1 Introduction

Economic load dispatch (ELD) is one of the prominent problems of the power systems domain [1]. It has attracted the attention of many researchers in the past few decades, and it is still a hot topic of research in the field of power systems. This problem aims to minimize the total generation cost in producing specific power demands while satisfying a set of constraints. A power plant can have multiple generating units (generators) with their respective cost coefficients and limits constraints. These generating units are treated as energy-producing resources in a power plant. The proper utilization of generating units is required to optimize fuel consumption and energy as fossil fuels used to generate electricity are limited in nature. The generating units can be appropriately utilized through an efficient optimization algorithm with the following characteristics.

  • The algorithm must produce the best scheduling strategy that meets a certain load demand.

  • The algorithm must satisfy all input constraints and work effectively on non-linear functions.

  • The algorithm must be capable of handling problems of high dimensions.

Various optimization algorithms have already been suggested to deal with ELD problems while achieving the aforementioned conditions. Still, none of them can defeat all other algorithms in all benchmark datasets. Optimization algorithms for handling ELD problems are being developed in the hope of getting a globally optimal solution while avoiding premature convergence. These algorithms use certain mathematical equations to perform exploitation and exploration and follow the same steps in uni-modal and multi-modal problems, even though they may have different requirements. An algorithm that effectively deals with uni-modal problems may not be effective in multi-modal problems; however, the converse is not valid. An algorithm designed to solve multi-modal problems will work efficiently in uni-modal problems. As multi-modal problems may have more than one global or local optimal solution, a thorough investigation of search space is required to target all optimal solutions. In general, algorithms encounter difficulty in achieving the desired objective if the dimension of the problem increases. An algorithm that performs well in lower-dimensional optimization problems may not work efficiently in higher-dimensional optimization problems. Therefore, combining a diversity-producing mechanism with a fast converging algorithm works more appropriately in high-dimensional optimization problems.

eld is a multi-modal high-dimensional constrained optimization problem with a large search domain and non-linear objective function. eld aims to determine the best schedule of generating units to minimize the total generation cost while meeting certain load demand under specified constraints [2, 3] with each generating unit producing electricity (power) in its generation range. In the conventional formulation of eld, some practical features such as ramp rate limits, prohibited operating zones, and valve point effect were not considered. However, in real power systems, these features are usually encountered. Therefore, neglecting these features may lead to inaccurate solutions of the eld problems [4, 5].

The large search domain and high dimensionalities of eld need an optimization algorithm with a powerful exploration method that helps in avoiding the problem of local entrapment during optimization. This is one of the np-hard problems that require more computational time during optimization. These problems may be solved by giving more attention to implementing an efficient diversity-preserving approach. In solving such problems, nature-inspired algorithms have gained a wide range of acceptability due to their population-based search techniques [6,7,8,9]. However, some nature-inspired algorithms do not contain much novelty. Authors in [10] have argued that the concepts used in the cuckoo search are similar to that of (μ + λ) evolutionary strategies. Authors in [11] have presented sufficient evidence that grey wolf, bat, and firefly algorithms are not novel. These algorithms have reiterated the ideas introduced first for particle swarm optimization. Some other research articles have highlighted valuable insight into optimization algorithms in [12,13,14]. Authors in [15] proposed a new dendritic neuron model for solving complex problems.

The presence of constraints makes it difficult to solve eld. These constraints are handled by using the respective penalty functions. Hence, an optimization algorithm should have the ability to handle these constraints by implementing a penalty function while solving eld. In such a case, optimization algorithms need more intelligent exploitation and exploration methods. An exploration method targets the promising solutions throughout the search space, whereas an exploitation method tries to search around the neighborhood of the good solutions found so far. A powerful exploration method improves the global search ability, while an intelligent exploitation method improves the convergence rate of the optimization algorithm. These two methods need to be balanced during the program execution to avoid the problems of random search and local entrapment.

In eld problems, the search domain for i th generator is given by the range \([P_{i}^{\min \limits },P_{i}^{\max \limits }]\). Here, \(P_{i}^{\min \limits }\) and \(P_{i}^{\max \limits }\) represent the lower and upper limits, respectively for ith generator. The number of generators in a power plant is treated as the dimension of the problem. In most of the datasets available in the literature, the number of generators is 6, 13, 15, 20, 40, 80, 140, and 160. In general, optimization algorithms perform well in solving eld problems with up to 20 generators. As the number of generators in a power plant increases, the optimization algorithms find it difficult to solve eld problems efficiently. To handle hard problems of different domains, recently, some meta-heuristics have been suggested. These algorithms use some natural phenomena to achieve the desired goal. Faris et al. [16] proposed a monarch butterfly optimization (MBO) algorithm. Moth search algorithm (MSA) is proposed for global optimization in [17]. Yang et al. [18] presented a hunger games search (HGS) algorithm for solving complex problems. An efficient approach based on the Runge Kutta method (RUN) is proposed in [19] for optimization problems. Tu et al. [20] suggested colony predation algorithm (CPA) for complex problems. Heidari et al. [21] proposed Harris hawks optimization (HHO) algorithm for hard problems. Modified versions of the HHO algorithm are used to solve data clustering problems in [22, 23].

Recently, the slime mould algorithm (SMA) [24] is proposed for solving complex problems. This algorithm implements the natural phenomenon of slime behavior during the foraging to the food source. SMA, being a new algorithm, there is a lot of scope of its performance improvement and its applicability in various real-world problems of science and engineering. Authors of the base paper of SMA and other researchers have argued the effectiveness of this algorithm in solving complex problems. These points could be the motivation for using SMA to tackle the ELD Problems. The replacement of random numbers by chaotic sequences improves the performance of an optimization algorithm [25]. In this paper, a chaotic sequence guided slime mould algorithm (CSMA) is presented for solving eld problems with prohibitive operating zones and ramp rate limits. In short, the originality and major contributions of this paper are summarized as follows.

  • Integration of merits of chaotic sequences generated by the logistic chaotic map into a fast converging algorithm.

  • Validation of efficacy of CSMA by five test cases with 6, 13, 15, 40, and 140 generators ELD problems.

  • Comparison of the performance of CSMA with seven recent state-of-the-art algorithms SMA, SSA, MFO, GWO, BBO, GOA, and MVO based on the experimental values.

  • Total generation cost for specified load demand is used as the main performance metric for comparing the performances of the algorithms.

  • Validation for the effectiveness of CSMA using three statistical tests: Friedman test, Iman-Davenport test, and Holm test.

The rest of the paper is organized as follows. We present the literature review in Section 2. Section 3 describes the mathematical formulation of eld and various constraints associated with it. Section 4 describes csma in detail. The description of the datasets and parameter setting is given in Section 5. The analysis of experimental results is given in Section 6. Section 7 highlights the conclusions and future research directions.

2 Literature review

Pothiya et al. [26] proposed an approach based on multiple tabu search (mts) algorithms to solve the dynamic economic dispatch problem with generator constraints. They used experimental data to show the efficacy of mts against genetic algorithms, simulated annealing, and particle swarm optimization (pso). Lin et al. [27] suggested an improved tabu search (its) algorithm for eld problems. They suggested the idea of a flexible memory system to avoid the problem of local entrapment. After the first proposal of tabu search for the eld problems, other variants have been suggested to speed up the performance during optimization. pso, on the other hand, has been applied with different variations to solve eld [28]. Another well-known and powerful algorithm to solve eld is based on differential evolution (de) [29].

Jayabarathi et al. [30] used the crossover and mutation operators combined with a grey wolf optimizer to solve eld. They claimed the efficacy of their proposal through experimental results. Pradhan et al. [31] integrated opposition-based learning into the basic gwo algorithm to improve the convergence rate and solution quality while solving eld. Elsakaan et al. [32] suggested an enhanced moth-flame optimizer (mfo) to solve non-smooth economic dispatch problems. They combined the merits of levy flight with mfo to achieve the desired goal. Mandal et al. [33] incorporated mutation and crossover operators of differential evolution in krill herd algorithm (kha) to solve eld. They used experimental data to argue for the importance of integrated operators in kha. Bulbul et al. [34] proposed an opposition-based krill herd algorithm for eld. Coelho et al. [35] proposed an improved harmony search (ihs) algorithm based on exponential distribution for eld. The involvement of exponential distribution with the harmony search algorithm yielded better performance for ihs. eld has also been solved by a tournament-based harmony search (ths) algorithm [36]. ths replaced the random selection process in the memory consideration operator with the tournament selection process to achieve the desired objective while improving the convergence rate. Pothiya et al. [37] suggested an ant colony-based optimizer for eld. They introduced the concept of the priority list, variable reduction, and zoom feature to improve the overall performance of the suggested approach. Elsayed et al. [38] presented an approach based on the social spider algorithm for solving the economic dispatch problem.

Recently, hybrid approaches have been widely used to solve various science and engineering optimization problems. These approaches leverage the merits of existing techniques to perform specific tasks efficiently. Various hybrid algorithms have been developed in the field of power systems. Bhattacharya et al. [39] proposed a hybrid algorithm by combining differential evolution with biogeography-based optimization (de/bbo) algorithm to solve convex and nonconvex eld problems considering transmission losses and constraints such as ramp rate limits, valve-point loading, and prohibited operating zones. In addition, various other hybrid algorithms have been proposed to solve a variety of eld problems, and the efficacy of the approaches has been justified based on experimental values and comparative performance analysis [40,41,42,43].

The use of chaotic sequences in place of random numbers improves the convergence rate and quality of solutions during optimization [44]. A chaotic sequence-based differential evolution algorithm for solving complex problems is proposed in [45]. Yang et al. [46] proposed an adaptive chaotic spherical evolution algorithm for optimization. Xu et al. [47] implemented chaotic local search into grey wolf optimizer to avoid the problem of local entrapment during the search process. By utilizing the evidence argued by many researchers, chaotic sequences are used in various optimization algorithms to solve real-world global optimization problems in science and engineering. Adarsh et al. [48] introduced a chaotic map-based bat algorithm for eld problems. They used chaotic sequences to enhance the performance of their suggested approach. They used experimental data to claim the effectiveness of their proposal. Arul et al. [49] proposed a chaotic self-adaptive differential harmony search (csadhs) algorithm to solve the dynamic economic dispatch problem. They replaced the pitch adjustment operator in the harmony search algorithm with a chaotic self-adaptive differential mutation operator to improve the searching ability with less computational cost. Lu et al. [50] proposed a chaotic map-based differential evolution for dynamic economic dispatch problems. Coelho et al. [51] integrated chaotic sequences and implicit filtering local search methods in pso to solve eld problems. A multi-population-based chaotic JAYA algorithm is proposed to solve eld problems in [52]. In [52], random numbers are replaced by chaotic numbers to improve the convergence rate of the JAYA algorithm. In addition, the population is divided into sub-populations to enhance diversity during optimization.

Zhao et al. [53] proposed a cuckoo search algorithm-guided approach by introducing a self-adaptive step size and neighbor-study strategies to improve the global search ability while solving the eld problems. Moreover, they proposed an improved lambda iteration strategy to create offspring. Mohammadi and Abdi [54] suggested a modified crow search algorithm guided approach for eld problems. They proposed an approach to capture optimal global solutions by introducing an adaptive adjustment of the flight length. A harmony search-based method is proposed in [55] to solve eld problems. In [55], the update process of the harmony search algorithm based on a greedy approach is replaced by another efficient method to enhance the global search capability of the algorithm during the search process. The effectiveness of the simplex search method is integrated into artificial algae algorithm to solve eld problems in [56].

Prakash et al. [57] proposed a quasi-oppositional self-learning teacher-learner-based-optimization algorithm to solve eld problems. Kaboli et al. [58] proposed an artificial cooperative search algorithm to solve eld problems. This algorithm tries to balance exploitation and exploration during the optimization to avoid the problem of stagnation and random search. Trivedi et al. [59] proposed an interior search algorithm to solve eld and combined economic emission dispatch (ceed) problems in microgrids. An Ameliorated gwo algorithm is presented in [60] to solve eld by synergizing the exploration and exploitation mechanism. In addition, an opposition-based learning approach is used to target the global optimal solution in [60].

Srivastava et al. [61] proposed an aggrandized class topper optimization algorithm for solving eld. A crow search algorithm guided approach for eld is presented in [62]. Some other approaches for eld are given in [63, 64]. A clustering-based cuckoo search approach for ELD problems is presented in [65]. Authors have shown the effectiveness of clustering in cuckoo search for solving ELD problems. A moth flame optimizer-guided approach for ELD problems is presented in [66]. Kamboj et al. [67] presented an approach based on a grey wolf optimizer for eld. A biogeography-based optimizer is proposed for ELD problems in [68]. Salp swarm algorithm (SSA) [69] is developed to solve complex problems. The main motivation of SSA is the swarming behavior of salps when navigating and foraging in oceans. After the first version of SSA, it is being applied to solve various real-world problems. A grasshopper optimization algorithm (GOA) [70] is presented for solving hard problems. The main inspiration of GOA is the behavior of grasshopper swarms. Mirjalili et al. [71] proposed a multi-verse optimizer (MVO) for solving challenging real-world problems. The authors have justified the competitiveness of MVO using experimental results.

3 Mathematical formulation and constraints

In this section, we present the basic concepts and mathematical formulation of eld and various constraints associated with it.

3.1 Total generation cost

The total generation cost with D generators is given by

$$ \begin{array}{@{}rcl@{}} \min F_{t} &=& \sum\limits_{i=1}^{D} F_{i}(P_{i}) \\ &=& \sum\limits_{i=1}^{D} \left( a_{i} {P_{i}^{2}} + b_{i} P_{i} + c_{i}\right) + e_{i} \\ &&\times |sin(f_{i} \times(P_{i}^{\min}-P_{i}))| \end{array} $$
(1)

where Ft and Fi are the total generation cost and cost function, respectively of i th generator. Pi is the power generated by i th generator, and ai, bi, ci, ei, fi are its cost coefficients.

3.2 Constraints

3.2.1 Power equality constraint

This constraint states that the power generated by all generating units should be equal to the sum of load demand and power loss. Mathematically,

$$ \sum\limits_{i=1}^{D} P_{i} = P_{D} + P_{L} $$
(2)

where PD is the load demand in a power plant. PL is the power loss that is computed using B-coefficients as follows:

$$ P_{L} = \sum\limits_{i=1}^{D} \sum\limits_{j=1}^{D} P_{i} B_{ij} P_{j} + \sum\limits_{i=1}^{D} B_{0i} P_{i} + B_{00} $$
(3)

where Bij, B0, B00 are the transmission loss coefficients.

3.2.2 Generation limits constraint

This constraint states that the i th generating unit can generate power between the lower and upper limits as follows:

$$ P_{i}^{\min} \leq P_{i} \leq P_{i}^{\max} $$
(4)

where \(P_{i}^{\min \limits }\) and \(P_{i}^{\max \limits }\) are the lower and upper limits of i th generator, respectively.

3.3 Ramp rate constraints

The ramp rate constraint restricts the operating range of the physical lower and upper limits to the effective lower limit \(P_{i}^{min}\) and upper limit \(P_{i}^{max}\), respectively. According to [72], the inequality constraints due to ramp rate limits for unit generation changes are given

  1. 1)

    as generation increases

    $$ P_{i} - {P_{i}^{0}} \leq UR_{i} $$
    (5)
  2. 2)

    as generation decreases

    $$ P_{i} - {P_{i}^{0}} \leq DR_{i} $$
    (6)

    where Pi and \({P_{i}^{0}}\) are the current and previous output, respectively. URi and DRi are the up ramp limit and down ramp limit, respectively, of the i th generator in MW/time-period.

3.4 Prohibited operating zone constraints

A power generator may have prohibited operating zones (poz s) due to the physical limitation of power plant components [73]. A generator with poz s has discontinuous input-output characteristics. Each generator with (Z − 1) poz s is characterized by Z disjoint operating sub-regions. The poz constraints are given by [74]:

$$ P_{iz}^{L} \leq P_{i} \leq P_{iz}^{U}, z = 1, 2, \cdots, Z $$
(7)

Note that \(P_{i1}^{L}\) = \(P_{i}^{\min \limits }\), \(P_{iZ}^{U}\) = \(P_{i}^{\max \limits }\). Z is the number of prohibited zones for each generator. The cost function of generator with prohibited zones is given in Fig. 1.

Fig. 1
figure 1

Cost function with prohibited operating zones

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Respective penalty functions handle all the constraints mentioned above during the program execution. The description of these penalty functions is given below.

3.4.1 Power balance penalty (pbp)

This penalty function is used to handle power equality constraint described in (2) and is defined as:

$$ PBP = |P_{D} + P_{L} - \sum\limits_{i=1}^{D} P_{i}| $$
(8)

In the ideal case, the value of pbp is zero.

3.4.2 Capacity limits penalty (clp)

This penalty function is used to handle generation limits constraint described in (4). Its mathematical formulation is

$$ \begin{array}{@{}rcl@{}} CLP &=& \sum\limits_{i=1}^{D} |P_{i} - P_{i}^{\min}| - \left( P_{i} - P_{i}^{\min}\right)\\ && + \sum\limits_{i=1}^{D} |P_{i}^{\max} - P_{i}| - \left( P_{i}^{\max} - P_{i}\right) \end{array} $$
(9)

In the ideal case, the value of clp is zero.

3.4.3 Ramp limits penalty (rlp)

This penalty function is used to handle ramp rate constraints given in (5) and (6). This penalty function is mathematically formulated as

$$ \begin{array}{@{}rcl@{}} RLP &=& \sum\limits_{i=1}^{D} |P_{i} - DR_{i}| - (P_{i} - DR_{i})\\ && + \sum\limits_{i=1}^{D} |UR_{i} - P_{i}| - (UR_{i} - P_{i}) \end{array} $$
(10)

3.4.4 Prohibited operating zone penalty (pozp)

This penalty function is used to handle the poz constraints given in (7). Its mathematical formulation is

$$ POZP = \sum\limits_{i=1}^{D} (P_{poz})^{2} $$
(11)

where

$$ P_{poz}= \left \{ \begin{array}{ll} \min(P_{i} - P_{iz}^{L}, P_{iz}^{U}-P_{i}), & P_{iz}^{L}\leq P_{i}\leq P_{iz}^{U} \\ 0, & \text{otherwise} \end{array} \right \} $$
(12)

where \(P_{iz}^{L}\) and \(P_{iz}^{U}\) are the lower bound and upper bound of the i th generator for the z th prohibited zone.

The penalty functions mentioned-above give either zero or non-zero values. The zero value of the penalty function indicates that the respective constraint is satisfied. In that case, the multiplication of penalty function value with any value of λ will be zero. Therefore, the total generation cost will remain the same. A non-zero value of the penalty function indicates that the respective constraint is not satisfied. The non-zero value can be treated as an error value. A solution that is unable to satisfy the constraint must be discarded. Since, in this work, we are dealing with a minimization problem. All optimization algorithms try to minimize the total generation cost of the generating units. Each penalty function value is multiplied by a constant value to magnify the error if it occurs. The resultant value is added to the total generation cost (fitness value). Hence, during the selection, the solutions that do not satisfy the constraints will not be selected and will not be able to create offspring.

The values presented in the Table 1 are chosen in such a way that a solution gets discarded if it does not satisfy the constraint. For example, if we consider test case 1, the values are λ1 = 1000, λ2 = 1000, λ3 = 100000, λ4 = 10000. These values are multiplied by the power balance penalty, capacity limits penalty, ramp limits penalty, and prohibited operating zone penalty, respectively. The resultant value is then added to the total generation cost. In such a case, the possibility of a solution getting discarded is very high if it does not satisfy the constraints. Test cases 2 and 4 are not given the data to calculate ramp limits and prohibited operating zone penalties. Therefore, there is a dash (-) corresponding to these values in Table 1. The total penalty (tp) is computed by:

$$ TP = \lambda_{1} \times PBP + \lambda_{2} \times CLP + \lambda_{3} \times RLP + \lambda_{4} \times POZP $$
(13)

where λ1, λ2, λ3, and λ4 are constants given in Table 1.

Table 1 Value of λ in different test cases
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The fitness function (1) with the above set of constraints (2), (4), (5), (6), (7) is considered as in (14) using the penalty function method.

$$ \begin{array}{@{}rcl@{}} \min F_{t} &=& \sum\limits_{i=1}^{D} F_{i}(P_{i}) \\ &=& \sum\limits_{i=1}^{D} (a_{i} {P_{i}^{2}} + b_{i} P_{i} + c_{i}) + e_{i} \\ &&\times |\sin(f_{i} \times (P_{i}^{\min} - P_{i}))| + TP \end{array} $$
(14)

We will use the above concepts to describe our algorithm in the next section.

4 Proposed algorithm

This section describes the proposed algorithm (CSMA) in detail. The optimization algorithms start their execution with the randomly initialized candidate solutions in the specified boundary as the initial search direction is not known in complex problems. Like other optimization algorithms, the proposed method starts with the randomly initialized population. Mostly, in the random initialization of solutions, the boundaries of the search domain and a minimum of one random value are used. The boundaries of the search domain depend on the problem taken under consideration. In this study, the lower and upper limits of generating units in the power plants decide the boundaries of the search domain. Each randomly initialized solution is a vector of size 1 × D. Here, D is treated as the dimension of the problem taken under consideration. In this work, D is the number of generating units in the power plant. The solutions for each of the D generators are randomly initialized by:

$$ P_{ji} = P_{i}^{\min} + r\times\left( P_{i}^{\max}-P_{i}^{\min}\right) $$
(15)

where i varies from 1 to D and j varies from 1 to N (N is the population size). \(P_{i}^{\min \limits }\) and \(P_{i}^{\max \limits }\) are the lower and upper limits, respectively, of the i th generator. Pji is the power generated by the i th generator at j th individual in the population, and r is a random value between 0 to 1. After random initialization, each solution needs to be evaluated using (14). Here, each solution can be treated as a slime mould. Power demand is passed in the program code during the optimization process. The algorithms are supposed to generate supplied power demand with the least generation cost. Meanwhile, it is also desirable to have minimum power loss and power balance penalty. In this study, all algorithms motivate costly generating units to produce minimum power to minimize the total generation cost.

Based on the total generation cost, the most appropriate scheduling of generating units can be identified for the first iteration, and relevant values can be saved for future use. Afterward, each solution needs to be updated to identify other possible generating units scheduling. The solution updating method depends on the algorithm taken into consideration. Different algorithms have different solution updating methods. These methods intentionally try to implement exploration and exploitation of solutions to achieve the desired goal. The mathematical expression for updating the position of slime mould is given in (16).

$$ X= \left \{ \begin{array}{cc} (P_{i}^{\min} + \alpha.(P_{i}^{\max}-P_{i}^{\min}), & r < z \\ X_{b}(t)+V_{b}.(W.X_{A}(t)-X_{B}(t)), & r < p \\ V_{c}.X(t), & r \ge p \end{array} \right \} $$
(16)

where Vb ∈ [−a,a] and Vc decreases linearly from 1 to 0. Xb and t represent the individual location with the highest odour concentration currently found, and current iteration, respectively. XA and XB are two randomly selected individuals from slime mould. W and X represent the weight and position, respectively of a slime mould. In this study, parameter z is set to 0.03. Here, it should be noted that each variable in (16) is in the form of a vector of size 1× D.

p is defined according to (17)

$$ p = \tanh |F(i) - BF| $$
(17)

where i ∈ 1, 2, ⋯ ,N, F(i) and BF are the fitness of i-th individual and the best fitness, respectively found so far. The value of a is calculated using (18):

$$ a = \text{arctanh}\bigg(-\Large(\frac{t}{T})+1\bigg) $$
(18)

where T represents the maximum number of iterations considered in this study. For the best N/2 solutions, W is calculated using (19).

$$ W = 1 + r.log\bigg(\frac{BF - F(i)}{BF - WF} + 1\bigg) $$
(19)

For the remaining N/2 solutions, W is calculated using (20).

$$ W = 1 - r.log\bigg(\frac{BF - F(i)}{BF - WF} + 1\bigg) $$
(20)

where BF and WF represent the best and the worst fitness values, respectively found in the current iteration.

In (16), α ∈ (0,1) is a chaotic sequence of size 1 × D created by the logistic chaotic map. This map is mathematically formulated as follows [75]:

$$ x_{t+1}=cx_{t}(1-x_{t}) $$
(21)

here, c = 4, and xt = 0.75.

The proposed method for solving eld problems is given in Algorithm 1.

figure b

4.1 Analysis of computational complexity

The proposed algorithm mainly consists of random initialization, fitness evaluation, sorting, and population update. The computational complexity of random initialization and fitness evaluation is O(N × D) and O(N), respectively. The computational complexity of sorting and population update is O(N log N) and O(N × D), respectively. In this study, the maximum iterations and number of independent runs are T and R, respectively. Therefore, the total computational complexity of the proposed algorithm is O(R × (N × D + T × N(1 + log N + D))).

In this study, the computational complexity of algorithms depends on N, T, R, D. These parameters are the same for all algorithms considered. In this work, the flow of the program execution is the same for all the algorithms, which are (1) the random initialization of population, (2) fitness evaluation, (3) creation of new solutions, (4) selection of the top solutions. Therefore, the computational complexity of the remaining algorithms will be similar to that of the proposed approach.

5 Datasets and parameter setting

We used five test cases with 6, 13, 15, 40, and 140 generators to compare the performance of the algorithms. Power demands and references of these test cases are given in Table 2. The cost coefficients, minimum and maximum power generation capacity of generating units, B-coefficients, and other relevant information of used datasets can be found in detail in respective references.

Table 2 Description of datasets used in this study
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5.1 Experimental setup

We compared the performance of the proposed approach against seven algorithms: slime mould algorithm (SMA) [24], salp swarm algorithm (SSA) [69], moth flame optimizer (MFO) [66], grey wolf optimizer (GWO) [67], biogeography-based optimizer (BBO) [68], grasshopper optimization algorithm (GOA) [70], and multi-verse optimizer (MVO) [71]. The individual parameters of these algorithms are set according to the respective paper. However, three parameters that are common for all algorithms are set as follow:

  • Population size (N) = 25.

  • Maximum iterations (T) = 600.

  • Independent runs (R) = 30.

For a fair comparison, all algorithms are implemented in MATLAB R2017a on a machine with 8gb of ram and a Core-i5 processor. All algorithms are executed over 30 independent runs. We compute a solution that meets the load demand with minimum generation cost in each run. In other words, we compute a solution with a minimum balance penalty. If two solutions have the same balance penalty, we save the solution with minimum generation cost. At the same time, we keep the total generation cost for the minimum balance penalty concerning iteration. Out of 30 independent runs, we stored the best solution for the respective algorithms and test cases.

6 Analysis of experimental results

This section presents the analysis of experimental results. We performed the analysis in two ways based on (1) the quantitative values of the power balance penalty, power loss, total generation cost, and (2) the statistical test. We conducted various experiments to identify the best possible method to solve eld problems. Tables 3, 4, 5, 6, 7 and 8 represent the comparison of experimental values of algorithms for test cases 1-5 eld problems on load demands of 1263MW, 1800MW, 2630MW, 10500MW, and 49342MW, respectively. These tables represent the optimal power output of each generating unit, total power output, power loss, power balance penalty, and total generation cost. The minimum values of total generation cost, power loss, and power balance penalty are desirable.

Table 3 Comparison of the experimental results for test case 1
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Table 4 Comparison of the experimental results for test case 2
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Table 5 Comparison of the experimental results for test case 3
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Table 6 Comparison of the experimental results for test case 4
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Table 7 Comparison of the experimental results for test case 5
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Table 8 Comparison of the experimental results for test case 5 cont...
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Analysis of Tables 3–8 prove the effectiveness of the proposed approach in solving eld problems. Table 4 shows that ssa crosses the boundary of the search domain in most of the cases, which is not desirable. In such cases, algorithms find difficulty in identifying the optimal power output of generating units. To avoid such problems, modular clamping could be more effective instead of boundary clamping. From Tables 5–8, it is easy to conclude that mfo crosses the search domain for some generating units. The reason could be the updating method adopted in mfo that motivates the solutions to cross the boundary range of the search domain.

Table 9 represents the ranking of each algorithm for different test cases and average ranking. This ranking has been calculated based on the total generation cost. An algorithm with the least generation cost got rank 1 (best). This table shows that csma outperforms sma in all test cases considered in this study. The reason behind the competitive performance of CSMA could be the inclusion of chaotic sequences generated by the logistic chaotic map. For test cases 4 and 5, csma got the first rank. These test cases correspond to 40 and 140 generators eld problems. In this study, the number of generators is treated as the dimension of the optimization problem. Therefore, it can be concluded that the proposed algorithm will perform well in higher dimensional optimization problems. The average ranking of mvo is 7, which is the worst among all approaches considered. In this work, goa got the second position.

Table 9 Ranking and average ranking of considered approaches based on total generation cost in different test cases
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In this study, each algorithm is executed 30 times. Each time, the maximum number of iterations is taken as 600. The best value of the total generation cost in each independent run is saved. Therefore, the best 30 values of total generation cost are obtained at the end of the program execution. These values are used in box plots. Figures 4b, 5 and 6b represent the box plots of algorithms for different test cases. A careful observation of these plots justifies the effectiveness of csma. On the other hand, an observation of total generation cost concerning iterations is also a method of comparing the performances of the algorithms. To do so, convergence curves of algorithms for all test cases have been plotted. Figures 2a, 3 and 4a show the convergence curves. In minimization problems, a high rate of decrease in objective values is desirable. The suggested algorithm has shown similar behavior during the program execution (Figs. 5 and 6).

Fig. 2
figure 2

Variation in total generation cost with respect to iterations for (a): test case 1, (b): test case 2

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Fig. 3
figure 3

Variation in total generation cost with respect to iterations for (a): test case 3, (b): test case 4

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Fig. 4
figure 4

(a): Variation in total generation cost with respect to iterations for test case 5, (b): Box plot for test case 1

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Fig. 5
figure 5

Box plot for (a): test case 2 (b): test case 3

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Fig. 6
figure 6

Box plot for (a): test case 4 (b): test case 5

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To validate the competitiveness of the proposed algorithm statistically, three statistical tests (Friedman test [79], Iman-Davenport test [80], Holm test [81]) have been performed at a 5% significant level. Friedman test is one of the nonparametric tests mathematically formulated as follows.

$$ FT=\frac{12}{nA(A+1)}\sum\limits_{j=1}^{A} {R_{j}^{2}}-3n(A+1) $$
(22)

Here, FT represents the Friedman test statistical value. n and A are the number of test cases and the number of algorithms, respectively. In this work, n = 5, and A = 8. Rj represents the sum of ranks for the j-th algorithm. The calculated Friedman test value is now compared with the table of Chi-square statistics by considering degree of freedom = 7(number of algorithms - 1) at a significance level α = 0.05. The p-value for a given Friedman test value can be calculated from https://www.socscistatistics.com/pvalues/chidistribution.aspx.

Iman-Davenport test

This test is derived from Friedman test [80] and mathematically expressed as follows.

$$ IDT=\frac{(n-1)\times FT}{n\times(A-1)-FT} $$
(23)

where IDT is the Iman-Davenport test statistical value. The null hypothesis got rejected as the statistical value is greater than the critical value. The p-value for a given IDT can be evaluated from https://www.socscistatistics.com/pvalues/fdistribution.aspx.

Holm test

This is one of the post hoc tests and mathematically expressed as follows.

$$ HT=(R_{i}-R_{j})/\sqrt{\frac{A\times (A+1)}{6n}} $$
(24)

Where HT is Holm’s test statistical value. Here, Rj is the average rank of the proposed algorithm, whereas Ri represents the average rank of the algorithm that is taken under consideration from the remaining algorithms. The p-value for a given statistical value of the Holm test can be calculated from https://www.socscistatistics.com/pvalues/normaldistribution.aspx. These tests are conducted on the total generation cost values that meet the specific load demand. In these tests, the rejection of the NULL hypothesis indicates a significant difference in the performance of considered algorithms. However, the non-rejected NULL hypothesis indicates that the algorithms perform comparable, i.e., statistically, there is no difference in the performance of the algorithms.

Table 10 shows the experimental values of Friedman and Iman-Davenport tests. The rejection of the null hypothesis for both cases indicates a significant difference among the performances of the algorithms considered in this work. A post hoc test (Holm test) is performed to check whether there is a significant difference between the proposed algorithm and the rest of the approaches. The best performing approach (csma) is considered a control algorithm to calculate the statistical values and p-values. Table 11 shows the values of the Holm test. The rejection of the NULL hypothesis for mvo, gwo, bbo indicates that csma performs statistically better than these algorithms. The NULL hypothesis for sma, ssa, mfo, and goa is not rejected which indicates that csma does not perform statistically better than these algorithms. However, the effectiveness of csma against these algorithms can be easily seen from Table 9.

Table 10 Friedman test (FT) and Iman-Davenport test (IDT) values based on total generation cost
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Table 11 Holm test (HT) values and p-values of different methods (CSMA is the control algorithm)
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Based on the experimental results and various comparative performance analyses, csma exhibits the best performance compared to other algorithms. The competitiveness and effectiveness of csma are due to its convergence rate and population diversity. csma can capture promising solutions from the entire search domain via its exploration and exploitation operators. The influential exploitation and exploration methods of an optimization algorithm avoid the problem of random search and local entrapment. In any optimization algorithm, the most important thing is the mechanism to update the position vectors in the search domain. Suppose the solutions of the population can visit a variety of locations from different sections of the search domain in a uniform fashion. In that case, the probability of getting the optimal global solution is very high.

In the proposed work, only a few parameters are needed to be adjusted. This could be one of the advantages of the suggested approach. In general, the performance of optimization algorithms degrades with an increase in dimensionalities. However, the experimental results indicate the superiority of the proposed algorithm in 40 and 140 generators ELD problems. This confirms that the suggested algorithm performs well in higher dimensional optimization problems. This could be another advantage of the proposed algorithm. The performance of the existing approaches decreases in higher dimensional optimization problems. Although the computational complexity of the considered algorithms is comparable, the suggested approach takes more execution time in some test cases. This could be the disadvantage of the proposed method. The computation for chaotic sequences for logistic map needs some execution time, which affects the overall computation time of the proposed algorithm. This might be the reason why the suggested algorithm takes more execution time in some test cases.

Although the suggested method outperformed other algorithms considered in this study in solving ELD problems, similar behavior in other complex applications is not guaranteed. To solve some special class of problems, algorithms needed to be adjusted according to the mathematical formulation of the objective function, set of constraints, dimensionalities, and search domain of the optimization problem. The proposed algorithm is modified to handle an ELD problem, a continuous optimization problem. Therefore, this algorithm might not be suitable for solving discrete optimization problems. This could be the limitation of the suggested approach. Some additional adjustment and parameter tuning needed to be done to solve discrete optimization problems.

7 Conclusions and future research directions

In this paper, the merits of chaotic sequences generated by a logistic chaotic map are integrated into a fast converging algorithm to solve economic load dispatch problems. The performance of the proposed approach is compared against seven recent state-of-the-art algorithms using five test cases. Based on the experimental values, it is easy to conclude that csma performs better than sma in all test cases. The integration of chaotic sequences could be the main reason for the effectiveness of csma during the optimization process. We further validated the efficacy of the suggested approach by conducting three statistical tests (Friedman test, Iman-Davenport test, Holm test). Again, the proposed method has shown its robustness in solving eld problems.

In the future, we would like to solve other real-world optimization problems such as electricity load forecasting, optimal controller placement problem in a software-defined network, and feature selection using CSMA. A multiobjective version of CSMA can be suggested to handle multiple conflicting objectives simultaneously.