A Cauchy-Gaussian Quantum-Behaved Bat Algorithm Applied to Solve the Economic Load Dispatch Problem

In this paper, a novel Cauchy-Gaussian quantum-behaved bat algorithm (CGQBA) is applied to solve the economic load dispatch (ELD) problem. The bat algorithm (BA) is an acknowledged metaheuristic optimization algorithm owing to its performance. However, the classical BA presents some weaknesses, such as premature convergence. To withstand the drawbacks of the BA, quantum mechanics theories and Gaussian and Cauchy operators are integrated into the standard BA to enhance its effectiveness. Since the economic load dispatch is a nonlinear, complex and constrained optimization problem, its main objective is to reduce the total generation cost while matching the equality and inequality constraints of the system. The validity of the CGQBA is tested on six standard benchmark functions with different characteristics. The numerical results indicate that the CGQBA is effective and superior to many other algorithms. Moreover, the CGQBA is applied to solve the ELD problems on various test systems including 3,6,20, 40,110 and 160 implemented generating units. The simulation results illustrate the strength of the CGQBA compared with other algorithms recently reported in the literature.


I. INTRODUCTION
Economic load dispatch (ELD) is one of the hot topics in the field of power system optimization [1]. The aim of the economic dispatch is to seek the most favorable allocation of generation units that reduces costs while meeting all the system's inequality and equality constraints [2]. Due to the excessive cost of power generation in fossil fuel power plants, the optimality of economic dispatch is advantageous in terms of saving money [3].
As an optimization problem, the economic dispatch (ED) problem consists of an objective function and various constraints [4]. In previous years, several methods known as conventional methods have been used to solve economic dispatch problems, such as linear programming [5], nonlinear programming [6], quadratic programming [7], dynamic The associate editor coordinating the review of this manuscript and approving it for publication was Ying Xu . programming [8], interior point programming [9], mixed integer programming [10], the Lagrangian relaxation algorithm [11], the decomposition technique [12], the branchand-bound method [13], the Newton-Raphson method [14], Lambda iteration [15] and the gradient method [16]. However, these classical methods experience difficulty when finding initial solutions and are prone to local optimal convergence [17]. The cost functions of generators were previously formulated as quadratic or piecewise quadratic functions based on the assumption that the incremental cost curves of the units are monotonically increasing piecewise-linear functions [18]- [20]. The practical ED is nonconvex, nonsmooth and nondifferentiable due to the presence of several constraints, such as prohibited operating zones, ramp rates, multifuel options, valve-point effects and transmission losses [21]- [24]. Conventional optimization techniques are not effective in solving the ED problem; therefore, researchers have developed new optimization techniques VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ known as ''metaheuristics'', which provide better solutions and overcome the demerits of conventional methods [25]. Moreover, they are applied to solve problems in many areas of engineering [26], [27], as well as other real-world optimization problems [28]- [30]. Various metaheuristic algorithms have been applied for solving ED problems: ant colony optimization (ACO) [31], grey wolf optimization (GWO) [32], the flower pollination algorithm (FPA) [33], the firefly algorithm (FA) [34], the social spider algorithm (SPA) [21], the cuckoo search algorithm (CSA) [3], the symbiotic organism search algorithm (SOS) [35], the bat algorithm (BA) [36], the particle swarm algorithm (PSO) [20], the imperialist competitive algorithm (ICA) [37], ant lion optimization (ALO) [38], the gravitational search algorithm (GSA) [39], the genetic algorithm (GA) [40], teaching-learning based optimization (TLBO) [41], the artificial bee colony algorithm (ABC) [42], and the differential evolution algorithm (DE) [43].
Some of these metaheuristic algorithms possess limitations when dealing with large complex systems in terms of finding the global optimal solutions, preventing local optimal and premature convergence [44]. To address these shortcomings, two solutions have been proposed to improve the performance of metaheuristics: modification and hybridization of algorithms [45].
A study by Elsayed et al. [46] suggested a modified social spider algorithm (MSSA) for solving the ELD problem, where the random walk in the basic SSA has been substituted by a chaotic sequence. An improved version of the adaptive differential evolution optimizer (ADE) has been applied for solving the nonconvex ED problem [47]. This presented algorithm shuns the problem of premature convergence and ameliorates the convergence speed. An elitist cuckoo search algorithm was examined in [48]. This variant of the CSA is based on modifying some parameters of the basic CSA (initialization), and it achieves remarkable success in solving the ELD problem. A variant of the charged search system (CSS) called the adaptive CSS (ACSS) was developed in [49], and the modifications performed in this algorithm focused on initialization and random walk. Consequently, the ACSS shows supremacy over the CSS in solving the ELD problem. A modified crow search algorithm (MCSA) was presented in [50], and the MCSA differs from the original CSA in terms of selecting the new crows and tuning the flight length. Phasor particle swarm optimization (PPSO) has been suggested for solving convex and nonconvex/nonsmooth ELD problems, as found in [51]. PPSO is endowed with convergence ability and higher performance due to the substitution of the control parameters of the basic PSO with phasor angles. The author in [52] suggested the solution of large-scale multifuel ED problems considering valve-point effects via a dual-population adaptive differential evolution (DPADE). In this study, a dual-population mechanism was used to enhance the searching capability, and an adaptive technology was utilized to elude the unsuitable parameters and to tune two parameters of great importance. An emended salp swarm algorithm (ESSA) for solving the economic emission power dispatch (EED) problem was suggested in [53]. In this study, the reproduction cycle of salp was integrated into the classical salp swarm algorithm (SSA) to prevent the algorithm from being caught in the local minimum and guarantee the diversity of the swarm. The balance between exploitation and exploration is a feature achieved by the proposed ESSA. The ESSA successfully handles single and multiobjective ED problems and outperforms several methods reported in the literature.
A continuous GRASP (greedy randomized adaptive search procedure) algorithm has been hybridized with differential evolution (DE) for solving ED problems [54]. The proposed algorithm (C-GRASP-DE) is vested with the aptitude for global searching and the ability to avoid local optimal stagnation. A hybrid grey wolf optimization (HGWO) was successfully applied for solving the ED problem in [55]. This HGWO combines the advantages of both GWO and the DE. The crossover and mutation operators of the DE are integrated into the classical GWO algorithm to enhance its efficiency in handling ED problems. A robust hybrid optimization technique was designed for solving the ELD problem with wind uncertainty in [56]. This approach exploits the merits of both the genetic algorithm (GA) and adaptive simulated annealing (ASA). The diversity of the population is preserved by utilizing the nonuniform mutation of the GA. Self-adaptive mutation and crossover frameworks are incorporated along with ASA with the aim of facilitating the selection of the best parameters. Short computational times and convergences rate are the most appealing characteristics of the proposed method.
The hybridization of three metaheuristic algorithms, ant colony optimization (ACO), artificial bee colony algorithm (ABC) and harmonic search (HS), is presented in [44]. In this hybrid algorithm, each algorithm fills its own role. The task of seeking the initial solution set is handled by the ACO algorithm. The ABC algorithm checks and enhances the solutions generated by the ACO algorithm, while the HS algorithm removes the mediocre solutions from the solution set and substitutes them with those of higher quality. The authors in [57] combined the modified PSO (MPSO) and genetic algorithm (MPSO-GA) to solve nonsmooth as well as nonconvex ED problems. The initialization is performed by the GA, and the results are conveyed to the MPSO. The exploration of all search spaces is not necessary in MPSO-GA since this work is assigned to the GA. The results reveal the supremacy of the MPSO-GA over both the MPSO and the GA. A hybrid optimization method that integrates PSO and termite colony optimization (TCO), known as HPSTCO, has been developed and applied for solving the dynamic economic dispatch (DED) problem [58]. In this HPSTCO algorithm, PSO iterations are tasked with global searching, while TCO iterations are assigned to explore the vicinity of the global solution. A study by [59] proposed the hybridization of competitive algorithms (ICAs) and sequential quadratic programming (SQP), known as HIC-SQP, for solving ELD problems with wind power. Although the ICA is a metaheuristic algorithm, it presents the disadvantage of being caught into local optima as the number of imperialists rises. The SQP has been employed to palliate this demerit, and the role of the SQP is to adjust the results of the ICA to improve its performance. A robust hybrid gravitational search algorithm (RHGSA) for handling ED problems was addressed in [60]. The efficiency of the classical GSA was improved thanks to the piecewise linear chaotic map that enhances the global search capability and the sequential quadratic programming that boosts the speed of the local search. The authors in [61] suggested a solution to the ED problem considering smooth cost function features by using a combination of lambda iteration and simulated annealing methods (MHLSA). In the proposed approach, the demerits of the SA, such as poor initialization and difficulty when performing global searches, are alleviated by using lambda iterations. Moreover, this method involves multiple searches to discover the locality with the best-suited global optimal solutions. The metropolis biogeography-based optimization-sequential quadratic programming (MpBBO-SQP) algorithm has been proposed to cope with the weakness of BBO [62]. A metropolis criterion of the simulated annealing (SA) algorithm is introduced in BBO to provide control over migrated individuals, thus improving the exploration quality. The solutions generated by MpBBO are adjusted by SQP to increase the performance of the MpBBO-SQP algorithm. 3, 13 and 40 generating units are used to test the validity of the proposed algorithm.
An enhanced bat algorithm (EBA) for solving the ED problem was presented in [63]. The modifications have been conducted on the classical BA, which contains the inertia weight, population distribution, pulse emission rate and loudness. A modified θ bat algorithm is presented in [64]. The proposed approach transforms the Cartesian search space into polar coordinates as a means for providing strong search capacity. In this method, three modifications are introduced into the BA: 1) Lévy flight and a genetic mutation operator are employed to increase the population diversity, and 2) the loudness parameters are tuned to accelerate the convergence rate. Moreover, an adaptive strategy is adopted to facilitate the selection of the best modification to avoid local optima.
A combination of PSO and the BA has been proposed for solving ED problems [65]. In this hybrid algorithm, the PSO integrates the frequency behavior of the bat algorithm to accelerate its velocity updates. The loudness of the bat algorithm is utilized to address boundary constraint violations as long as the solution improves. The authors in [66] suggested the multiobjective chaotic bat algorithm (MOCBA) to handle the EED problem. The chaotic map is introduced to modify both the loudness and the pulse emission rate with the purpose of preventing premature convergence. A Pareto optimal front has been employed to facilitate the simultaneous minimization of fuel and emission.
Most of the abovementioned papers reveal that the balance between exploration and exploitation is extremely important. A novel algorithmic framework for solving economic load dispatch problem is proposed in this paper. To the best of the authors' knowledge, the Cauchy-Gaussian quantum -behaved bat algorithm has never been applied for solving the ELD problem. The quantum theory is introduced into classical BA to ameliorate the searching capability. Moreover, Gaussian and Cauchy mutations are used to improve the population diversity and to enable the algorithm to escape from local optima. To validate the proposed algorithm, the CGQBA is tested on six benchmark functions and applied to solve ELD problems containing 3, 6, 20, 40, 110 and 160 generating units. The rest of the paper is organized as follows: Section II provides details about the formulation of the ELD problem. Section III explains the Bat Algorithm (BA) and the Cauchy-Gaussian quantum-behaved bat algorithm (CGQBA). Section IV portrays various test systems and the simulation results of the proposed algorithm in comparison with other well-known algorithms reported in the literature. Finally, in Section V, the conclusion of the paper is drawn.

II. ED PROBLEM FORMULATION
This section explains the practical formulation of the ED problem with an objective function subject to constraints [49], [67], [68].

A. OBJECTIVE FUNCTIONS
The quadratic fuel cost function of the thermal units is minimized according to the following expression: where N g is the total number of generating units, F j P j is the fuel cost of the j th generating unit (in $/hr), P j is the power generated by the j th generating unit in MW, and a j , b j and c j are the cost coefficients of the j th generator.
For the case that takes the valve-point loading effect into consideration, the objective function is expressed as follows: where e j and f j are the constants for the valve-point effects of generators. VOLUME 9, 2021 For the case where multiple fuel options are presented, the fuel cost of the j th generator is given by Every generator with k fuel options contains k discrete regions.

B. OPTIMIZATION CONSTRAINTS
The equality and inequality constraints of the ED problem are the power-balance equality and power generation limits, which are described by the following two equations: where P j , P D , P L , P min j and P max j are the generation of the j th generator unit (in MW), the total power demand (in MW), and the minimum and maximum power generation limits of the j th generator, respectively. P L represents the line losses (in MW), and its value is obtained using B coefficients, given by: where P i and P j represent the power injection at the i th and j th buses, respectively, and B ij indicates the loss coefficients, which are frequently assumed to be constant under normal operating conditions.

C. PRACTICAL OPERATING CONSTRAINTS OF GENERATORS 1) POZs (PROHIBITED OPERATING ZONES)
Because of the operation of the steam valve or vibrations in the shaft bearings, the operating zones are considered. In practice, operations in such areas must be prevented to attain the best fuel economy [69].The feasible operating zones of unit j are formulated as follows: where n j , P l jk , P u jk are the number of prohibited zones and the lower and upper power outputs of the k th prohibited zone of the j th generator, respectively.

2) RAMP RATE LIMITS
The physical limitations of shutting down and starting up generators restrict ramp rate limits, which are formulated by the following two conditions: Limitation of the increase in generation: Limitation of the decrease in generation: where P 0 j , P j , UR j , DR j are the previous and current power outputs and the downramp and the upramp limits for the j th generator, respectively.
Combining (8) and (9) with (5) leads to the following generation limits: (10) in which and Combining this with (2), the ED problem can be mathematically described as follows:

III. THE BAT ALGORITHM A. THE BASIC BAT ALGORITHM
The bat algorithm (BA) was designed by Yang in 2010 [70] and was inspired by the echolocation behavior of microbats while seeking prey, foraging, and avoiding obstacles [71]. The echolocation characteristics of microbats are modeled via three rules as described in [71]: 1) Each microbat uses echolocation to approximate the distances between prey and neighborhoods. 2) Flying is performed to look for prey and is done at random with velocity V i at position X i with a predetermined frequency f min , varying wavelength λ and loudness A 0 . Bats can spontaneously tune the wavelengths   (or frequencies) of their emitted pulses and tune the rate of pulse emission r 1 ∈ [0, 1] based on the vicinity of their objective.
3) The loudness is supposed to change from a large (positive) A 0 to the least constant value A min . Each bat i possesses a position X i , a velocity V i and a frequency f i in d− dimensional space, and these characteristics should be updated iteratively towards the current best position as follows: where r 1 , f min , f max and f i are a uniformly distributed random number in the range [0, 1] ; the minimum tolerable frequency, the maximum tolerable frequency, and the frequency of the i th bat, respectively. As given in [71], the current ED problem assumes that the values of f min and f max are set to 0 and 100, respectively. t is the current iteration number, and X best is the location (solution) that possesses the best fitness in the current population. At initialization, V i is assumed to be 0. Each bat owns a new solution that can be generated locally through a random walk as follows: where ε is a random number uniformly drawn from [0, 1] and A i (t) is the loudness.
Once the bat has discovered its prey, the loudness continues to decline while the pulse rate emission continues to rise. The loudness A i and the pulse emission rate R i are iteratively updated as follows: where A i (0) ∈ [1, 2] and R i (0) ∈ [0, 1] are randomly generated within their respective limits. For the sake of simplicity, we set α = γ = 0.9, as in [72]. The pseudocode of the bat algorithm is written as follows [71]: Initialize the bat population X i (i = 1, 2, . . . , N ) and V i ; Define the pulse frequency f i , pulse rate R i and loudness A i ; while (t < T ) do Generate new solutions by adjusting the frequency and updating the velocities and positions using equations (14)- (16); if (rand > R i ) then Select a solution among the best solutions randomly; Generate a local solution around the selected best solution using equation (17); then Accept the new solutions; Increase R i and reduce A i using equations (18) and (19); end if Rank the bats and find the current X best ; t = t + 1; end while Output the best solution X best

B. QUANTUM-BEHAVED BAT ALGORITHM
The quantum-behaved bat algorithm (QBA) is inspired by [73]- [75]. Some variants of the quantum-behaved bat algorithm are addressed in [76]- [78]. In [76], the frequency equation of the proposed algorithm includes the bats' capability of self-adaptive compensation for Doppler effects in echoes. Moreover, the algorithm formulates the bats' habitat selection as the selection between their quantum behaviors and mechanical behaviors. In [77], the presented algorithm possesses its own way of generating a new solution different from that of the original BA. The position of each bat is determined by both the current optimal solution and the mean best position, and the incorporation of quantum-behaved bats enables improvement of the population diversity and prevents the bats from falling into local minima. The improved version of [77] is addressed in [78]. In our paper, quantum theory is applied to the bat algorithm, and then two mutation operators, Gaussian and Cauchy, are incorporated. In the QBA, the bats possess quantum behavior, and their positions are updated as follows: where both u and k are random numbers in the range [0,1] generated by the uniform distribution and β is the contraction-expansion coefficient, which can be adjusted for the sake of controlling the convergence speeds of the algorithms. It is defined as where β 0 and β 1 are the initial and final values of β, respectively. VOLUME 9, 2021 T is the maximum number of iterations, and t is the current iteration number.

M best (t)
= M best,1 (t) , M best,2 (t) , . . . , M best,d (t) where M best is the mean best position and represents the mean of all the best positions P i (t) of the population, P i (t) represents the current best position of the i th bat. N denotes the size of swarm, and d indicates the dimension of the problem. The pseudocode of the quantum-behaved bat algorithm is shown as follows: Initialize the bat population X i (i = 1, 2, . . . , N ) and V i ; Define the pulse frequency f i , pulse rate R i and loudness

If (rand > R i )
Generate a local solution around the selected best solution using equation (17); then Accept the new solutions; Increase R i and reduce A i using equations (18) and (19); end if Rank the bats and find the current X best ; t = t + 1; end while Output the best solution X best

C. THE CAUCHY-GAUSSIAN QUANTUM-BEHAVED BAT ALGORITHM
Gaussian, Cauchy and exponential probability distributions are more effective than uniform probability functions in terms  of generating random numbers to update the velocity equation of the classical PSO [79]. Inspired by [80]- [82], in which two or more mutation operators are combined, we find that the incorporation of both Gaussian and Cauchy operators into the quantum-behaved bat algorithm improves its performance when applied to the ELD problem. The explanation of these operators is given below: First, the Gaussian mutation operator is applied to the quantum bat algorithm. The one-dimensional Gaussian density function is given by the following equation [83], [84]: For µ = 0 and σ = 1, the Gaussian distributed function is given by equation (24) [83], [84]: In this section, we follow the same line of study as in [79]. The generation of random numbers is achieved by using the absolute value |·| of the Gaussian probability distribution with zero mean and unit variance, i.e., |N(0, 1)|, and then mapping to a truncated signal given by G = 0.33= |N(0, 1)|. The combination of the QBA with the Gaussian mutation operator is expressed by equation (25). Note that the parameter β of equation (20) has been substituted by G as indicated in equation (25).
The pseudocode of the Gaussian quantum-behaved bat algorithm is shown as follows: Initialize the bat population X i (i = 1, 2, . . . , N ) and V i ; Define the pulse frequency f i , pulse rate R i and loudness A i ; while (t < T ) do Generate new solutions by adjusting the frequency and updating the velocities and positions using equations (14)- (16); if (rand < p m ) Generate new solutions using equations (22) and (25); Generate a local solution around the selected best solution using equation (17); then Accept the new solutions; Increase R i and reduce A i using equations (18) and (19); end if Rank the bats and find the current X best ; t = t + 1; end while Output the best solution X best VOLUME 9, 2021  Second, the Cauchy mutation possesses the ability to escape from local optima [83] and it is applied to enhance the Gaussian quantum-behaved bat algorithm. The definition of the one-dimensional Cauchy density function is given by the following equation [83], [84]: x ∈ [−∞, ∞], and k > 0 is the scale factor. Then, the Cauchy distributed function is defined by equation (27): The new candidates are generated by the following equation: where C is a random number of Cauchy distributions in the range [0,1] The pseudocode of the Cauchy-Gaussian quantumbehaved bat algorithm is shown as follows: Initialize the bat population X i (i = 1, 2, . . . , N ) and V i ; Define the pulse frequency f i , pulse rate R i and loudness A i ; while (t < T ) do Generate new solutions by adjusting the frequency and updating the velocities and positions using equations (14)- (16); if (rand < p m ) Generate new solutions using equations (22) and (25); End if If (rand > R i ) Generate a local solution around the selected best solution using equation (28); end ifx if (rand < A i && f X i < f X best ) then Accept the new solutions; Increase R i and reduce A i using equations (18) and (19); end if Rank the bats and find the current X best ; t = t + 1; end while Output the best solution X best The flowchart of the Cauchy-Gaussian quantum-behaved bat algorithm is given in Fig. 1.

D. IMPLEMENTATION OF CGQBA TO SOLVE ED PROBLEM
Step 1: Initialize the population of bats which are bat position X i and velocity. In this case, X i corresponds to the power P i generated by the i th generator whereas n is defined as the number of generators. The value of X i is randomly generated within the clearly defined boundaries P min , P max , and the initial value of V i is set to zero.
Step 2: Initialize frequencies f i , pulse rates R i and the loudness A i for each bat.
Step 3: Fix the maximum number of iterations Step 4: Calculate the fitness values of all the bats utilizing the objective function in Equation (1) Step 5: Generate the new solution by using Equation (

22) and (25)
Step 6: Generate local solution in the vicinity of the best solution using Equation (28) Step 7: Update both R i and A i using Equation (18) and (19), respectively.
Step 8: Verify if all the constraints are respected Step 9: Repeat steps 1 to 8 until the maximum iteration is achieved.

IV. RESULTS AND DISCUSSION
The performance of the GQBA is tested on seven different test systems, including 3-, 6-, 20-, 40-, 110-and 160-unit systems. The comparison between the achieved results for the proposed algorithm after 50 independent trial runs and the results of the recently published algorithms for each test system are reported in their respective Tables. The abbreviations of those algorithms are alphabetically ordered in Table 1. The number of bats is set to 20 for each of the test systems, and the maximum number of iterations is 1000. For the sake of simplifying the comparison, the best fuel costs among the results are organized in ascending order. MATLAB is used VOLUME 9, 2021 to implement the programs on a personal computer with a 2.16 GHz processor and 4 GB RAM running on Windows 10.

A. BENCHMARK FUNCTION VALIDATION
Six benchmark functions are studied in this section to investigate the performance of the proposed CGQBA. Data for the benchmark functions are taken from [85] and are described in Table 2. The proposed CGQBA is applied to the aforementioned benchmark functions, and the mean and standard deviation of the results are provided in Table 3. Benchmark function data.

B. TEST SYSTEM 1
This system consists of three generators with a load demand of 850 MW. In this system, the constraints and valve-point load effects are taken into account, whereas the transmission losses are neglected. The system data are taken from [86]. Optimal generations and costs obtained by the QBA, GQBA and CGQBA for Test System 1 are presented in Table 4. Table 4, both the GQBA and CGQBA successfully achieve the best solution for the system, which is $8234.071766/hr.

As shown in
A comparison of the statistical results of the QBA and GQBA. The CGQBA and the algorithms available in the literature, the CE-SQP [87], BA [63], [88], NRHS, NTHS, NPHS, NGHS [89], BSA, GA-API, GA-PS-SQP, PS, and GA [90], are provided in Table 5. Since the system size is small, the results show that a large number of algorithms converge to the same optimal solution.
The convergence characteristics of the PSO, BA, QBA, GQBA and CGQBA algorithms are illustrated in Fig. 2. The figure reveals that the CGQBA performs better than other methods because it converges to the optimal solution in early iterations.

C. TEST SYSTEM 2
This system comprises six generating units supplying a load demand of 283.4 MW. Transmission losses are included. The data are taken from [20]. Table 6 provides the optimal generations and costs obtained. The best fuel cost and the corresponding transmission loss achieved by the CGQBA are $924.90309 /hr and 10.8994507 MW, respectively. Table 7 presents a comparison of the statistical results of the GQBA, QBA and the other reported algorithms (the BSA, MSG-HS, PSO, NSOA, GA-APO, GA [90], FMILP [91] and ACS [92]). It is shown that the proposed CGQBA yields the best fuel cost compared to those obtained by these algorithms. Fig. 3 shows the convergence behavior of the generation cost versus the iteration number for the PSO, BA, QBA, GQBA and CGQBA algorithms. It is seen in Fig. 3 that both the GQBA and CGQBA obtain better convergence quality when compared to other methods, but the CGQBA achieves the more optimal solution than that of the GQBA.

D. TEST SYSTEM 3
This system consists of twenty generators with a load demand of 2500 MW. Transmission losses are considered in this system. The data are taken from [93]. As shown in Table 8, the best fuel cost and the corresponding transmission loss obtained by the CGQBA are $62455.413/hr and 84.066567 MW, respectively.   The CGQBA yields the lowest cost in comparison with those of the other methods (the CKH [18], GSO, CBA [67], FMILP [91], λ-Iteration, HM [93], BSA, BBO [94], CQGSO [95], BLPSO [96] BA, EBA [97] and ADE-MMS [98]), as seen in Table 9.
The convergence characteristics of the PSO, BA, QBA, GQBA and CGQBA for Test System 3 are illustrated in Fig. 4. It is shown that the CGQBA obtains the best convergence property compared to the other methods as it converges to the optimal solution earlier.
The statistical results reveal that the CGQBA can compete with many optimization methods; only the PI-CBA, HAAA and AGWO perform better than the CGQBA.   From Fig. 5, the CGQBA performs better than the GQBA in early iterations in terms of convergence, but the GQBA becomes superior as the number of iterations increases. However, the optimal solution is finally achieved by the CGBA.

F. TEST SYSTEM 5
A system with 40 generating units meeting a load demand of 10500 MW is considered. This system incorporates the valve-point loading effects, and the transmission loss is considered. The data are given in [146] and [152].  MCSA [50], OGWO [69], ACS [92], AAA, HAAA [99], OIWO [104], BBO, DE/BBO, ORCCRO [111], OKHA [112] and KHA-IV [113], the CGQBA provides a better performance, as shown in Table 13. Fig. 6 shows the convergence behavior of the PSO, BA, QBA, GQBA and CGQBA for Test System 5. As depicted in Fig. 6, the CBGQBA exhibits strong convergence in the beginning when compared to the other algorithms. The GQBA becomes better in later iterations, but finally, the optimal solution is achieved by the CGQBA.

G. TEST SYSTEM 6
This test system consists of 110 generating units with quadratic cost behavior. The load demand is 15000 MW, and valve-point loading effects are taken into consideration. The system data are taken from [104]. Table 14 provides the results achieved by the QBA, GQBA and CGQBA. The best fuel cost obtained by the CGQBA is $197853.82/hr.
From the results obtained in Table 15, it is shown that the CGQBA performs better than the algorithms in recently cited works. The convergence characteristics of the PSO, QBA, GQBA and CGQBA are illustrated in Fig. 7. It is revealed that the CGQBA avoids being trapped into local optima and achieves the optimal solution at the end.

H. TEST SYSTEM 7
This test system comprises 160 generating units meeting a load demand of 43200 MW, and it is obtained by duplicating the 10-unit system 16 times.   The system contains multiple fuel options and incorporates valve-point loading effects. The data are adopted from [115]. The presentation of the optimal values and costs achieved by the QBA, GQBA and CGQBA are provided in Table 16. The best fuel cost obtained by the CGQBA is $9994.3235/hr. Table 17 shows the comparison of statistical results of the CGQBA and the other recently reported algorithms (the ACSS [49], DPADE [52], CBA, CGA_MU, IGA_MU [67], ADE-MMS, SADE, MBDE, IMSaDE [98], PI-CBA [106], BBO, DE/BBO, ORCCRO [111], RCCRO [116] and CSA [117]).
According to the results, the CGQBA provides satisfactory results even if there are some algorithms that perform better than it. The most eminent algorithms that outperform the CGQBA are the ACSS, DPADE, and ADE-MMS.  Fig. 8 that there is an alternating pattern in terms of which algorithm is best based on convergence between the QBA, GQBA and CGQBA. In early iterations, the QBA and CGQBA are better than the GQBA. In the following iterations, the GQBA becomes better than the QBA and CGQBA. The superiority of the CGQBA over the QBA and GQBA is proven in late iterations as it converges to the optimal solution.

V. CONCLUSION
In this paper, a proposed Cauchy-Gaussian quantum-behaved bat algorithm is successfully applied for solving the ELD problem. Quantum mechanics theories and the Gaussian and Cauchy operators are integrated into the classical bat algorithm to improve its performance. First, the bat algorithm guarantees quantum behavior by incorporating quantum mechanics theories. Second, the Gaussian and Cauchy probability distributions are applied to the QBA in place of a uniform distribution to avoid the premature convergence that persists in the QBA and to balance exploitation and exploration.
To demonstrate the feasibility of the proposed method, we compare the GQBA, QBA and the other optimization methods reported in the literature based on the different test systems possessing 3, 6, 20, 40, 110 and 160 units, as illustrated in the Tables. According to the results, it can be seen that the CGQBA outperforms or can compete with many methods recently reported in the literature. Moreover, the CGQBA is proven to tackle small-, medium-and large-scale problems.
For future research, we will try using the combination of two or more operators and varying them in the pursuit of the most effective optimization algorithm. Moreover, we will study some of the more complex problems: dynamic economic/emission dispatch (DEED), combined heat and power (CHP), combined heat and power economic dispatch (CHPED), combined heat and power economic emission dispatch (CHPEED) and combined cooling, heating and power (CCHP) in the presence of renewable energy (photovoltaic and wind energy).
QI JIA (Member, IEEE) received the B.S. and M.S. degrees from Northeast Electric Power University, Jilin, China, where he is currently pursuing the Ph.D. degree with the School of Electrical Engineering. His current research interests include renewable energy generation, stability analysis, and control of grid-connected renewable generation.
SHANFENG ZHANG is currently pursuing the M.S. degree with the School of Electrical Engineering, Northeast Electric Power University, Jilin, China. His current research interests include renewable energy generation and new energy generation actively participates in grid frequency regulation.
CHRISTOPHE BANANEZA received the B.S. degree in electrical power engineering from the National University of Rwanda, Huye, Rwanda, in 2011, and the M.S. degree in electrical engineering from the College of Energy and Electrical Engineering, Hohai University, Nanjing, China, in 2019.
He is currently an Assistant Lecturer with the IPRC-North/College of Tumba, Rwanda. His research interest includes power systems.