1 Introduction

The term optimization in mathematics and engineering refers to the process of finding the best solution for a particular problem, subject to a set of constraints. Mathematical optimization techniques, which borrowed ideas from fundamental geometry and calculus, were the tools to solve engineering optimization problems till the mid-eighties. These mathematical optimization techniques are deterministic in nature and some of them are gradient based methods which employ derivation of the search space. This increases complexity of the problem and often leads to entrapment into local optima.

With the rapid improvement in computational efficiency in the last few decades, metaheuristic algorithms have been widely utilized as the primary tool for solving real world engineering optimization problems. The metaheuristic algorithms are problem independent and driven by stochastic operators that assist them to avoid local optima. Evolutionary algorithms (EAs) are stochastic, nature inspired metaheuristic algorithms that evolve a randomly-picked solution set from the search space of the problem toward the global best solution in an iterative manner. Some of the widely used nature inspired metaheuristic algorithms include genetic algorithm (GA) (based on Darwin’s theory of evolution) [1], particle swarm optimization (PSO) (which mimics the foraging behavior of a swarm of birds or fishes) [2], differential evolution (DE) (based on evolution theory) [3], artificial bee colony (ABC) algorithm (that depicts the foraging behavior of honey bees) [4] and ant colony optimization (ACO) (inspired from the behavior of real ant colonies searching for food) [5]. These algorithms have grown interest among the researchers since the last two decades by offering good quality solutions to science and engineering optimization problems. In contrast to the success of these algorithms, there also exist a handful of new metaheuristics which have been developed in the recent years and this development is driven by the shortcoming of the no free lunch (NFL) theorem [6]. According to the NFL theorem, there is no such optimization technique that can solve all kinds of optimization problems. If any optimizer performs well for a set of problems, then there is no guarantee that it will offer good results for a different set of optimization problems. This inspires researchers to develop and propose novel metaheuristic algorithms for global optimization and also sets up a new line of research. Some of the recently developed nature inspired metaheuristics are firefly algorithm (FA) [7], league championship algorithm [8], gravitational search algorithm (GSA) [9], bat algorithm (BA) [10], cuckoo search (CS) [11], teaching-learning based optimization (TLBO) [12], krill herd algorithm [13], mine blast algorithm [14], symbiotic organisms search (SOS) [15], antlion optimizer (ALO) [16], water wave optimization [17], moth flame optimization (MFO) [18], sperm whale algorithm [19], multi-verse optimization (MVO) [20], whale optimization algorithm (WOA) [21] and crow search algorithm (CSA) [22]. Beside this, a large number of research works have been published in the last few years which improve the effectiveness of an existing metaheuristic algorithm by either integrating an ingenious search strategy into its original framework (see [23,24,25,26,27,28]) or by hybridizing two or more metaheuristics (refer [29,30,31,32]).

Population based metaheuristic algorithms commence the search process in two fundamental phases, namely, exploration and exploitation. In exploration phase, an extensive stochastic search is carried out over the entire search space which improves the diversity of the solution. The exploitation phase is followed by the exploration phase and it aims to improve the quality of the solution by continuing local search around the promising regions of the search space, already obtained in the exploration phase. It is very important to keep a good balance between exploration and exploitation while designing an algorithm because any improper handling of these phases may produce sub-optimal solution leading to stagnation into the local optima.

Recently, a nature inspired metaheuristic algorithm (i.e., ALO [16]) has been proposed by Mirjalili which is based on the hunting behavior of antlion larvae. The ALO algorithm has offered better results in terms of solution accuracy and convergence mobility as compared to some other popular methods like PSO, GA, FA, BA and CS [16] in solving global optimization problems. Also, the ALO algorithm has successfully been utilized to solve some engineering optimization problems (such as, design of ship propeller [16], wind based hydrothermal scheduling [33, 34], automatic generation control [35], design of linear discrete filters [36] and so on).

Motivated by the NFL theorem [6] and the previous researches for enhancing the performance of an existing metaheuristic, a new modified ALO algorithm has been proposed in the present work which is designated as quasi-oppositional chaotic ALO (QOCALO). The proposed QOCALO algorithm is having a quasi-opposition based learning (QOBL) strategy and a chaotic local search (CLS) inside the framework of the original ALO. The concept of QOBL helps to explore new areas of the search space and provides better exploration. On the other hand, CLS directs the search process around the most promising areas of the search space which provides better exploitation. Thus, a better balance between exploration and exploitation holds in case of the proposed QOCALO algorithm which makes this newly developed algorithm more robust as compared to its original counterpart. The performance of the proposed algorithm is validated by employing it for solving nineteen mathematical benchmark functions which include seven unimodal, six multimodal and six composite benchmark functions. The effectiveness and the superiority of the proposed algorithm have been established by comparing the performance of the proposed QOCALO with the basic ALO algorithm and some other recently developed nature inspired metaheuristics, such as MFO, MVO, WOA and CSA.

The main contribution of this paper lies in the framing of a new and robust variant of ALO algorithm which is capable to solve global optimization problems more effectively than the original ALO in terms of solution accuracy and convergence mobility. The proposed QOCALO algorithm has been utilized to solve three real world engineering optimization problems, such as, (a) the placement and sizing problem of distributed generators (DGs) in radial distribution networks, (b) the congestion management (CM) problem in power transmission system and (c) the optimal design problem of pressure vessel.

The rest of this paper is organized as follows: In Section 2, ALO algorithm is briefly described. Section 3 illustrates the proposed QOCALO algorithm. The simulation results of benchmark test functions are presented and discussed in Section 4. In Section 5, the placement and sizing problem of DGs has been solved by the proposed QOCALO algorithm. The CM problem has been solved using the proposed algorithm in Section 6. In Section 7, the proposed QOCALO algorithm is utilized to determine the optimal design of pressure vessel. Finally, the conclusions of the present work are drawn in Section 8 along with focusing on some future research directions.

2 The ALO algorithm

The ALO algorithm [16] is proposed by Mirjalili in 2015. The ALO is a population based stochastic search algorithm which mimics the predatory behavior of antlions in the nature. The antlions (or doodlebugs) belong to the Myrmeleontidae family of insects and Neuroptera order. The average lifespan of antlions may take up to three years, out of which only 3-5 weeks are spent in adulthood and the rest of the lifetime is spent as larvae. The antlions carry out a unique process of hunting in their larvae stage and preferably hunt for ants, which leads to their unique name. The antlion larva digs conical pits in the sand with the help of its massive jaw and waits at the bottom of the pit for the prey to be trapped in the sand pits. The edges of the sand pits are sharp enough to make insects (rather ants) fall down at the bottom of the pit. Once a prey falls inside the trap, the antlion throws sands out of the pit toward the prey such that the prey fails to escape from the pit and slides at the bottom of the pit. The prey is then consumed by the antlion and the leftovers are thrown away from the pit. In this way, the antlion prepares the pit for its next hunt [16]. In case of the ALO algorithm, the ants behave as the search agents, which move over the entire search space, whereas the antlions hunt them and become fitter.

During the course of optimization of ALO algorithm, the positions of ants and antlions over the search space are stored in the following two matrices, namely, P a and P a l (see (1) and (2))

$$ P_{a} =\left[ {{\begin{array}{cccc} {a_{1,1}} & {a_{1,2}} & {\cdots} & {a_{1,d}}\\ {a_{2,1}} & {a_{2,2}} & {\cdots} & {a_{2,d}}\\ {\vdots} & {\vdots} & {\vdots} & \vdots\\ {a_{n,1}} & {a_{n,2}} & {\cdots} & {a_{n,d}}\\ \end{array}}} \right] $$
(1)
$$ P_{al} =\left[{{\begin{array}{cccc} {al_{1,1}} & {al_{1,2}} & {\cdots} & {al_{1,d}}\\ {al_{2,1}} & {al_{2,2}} & {\cdots} & {al_{2,d}}\\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {al_{n,1}} & {al_{n,2}} & {\cdots} & {al_{n,d}}\\ \end{array}}} \right] $$
(2)

where P a stores the positions of ants over the search space, d is the number of decision variables and P a l is the matrix consisting of the positions of antlions hiding in the search space. The number of ants and antlions in the search space are equal and are represented by n. The position matrices (i.e. P a and P a l ) are utilized to evaluate the objective function and the fitness values are stored in the following two vectors, namely, F a and F a l (refer (3) and (4))

$$ F_{a} =\left[ {\begin{array}{c} {\begin{array}{c}{f\left( {\left[ {a_{1,1} ,a_{1,2} ,{\cdots} ,a_{1,d} } \right]} \right)}\\ {f\left( {\left[ {a_{2,1} ,a_{2,2} ,{\cdots} ,a_{2,d} } \right]} \right)}\\ \vdots\\ \end{array}}\\ f\left( {\left[ {a_{n,1} ,a_{n,2} ,{\cdots} ,a_{n,d} } \right]} \right)\\ \end{array}} \right] $$
(3)
$$ F_{al} =\left[ {\begin{array}{c} {\begin{array}{c} {f\left( {\left[ {al_{1,1} ,al_{1,2} ,{\cdots} ,al_{1,d}}\right]}\right)}\\ {f\left( {\left[ {al_{2,1} ,al_{2,2} ,{\cdots} ,al_{2,d} } \right]} \right)}\\ \vdots\\ \end{array}}\\ f\left( {\left[ {al_{n,1} ,al_{n,2} ,{\cdots} ,al_{n,d} } \right]} \right)\\ \end{array}} \right] $$
(4)

where F a stores the fitness value of each ant while F a l stores the fitness value of each antlion and f denotes the objective function.

The optimization process of the ALO algorithm is governed by six operating steps which are, basically, the adaptation of the real hunting deeds of antlions found in nature. The step-by-step operations of the ALO algorithm are discussed in the following six sub-sections.

2.1 Creation of random walks for ants

In the ALO algorithm, it is considered that the ants move freely over the entire search space in search for food until they get trapped by antlions. As the movement of the ants is stochastic in nature, it is modeled based on random walk and may be represented by (5) [16]

$$\begin{array}{@{}rcl@{}} X(t)&=&[0,cumsum(2r(t_{1})-1),cumsum(2r(t_{2})-1),\cdots,\\ &&cumsum(2r(t_{\max})-1)] \end{array} $$
(5)

where t is the step of random walk (or current iteration), t max is the maximum iteration, cumsum computes the cumulative sum and r(t)is a function defined in (6).

$$ r\left( t \right)=\left\{ {\begin{array}{l} 1\,\,if\,\,rand>0.5 \\ 0\,\,otherwise \\ \end{array}} \right. $$
(6)

Here, rand is a uniformly distributed random number in the interval [0,1]. At each step of the optimization process, the positions of the ants are updated with the help of this random walk. In order to maintain the random walks inside the search boundaries, (5) is normalized to (7)

$$ {X_{i}^{t}} =\frac{\left( {{X_{i}^{t}} -\alpha_{i} } \right)\times \left( {{\delta_{i}^{t}} -{\gamma_{i}^{t}} } \right)}{\left( {\beta_{i} -\alpha_{i} } \right)}+{\gamma_{i}^{t}} $$
(7)

where \({X_{i}^{t}}\) is the ant position in the i-th dimension of a d-dimensional space at the t-th iteration, α i and β i are the minimum and the maximum of random walks, respectively, for the i-th variable, \({\delta _{i}^{t}}\) and \({\gamma _{i}^{t}}\) are the upper and the lower bounds of the i-th variable at the t-th iteration, respectively.

2.2 Building of traps using roulette wheel

It is assumed that each ant is to be trapped by only one selected antlion. Also, the fitter the antlion is, the higher the chance of hunting ants as the fitter antlion makes a better trap. Hence, the selection of the antlion is done using a roulette wheel operator which is based on the fitness of the antlions during the optimization process [16].

2.3 Entrapment of ants in antlion’s pit

As per the previous discussion, the random walks of the ants get affected by the antlion traps. To model this mathematically, the boundary of ant movement is adjusted in each iteration, such that the ant moves in a hyperspace around the selected antlion trap. The upper and lower bounds of the ant dimension are computed in each iteration following (8) and (9) [16]

$$ \delta^{t}=antlio{n_{j}^{t}} +\delta^{t} $$
(8)
$$ \gamma^{t}=antlio{n_{j}^{t}} +\gamma^{t} $$
(9)

where δ t and γ t are the upper and the lower boundaries of all ant dimensions at current iteration t and \(antlio{n_{j}^{t}}\) is the position of the j-th selected antlion at the current iteration t. Thus, the ant movement is restricted in a d-dimensional hyperspace with a boundary of (γ t,δ t), using (8) and (9), in order to entrap the ants in the antlion pits.

2.4 Sliding of ants toward the antlion

When an antlion realizes that an ant has fallen into its pit, it throws sands toward the ant, such that the ant never escapes and slides down toward the antlion waiting at the bottom of the pit. Hence, the radius of the random walk of ants should be adaptively decreased to mathematically model this behavior of antlions and it is governed by the following three equations [16]:

$$ \gamma^{t}=\frac{\gamma^{t}}{I} $$
(10)
$$ \delta^{t}=\frac{\delta^{t}}{I} $$
(11)
$$ I = 10^{\omega}\cdot \frac{t}{t_{\max}} $$
(12)

where ω is a constant that takes up a value between 1 and 6 depending on (13).

$$ \omega =\left\{ {\begin{array}{l} 2\,\,\,if\,\,t>0.1\times t_{\max } \\ 3\,\,\,if\,\,t>0.5\times t_{\max } \\ 4\,\,\,if\,\,t>0.75\times t_{\max } \\ 5\,\,\,if\,\,t>0.9\times t_{\max } \\ 6\,\,\,if\,\,t>0.95\times t_{\max } \\ 1\,\,\,otherwise \\ \end{array}} \right. $$
(13)

From (10) – (13), it is clear that with an increase in the current iteration, the value of I is increased and, hence, the radius of the random walk of ants is reduced. This ensures shrinking of search space gradually and offers better exploitation.

2.5 Catching of ants and rebuilding the pit

The final stage of hunt occurs when the ant reaches at the bottom of the antlion’s pit. At this stage, the antlion catches the ant and drags it inside the sand to devour it. For the sake of mathematical modeling of this stage, it is assumed that the ingestion of ant by the antlion takes place when the ant becomes fitter than its corresponding antlion. The antlion then updates its position to the latest position of the hunted ant, i.e. improves its fitness and, hence, the chance of catching a new prey is increased. This process of catching ants is guided by (14) for the minimization of the objective function [16]

$$ antlio{n_{j}^{t}} =an{t_{i}^{t}} \,\,\,if\,\,f\left( {an{t_{i}^{t}} } \right)<f\left( {antlio{n_{j}^{t}} } \right) $$
(14)

where \(an{t_{i}^{t}} \) is the position of the i-th ant at the t th iteration and \(f\left ({an{t_{i}^{t}}}\right )\) is its corresponding fitness value. This is analogous to the rebuilding of the pit to enhance the chance of hunting.

2.6 Application of elitism

The ALO algorithm applies elitism to its search strategy by preserving the best solution (i.e. the fittest antlion) obtained at each generation. This fittest antlion (or elite antlion) is assumed to influence the movement of each ant. Hence, it is considered that each ant is about to take a random walk around the antlion selected by the roulette wheel and the elite antlion simultaneously and this may be represented by (15) [16]

$$ an{t_{i}^{t}} =\frac{RW_{antlion}^{t} +RW_{elite}^{t} }{2} $$
(15)

where \(RW_{antlion}^{t} \) represents the random walk around the antlion selected by roulette wheel at the t-th iteration and \(RW_{elite}^{t} \) represents the random walk around the elite antlion at the t-th iteration. The pseudo-code of the ALO algorithm is provided in Algorithm 1 [16].

figure c

3 The proposed QOCALO algorithm

The framework of the proposed QOCALO algorithm of the present work is formulated by simultaneous integration of QOBL and CLS strategies into the basic ALO algorithm. The different components of the proposed algorithm are discussed in the following sub-sections.

3.1 QOBL: a concept

Conventional metaheuristic algorithms begin the search process using a set of randomly generated initial solutions and progress toward the global best one. Hence, the rate of convergence depends on the distance between the initial solution set or the initial population and the global optimum solution. If the randomly generated solution is too far away from the global optimum one, then the algorithm takes considerably longer time to converge and, hence, the convergence rate becomes very poor. To avoid this problem and to improve the convergence rate, Tizhoosh [37] has suggested for consideration of both the randomly generated solutions and their opposite solutions simultaneously. According to [38], the random guess is far away from the global optimum than its opposite guess for 50% cases. Hence, starting the search process with an initial solution set that consists of the best of the two sets (i.e. the randomly generated solution set and its opposite solution set) helps to improve the convergence mobility. The attributes of QOBL are defined as follows.

3.1.1 Opposite point

Let x be a real number in one-dimensional search space with search interval [a,b]. Then, the corresponding opposite number (o x) is defined by (16)

$$ ox=a+b-x $$
(16)

In case of a d-dimensional search space, let X = (x 1,x 2,...,x d ) be a candidate solution, where (x 1,x 2,...,x d ) ∈ R and x i ∈ [a i ,b i ] ∀i ∈ {1,2,...,d}. The opposite point of X is considered as O X = (o x 1,o x 2,...,o x d ) and is defined by (17)

$$ ox_{i} =a_{i} +b_{i} -x_{i} $$
(17)

3.1.2 Quasi-opposite point

For one-dimensional search space, the quasi-opposite number (q o x) is defined as a random number between the center of the search space \(\left ({\frac {a+b}{2}} \right )\) and the opposite number (o x), as in (18)

$$ qox=rand\left( {\left( {\frac{a+b}{2}} \right),ox} \right) $$
(18)

Again, for d-dimensional search space, the quasi-opposite point of X is considered as Q O X = (q o x 1,q o x 2,...,q o x d ) and is, mathematically, defined by (19).

$$ qox_{i} =rand\left( {\left( {\frac{a_{i} +b_{i} }{2}} \right),ox_{i} } \right) $$
(19)

The quasi-opposite point has a higher chance to be closer to the global optimum as compared to the opposite point [39] and, hence, QOBL is more capable to improve the convergence rate. Recently, QOBL has been applied in different metaheuristic algorithms to improve its performance and some of them are quasi-oppositional DE [39], quasi-oppositional harmony search algorithm (HSA) [40], quasi-oppositional TLBO [41] and quasi-oppositional group search algorithm [42].

3.1.3 QOBL based population initialization

The initial population of the conventional metaheuristic algorithms is generated randomly without having any prior knowledge of the solution space. In this case, the QOBL based population initialization may achieve fitter candidate solutions as the simultaneous consideration of the randomly generated initial positions and their quasi-opposite positions improves the quality of the initial population and accelerates the search process by exploring the powerful regions of the search space. The pseudo-code of QOBL based population initialization is presented in Algorithm 2.

figure d

3.1.4 QOBL based generation jumping

The evolutionary process of the algorithm may be forced to jump to a new candidate solution by applying the concept of QOBL based generation jumping. This quasi-opposite population jumping has higher probability to produce fitter candidate solution than the existing one [39]. The QOBL based generation jumping is associated with a parameter (namely, jumping rate (j r )) which decides whether to remain at the position of the candidate solution generated by the search process of the algorithm or jump over to the quasi-opposite position of the candidate solution. The process of QOBL based generation jumping is depicted in Algorithm 3.

figure e

3.2 CLS based optimization

To improve the performance of ALO in terms of solution quality, a CLS based search strategy has been incorporated in the present work. The CLS also helps to prevent the optimizer to be trapped into local optima. In recent years, a wide variety of CLS based metaheuristic algorithms has been proposed and some of them are CLS based PSO [43,44,45,46,47,48], chaotic binary PSO [49, 50], CLS based DE [51,52,53], chaotic differential bee colony optimization [54], chaotic bee colony algorithm [55, 56], chaotic HSA [57], chaotic TLBO [58] and CLS based SOS [59, 60]. In contrast to the performance of these algorithms, the effectiveness of the CLS strategy to improve the search performance of the metaheuristic algorithms may be justified.

The CLS paradigm of the present work is applied to the global best solution obtained from the traditional ALO algorithm. This is done so as there is higher probability to achieve better solution in the vicinity of the global optimum solution [44]. The CLS intensifies the search process toward a more promising region and, thus, improves exploitation. The CLS is stopped when a better solution is found or the local search limit is reached.

3.2.1 Chaotic map

Chaos is a non-linear, dynamic and deterministic system which is very sensitive to its initial condition [61]. Due to some special characteristic features (like ergodicity and non-repetition), the chaotic system has the potential to carry out search process at a higher speed than the normal stochastic search which is probabilistic in nature [50]. There is a wide variety of chaotic maps available in the literature (see [62, 63]). In the present work, the well-known logistic map is utilized to carry out the CLS. The logistic map is, mathematically, formulated as in (20) [51]

$$ Ch_{k + 1} =\mu \times Ch_{k} \times \left( {1-Ch_{k} } \right)\,,\,\,\,\,Ch\in \left( {0,1} \right),\,\,\,Ch\ne 0.25,\,0.5\,,0.75\, $$
(20)

where C h k is the chaotic variable at the k-th generation. The initial value of the chaotic sequence (i.e. C h 0) is a random number generated by using rand function. The behavior of the logistic map is controlled by the parameter μ and it is depicted in Fig. 1. At μ = 4, the logistic function shows thorough chaotic behavior and, hence, the value of μ is set to 4. The chaotic sequence generated by the logistic map over 100 numbers of iterations is shown in Fig. 2.

Fig. 1
figure 1

Bifurcation diagram of the logistic map

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Fig. 2
figure 2

Distribution of logistic map over 100 numbers of iterations

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3.2.2 The proposed CLS strategy

The CLS strategy is, generally, employed to refine the quality of the previous best solution. In the present work, a new candidate solution is generated in the neighborhood of the previous best solution following (21)

$$ p_{i,j} =p_{g,j} +Ch\times \left( {p_{k1,j} -p_{k2,j} } \right) $$
(21)

where p i,j is the j-th dimension of the i-th solution generated in the vicinity of the previous best solution, p g,j is the j-th dimension of the previous best solution, C h is the chaotic number generated by the chaotic sequence expressed in (20), k1 and k2 are two mutually exclusive integers randomly chosen from {1, 2, ..., n} (n being the population size). Thus, the CLS strategy, guided by (21), well maintains the population diversity as it produces enough random solutions. The detailed process of the proposed CLS strategy is presented in Algorithm 4.

figure f

3.3 The proposed algorithm

The proposed algorithm is constructed by integrating both the QOBL and the CLS based search strategies into the basic structure of the original ALO algorithm. The search process of the proposed QOCALO algorithm is presented in detail in Algorithm 5.

figure g

4 Experimental results pertaining to function optimization problem

4.1 Details of test functions and experimental setup

The optimization capability of the proposed QOCALO algorithm is investigated on nineteen benchmark test functions. These test functions may be grouped into three categories, viz. unimodal, multimodal and composite functions. Unimodal functions have a single optimum and, hence, they are useful for examining the exploitation and the convergence behavior of the algorithm [16]. On the other hand, multimodal functions have multiple optima, one of which is global optimum and the rest are local optima. The optimization algorithm should converge to the global optimum avoiding all the local optima. Hence, the multimodal functions are benchmarked to examine the exploration of the algorithm. Finally, the composite functions are the rotated, shifted, combined and biased versions of other unimodal and multimodal test functions. The composite functions have a large number of local optima and the shape of search space is different in different regions of the search space which resembles the real search space for practical problems [16]. Therefore, the composite functions are benchmarked to check whether an algorithm is capable of maintaining a proper balance between exploration and exploitation. The details of the unimodal, the multimodal and the composite test functions are listed in Tables 1, 2 and 3 [16].

Table 1 Detailed description of unimodal benchmark functions
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Table 2 Detailed description of multimodal benchmark functions
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Table 3 Detailed description of composite benchmark functions
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The simulations are carried out in MATLAB 2013a computing environment on a 2.3 GHz core i5 personal computer with 3.86 GB RAM. Due to the stochastic nature of the evolutionary optimization techniques, each test function is tested for 100 independent trial runs to minimize the statistical error and the best, the worst, the mean and the standard deviation of the results are obtained over the trial runs. The proposed algorithm is tested on 10, 30 and 200-dimensional versions of the unimodal and multimodal test functions to establish the scalability of the proposed algorithm.

4.2 Non-parametric statistical test set up

The metrics, such as the mean and the standard deviation, compare the overall performance of the algorithms while statistical tests verify the superiority of an algorithm over the others by considering the results of each trial run. The statistical tests are necessary as the metaheuristic algorithms are stochastic in nature. In the present work, a non-parametric pair-wise Wilcoxon rank-sum test is performed to verify whether the simulation results of the proposed QOCALO are significantly superior to the simulation results of other reported algorithms. The null hypothesis of this test states that the median of the difference of the two samples is zero (i.e. the solution sets of the algorithms are not statistically different). The test is implemented at a significance level of 5% and a p-value less than 0.05 points toward a significant difference between the two samples (i.e. the null hypothesis is rejected).

4.3 Parameter setting

In the present work, the optimization capability of the proposed QOCALO algorithm is tested by comparing its performance with the other recently developed nature inspired algorithms such as MFO, MVO, WOA, CSA and the original ALO. Each trial run employs 30 antlions to carry out the search process over 1000 iterations for 10 and 30-dimensional test problems. The number of antlions and iterations are considered as 100 and 5000, respectively, for 200-dimensional test problems. These settings are considered in line with [16] to maintain a fair comparison between the proposed QOCALO and the basic ALO. The parameters of the other reported algorithms are considered as of their original published works (see [18, 20,21,22]).

The parameter j r of the proposed algorithm controls the convergence rate and, hence, it is to be carefully selected. A higher value of j r may reduce the population diversity at a very faster rate and leads to premature convergence. An experiment has been conducted to set the value of j r . Figure 3 shows the variation in the number of iterations required for convergence with varying j r for the function F4. It may be observed from Fig. 3 that, a j r value beyond 0.6 results in premature convergence. Hence, the value of j r is set to 0.6.

Fig. 3
figure 3

Variation in number of iterations required for convergence with varying j r

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Another parameter of the proposed algorithm is the CLS limit (K), which is also set after conducting an experiment on function F5 with 30 dimensions. In this experiment, all the parameters of the proposed QOCALO algorithm are considered to be the same as the previous experiment while j r is considered as 0.6 and K is varied as in Fig. 4. From Fig. 4, it is observed that the proposed algorithm yields the minimum fitness value at K = 10. Therefore, the K value is set to 10.

Fig. 4
figure 4

Variation in fitness value with varying K

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4.4 Simulation results for unimodal functions

The performance of the proposed QOCALO in optimizing unimodal functions is compared to the performances of the other reported algorithms and the detailed simulation results are listed in Tables 4, 5 and 6. The algorithms are ranked on the basis of their performance to optimize the set of seven unimodal functions. The algorithm that produces the minimum mean fitness value is considered to have rank 1 while in case of a tie equal ranks are given to the respective algorithms. It may be observed from Tables 4–6 that the proposed algorithm stands second to only MFO for function F6 with 10 dimensions while it shares the first rank with WOA for the 200 dimensional functions (F1 and F2) but for 30-dimensional case the proposed algorithm stands first for all the test functions. Hence, the proposed QOCALO dominates other algorithms in optimizing unimodal functions as it attains an overall rank 1 for all the three different dimensions. The p-values obtained from the pair-wise Wilcoxon rank-sum test for the unimodal functions with three different dimensions are listed in Table 7. It may be noted that the p-values are much less than 0.05 for almost all the instances which statistically establishes the superiority of the proposed QOCALO algorithm over the others. The ‘ + ’ sign implies that the proposed QOCALO algorithm is significantly better than the other algorithm, ‘-’ sign indicates that the other algorithm is significantly better than the proposed QOCALO and ‘=’ sign signifies that there is no significant difference between the proposed QOCALO and the other algorithm. From Table 7 it may be observed that the proposed algorithm straightly outperformes MFO, MVO and CSA for all the unimodal functions for all the three different dimensions while it shows similar performance with ALO for the function F6 and with WOA for the functions F1, F2 and F7 with 200 dimensions. The comparative convergence profiles of the proposed QOCALO and the other compared algorithms are depicted in Figs. 5, 6 and 7. From the observation tables and convergence profiles it may be concluded that the proposed algorithm shows better performance in optimizing unimodal test functions which points to better exploitation and faster convergence behavior of the proposed algorithm as compared to other reported algorithms.

Table 4 Comparative results for 10 dimensional unimodal test functions
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Table 5 Comparative results for 30 dimensional unimodal test functions
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Table 6 Comparative results for 200 dimensional unimodal test functions
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Table 7 Results of Wilcoxon rank-sum test for unimodal test functions
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Fig. 5
figure 5

Comparative convergence profiles of fitness function value for 10 dimensional functions a F3, b F5, c F9 and d F10

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Fig. 6
figure 6

Comparative convergence profiles of fitness function value for 30 dimensional functions a F4, b F6, c F12 and d F13

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Fig. 7
figure 7

Comparative convergence profiles of fitness function value for 200 dimensional functions a F3, b F6, c F10 and d F13

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4.5 Simulation results for multimodal functions

The comparative simulation results of the proposed QOCALO and the other algorithms for multimodal test functions are provided in Tables 8, 9 and 10. From these tables it may be observed that the proposed algorithm gets an overall rank 1 for 10 and 30 dimensional test functions while it stands second to WOA for 200 dimensional functions. The p-values of the pair-wise Wilcoxon rank-sum test for the multimodal test functions are reported in Table 11. From Table 11, it may be noted that the proposed QOCALO produces significantly better results than MFO and MVO while it completely outperforms ALO and CSA by beating them in 17 instances. The only competitor to the proposed algorithm in this section is the WOA as it produces better results than the proposed algorithm for three instances and similar results for seven instances. Hence, it may be stated that the proposed algorithm has better capabilities to optimize lower and medium dimensional multimodal functions as compared to other reported algorithms but the performance deteriorates slightly when it comes to optimize very high dimensional problems. Considering the properties of the multimodal test problems and the performance of the proposed algorithm, it may be inferred that the proposed QOCALO algorithm exhibits a better exploration over the search space. The comparative convergence profiles for some selected test functions are depicted in Figs. 5–7, which indicate faster convergence mobility of the proposed algorithm.

Table 8 Comparative results for 10 dimensional multimodal test functions
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Table 9 Comparative results for 30 dimensional multimodal test functions
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Table 10 Comparative results for 200 dimensional multimodal test functions
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Table 11 Results of Wilcoxon rank-sum test for multimodal test functions
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4.6 Simulation results for composite functions

The simulation results for the composite test problems are tabulated in Table 12. These results are quite far from the global minima as compared to the previous cases and this is due to the complexity involved in the composite test functions. However, the proposed QOCALO algorithm yields better results in optimizing composite test problems compared to other reported algorithms. The proposed algorithm attains an overall rank 1 in solving composite test problems compared to its original counterpart (i.e. ALO) which stands second. The p-values obtained from the pair-wise Wilcoxon rank-sum test are reported in Table 13 which also shows the superiority of the proposed algorithm over other reported algorithms. From Table 13 it may be observed that the proposed algorithm produces significantly better results than MFO, WOA and CSA for all the six composite test functions. The proposed algorithm produces significantly better results than the original ALO for four functions while it produces statistically similar results for the functions CF2 and CF4. These results indicate that the proposed QOCALO algorithm is capable of maintaining a better balance between exploration and exploitation compared to original ALO and other reported algorithms. The comparative convergence profiles for the composite test functions are presented in Fig. 8, which shows that the proposed QOCALO algorithm offers the best convergence mobility among all the reported nature inspired algorithms.

Table 12 Comparative results for composite test functions
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Table 13 Results of Wilcoxon rank-sum test for composite test functions
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Fig. 8
figure 8

Comparative convergence profiles of fitness function value for composite functions a CF3 and b CF5

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5 Solution of placement and sizing problem of DGs in radial distribution network

The aim of this section is to check whether the proposed QOCALO algorithm is capable to solve real world constrained engineering optimization problems rather than mathematical test problems. In pursuit of this, the problem of optimal placement and sizing of DGs (as first application of the proposed QOCALO algorithm) is solved using the proposed QOCALO algorithm. The considered problem is a power system optimization problem and is a topic of research since the last decade. Various metaheuristic algorithms have been utilized by the researchers to solve this problem and some of these reported algorithms are GA [64,65,66], PSO [67, 68], DE [69], ACO [70], ABC [71] and GA-PSO [72].

5.1 Mathematical formulation of the problem

The multi-objective problem of finding the optimal locations and sizes of real power DGs is associated with the reduction of real and reactive power loss indices and improvement of voltage profile while keeping the line MVA flow within its specified limit. The detailed mathematical formulation of the objective functions is provided in the following seven sub-sections.

5.1.1 Real and reactive power loss indices

The real and reactive power loss indices of the system are defined in (22) and (23), respectively [66]

$$ ILP=\frac{P_{LDG} }{P_{L} } $$
(22)
$$ ILQ=\frac{Q_{LDG} }{Q_{L} } $$
(23)

where P L and Q L are the total real and reactive power losses of the distribution system without DGs while P L D G and Q L D G are the total real and reactive power losses of the system after installing DGs. The real power loss (P l o s s ) and the reactive power loss (Q l o s s ) are calculated, in order, using (24) and (25)

$$ P_{loss} =\sum\limits_{i = 1}^{nbr} {R_{i} \left| {I_{i} } \right|^{2}} $$
(24)
$$ Q_{loss} =\sum\limits_{i = 1}^{nbr} {X_{i} \left| {I_{i} } \right|^{2}} $$
(25)

where R i and X i are the resistance and reactance of the i-th branch; I i is the branch current of the i-th branch and nbr is the total number of branches in the network.

5.1.2 Voltage profile index

Installation of DGs in the distribution system greatly improves the voltage at each node of the system (except the first node). The voltage profile index is defined as

$$ IVD=\underset{i = 2}{\overset{n}{\max}}\left( \frac{\left|{\overline{V}_{no\min al}}\right|-\left|{\overline{V}_{i}}\right|}{\left|{\overline{V}_{no\min al}}\right|}\right) $$
(26)

where \(\overline {V}_{no\min al}\) is 1.03 p.u. for 38-node system and 1.00 p.u. for 69-node system [73]; \(\overline V_{i}\) is the voltage at the i-th node of the system and n is the total number of nodes.

5.1.3 MVA capacity index

Incorporation of DGs in distribution networks, significantly, changes the power flow in different sections of the network. In order to avoid overloading of the lines, it is very important to keep the line MVA flow within its maximum allowable limits. Hence, the MVA capacity index is defined in (27)

$$ IC=\underset{i = 1}{\overset{nbr}{\max}}\left( {\frac{\left|{\overline{S}_{ij}} \right|}{\left|{\overline{CS}_{ij}}\right|}}\right) $$
(27)

where \(\overline S_{ij}\) is the MVA flow in the line connecting nodes i and j while \(\overline {CS}_{ij}\) is the MVA capacity limit of the line connecting nodes i and j.

5.1.4 Voltage stability index (VSI)

The VSI of distribution system is defined as in (28) [74]

$$\begin{array}{@{}rcl@{}} VSI(m2)&\,=\,&\left| {V(m1)} \right|^{4}\!-4[P(m2)x(jj)\!-Q(m2)r(jj)]^{2}\\ &&\!-4[P(m2)r(jj)\!+Q(m2)x(jj)]\left| {V(m1)} \right|^{2} \end{array} $$
(28)

where m1 is the sending end node, m2 is the receiving end node, jj is the branch number, P(m2) is the real power load fed through node m2, Q(m2) is the reactive power load fed through node m2, r(j j) is the resistance of the branch jj, x(j j) is the reactance of the branch jj and V (m1) is the voltage at node m1.

For stable operation of the system, the value of VSI should be greater than zero. The node having the lowest value of VSI (V S I min) is the weakest node of the system in terms of voltage stability and is more vulnerable to voltage collapse. Therefore, the objective should be the improvement of the value of V S I min.

5.1.5 Multi-objective problem formulation

In the present work, the problem of placement and sizing of DGs is considered as a multi-objective optimization problem taking into account all the separate objective functions, as discussed in the previous sub-sections. The fitness function of the current problem may be defined as of (29) [73]

$$\begin{array}{@{}rcl@{}} MOF&=&\alpha_{1} \cdot ILP+\alpha_{2} \cdot ILQ+\alpha_{3} \cdot IVD\\ &&+~\alpha_{4} \cdot IC+\alpha_{5} \cdot \left( {\frac{1}{VSI_{\min }}} \right) \end{array} $$
(29)

where \(\sum \limits _{k = 1}^5 {\alpha _{k} = 1\,\,\wedge \,\,\alpha _{k} \in \left [ {0,1} \right ]} \). The values of these weighting factors (α k ) are taken from [73].

The fitness function is minimized subject to a variety of operational constraints to be satisfied. These operational constraints are listed in the Sections 5.1.6 and 5.1.7

5.1.6 Equality constraint

The equality constraint of the current problem is defined in (30)

$$ P_{SS} =\sum\limits_{i = 2}^{n} {P_{D} (i)+\sum\limits_{j = 1}^{nbr} {P_{loss} (j)-\sum\limits_{k = 1}^{ndg} {P_{DG} (k)}}} $$
(30)

where P D (i) is the real power demand of the i-th node, P l o s s (j) is the real power loss of the j-th branch, P D G (k) is the real power generated by the k-th DG, ndg is the total number of DGs and P S S is the total real power delivered by the sub-station.

5.1.7 Inequality constraints

The inequality constraints of the current problem are defined in (31)–(33)

$$ P_{DG}^{\min} \le P_{DG} \le P_{DG}^{\max} $$
(31)
$$ V_{i}^{\min} \le V_{i} \le V_{i}^{\max} $$
(32)
$$ S_{ij} \le S_{ij}^{\max} $$
(33)

where S i j is the thermal capacity of the line connecting nodes i and j. The minimum and the maximum values of V i for 38-node system are 0.95 p.u. and 1.03 p.u., respectively, while the same for 69-node system are, in order, 0.90 p.u. and 1.00 p.u.

5.2 Simulation results and discussion

The current multi-objective problem is solved individually using the proposed QOCALO algorithm, its original counterpart (i.e. ALO) and other recently developed nature inspired algorithms such as MFO, MVO, WOA and CSA and the performances are compared to those yielded by the chaotic ABC (CABC) algorithm available in [73]. The simulations are carried out in the same computing environment with the same algorithmic parameters as in the case of function optimization problems. The algorithms are run for 100 independent trial runs and each run is associated with 100 numbers of iterations.

5.2.1 Case study 1: 38-node system

The effectiveness of the proposed QOCALO is tested by implementing it on the 38-node radial distribution system. The 38-node radial distribution network consists of 38 nodes and 37 branches. The details of branch and load data for this test system may be found in [66]. The system has the base values of 100 MVA and 23 kV. The total real and reactive power losses of this system without the installation of DGs are 0.20206 p.u. and 0.13474 p.u., respectively. In the current problem, three real power DGs with unity power factor are considered to be incorporated in the distribution network. The sizes of the DGs are considered in the range 0 - 1.2 p.u. [73].

The simulation results yielded by the proposed QOCALO and the other algorithms for this test system are provided in Table 14. From Table 14, it may be observed that the proposed QOCALO yields better values of real and reactive power losses after successfully integrating three DGs into the system, as compared to the basic ALO, MFO, MVO, WOA and CSA. Also, the proposed QOCALO algorithm successfully minimizes the values of the indices ILP and ILQ, as compared to the basic ALO, MFO, MVO, WOA, CSA and CABC [73]. It may further be noted that the proposed QOCALO yields better value of V S I min than the basic ALO, MVO and CSA. The line MVA flows are within their respective limits for both the proposed QOCALO and the other reported algorithm. Finally, the proposed algorithm produces better overall fitness value compared to the other recent algorithms except MFO and CABC. Hence, it may be stated that the proposed QOCALO algorithm successfully finds the optimal locations and sizes of DGs while satisfying all the operating constraints and performs better than the basic ALO and some of the other recently developed algorithms. The comparative convergence profile of the proposed QOCALO and the other reported algorithms is shown in Fig. 9.

Table 14 Comparative results of optimal DG placement for 38-node system offered by different algorithms
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Fig. 9
figure 9

Comparative convergence profiles of fitness function value for 38-node system pertaining to DG placement and sizing problem

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5.2.2 Case study 2: 69-node system

The 69-node radial distribution system has 69 nodes and 68 branches. The details of system branch and load data for this test system may be found in [73]. The base values for this system are 100 MVA and 12.66 kV. The total real and reactive power demand for this test system are 3.8021 MW and 2.6945 MVAr, respectively. The real and reactive power losses of the system before installation of DGs are 224.97 kW and 102.12 kVAr, respectively. Alike the previous case, in this case also three numbers of real power DG units are incorporated into the system to minimize the power loss and to improve the voltage profile. The DGs are operated at unity power factor and the sizes of the DGs are in the range 0 - 1.2 MW [73].

The simulation results for this test system are given in Table 15. From Table 15, it may be noted that the proposed QOCALO algorithm yields lesser values of the indices ILP and ILQ as compared to the basic ALO and the other reported algorithms except CABC. Both the real and the reactive power losses of the system offered by the proposed QOCALO are lesser than the losses offered by the original ALO and other reported algorithms except CABC. Also, it may be found from Table 15 that the proposed algorithm yields the best values for the indices IVD and V S I min as compared to the basic ALO, CABC [73] and other reported algorithms. In the context of solving the current multi-objective optimization problem, the proposed QOCALO algorithm produces minimum value of the fitness function compared to other algorithms referred therein. Hence, it may be remarked that the proposed QOCALO algorithm performs better than its original counterpart and produces competitive results to other reported algorithms in solving the current problem. A comparative convergence profile of fitness function value for the considered case is portrayed in Fig. 10.

Table 15 Comparative results of optimal DG placement for 69-node system offered by different algorithms
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Fig. 10
figure 10

Comparative convergence profiles of fitness function value for 69-node system pertaining to DG placement and sizing problem

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6 Solution of CM problem in power transmission system

The aim of this section is to minimize the congestion cost in power transmission system by rescheduling the real power outputs of the generators while satisfying all the operating constraints. This may be treated as the second real world engineering optimization application of the proposed algorithm. The considered power system optimization problem is solved using the proposed QOCALO algorithm and the simulation results are compared to those offered by the original ALO and some other recently reported algorithms, such as, BA, chaotic BA, flower pollination algorithm (FPA) and FA [75].

6.1 Mathematical problem formulation

The transmission CM problem may be formulated as in (34) [76]

$$ \text{Minimize } C =\sum\limits_{j\varepsilon N_{g} } {(C_{j} {\Delta} P_{Gj}^{+} +D_{j} {\Delta} P_{Gj}^{-} )} {\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\$/\mathrm{h}{\kern 1pt}{\kern 1pt}{\kern 1pt} $$
(34)

where C represents the total CM cost ($/h), C j and D j represent the increment and decrement price bids ($/MWh), respectively, submitted by the generating company for j-th generator, \({\Delta } P_{Gj}^{+}\) and \({\Delta } P_{Gj}^{-}\) represent the real power increment of j-th generator (MW) and the real power decrement of j-th generator (MW), respectively and N g is the total number of generators.

6.1.1 Equality constraints

The equality constraints of the current problem are stated in (35)–(38) [75, 76]

$$ P_{Gk} -P_{Dk} \,=\,\left| {V_{j} } \right|\left| {V_{k}} \right|\left| {Y_{kj} } \right|\cos (\delta_{k} -\delta_{j} -\theta_{kj} ){\kern 1pt}{\kern 1pt};{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\,{\kern 1pt}{\kern 1pt}{\kern 1pt}j = 1,2,......,N_{b} $$
(35)
$$ Q_{Gk} -Q_{Dk} \,=\,\left| {V_{j} } \right|\left| {V_{k} } \right|\left| {Y_{kj} } \right|\sin (\delta_{k} -\delta_{j} -\theta_{kj} ){\kern 1pt}{\kern 1pt}{\kern 1pt};{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}j = 1,2,.....,N_{b} $$
(36)
$$ P_{Gk} =P_{Gk}^{C} +{\Delta} P_{Gk}^{+} -{\Delta} P_{Gk}^{-} {\kern 1pt};{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}k = 1,2,.....,N_{g} $$
(37)
$$ P_{Dj} =P_{Dj}^{C} {\kern 1pt};{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}j = 1,2,.....,N_{d} $$
(38)

where P G k and Q G k are the real and reactive power generated at bus k respectively; P D k and Q D k are the real and reactive load at bus k, respectively; V j and V k are the voltages at busses j and k, respectively; δ j and δ k are the bus voltage angles of busses j and k, respectively; 𝜃 k j is the admittance angle of the line connected between busses k and j; N b and N d are the total number of busses and loads, respectively; \(P_{Gk}^{C} \) and \(P_{Dj}^{C} \) are the real power produced by generator k and the real power consumed by load bus j, respectively, as obtained by the market clearing procedure.

6.1.2 Inequality constraints

The inequality constraints associated with the current problem are stated in (39)–(43) in line with [75, 76]

$$ P_{Gk}^{\min } \le P_{Gk} \le P_{Gk}^{\max } ,{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\forall k\,\in Ng $$
(39)
$$ Q_{Gk}^{\min } \le Q_{Gk} \le Q_{Gk}^{\max } ,{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\forall k\,\in Ng $$
(40)
$$ (P_{Gk} -P_{Gk}^{\min } )=\!{\Delta} P_{Gk}^{\min } \!\le {\Delta} P_{Gk} \le {\Delta} P_{Gk}^{\max } =(P_{Gk}^{\max } -P_{Gk} ) $$
(41)
$$ V_{n}^{\min } \le V_{n} \le V_{n}^{\max } ,{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\forall n\,\in N_{b} $$
(42)
$$ P_{ij} \le P_{ij}^{\max } $$
(43)

where the superscripts min and max represent the minimum and the maximum values of the respected variables, respectively.

6.1.3 Formulation of fitness function

The fitness function (FF) for achieving the desired minimum CM cost is given by (44)

$$ FF=C+P $$
(44)

where, P is a penalty function, given by the following equation

$$ P=pf_{1} \sum\limits_{i = 1}^{N_{l} } {P_{l} +pf_{2} \sum\limits_{j = 1}^{N_{b} } {P_{v} +pf_{3} } } \sum\limits_{k = 1}^{N_{g} } {P_{s} } $$
(45)

where, p f 1,p f 2 and p f 3 are penalty factors which are user defined and N l is the total number of transmission lines. The values of p f 1,p f 2 and p f 3 are taken as 10,000 [75, 76].

The terms P l , P v and P s are described as of (46)–(48).

$$ P_{l} =\left\{ {\begin{array}{l} 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,P_{ij} \le P_{ij}^{\max } \\ \left( {P_{ij} -P_{ij}^{\max } } \right)^{2}\,\,\,\,\,\,\,\,if\,P_{ij} >P_{ij}^{\max } \\ \end{array}} \right. $$
(46)
$$ P_{v} =\left\{ {\begin{array}{l} 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,V_{n}^{\min } \le V_{n} \le V_{n}^{\max } \\ \left( {V_{n}^{\min } -V_{n} } \right)^{2}\,\,\,\,\,\,\,\,\,\,\,if\,\,V_{n} <V_{n}^{\min } \\ \left( {V_{n} -V_{n}^{\max } } \right)^{2}\,\,\,\,\,\,\,\,\,\,\,if\,\,V_{n} >V_{n}^{\max } \\ \end{array}} \right. $$
(47)
$$ P_{s} =\left\{ {\begin{array}{l} 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,P_{Gk}^{\min } \le P_{Gk} \le P_{Gk}^{\max } \\ \left( {P_{Gk}^{\min } -P_{Gk} } \right)^{2}\,\,\,\,\,\,\,\,if\,P_{Gk} <P_{Gk}^{\min } \\ \left( {P_{Gk} -P_{Gk}^{\max } } \right)^{2}\,\,\,\,\,\,\,if\,P_{Gk} >P_{Gk}^{\max } \\ \end{array}} \right. $$
(48)

6.2 Simulation results and discussion

The CM problem of power transmission system is solved by the proposed QOCALO algorithm and the simulation results are compared with the results reported in [75]. The simulations are carried out in MATLAB with the same system configuration as in the previous cases. The number of antlions is considered to be 40 in line with [75], the value of j r is set to 0.6 and the value of K is equal to the dimension of the considered problem.

6.2.1 Case study 1: Modified IEEE 30-bus system

The modified IEEE 30-bus system has six generator busses, twenty-four load busses and forty-one transmission lines. The total real and reactive power loads for this test system are 283.4 MW and 126.2 MVAr, respectively. The system data for the considered test system is taken from [75]. The details of contingencies considered for this case study are provided in Table 16.

Table 16 Details of simulated test cases for CM problem
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In case 1A, congestion has occurred in the lines 1-7 and 7-8 due to the outage of the line 1-2. The real power flows in these congested lines are 147.463 MW and 136.292 MW, respectively, while the maximum allowable limit is 130 MW for both the lines. For secure operation of the system, the line overloads should be alleviated. In the present study, a QOCALO based CM scheme has been adopted to alleviate the line overloads. The simulation results for this case have been provided in Table 17. From Table 17, it may be observed that the proposed QOCALO provides the minimum CM cost compared to other metaheuristic methods. The amount of real power rescheduling (ΔP G ) as yielded by the proposed method for different generators has also been provided in Table 17. A positive value of ΔP G suggests for increment in generation while a negative value suggests for decrement. The power flow in the previously congested lines comes under its specified limit after the application of the proposed QOCALO based CM technique (refer Table 16).

Table 17 Comparative results of CM problem for modified IEEE 30-bus system offered by different algorithms
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In case 1B, congestion is created in the system by considering the outage of the line 1-7 along with the increment in real and reactive power loads by 50%. As a result of this, the lines 1-2, 2-8 and 2-9 get overloaded as the power flows in these lines exceed their specified limit (refer Table 16). The QOCALO based CM technique is applied for relieving these line overloads and the simulation results are tabulated in Table 17. From Table 17, it may be noted that the proposed method yields the minimum value of CM cost as compared to other reported algorithms. Also, it may be observed that the power flows in the previously congested lines have been reduced and come under their specified limit (refer Table 16). Hence, it may be concluded that the proposed method successfully clears out the line congestions in both the cases and also outperforms other algorithms in terms of solution quality.

The comparative convergence profiles of the proposed QOCALO and the basic ALO algorithms in solving the CM problem for the modified IEEE 30-bus system are portrayed in Fig. 11.

Fig. 11
figure 11

Comparative convergence profiles of fitness function value for a case 1A and b case 1B pertaining to CM problem

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6.2.2 Case study 2: Modified IEEE 57-bus system

The modified IEEE 57-bus system has seven generator busses, fifty load busses and eighty transmission lines. The total real and reactive power loads are 1250.8 MW and 336 MVAr, respectively. The details of the system data may be found in [75]. The two different cases have been considered for this case study and the details are provided in Table 16.

In case 2A, the congestion is created in the system by considering the power flow limits of the lines 5-6 and 6-12 as 175 MW and 35 MW, respectively, instead of their original power flow limits of 200 MW and 50 MW. Thus, the lines 5-6 and 6-12 get overloaded at the base condition, as the power flow in those lines exceeds the new specified limit (refer Table 16). To relieve this line overload, the QOCALO based CM technique is applied and the simulation results are provided in Table 18. From Table 18, it may be noted that the proposed QOCALO yields the minimum cost of CM as compared to other reported algorithms. The power flows in the previously congested lines also come under their specified limit after successful implementation of the QOCALO based CM method (refer Table 16).

Table 18 Comparative results of CM problem for modified IEEE 57-bus system offered by different algorithms
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In case 2B, the congestion is created in the system by reducing the capacity of the line 2-3 from 85 MW to 20 MW. As a result of this, the line 2-3 becomes overloaded as the power flow in this line is 37.048 MW which is greater than its new specified limit (refer Table 16). The simulation results obtained after applying the proposed QOCALO based CM technique are provided in Table 18. From Table 18, it may be noted that the proposed QOCALO yields the minimum value of CM cost as compared to other reported algorithms. Also, the power flow in the previously congested line is reduced to be settled under its specified limit (refer Table 16). Thus, the proposed QOCALO algorithm outperforms its original counterpart in solving the transmission CM problem in terms of quality of solution.

The comparative convergence profiles of the proposed QOCALO and the basic ALO algorithms in solving the CM problem for the modified IEEE 57-bus system are portrayed in Fig. 12.

Fig. 12
figure 12

Comparative convergence profiles of fitness function value for a case 2A and b case 2B pertaining to CM problem

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7 Solution of pressure vessel design problem

The pressure vessel design problem is a well known design optimization problem and is considered as the third real world engineering optimization application of the proposed QOCALO algorithm. The objective of this problem is to find out an optimal design of the pressure vessel with the least fabrication cost. The details of the problem may be found in [11]. This design optimization problem can be formulated as follows:

$$\text{Consider } X=\left[ {x_{1} \,x_{2} \,x_{3} \,x_{4}}\right] $$
$$\begin{array}{@{}rcl@{}} \text{Minimize } f\left( X \right)&=&0.6224x_{1} x_{3} x_{4} + 1.7781x_{2} {x_{3}^{2}}\\ &&+ 3.1661{x_{1}^{2}} x_{4} + 19.84{x_{1}^{2}} x_{3} \end{array} $$
(49)
$$ \text{Subject to } \left.{\begin{array}{l} g_{1} \left( X \right)=-x_{1} + 0.0193x_{3} \le 0 \\ g_{2} \left( X \right)=-x_{2} + 0.0095x_{3} \le 0 \\ g_{3} \left( X \right)=-\pi {x_{3}^{2}} x_{4} -\frac{4}{3}\pi {x_{3}^{3}} + 1296000\le 0 \\ g_{4} \left( X \right)=x_{4} -240\le 0 \\ \end{array}} \right\} $$
(50)

The variable range may be specified as in (51) [11].

$$ \left. {\begin{array}{l} 1\times 0.0625\le x_{1} ,x_{2} \le 99\times 0.0625 \\ 10\le x_{3} ,x_{4} \le 200 \\ \end{array}} \right\} $$
(51)

This problem is solved using the proposed QOCALO algorithm and the simulation results are compared to other metaheuristic algorithms, such as, MFO, GSA, PSO, GA, DE and ACO. The number of antlions is considered to be 40, the value of j r is set to 0.6 and the value of K is equal to the dimension of the considered problem. The comparative simulation results are reported in Table 19. From Table 19, it may be easily observed that the proposed QOCALO yields the minimum fabrication cost as compared to other reported algorithms. Hence, it may be concluded that the proposed QOCALO outperforms its original counterpart and other popular swarm and evolution based algorithms in solving this highly constrained engineering design optimization problem.

Table 19 Comparative results of pressure vessel design problem offered by different algorithms
Full size table

8 Conclusion and scope of future work

In the present article, a novel QOCALO algorithm is proposed for solving global optimization problems. The proposed algorithm is designed by combining the QOBL based search and the CLS based search strategies. The inclusion of QOBL based search technique improves the quality of the initial solution set and accelerates the search process. It greatly improves the convergence rate of the proposed QOCALO algorithm over its original counterpart. Also, the QOBL diversifies the search direction by exploring powerful regions of the search space which offers better exploration. On the other hand, the CLS guides local search around the global best solution which is the most promising region in the search space. It provides better exploitation and the quality of solution is also improved. It may be observed from the simulation results that the proposed QOCALO algorithm performs much better in optimizing all the three types of mathematical benchmark test functions (viz. unimodal, multimodal and composite) compared to its original counterpart. Hence, it may be concluded that a fair balance between exploration and exploitation is maintained in the proposed algorithm which makes it robust. The proposed QOCALO has also been utilized for solving three different real world engineering optimization problems. The simulation results show the superiority of the proposed QOCALO algorithm over the original ALO and the other reported algorithms. Hence, it may be concluded that the proposed algorithm is capable of solving real world engineering optimization problems with unknown search spaces. Thus, the proposed QOCALO may become a promising optimizing tool and may be further utilized to solve different non-linear optimization problems in different fields of science and engineering studies.