. ( 16).
Equation 16 was proposed in CSA to ensure optimal convergence by decelerating the search velocity and improving the key features of CSA.Big values for τ result in a local search in the proximity of x k i, j , while tiny values lead to global search which is far away from x k i, j .The parameter η, as a random number between 0 and 1, was adequately configured to select between the aforementioned movements.Therein, it is divided into intervals to select between strategies of movements.As such, the leaders' position is updated on the basis of random values produced by the use of the η operator.Specifically, η ≥ 0.1 denotes that the leaders are aware of the food's location in the surrounding environment.In this, they change their position as per the observation of a food source in the search domain using one of the five locomotion paradigms presented above.Anyway, when η < 0.1, the leaders use Eq. ( 15) to randomly change their location in the search domain in several ways and regions during foraging.
The followers' position in CSA can be updated as follows: where x k i, j and xk i, j refer to the former position of the followers and the current position of the leaders at dimension j, respectively.
In the CSA algorithm, the solutions are evaluated for each new position of each capuchin on the basis of a pre-defined fitness norm.The optimization steps carried out with CSA can be undertaken through iterative loop steps.In this, the new position of the capuchins can be assessed and amended.

Algorithm 1
The pseudo code of the capuchin search algorithm.
1: Initialize the control parameters of CSA: k, K , n, P ef , P b f , g, ub j , lb j 2: η and r 3 are random values in the range of [0, 1] 3: Initialize the positions and velocity of all capuchins 4: Evaluate the fitness of each solution 5: Define θ as given in Equation 10. 6: while (k < K ) do 7: Update χ and τ using Eqs.8 and 16, respectively.8: Update the capuchins' velocity using Equation 79: Update the position of the leaders using Equations 9,11,12,13,14 and 15 10: Update the position of the followers using Equation 1711: Assess the fitness of all capuchins 12: Update the global and local solutions of all capuchins 13: k ← k + 1 14: end while 15: Return the best global solution These steps are to be reiterated at each iteration until the point of convergence is attained, by which the look for convergence ceases when a pre-established termination norm is met.The pseudo-code of the basic CSA algorithm can be concisely given as shown in Algorithm 1.
As CSA has assured its credibility and speed of convergence in solving many engineering optimization problems (Braik et al. 2021), we concluded that CSA could be a suitable optimization algorithm for solving the ELD problems in power systems.However, as a newly mature algorithm, CSA holds shortcomings in some points.The first point concerns with the bounded exploration feature that may degrade the diversity of solutions and in some cases get stuck in local solutions.Also, there is a bounded exploitation feature with respect to the search in promising search areas, which may fall apart in ameliorating the quality of the solution.To mitigate these issues, the power of CSA in both exploration and exploitation features has to be heightened before embarking to solve ELD problems of interest.Therefore, a hybridization of CSA with gradient-based optimization (GBO) described in the next subsection was conducted to enhance each of the aforementioned features as well as to balance them.

Gradient-based optimizer
The GBO optimizer was evolved based on the simulation of gradient-based Newton's approach (Ahmadianfar et al. 2020).It relies on two core operators that used to update the solutions, each with its own task.The first operator is the gradient search rule (GSR) which is utilized to boost exploration in GBO, while the second one is the local escaping operator (LEO), which is used to promote the exploitation feature.

Initialization phase
The first process in GBO is to randomly generate an initial population using a uniform random distribution.The population contains a number of N search agents in a d-dimensional search space.This process is achieved using the following mathematical formula: where i = 1, 2, . . ., N , lb and ub are the lower and upper limits of the decision variables in the search space, and rand refers to a random number produced in the interval [0, 1].

Gradient search rule phase
The GBO algorithm updates the position of the search agents during optimization on the basis of the gradient specific direction.To ensure and promote a balance between exploration and exploitation in the search space in order to arrive at near or optimal global solutions during optimization, an important factor in GBO, denoted as χ 1 , was defined as follows Ahmadianfar et al. (2020): where the component α can be defined as shown below: where β can be defined as given below: where β min = 0.2, β max = 1.2, k refers to the current iteration, and K stands for the total number of iterations.
It is clear from Eqs. ( 19) and ( 20) that to strike a balance between exploration and exploitation of GBO, the parameter ρ 1 is updated on the basis of the sine function α in Eq. ( 20).The GBO algorithm is then empowered to avoid local subareas, where the GSR operator can be applied in GBO to update the solutions as follows: where ρ 1 is drafted as given by Eq. ( 19), x best and x worst represent the best and worst solutions got so far, and the component x is defined as shown below: where rand(1 and step stands for a step size defined by x best and x k r 1 as given by Eq. (24).
where x r 1 is A randomly chosen integer from [1, N ] and to ensure that x alters throughout the iterative process, and δ is defined as shown in Eq. (25).
where x r 1 , x r 2 , x r 3 , and x r 4 are different integers chosen randomly from [1, N ] where The concept of the GSR operator in the GBO algorithm presents stochastic behavior throughout the iterative process and thus enhances the exploration behavior with fleeing from local optima.
The direction of movement (DM) in GBO was used to converge around the solution area x i .In this term, the best vector is used to move the current vector (x i ) in the direction of (x best − x i ).DM is defined using the following formula: where rand stands for a uniform distributed number within the interval [0, 1] and ρ 2 is a random parameter employed to adjust the step size of each search agent, which is defined as follows: Lastly, based on the operators GSR and DM, Eqs.22 and 26, respectively, can be used to iteratively update the position of the current vector (x k i ).
where x1 k i is the new vector produced by the update of x k i , where x1 k i can be reformulated as follows: where yp k i and yq k i , respectively, represent y i + x, and y i − x, y i represents the mean of the current solution x i and z i+1 , where z i+1 can be computed as follows: where randn is a random solution vector of dimension d, x worst and x best , respectively, identify the worst and best solutions, and x is defined by Eq. ( 23).Based on the above mathematical formula, when replacing the best solution x best with the current solution x k i , X 2 k i can be collected as follows: Equation 29 was defined to enhance the global search ability during the exploration stage, while Eq. ( 31) was utilized to enhance the local search ability during the exploitation stage.Finally, a new solution k + 1 can be obtained at iteration on the basis of the positions x1 k i , x2 k i and x3 k i .This is basically defined as given by Eq. ( 32).
where the parameters r a and r b stand for uniformly formed random numbers in the range [0, 1], and x3 k i can be identified as follows:

Local escaping operator phase
The LEO operator was proposed in GBO to expand its exploitation capacity.Specifically, LEO can influentially update the current solutions of GBO by assisting it to avoid local optima solutions and speeding up the convergence process.This is attained through the updating the solution X k i as per Eq. ( 34).
where pr is a probability value, rand represents a random number uniformly distributed between 0 and 1, f 1 represents a uniformly distributed random number created in the range of [−1, 1], f 2 is a random number generated with a normal distribution with a standard deviation of 1 and a mean of 0, and u 1 , u 2 , and u 3 specify three random values defined as shown in Eqs.34, 34 and 34, respectively.
where rand and μ 1 represent two random numbers defined in the range ∈ [0, 1].Equations 34, 34 and 34 that are defined for u 1 , u 2 , and u 3 can be simply expressed as given in Eqs.34, 35 and 36, respectively.
where L 1 denotes a binary variable that can be either 0 if Finally, the new solution xm k can be constructed as follows: where x k p indicates a solution chosen randomly from the population, μ 2 stands for a random number formed in the range ∈ [0, 1], and x rand denotes a random solution produced using Eq.(37).
Briefly, the key pseudo-code steps of the GBO algorithm are summed up in Algorithm 2.

Hybrid ICSA for ELD problems
This section describes the proposed hybrid improved capuchin search algorithm with gradient-based optimizer, referred to as HICSA, presented to solve the demanding ELD problem.The proposed HICSA algorithm works in two stages: i) The velocity and position update models of the standard CSA were improved using a memory concept and two adaptive functions for the social and cognitive parameters of CSA, wherein this approach is named as ICSA, and ii) a hybridization of the ICSA algorithm with GBO was implemented using the new positions and the best global solutions got by ICSA for all capuchins at each iteration level of the GBO algorithm.
Algorithm 2 A pseudo-code summarizing the key steps of the GBO algorithm.
1: Initialize the control parameters of GBO: N , pr, T , η 2: Generate the initialize population of GBO using Equation 183: Evaluate the position of the initial population using the fitness function f ( * ) This subsection formulates the set of improvements and benefits crafted to the classical CSA model.Also, an explanation of the differences between the proposed ICSA and standard CSA is provided in this section.The amendments made to CSA are intended to reinforce the exploration and exploitation features of it on top of maturing a proper balance between these features as described below.

Velocity updating model
In the proposed ICSA algorithm, purposeful modifications were made to the velocity of capuchins in the basic CSA during foraging, wherein this velocity was modeled in ICSA as given by Eq. (38).
where r 1 and r 2 are two random values created from a standard normal distribution between 0 and 1, ν i is the ith index of the capuchins' memory reaching the best position and is defined by Eq. ( 39), c 1 and c 2 are two adaptive acceleration parameters defined in Eqs.41 and 42, respectively, and χ is the adaptive inertia weight used to control the effect of the former velocity on the capuchin movement specified in Eq. (40).
where rand (1, n) is a vector of independently generated random values within the range [0, 1].Equation 39 was proposed in ICSA to flesh out the memory concept of the capuchins to keep track of the potential positions of food sources and other capuchins in the population.It is widely known that capuchins have memories that help them follow leaders to guide them to find food sources in the surrounding environment.The followers roam the search space and attempt to follow other followers and leaders to assist them locate a source of food at iteration time k.
With specific details, upon iteration k, the position of leaders looking for food at dimension j is assigned to x(ν) k i, j .It is usual for leaders and followers to navigate the environment in search of the best places with the best food sources.In this, there is no doubt that every follower keeps in its memory the place of the leaders or other followers that arrive at the place of a memorized food source.This process may lead to an enhancement in the exploitation feature of CSA, and thus, capuchins may find their way to the best solution.Also, during the population-based process, ICSA maintains all the fitness values of the best global solution to rapidly converge in time to the global optimum solution.
Comparing the proposed ICSA with the basic CSA, the ICSA takes advantage of the inertia weight χ in balancing global and local search abilities.This value should be large in the case of exploration and small in the case of exploitation.However, it is not necessarily right to lessen χ absolutely with time.41) where v min and v max are constant values used to manage exploration and exploitation features of ICSA where their values are 0.5 and 1.5, respectively.The values of v min and v max were attentively picked to carry out proper exploration and exploitation aspects in the search domain by helping the capuchins to search globally and effectively for food sources.This is because their values are fundamental and have a significant impact on the overall performance of ICSA.As shown in Eq. ( 41), the parameter c 1 is given as a function of iterations specifically regressing with the number of iterations as shown in Fig. 1.On the other hand, the parameter c 2 as modeled in Eq. ( 42) and graphically displayed in Fig. 2 grows along the path of iterations.The benefits of the coefficients c 1 and c 2 play a role in ameliorating efficiency and speeding up the convergence toward the global optimum solutions, which are then critical to the proper success of ICSA.In this, a larger value for c 1 and a smaller value for c 2 were initially set and progressively flipped during the optimization process.In this light, a variety of values are generally used for these parameters at each iteration loop in order to swap between the exploration and exploitation aspects of ICSA during its iterative process.This may lead to a better performance level for this algorithm.Unlike the standard CSA algorithm, it uses constant values for c 1 and c 2 as shown in Eq. (7).Simply put, parameter c 1 represents the "self-cognition" which attracts the capuchins to their own best former position, helping to scout diverse local niches and preserve the diversity of the capuchins.Parameter c 2 represents the "social influence" that drives the capuchins to approach the current best global area, aiding in rapid convergence.In sum, the time-varying acceleration factors c 1 and c 2 in combination with the stochastic parameters r 1 and r 2 in Eq. ( 38) control the random effect of the cognitive and social parts on the velocity of capuchins in searching for food at each iteration loop of the proposed ICSA algorithm.Compared to the velocity updating model of CSA, the parameters c 1 and c 2 have fixed values during the execution phase of CSA.This shows that exploration and exploitation aspects of the basic CSA are dependent on fixed values for these parameters.This has an impact on the CSA's search behavior, which could lead to modest exploration and exploitation in the absence of a strict structure.

Position hierarchy updating model of ICSA
The positioning update model of ICSA algorithm is amended over that of the basic CSA algorithm.In the proposed ICSA algorithm, a supplemental positioning updating model in a hierarchy approach was proposed.In this respect, the positions of the capuchins (i.e., leaders and followers), after applying Eqs. 9, 11, 12, 13, 14, 15 and 17, are then updated in the search space as per Eq. ( 43).
where rand stands for random numbers generated within the range [0, 1].
In the position hierarchy model of the capuchins in ICSA, the parameters c 1 and c 2 are adaptively updated inside each iteration to achieve promising exploration and exploitation models in the initial and subsequent stages of ICSA along with rendering an adequate convergence process.As per this, four strategies can be deduced according to the likely variations of c 1 and c 2 in Eq. ( 43), as described as follows: • Strategy A -increasing c 2 and decreasing c 1 in an exploration case: It is substantial to explore as many optima as conceivable in the exploration case.Hence, increasing c 2 and decreasing c 1 can assist capuchins to explore individually and realize their own best historical locations, rather than huddle around the current best leader that is probable to be correlated with a local optimal.• Strategy B -decreasing c 1 lightly and increasing c 2 lightly in an exploitation case: In this case, the capuchins make use of local information and aggregation toward potential local optimum positions denoted by the previous best position of each capuchin.Hence, decreasing c 1 little by little and retaining a relatively large value can corroborate that the search and exploitation about the best capuchin arrived at the best position of the food source.Meanwhile, the globally best capuchin does not always identify the global optimum area at this point yet.Hence, increasing c 2 gradually and keeping a small value can avert the deceit of a local optimal value.Moreover, the exploitation case is more probably to take place after an exploration case and before a convergence case.Thus, the directions of c 1 and c 2 have to be moderately changed from the exploration case to the convergence case.• Strategy C -decreasing c 1 slightly and increasing c 2 slightly in a convergence case: In the case of convergence, the capuchins appear to locate the globally optimum area, and thus, the impact of c 2 should be underlined to direct other followers to the potential globally optimal area.Thus, the value of c 2 should be expanded.On the other side, the value of c 1 should be lessened to allow the entire population of capuchins to converge rapidly.• Strategy D -decreasing c 1 and increasing c 2 in a jumping-out case: When the globally best capuchin jumps out of a local optimum toward a better optimum, it is probably to be far from the crowded cluster.Once this new area was found by a capuchin that becomes the (potentially new) leader, the others should follow it and move to this new area as quickly as possible.A large c 2 in combination with a relatively small c 1 assists to realize this aim.
Briefly, the hierarchy position updating model of the ICSA algorithm implemented by Eqs. 9 to 17 and Eq. ( 43) was proposed to further enrich the exploration and exploitation aspects of the basic CSA algorithm.This is also to obtain a dependable convergence process toward the global optimum and to heighten the potential of CSA to deal with the challenging ELD problems under study.

Iterative-level hybridization of CSA with GBO
Iteration-level hybridization is a modest approach to implementing successive iterative optimization algorithms that aims to boost the performance score of each native algorithm (Braik et al. 2023).Here, the hybridization approach, referred to as HICSA, is performed at two iterative levels: (1) The proposed ICSA algorithm is used to explore the most promising search areas in the search space, and then, (2) GBO is further used to exploit the previously limited area of optimization to select plausible solutions.
In the proposed HICSA algorithm, the current and global best solutions of the search agents of GBO are initialized with the current and global best solutions of the capuchins of ICSA.In this regard, the positional vector of the capuchins of HICSA can be defined as follows: where the parameters of Eq. ( 44) are defined in Sects.4.1 and 4.2.
As it is expected from HICSA, the global and local searches during the exploration and exploitation phases, respectively, offered by both ICSA and GBO promote a balance between these two phases.Further, the convergence process of the population is anticipated to be improved throughout the course of iterations of the proposed HICSA algorithm.As a result, the proposed HICSA algorithm is expected to perform better than the parent algorithms.

Repair process
A repair process has been proposed to be used by the proposed HICSA to repair infeasible search agents, particularly when the feasibility of the generated solution by this algorithm is not retained.In this, the proposed HICSA evokes this repair process to deal with the offending constraints.The pseudo-code for this repair can be functionally summarized as shown in Algorithm 3.

Implementation of the proposed HICSA
In the proposed HICSA algorithm, the exploration and exploitation features of the GBO algorithm were combined with those of the ICSA algorithm in the initial and final stages of the optimization process.Algorithm 4 summarizes the pseudo-code of the proposed HICSA algorithm.
Algorithm 4 divulges that HICSA commences with approximation of the global optimum solutions by creating a population of capuchins with stochastic positions.During the iterative procedures of HICSA, the capuchins use Eqs. 9, 11, 12, 13, 14, 15, 17 and 44 to continuously explore and exploit the search space during foraging.Thereafter, the solutions are assessed using a pre-defined cost function, which is recomputed inside the optimization process to locate the capuchin with the best cost value.The adaptive parameters of HICSA such as χ , c 1 , and c 2 that are iteratively updated over the path of iterations contribute greatly to devising a good balance between exploitation and exploration of HICSA.
In opposition to the mathematical model of CSA which in some cases may lead to tripping into local optimal solutions, the capuchins' positions in HICSA are consistently updated Algorithm 3 A pseudo-code showing the repair process's steps in HICSA.

29:
end if 30: end for 31: end for with the use of its adaptive parameters, where the capuchins move little by little toward food sources.The amendments conducted in the proposed HICSA algorithm may impede it from smooth stagnancy into local solutions, and eventually find correct evaluation of the best solutions ever got during optimization.In short, the proposed HICSA is anticipated to impose its power to tackle the challenging ELD problems as can be read from the results in Sect.6.

Experiments results and discussion
This section describes the experiments conducted on the optimization of ELD problems in power systems.Experimental settings and evaluation of the six cases of the ELD problems considered in this work are presented in this section.Then, the convergence analysis results of the basic CSA proposed HICSA are presented and discussed in depth

Parameter settings
The settings of the parameters of the basic CSA algorithm were assigned as advisable in their native paper.The settings of the parameters of the proposed HICSA algorithm were established after careful analysis and testing on several test cases of the ELD problem to regulate the exploration and exploitation features.The parameter settings of the basic CSA and the proposed HICSA utilized to solve six test cases Algorithm 4 A pseudo code describing the proposed HICSA algorithm.
As presented in Table 1, the number of capuchins (i.e., population size) and the maximum number of iterations for the basic CSA and proposed HICSA algorithms were set to 100 and 100,000, respectively.These algorithms were carried out 30 independent runs for all of the test systems considered in this work.This is in order to give an idea of the stability of the CSA and HICSA algorithms.It should be pointed out that these settings were adopted after conducting a set of experiments.The full experiments of the basic CSA and pro- The next subsection presents the evaluation results of the basic CSA, the proposed HICSA, and other algorithms in solving the ELD problems under study.

Evaluation of HICSA on ELD problems
The appropriateness and feasibility of the proposed HICSA algorithm were exerted in grappling 6 different test cases of ELD problems in a power system, with each test case having a different power generating unit.In this paper, the transmission losses of the power system units were neglected in all test cases to satisfy the constraints of the transmission capacity.This was implemented by setting the P L coefficient in Eq. ( 4) to a value of 0. The six test systems of ELD problems touched upon in this work can be briefly described as presented in Table 2.
The test systems (TSs) characterized in Table 2 can be classed into five categories in this manner: TS1 is classed as an extremely small-scale ELD problem; TS2 and TS3 are classed as tiny-scale ELD problems; TS4 is classed as a medium-scale ELD problem; TS5 can be considered a largescale ELD problem, and finally, TS6 is a highly large-scale ELD problem.All datasets of the above test systems for 3-, 13-, 40-, 80-, and 140-unit generating systems (x min i , x max i , a i , b i , c i , e i , f i ) were selected from Ullah et al. (2012), see also Appendix A.
Remarkably, there is an extensive literature that includes a range of meta-heuristics used to solve ELD problems under different kinds of constrains with a range of various power system units.Therefore, a comparison was presented between the solution quality and computational efficiency of the basic CSA and the proposed HICSA to those of other methods that reported the best performance in the literature.In all of the 6 test systems considered in this work, valvepoint effect, power balance, and generation limits are the only features considered by other related methods reported in the literature.As presented in Sect.3.2, the power output limits and power balance constraints are only those that were considered in solving the ELD problems under study.This is consistent with what has been used by other comparative methods as mentioned in the relevant literature.The results of the comparative algorithms listed in all of the tables below were gathered from their original references.
The following subsections describe the evaluation results of the proposed algorithm and other competitors on the above test regimes of ELD problems, with the best results tabulated in the tables in these subsections highlighted in bold.This is to exhibit their importance among the other results.In these tables, "NA" indicates that the results are not available in the relevant references.Further, the results of the proposed and competing algorithms are presented in terms of mean and standard deviation (STD) values.

TS1: test system 1
The electric power system considered in this case study involves a 3-unit generator with valve-point effect for ELD problem.The expected load demand for this system is 850 MW, while the data for all generator units can be found in Walters and Sheble (1993).The best solutions obtained by implementing the proposed HICSA together with the basic CSA are presented in Table 3. Besides, the total fuel cost of the best solutions, the average total fuel costs, and standard deviation results are also summarized in the same table.The total number of iterations needed by the two algorithms to achieve the best solution (i.e., the optimum solution) along with the average computational times to run the two algorithms in all iterations is also recorded in the same table.
It is clear that the proposed HICSA's performance is similar to that of the standard CSA in obtaining the same total fuel cost, mean costs, and STD outcomes.This is due to the simplicity of the test system undertaken in this case study, and hence, not much effort is required to reach optimal results.
It is worth noting that the CSA algorithm needs 5238 iterations out of 100,000 iterations to achieve the best solution, The best results are highlighted in bold while the HICSA algorithm needs 226 iterations to achieve the best solution.This proves the effectiveness of the proposed HICSA algorithm by finding the right balance between exploration and exploitation abilities during the search process, and thus achieving the best results with the lowest number of iterations.
Reading the results shown in Table 3 once more time, it can be seen that the CSA algorithm is faster than the proposed HICSA algorithm by getting the lowest average computational times consumed by each algorithm to finish all iterations in each run.It should be noted that the computational time is not as hard significant constraint as the others.
The performance levels of the proposed algorithm were then compared with those of the basic CSA and other rival algorithms that were examined for solving the same test cases of ELD problems.Table 4 shows the total fuel cost of the best solutions (i.e., the best results) and the average total fuel costs (i.e., the mean results) of the proposed HICSA, the basic CSA, and other 17 comparative algorithms.
It can be seen that the proposed HICSA ranks first by having the minimum total fuel costs of 8,234.07$/h, as achieved by 16 out of 18 other comparative methods.In addition, the results of HICSA and some other comparative algorithms are better than those of NSS and GA-PS-SQP algorithms.The best results are highlighted in bold

TS2: test system 2
This test system of the ELD case study considered in this study comprises a generator with 13-unit and valve-point effect.Detailed datasets of the different generating units of this test system were obtained from Walters and Sheble (1993).Table 5 shows the best solutions, total fuel costs, average total fuel costs, and STD results obtained by executing the proposed HICSA and standard CSA algorithms over 30 independent runs.In addition, the total number of iterations to achieve the best solution as well as the average computational times to run the two algorithms in all iterations is also recorded in the same table.
From Table 5, it can be seen that the performance of HICSA significantly outperforms the standard CSA by obtaining the best solution with the lowest total fuel cost (17,960.37$/h), with the difference between HICSA and CSA equal to 7.99 $/h.This proves the efficiency of the proposed HICSA in navigating the search space of the current state of the ELD problem and thus avoiding local optima.Reading Table 5 again, it can be noticed that CSA is more stable than HICSA in line with the STD results.
It can be observed that the CSA algorithm demands 34,802 iterations to achieve the best solution, while the proposed HICSA algorithm demands 85,924 iterations to achieve the best solution.Remember that the optimal solution obtained by the proposed HICSA algorithm is better than the solution of the CSA algorithm in terms of the total fuel cost.On the other hand, the CSA algorithm appears to be faster than the proposed HICSA based on the average computational times each algorithm consumes to finish all iterations in each run.Notably, the computational time is not a hard constraint like others used to favor between algorithms to select the best one among them.
The experimental results obtained by the proposed HICSA algorithm and the standard CSA algorithm were compared with those of twenty-nine methods published in the literature.The total fuel cost of the best solution (i.e., the best cost) and the mean total fuel costs (i.e., the average cost) are appeared in Table 6.
It is evident that the proposed HICSA as well as other seven comparative methods (i.e., DHS, HAAA, IHS, Island-HS, MPDE, NUHS, and THS) ranked first.This is because these methods succeeded in obtaining the solution with the lowest total fuel cost (17,960.37 $/h).Moreover, the performance of the proposed HICSA is better than the performance of the remaining comparative methods shown in Table 6.Anyway, the standard CSA is ranked 23rd by having the best solution with a total fuel cost of 17,968.36$/h.Then, the solution with the worst fuel cost is got by MSL, while the difference between the results of HICSA and MSL is 198.31 $/h, which is relatively large and significant.This system involves a 13-unit generator with valve-point effect.The datasets of this test case were retrieved from Walters and Sheble (1993).The best solutions, total fuel costs, mean fuel costs, and STD results obtained by the proposed HICSA algorithm and the basic CSA algorithm are provided in Table 7. Furthermore, the total number of iterations to reach the best solution and the average computational times to run the two algorithms are shown in the same table.
It is clear from Table 7 that the proposed HICSA is superior to the standard CSA by having the best solution with the lowest total fuel cost (24,164.05$/h).There is a difference of 4.76 $/h between HICSA and CSA where this difference is relatively large and somewhat significant.Again by reading Table 7, one can notice that the performance of the proposed HICSA is more robust and stable than that of the basic CSA as per the average costs and STD outcomes.This corroborates the efficiency of HICSA in navigating the search space of this case study, thus escaping the local optima problem.It is important to note that the proposed HICSA achieved the best solution within 42,363 out of 100,000 iterations, while CSA got the best solution within 60,575 iterations.Remember that the best solution obtained by the proposed HICSA is better than that of the CSA in terms of total fuel cost.The CSA algorithm seems faster than the proposed HICSA according to average computational times and this is not significant to determine which algorithm has better performance in the ELD domain.
Table 8 shows a comparison of the results obtained by HICSA, CSA and other 17 comparative methods reported in the literature.
The results summarized in Table 8 are presented in terms of the total fuel cost of the best solution (i.e., minimum cost), average total fuel cost (i.e., mean cost), and the rankings of the algorithms.In general, the algorithm with the lowest total fuel cost is the best and ranks first.The proposed HICSA, IODPSO-G, and Island-HS ranked first by getting a solution with the lowest total fuel cost (24,164.05$/h).Moreover, the performance of the proposed HICSA is better than the remaining algorithms in terms of the best results.In any case, the basic CSA is ranked tenth, while HCASO is ranked last.

TS4: test system 4
The electric power system considered in this case study consisted of a 40-unit generator with valve-point effect, where the detailed datasets of the various generating units for this test case are widely available in Sinha et al. (2003).Table 9 shows the best solution obtained by HICSA and CSA along with the total fuel cost of the best solution, average fuel costs and STD outcomes.The number of iterations to reach the best solution by the two algorithms and the average compu-tational times to run the two algorithms are displayed in the same table.
It can be seen from Table 9 that the proposed HICSA performs better than the standard CSA by having minimal results in terms of the total fuel cost and average costs.Further, the proposed HICSA algorithm is more stable than the CSA algorithm as it reported minimal STD values.The CSA algorithm needs 98,065 out of 100,000 iterations to achieve the best solution, while the proposed HICSA algorithm needs 99,538 iterations to obtain the best solution.The best solution got by the proposed HICSA algorithm is better than that of the CSA algorithm in terms of the total fuel cost but with a higher number of iterations.Finally, when reading the results recorded in Table 9 again, it can be noticed that the CSA algorithm appears to be faster than the proposed HICSA as per the average computational times.
The experimental results obtained from the proposed HICSA and the standard CSA, in terms of the total cost of the best solution and mean fuel costs, are compared with other fifty-two comparative methods as shown in Table 10.The average rankings of the algorithms are also provided in Table 10.
Interestingly, it can be seen that the proposed HICSA ranked first by achieving a solution with the lowest total fuel cost (121,369.43$/h).Thereafter, HAAA ranked second by acquiring a solution with the second lowest total fuel cost (121,403.70$/h).The difference between the results of HICSA and HAAA is 34.27 $/h, which is large and statistically significant.Anyhow, the standard CSA is ranked 40th, while QGSO is ranked last with the worst results.Based on these findings, the proposed HICSA is able to efficiently navigate the search space of this test case of the ELD problem by creating the right balance between exploration and exploitation behaviors.This leads it to avoid the local optima problem and thus achieve superior results.

TS5: test system 5
The electrical system exploited in this case study consisted of an 80-unit generator, with the datasets of the different generating units taken by doubling the data of the previous TS4 test system.As mentioned earlier, this test system is categorized as a large-scale ELD problem, so the search space of this system is massive with lots of local optimums.Therefore, the optimization method in this case demands a lot of effort to achieve satisfactory results.The best solutions achieved by running the proposed HICSA and standard CSA are presented in Table 11.In addition, the total fuel cost of the best solution, average fuel costs, STD results, the total number of iterations to reach the best solution, and average computational times to run the two algorithms 30 times are also presented in this table.The best results are highlighted in bold From Table 11, one can clearly notice that the proposed HICSA algorithm performs better than the basic CSA algorithm by getting the solution with the lowest total fuel cost (242,477.23$/h).There is a difference of 1488.48 $/h between the average fuel costs of HICSA and CSA, which is statistically significant.These results indicate that the proposed HICSA algorithm is more powerful than the basic CSA algorithm.With regard to the STD outcomes, it can be given a note that HICSA is more stable than CSA by arriving at the lowest STD outcomes.Thereafter, the proposed HICSA used 98,932 iterations to obtain the best solution, while the CSA used 99,906 iterations to achieve the best solution.However, CSA seems to be more rapid than the proposed HICSA by having the lowest average computational times.Again, the computational times are not an extreme case for determining which algorithm has better performance in the domain of ELD.
The minimum total fuel cost and average fuel costs of the proposed HICSA and standard CSA are compared with those of other eighteen comparative methods as shown in Table 12.Definitely, it can be noticed that HICSA ranked first by having the minimum fuel cost results (242,477.23$/h).The CLCS-CLM was ranked second by achieving the second minimum fuel costs (242,794.73$/h).The difference between the fuel costs obtained by HICSA and CLCS-CLM is 317.5 $/h, which is large and statistically significant.The standard CSA ranked 16th, while the SCA ranked last with the worst results.

TS6: test system 6
This test case is a very large-scale electrical power system consisting of a 140-unit generator.The datasets of the various generating units of this test system for an ELD problem are available in Park et al. (2009).In this case of the ELD problem, the search space is quite huge with lots of local optimums.Therefore, there was a large need for an ingenious optimization method with a sufficiently adequate balance between exploration and exploitation features to reach sat-isfactory results.This is the motivation to promote the CSA algorithm to obtain the proposed HICSA algorithm.
Table 13 provides a summary of the best solutions, total fuel costs of the best solutions, average total fuel costs, and STD results got from the proposed HICSA and basic CSA.The total number of iterations to reach the best solution along with the average computational times over 30 independent runs is also mentioned in Table 13.
Attractively, it can be perceived from Table 13 that HICSA performs considerably better than CSA by coming up with a solution with the lowest total fuel cost (1,559,693.34 $/h).The difference between them is 8567.88$/h, where this difference is very large and statistically significant.Furthermore, HICSA appears to be more robust and stable than CSA consistent with the mean costs and STD outcomes.It is important to observe that the proposed HICSA requires 92,330 out of 100,000 iterations to achieve the best solution, while CSA requires 98,782 iterations to receive the best solution with higher total fuel cost.However, the CSA algorithm appears to be faster than the proposed HICSA algorithm in accordance with the average computational times obtained.
To assess the efficiency of the proposed HICSA algorithm, its results are compared with the basic CSA and other thirteen comparative algorithms as presented in Table 14.
It can be apparently seen that the proposed HICSA is put first by fulfilling a solution with the lowest total fuel cost (1,559,693.34 $/h).Then, HAAA was ranked second by having a solution with the second minimum total fuel cost (1,559,710.00$/h).The difference between HICSA and HAAA is 16.65 $/h, which is relatively small and statistically non-significant.On the second hand, the basic CSA is ranked eighth, while CTPSO, CSPSO, CCPSO, and COPSO ranked last by performing worst.
A noteworthy to be mentioned is that all the comparative methods have not reported the computational times in their original works for all of the 6-test system considered.In accordance with this, the computational times for these methods were not calculated or estimated in this work as this a crude process and may be imprecise.Hence, we accepted the computational results of these methods without implementing them and ignored the computational times, and the computational times were calculated only for the methods developed in this work.

Convergence results and analysis
Again, after reading the results in depth in Tables 4,6,8,10,12, and 14, one can underline, once again, that the proposed HICSA's performance score is better than the basic CSA in five of the six test cases of the ELD problems, namely TS2, TS3, TS4, TS5, and TS6, while both of HICSA and CSA behaved similarly in the first test case, which is TS1.As mentioned earlier, the TS1 test case study consists of a generator with 3 units; so it is a simple case study and does not need a highly efficient optimization method to get the optimal results.Still and all, in the comparison made between the proposed HICSA and the other fifty-four comparative methods mentioned in the literature, one can draw up the following finales: HICSA outstripped all other competitors in TS4, TS5, and TS6, where these test cases were classed as medium-scale, large-scale, and very large-scale ELD problems, respectively.On top of that, the performance of the proposed HICSA is analogous to some other promising rivals Fig. 3 The of the total fuel costs of the proposed HICSA and basic CSA, over 30 independent runs, in different ELD problems 123 Fig. 4 The convergence characteristic curves of HICSA and CSA in different ELD problems by reporting the best published results in the remaining test cases categorized as small-scale ELD problems.This ascertains the competence of the proposed HICSA algorithm in solving test cases in power systems by exploring the search space of ELD problems and thereby realizing highly competitive results.This is done by striking a proper balance between exploration and exploitation aptitudes during optimization.
The notched box-plots used to visualize the distributions of total fuel costs of the proposed HICSA compared to the standard CSA in different test power systems for ELD problems are displayed in Fig. 3.In Fig. 3, a small gap between the best, average, and worst results indicates the stability and strength of the algorithm.It can be ascertained from the plots in Fig. 3 that the proposed HICSA is more robust and stable than the basic CSA in four test systems of ELD problems (i.e., TS3, TS4, TS5, and TS6).This means that HICSA is adept at achieving roughly the same results over 30 independent runs in these test cases.What is more, the performance of HICSA is similar to that of CSA for TS1, in which they reached the same results over the course of 30 pre-selected independent runs.Last but not least, the basic CSA looks like to be more stable than the proposed HICSA for TS2, while the results of HICSA are better than those of CSA.
Figure 4 depicts the convergence behavior of the proposed HICSA against the standard CSA for all test systems of ELD problems, where x-axis and y-axis represent the number of iterations and the total fuel costs, respectively.
In Fig. 4, it should be noted that the convergence property of HICSA and CSA represents the behavior of the best solution got by each algorithm.It could also be seen that the regression of HICSA and CSA is very high in the first half of iterations of all test cases.In depth, it can be perceived that the slope of the proposed HICSA is better than its counterpart algorithm in four test cases (i.e., TS2, TS4, TS5, and TS6).As the slope of HICSA and CSA is approximately the same for TS3 after 60,000 iterations, thus both reached nearly the same results.Finally, the characteristic behavior of HICSA and CSA is almost the same for TS1 in the initial stages of the search process (after 6000 iterations).This points out the simplicity of the system considered in this test case.

Conclusion and future works
This paper has presented the use of a hybridization of two meta-heuristics, namely capuchin search algorithm (CSA) and gradient-based optimizer (GBO), to solve challenging economic load dispatch (ELD) problems.The adaptability of this approach, referred to as hybrid improved CSA (HICSA), is shown in solving ELD problems for six different test cases with a generator of various units and scales with simple and rugged search spaces and for different numbers of constraints and loading conditions with varied degrees of complexity.A memory feature was added to the search agents of the proposed HICSA to solidify both local and global searches.Then, new adaptive functions were applied in the iterative loops of HICSA to implement both social and cognitive parameters of the capuchins' velocity.At last, the hybridization approach was implemented to get about to the most propitious areas of the search space and intensify the exploitation of the most promising possible solutions.The adequacy of HICSA in solving ELD problems was revealed by solving 6 test systems for a generator with 3, 13, 13, 40, 80, and 140 generating units.A comparison of HICSA with CSA and other optimization algorithms proved that HICSA is a powerful optimization algorithm that surpasses many corresponding promising ones.
Future research would focus on expanding the HICSA algorithm as a multi-objective optimization method to han-dle ELD problems as multi-objective optimization problems.Another area of future study will focus on verifying HICSA's efficacy in tackling other optimization problems of similar kinds, such timetabling and fellowship scheduling problems.The solution quality and convergence behavior of CSA may also be thought to be improved by hybridizing it with other meta-heuristics.In a broader perspective, extending the HICSA method to deal with other ELD problems with very large numbers of generating units could be a further study.Further work will need to reduce the computational time of the proposed HICSA in order to outperform other rival algorithms.

Fig. 1 A
Fig.1A cognitive parameter defined to provide a sensible balance between exploration and exploitation features of ICSA 4: Identify the best and worst solutions: x k best , x k best 5.1 Improved CSA (ICSA)

Table 1
Parameter settings for the CSA and HICSA algorithms used to implement ELD problems for several unit systems

Table 2
Settings

Table 3
The optimal results of TS1 obtained by the basic CSA and

Table 13
The optimal results of TS6 obtained by the basic CSA and proposed HICSA algorithms

Table 16
Generating unit data for Test Systems 2 and 3(13-unit generator system)

Table 18
Generating unit data for Test

Table 19
Generating unit data for Test