Energy Management Strategy for Optimal Sizing and Siting of PVDG-BES Systems under Fixed and Intermittent Load Consumption Profile

1. Introduction

In recent years, the deployment of photovoltaic (PV) systems has gained a significant attention in dealing with energy balance between production and consumption due to its positive effects in terms of pollution reduction, free availability and electricity generation. Therefore, there has been an increasing interest in the integration of distributed generation (DG) in power grids because of the advances in renewable energy technologies. Nevertheless, unpredictable and intermittent PV system production levels can threaten power grid safety and quality. Indeed, the penetration of PV based DG (PVDG) units in distribution networks converts them from passive to active networks, which leads to bringing about bidirectional power flows in the network, i.e., from the source to consumption points and vice-versa. This phenomenon, called reverse power flow, happens when the power output of DG becomes more than the local load due to the randomness characteristic of the renewable resources and loads. The aforementioned problem progressively increases the utility grid’s damage potential [1], such as voltage and frequency fluctuations, high power losses, and so on. Actually, this issue can be solved by adding energy storage systems (ESSs) such as battery energy storage (BES) systems to the point of connection. To clarify, the storage device is used to store energy when the power produced by the PV system is higher than the load needed and to provide energy when the power generated by these systems is not enough to cover all load demand. In other words, the excess power will be injected directly into these devices instead of sending it to the grid, and the lack of power will be taken from the BES instead of extracting it from the grid. As a result, the electrical network intervention will be eliminated and reduced, and the potential grid damage will decrease further. However, the importance of the BES system proportionally increases with the PV integration level. Thus, choosing the optimal capacity and allocation of the PV-BES system should be carefully considered to enhance and improve the power grid’s efficiency and reliability. As an elucidation, the determination of the best position and size of the PVDG-BES system could be obtained through grid power loss reduction, system reliability improvement and system voltage profile amelioration. Nonetheless, due to the intermittent and unpredictable characteristics of renewable resources and the significant variation of the daily load consumption profiles, it is difficult to assess the behavior and performance of a PVDG system. Subsequently, it is not easy to effectively design and size it. Accordingly, it is essential to emphasize that using an optimal strategy to correctly manage the system power flow between all grid-connected PVDG-BES system components is considered an efficient step to enhance system reliability and security in terms of power loss reduction and load satisfaction guarantee. In fact, all these important points are covered in this research study.

1.1. Literature Review

Reducing power losses and improving the voltage profile are becoming priorities for distribution network operators [2]. Indeed, minimizing real power losses can lead to a reduction in power distribution costs, and it also allows us to increase the transit margins of power flows and improve the voltage profile. From the literature review, it has been shown that the appropriate location and setting of DGs can be solutions to these issues [2,3]. Advances in PV technology have made PV systems more competitive with other renewable energy resources. Within this context, various initiatives and approaches have been carried out and developed for the problem of optimal siting and sizing of the PVDG system [4,5]. Generally, this problem has been described as an optimization problem where active power losses and bus voltage deviation have been frequently suggested to be objective functions [5]. Various optimization techniques ranging from classical techniques [6,7] to meta-heuristic techniques [8,9] have been suggested for solving this kind of problem. In [6], a classical method called the Lagrangian-based approach has been applied for optimal locations of DGs where stability and economic aspects have been considered. Gautam and Mithulananthan [7] have developed two methodologies based on the Lagrangian multiplier associated with the problem equality constraints describing the real power balance at each node. Moreover, two different objective functions based on maximizing the profit of DG owners and social welfare, subject to various operating constraints, have been included in the problem formulation.

Recently, meta-heuristic techniques such as genetic algorithm (GA) [8], particle swarm optimization (PSO) [9], artificial bee colony (ABC) [10], firefly algorithm (FA) [11], simulated annealing (SA) [12], ant colony optimization (ACO) [13] and gravitational search algorithm (GSA) [14] have gained popularity, among other conventional techniques for solving problems linked to the optimal placement and sizing of renewable energy-based DG and ESSs in distribution networks. This is due to their flexibility in dealing with complex nonlinear problems compared to traditional methods.

In recent years, GAs have gained significant attention when it comes to dealing with complex and non-convex optimization problems that include continuous and/or discrete decision variables. GA is inspired by the biological evolution principle. It starts by generating the initial population of individuals, also called candidates, in a random manner. Then, at each generation, genetic operators, which are selection, crossover and mutation, are applied to update solutions according to certain probabilities [15]. Contrary to other population-based optimization methods, the GA is considered more straightforward and less complex [8]. Thanks to their capacity to produce efficient solutions for complex optimization problems, GAs have been applied in various engineering domains, notably, for the optimal location and sizing of DGs and ESSs. This problem is addressed from three important points of view regarding technical [16,17,18], economical [19,20] and environmental [21] aspects. In [16], a new method has been depicted to optimally allocate a PV system with the aim of reducing grid power losses. To reach the required outcome, the GA has been used as an optimization technique, and three load consumption levels have been implemented. The main goal of this approach is to analyze the effect of intermittent load profiles on power loss minimization. However, this technique has the limitation of ignoring a BES system introduction. Indeed, annually, monthly and hourly simulations have been carried out in [17] to effectively allocate a grid-connected PV system in a distribution network given the minimum of power losses using the GA. In fact, the introduction of a BES unit has not been considered in this study. Besides, authors in [18] investigated the grid total losses issue with the aim of obtaining the optimal PV-BESS size and location in the standard IEEE 33 nodes test feeder using the GA. The main objective of this technique is to discuss and highlight the importance of introducing a BES system for power loss reduction. However, neglecting the power flow management strategy implementation is considered the major drawback of this approach. In [19], a novel approach has been proposed with the aim of optimally sizing and allocating a PV system feeding a specific load, which is a faculty in Karabuk University. This method has the benefit of enhancing the system reliability by reducing the total net present cost (TNPC), which is based on the loss of power supply probability (LPSP) parameter variation. To reach these targets, the GA has been used, taking into account the LPSP constraint satisfaction. Nevertheless, no significant attention has been offered to energy management strategy application in this study. Besides, authors in [20] have investigated a new approach in order to determine the best allocation and size of PV-BES systems. The cost of energy (COE) minimization subjected to financial and LPSP constraints is considered as the main contribution of this study. In fact, this research work has been tested by a proposed energy management strategy. Last but not least, authors in [21] have proposed a new techno-economic approach to optimally size and introduce a hybrid (PV-BES-wind) system using the GA algorithm. The major benefit of this study is considered to be based on financial and environmental concerns to enhance system reliability. This amelioration is established by reducing the COE, the net present value (NPV) parameter and CO2 emissions. However, both the load demand satisfaction by determining the optimal LPSP parameter and technical condition implementation are not taken into consideration in this research work.

Nevertheless, various research studies have proved that despite its positive conception characteristics and its advantages, GA presents some downsides in terms of speed convergence, slowness and local solution determination. This can be explained by the long search time due to solution repetition during the system simulation process [22]. To tackle the aforementioned downsides of the classical GA, researchers proposed and introduced novel approaches combining GA with other sophisticated algorithms in such a way that the advantages of the involved algorithms are kept [22]. As a result, the shortcomings of the classic GA can be alleviated by benefiting from the advantages of other algorithms.

Several combinations of GA with other meta-heuristic algorithms have been presented to deal with hybrid renewable energy sizing and energy management problems. In [23], a hybrid GA combined with PSO (GA-PSO) has been applied and presented to optimally size an off-grid building with a hybrid PV-wind-BES system. In this paper, a comparison study between GA-PSO, multi-objective PSO technique and HOMER software has been discussed and analyzed. Simulation results have presented the strengths and weaknesses of each technique. A new method for sizing a hybrid BES (HBES) has been proposed in [24], with the aim of smoothing the fluctuation of wind output power. The main objective of this research work has been to determine the energy capacities and the capacity of HBES to reduce the system total cost (STC) per day using an energy management strategy. To reach this goal, a hybrid parallel (PSO-GA) has been presented as an optimization algorithm. Simulation results have proven the efficiency of the proposed method. Moreover, authors in [25] have investigated an artificial neural network ANN combined with GA (ANN-GA) to optimally size a hybrid renewable system located in Spain. Based on this new technique, the proposed system has been technically and economically evaluated. In [26], a new approach has been proposed and developed for renewable sources integration with electric vehicle presence in order to satisfy loads demand and ameliorate voltage profile where a fuzzy GA (F-GA) technique has been implemented to solve this optimization problem. The obtained results have shown the effectiveness of the F-GA in terms of active and reactive power loss reduction.

Unfortunately, these stochastic hybrid techniques have been criticized by different studies due to the random characteristics of their phases. For instance, in GA, various random numbers are involved in the initialization, crossover and mutation phases; therefore, it is not possible to predict the next system state using the previous one. Moreover, the number of repeated solutions and iterations will be increased during the system process, and an optimal local solution can be provided.

Recently, several research studies have modified the original stochastic techniques by including chaos theory in their optimization processes, such as chaotic PSO [27], chaotic GSA [28], chaotic differential evolution [29], chaotic honey badger algorithm [30], chaotic ABC [31] and chaotic GA [32]. As a matter of fact, chaos systems are defined as deterministic nonlinear dynamic systems that are sensitive to initial conditions and their operating parameters [30]. In chaos systems, state variables are bounded and determined without repetition. These proprieties of chaos have encouraged researchers to incorporate them in the stochastic optimization algorithms, hoping to improve the solution quality and avoid the convergence of these algorithms into local optima. For instance, in [32], a chaotic crossover has been integrated into the classical GA instead of using where candidate solutions have been represented by Gray codes. Another new chaotic GA has been proposed in [33] where chaotic sequences generated by chaotic maps have been used to generate the initial population instead of the random process. However, genetic operators have been kept the same as in the traditional GA.

Against this background, a new chaos-based GA is proposed in this study. In this proposed optimization technique, different chaotic maps are combined with the classical GA hoping to escape from the local optima and enhance its convergence characteristics. Chaotic maps are used in the population initialization, crossover and mutation phases of the GA to generate chaotic sequences that will be employed instead of random numbers. The effectiveness of the suggested optimization method is firstly tested using various unimodal and multimodal benchmark functions; then, it is applied for solving the optimal sizing and emplacement of PVDG-BES systems in a distribution network.

1.2. Contributions

As can be seen, the first finding of the mentioned literature review is that, although the existence of a large number of research works related to the sizing of PVDG systems area, limited efforts have been presented to optimally size and place DG system with BES integration in the medium distribution network. Adding to that, obtaining the best sizing and placement of the DG system with an energy storage device, under intermittent load consumption profile and variable atmospheric conditions, has not been covered in depth. Moreover, not enough details have been presented yet to discuss and analyze the use of optimal power flow management strategy, for the determination of the optimal capacity and size of PVDG-BES system in electrical distribution networks.

The main objective of this research paper is to find the optimal allocation and size of a PVDG system coupled with a storage unit based on an optimal power flow management strategy. This strategy leads to obtaining better results in terms of total power loss reduction and grid quality improvement. In fact, when the power flow between all system components is correctly managed, grid intervention will be reduced, and subsequently, reverse power flow will be avoided.

Thus, the novelty and the contribution of this work, compared to other techniques discussed and presented in the aforementioned literature review, can be summarized as follows.

• A novel methodology to find the optimal site and size of PVDG-BES systems is proposed. The proposed methodology is based on an optimal power flow management strategy. In this strategy, the optimal placement and sizing of PVDG-BES are represented by an optimization problem where total real power losses are considered as the objective function. Several equality and inequality constraints, such as power flow equations, node voltage limits, PVDG capacity limits and LPSP constraints are taken into account. Since power flow equations are nonlinear, a Newton–Raphson-based method is adopted in this study for solving the power flow problem.

• Due to the problem’s complexity, a new meta-heuristic optimization method hybridizing the classical genetic algorithm with different chaotic maps is developed and applied for solving this optimization problem. In the proposed optimization method, chaotic maps are used instead of random numbers involved in the population initialization and genetic operators. The performance of this chaos-based optimization method is verified by using various unimodal and multimodal benchmark functions.

• The applicability and robustness of the proposed strategy are also validated using the IEEE 14-bus distribution network under fixed and intermittent load profiles.

1.3. Paper Organization

The remainder of this paper is structured as follows. Section 2 presents the problem formulation. The proposed optimization strategy is developed in Section 3. Then, Section 4 exhibits the numerical validation with different benchmark functions. The simulation results and discussion are presented in Section 5. The conclusion is presented in Section 6.

2. Problem Formulation

Generally, the optimal sizing and placement of PVDG in electrical distribution networks including ESSs are considered an optimization problem that should be solved in order to realize grid stability and power loss reduction. Nevertheless, to solve any optimization problem, an objective function is important to be used and formulated to reach the feasibility of solutions [21]. To do this, this section is devoted to developing a mathematical description for the problem of optimal sizing and placement of PVDG in presence of BES (PVDG-BES). To provide a better understanding, Figure 1 presents a simplified diagram of the studied PVDG-BES system connected to a distribution line connecting nodes i and i + 1.

For more precision, PV and BES units are located at the same node to avoid power losses in case of the BES charge state [22]. The PVDG-BES optimal location and capacity will be determined, taking into account the total power losses in the system lines. Note that the best locations of PVDG-BES systems are selected from the set of load bus numbers. In fact, the integration of DG systems as close as possible to energy consumption will reduce energy distribution losses [34].

In Figure 1, $V_i$ and $V_{i+1}$ are magnitudes of voltages at nodes i and i + 1, respectively. $R$ and $X$ are the resistance and reactance of the line (i, i + 1), respectively. $P_{DG\_i}$ and $P_{Load\_i}$ are power produced by PVDG system and that consumed by loads, respectively. $P_{bat\_char}$ is the power stored in the BES during the charge state. $P_{bat\_dis}$ is the power taken from the BES during the discharge state. $P_{losses\_i}$ is power losses in the line (i, i + 1).

2.1. Mathematical Model of PVDG System

This research study uses the Conergy PowerPlus PV panel type [35], which is taken from the desalination solar station of “Ben Guardan”, located in southeast Tunisia. It is composed of 60 polycrystalline cells, where the values of their parameters are determined for the standard test conditions (STC) corresponding to a temperature $T=25°\mathrm{C}$ and an irradiation $G=1000\mathrm{W}/{\mathrm{m}}^2$ [35]. The value of the current generated by this system can be expressed as follows [36].

(1)$I=N_p\times I_{ph}-N_p\times I_{ph}-I_0\left(exp\left(q\times \frac{V_{pv}+I{R}_s}{N_{s}aKT}\right)-1\right)$

(2)$I_{ph}=G\left(I_{sc}+K_I\left(T-T_r\right)\right)$

where $N_p$ and $N_s$ represent the parallel and series number of PV module cells, respectively. $I_{ph}$ and $V_{pv}$ represent cells current and voltage, respectively. $I_0$ is the diode saturation current; $R_s$ is the series cell resistance in ohms (?). T and G are the temperature in kelvin and the irradiation in W/m², respectively. a and $K_I$ represent the ideal factor of PV system and the temperature coefficient of the cell short current, respectively. K is the Boltzman constant ($K=1.38\times {10}^{-23}\mathrm{J}/\mathrm{K}$), and q is the elementary electron charge ($q=1.6\times {10}^{-19}$ C).

According to Equations (1) and (2), it can be noted that the current injected by the PV system depends on weather conditions such as temperature and irradiation. This implies that when these conditions change, the output power of the PV system $P_{DG\_i}$ will change. This power depends on the current I and the voltage $V_{DG}$ produced by the DG system.

(3)$P_{DG\_i}=I\times V_{DG}$

2.2. Mathematical Model of BES

Due to its ease of installation and low maintenance cost compared to other storage units such as lithium-ion (Li-ion) batteries [37], the lead acid (LA) battery is used in this paper. This element can play an important role in the power loss issue [38,39], and it is characterized by an important parameter known as SOC (State of Charge), which can describe and present its level of charge [40].

The equivalent circuit of the CIEMAT (Centro de Investigaciones Energéticas, Mediombientalesy Technologicas) LA battery is presented in Figure 2. The expressions of the battery voltage V_bat is given in Equation (4) [40].

(4)$V\_bat={(n}_b\times E\_bat)+{(n}_b\times R\_bat\times I\_bat)$

Due to the difficulty of directly calculating this parameter, SOC estimation is considered a challenge for many researchers aiming to improve battery efficiency by increasing its lifetime and determining the optimal battery energy management. Several methods are used for SOC estimation. In this paper, the ampere-hour integral method is used because it is considered the most common method for determining the SOC [41]. It is based on knowing the initial state of charge ${SOC}_0$ at time t = 0 and integrating the battery current during a corresponding time. The expression of this parameter is given in Equation (5).

(5)$SOC={SOC}_0-?\frac{I\_bat}{C\_bat}dt$

In fact, $n_b$ is the number of series cell batteries, and $E\_bat$ and $R\_bat$ are the electromotive force and the internal resistance of one battery cell, respectively. Then, $I\_bat$ is the battery current. $C\_bat$ is the battery storage capacity.

2.3. Objective Function

The PVDG system is connected to bus i through a DC/DC boost converter and a DC/AC inverter. The first converter is used to increase the power produced by this system and to control the charge and discharge of the ESS. The second converter is utilized in order to convert the DC power output of the PVDG system into AC form. To feed the point of consumption by resources, the power is transferred through distribution lines. However, the more the power demand increases, the more the line heat dissipation due to its resistance R increases. The total real power losses in distribution lines can be calculated as follows [42].

(6)$P_{loss}={\displaystyle \underset{i=1}{\overset{N}{?}}}{R}_i\times {I_i}^2={\displaystyle \underset{i=1}{\overset{N}{?}}}{R}_i\times \frac{{P_i}^2+{Q_i}^2}{{V_i}^2}$

where N is the number of line sections. Note that the ith line section is the line connecting nodes i and i + 1. $I_i$ is the current in the ith line. $V_i$ represents the voltage at the ith node. $R_i$ is the resistance of the ith line. $P_i$ and $Q_i$ are active and reactive powers at the ith node and can be expressed as follows [43].

(7)$P_i={?}_{j=1}^m\left|V_i\right|\left|V_j\right|\left|Y_{N_{ij}}\right|\cos ?\left(a_i-a_j-{?}_{ij}\right),i=1\dots m$

(8)$Q_i={?}_{j=1}^m\left|V_i\right|\left|V_j\right|\left|Y_{N_{ij}}\right|\mathrm{s}\mathrm{i}\mathrm{n}?(a_i-a_j-{?}_{ij}),i=1\dots m$

where m is the nodes number. $\left|V_i\right|$ and $a_i$ are the magnitude and phase angle of $V_i$. $\left|Y_{N_{ij}}\right|$ and ${?}_{ij}$ are the magnitude and angle of the (i, j) element of the bus admittance matrix $Y_N$, respectively.

In fact, the real power losses in distribution systems can be minimized by optimal sizing and placement of DGs. Within this context, the main goal of this research study is to find the optimal location and size of a grid-connected PVDG-BES system according to the loss of power supply probability (LPSP). To do this, the problem is converted into an optimization problem aiming to minimize the system real power losses given in Equation (5), subject to various operating constraints, such as power flow constraint, PVDG capacity, BES capacity and LPSP parameter constraint.

• Power flow equality constraints

When only a PVDG system is integrated into the power grid, at the sending end of branch i, the real power flow equations can be formulated as follows [44].

(9)$P_i+P_{DG\_i}=P_{loss\_i}+P_{load\_i}$

where $P_{DG\_i}$ and $P_{load\_i}$ are the power output of the PVDG system and load at the ith node, respectively.

However, when the storage energy unit is added, the expression of the power balance constraint is given in (10) in the case of battery discharging state, and it is shown in (11) in the case of battery charging state [45].

(10)$P_i{+P}_{DG\_i}+P_{bat\_dis}=P_{loss\_i}+P_{load\_i}$

(11)$P_i+P_{DG\_i}-P_{bat\_char}=P_{loss\_i}+P_{load\_i}$

• Bus voltages constraints

The voltage at all network nodes should be bounded by the permissible bounds, as described in Equation (12) [46].

(12)$V_{min}=V_i=V_{max}$

where $V_{min}$ and $V_{max}$ are minimum and maximum of bus voltages, respectively. In this study, $V_{min}=0.9\mathrm{p}\mathrm{u}$ and $V_{max}=1.1\mathrm{p}\mathrm{u}$.

• PVDG power constraints

To improve grid quality and security, the PVDG system should produce an amount of power that can be bounded by the acceptable bounds, which are given in Equation (13).

(13)$P_{DG,min}=P_{DG}=P_{DG,max}$

where $P_{DG,min}$ and $P_{DG,max}$ are minimum and maximum limits of the power output of the PVDG system, respectively. In this study, $P_{DG,min}=1$ MW and $P_{DG,max}=5$ MW.

• BES power constraints

For the same reason as the PVDG system, the power that the BES unit can store should be bounded, as is described in the following equation.

(14)$P_{BES,min}=P_{BES}=P_{BES,max}$

where $P_{BES,min}$ and $P_{BES,max}$ are minimum and maximum of $P_{BES}$, respectively. These limits are fixed as follows.

$P_{BES,min}=1$ MW.

$P_{BES,max}=4$ MW.

• LPSP parameter constraint

LPSP describes the system ability to satisfy and to cover load demand for a daily 24 h, and it is used to test the system reliability [19]. The expression of this parameter is given in Equation (15).

(15)$LPSP=\frac{{\displaystyle {?}_{ti=1}^T}\left(P_{load}\left(t_i\right)-\left(P_{DG}\left(t_i\right)+P_{bat-dis}\left(t_i\right)\right)\right)}{{\displaystyle {?}_{i=1}^T}{P}_{load}\left(t_i\right)}$

where $T=24$ and $ti?\left\{1,\dots ,24\right\}$.

Note that $LPSP=0$ means that the system under study is reliable and loads are satisfied, whilst $LPSP=1$ implies that the system is not reliable and cannot cover all loads demand [19].

(16)$0<LPSP<1$

3. Proposed Optimization Strategy

3.1. Classical Genetic Algorithm

In this study, an improved version of GA is implemented for best location and sizing of PVDG-BES system in a distribution network. GA is considered as the most powerful algorithm and technique that deals with discrete and continuous optimization problems [47]. GA, which was discovered by John Holland [8] in 1975, mimics the Darwinian concept describing the evolution process of a biological organism’s population over generations. It is defined as a classical randomized search algorithm that uses random numbers during the optimization process. Indeed, the main principle of GA method is based on four main steps, which are population initialization, selection, crossover and mutation. It begins with the random generation of the initial population, which is formed by a group of individuals named chromosomes, as given in Equation (17). In order to reduce the CPU time, real-coded numbers can be used for optimization variables.

(17)$X_j^i=X_j^{min}+a\left(X_j^{max}-X_j^{min}\right),i=1,?,\left|POP\right|\mathrm{a}\mathrm{n}\mathrm{d}j=1,?,D$

where $a?\left(\mathrm{0,1}\right)$ is a uniformly distributed random number. $\left|POP\right|$ is the population size, and D is the number of optimization variables. $X^i=\left[X_1^i,X_2^i,?,X_D^i\right]$ is the ith solution. $X_j^{max}$ and $X_j^{min}$ are the upper and lower limits of the jth optimization variable.

Each individual from the population is evaluated by using a fitness function that represents the objective function. In GA, individuals evolve during successive generations by applying genetic operators. To do this, at each generation, multiple couples of individuals (solutions) are selected from the population in a stochastic manner according to their fitness values. Then, a crossover operator is applied for each couple of individuals $X^i$ and $X^j$ to generate two new individuals, ${\tilde{X}}^i$ and ${\tilde{X}}^j$, called offspring solutions. Note that a non-uniform arithmetic crossover with a probability $p_{cr}$ is adopted in this study as follows [48].

(18)$\left\{\begin{array}{c}{\tilde{X}}^i=\mu X^i+\left(1-\mu \right)X^j\\ {\tilde{X}}^j=\mu X^j+\left(1-\mu \right)X^i\end{array}\right.$

where $\mu ?\left(\mathrm{0,1}\right)$ is a uniformly distributed random number.

After that, the new solutions are randomly perturbed in the mutation phase according to a pre-specified mutation probability, $p_{mu}$. In the proposed technique, the non-uniform mutation is applied [48]. Therefore, each variable $X_k^i$ of the solution vector $X^i$ can be updated according to the following equation.

(19)${\tilde{X}}_k^i=\left\{\begin{array}{c}{X}_k^i+?\left(g,X_k^{max}-X_k^i\right),\mathrm{if}t=0\\ X_k^i-?\left(g,X_k^i-X_k^{min}\right),\mathrm{if}t=1\end{array}\right.$

where $?\left(g,z\right)=z\left(1-r^{{\left(1-\frac{g}{G_{max}}\right)}^{ß}}\right)$. $t$ is a random binary number. $r?\left(\mathrm{0,1}\right)$ is a uniform random number. $ß$ is a shape parameter. g and $G_{max}$ are the current generation and the maximum number of generations.

After updating solutions using genetic operators, a greedy selection between parent solutions and offspring solutions is performed to select which solutions will survive in the next population. To be more specific, the pseudo-code of the greedy selection between two solutions $X^i$ and ${\tilde{X}}^i$ can be as presented in Algorithm 1. Moreover, Figure 3 shows the flowchart of the classical GA.

Algorithm 1 Pseudo-Code of the Greedy Selection Mechanism

if $fitness\left(X^i\right)<fitness\left({\tilde{X}}^i\right)$ $X_{new}=X^i$ else $X_{new}={\tilde{X}}^i$ End

3.2. Chaos Theory

It is worth noting that despite its benefits in terms of conception simplicity and dealing with complex problems, the GA method presents some limitations, such as slow convergence to the optimal solution and local solution determination. Like other stochastic techniques, classical GA has been criticized in various research studies due to its premature convergence and global search inability. One of the main causes of these drawbacks is linked to the use of random numbers in the phases of these algorithms. Various studies and analyses of chaos theory-based optimization techniques have shown that chaos features, such as ergodicity and sensibility to small changes in initial conditions, can effectively improve the effectiveness of stochastic techniques and avoid their convergence into local optima [49].

A chaotic system is a deterministic discrete-time dynamical system that is mathematically described by chaotic sequences $\left\{x_i\right\}$ as follows.

(20)$x_{i+1}=f\left(x_i\right);0<x_i<1,i=\mathrm{1,2},?$

where $x_i$ is the state variable and i is the iteration number.

In this regard, four chaotic maps are adopted for improving GA performance and escape from local optima. In this new chaotic-based GA (NCGA), the chaotic maps, which are the logistic map, tent map, sine map and Henon map, are embedded with the initialization, crossover and mutation phases of the classical GA. These maps are presented below.

3.2.1. Logistic Map

This chaotic map is derived from the differential equation describing the population growth. It has gained significant attention from researchers due to its simplicity and ease of implementation [27]. It can be described as follows.

(21)$x_{i+1}={a\times x}_i\left(1-x_i\right),i=0,1,2,\dots$

where $x_i$ is defined as the ith value of the logistic map. The initial condition $x_0?\left(\mathrm{0,1}\right)$ should be different from 0, 0.25, 0.5, 0.75 and 1 to have better results [27]. a is the bifurcation parameter and should be between 3.569945672 and 4 to reach a chaotic behavior and generate better solutions [27].

3.2.2. Tent Map

A tent chaotic map is distinguished by its excellent ergodicity and convergence rate among various employed chaotic maps [50]. It can be employed instead of pseudo-random number generators to ameliorate performances of optimization algorithms in terms of jumping out of the local optimum. The tent map can be described as follows.

(22)$x_{i+1}=\left\{\begin{array}{l}?x_i,if0=x_i=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ ?\left(1-x_i\right),if\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.=x_i=1\end{array}\right.,i=\mathrm{0,1},2,\dots$

where $x_i$ is the ith value of the state variable of the tent map. $?$ is the control parameter of the tent map, which should be more than in the interval $\left(\mathrm{1,2}\right)$ [50].

3.2.3. Sine Map

A chaotic sine map is a unimodal map. It is inspired by the sine function, and it can be mathematically written as follows [51].

(23)$X_{i+1}=?sin?\left(p{X}_i\right),i=\mathrm{0,1},2,\dots$

where $x_i?\left(\mathrm{0,1}\right)$ is the ith value of the state variable of the sine map and $??\left(\mathrm{0,1}\right)$ is its control parameter. Generally, $?$ is selected to be 1 to have a chaotic behavior.

3.2.4. Henon Map

A henon map is a 2-dimensional chaotic map that is considered as a numerical series composed by chaotic maps, described as given below [30].

(24)$\left\{\begin{array}{c}{x}_{i+1}=\end{array}\right.$