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

Abstract

Advances in PV technology have given rise to the increasing integration of PV-based distributed generation (PVDG) systems into distribution systems to mitigate the dependence on one power source and alleviate the global warming caused by traditional power plants. However, high power output coming from intermittent PVDG can create reverse power flow, which can cause an increase in system power losses and a distortion in the voltage profile. Therefore, the appropriate placement and sizing of a PVDG coupled with an energy storage system (ESS) to stock power during off-peak hours and to inject it during peak hours are necessary. Within this context, a new methodology based on an optimal power flow management strategy for the optimal allocation and sizing of PVDG systems coupled with battery energy storage (PVDG-BES) systems is proposed in this paper. To do this, this problem is formulated as an optimization problem where total real power losses are considered as the objective function. Thereafter, a new optimization technique combining a genetic algorithm with various chaotic maps is used to find the optimal PVDG-BES placement and size. To test the robustness and applicability of the proposed methodology, various benchmark functions and the IEEE 14-bus distribution network under fixed and intermittent load profiles are used. The simulation results prove that obtaining the optimal size and placement of the PVDG-BES system based on an optimal energy management strategy (EMS) presents better performance in terms of power losses reduction and voltage profile amelioration. In fact, the total system losses are reduced by 20.14% when EMS is applied compared with the case before integrating PVDG-BES.

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].

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

(1)

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

(2)

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 (Ω). 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.

$$P_{DG\_i}=I\times V_{DG}$$

(3)

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].

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

(4)

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).

$$SOC={SOC}_0-\int \frac{I\_bat}{C\_bat}dt$$

(5)

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].

$$P_{loss}={\displaystyle \sum _{i=1}^N}{R}_i\times {I_i}^2={\displaystyle \sum _{i=1}^N}{R}_i\times \frac{{P_i}^2+{Q_i}^2}{{V_i}^2}$$

(6)

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].

$$P_i={\sum }_{j=1}^m\left|V_i\right|\left|V_j\right|\left|Y_{N_{ij}}\right|\cos \left({\alpha }_i-{\alpha }_j-{\theta }_{ij}\right),i=1\dots m$$

(7)

$$Q_i={\sum }_{j=1}^m\left|V_i\right|\left|V_j\right|\left|Y_{N_{ij}}\right|\mathrm{s}\mathrm{i}\mathrm{n}({\alpha }_i-{\alpha }_j-{\theta }_{ij}), i=1\dots m$$

(8)

where m is the nodes number. $\left|V_i\right|$ and ${\alpha }_i$ are the magnitude and phase angle of $V_i$. $\left|Y_{N_{ij}}\right|$ and ${\theta }_{ij}$ 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].

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

(9)

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].

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

(10)

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

(11)

• Bus voltages constraints

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

$$V_{min}\le V_i\le V_{max}$$

(12)

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).

$$P_{DG,min}\le P_{DG}\le P_{DG,max}$$

(13)

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.

$$P_{BES,min}\le P_{BES}\le P_{BES,max}$$

(14)

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).

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

(15)

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

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].

$$0<LPSP<1$$

(16)

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.

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

(17)

where $\alpha \in \left(\mathrm{0,1}\right)$ is a uniformly distributed random number. $\left|POP\right|$ is the population size, and D is the number of optimization variables. $X^i=\left[X_1^i,X_2^i,\cdots ,X_D^i\right]$ 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].

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

(18)

where $\mu \in \left(\mathrm{0,1}\right)$ 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.

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

(19)

where $∆\left(g,z\right)=z\left(1-r^{{\left(1-\frac{g}{G_{max}}\right)}^{\beta }}\right)$. $\tau$ is a random binary number. $r\in \left(\mathrm{0,1}\right)$ is a uniform random number. $\beta$ 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.

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

(20)

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.

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

(21)

where $x_i$ is defined as the ith value of the logistic map. The initial condition $x_0\in \left(\mathrm{0,1}\right)$ 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.

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

(22)

where $x_i$ is the ith value of the state variable of the tent map. $\vartheta$ 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].

$$X_{i+1}=\rho sin\left(\pi X_i\right),i=\mathrm{0,1},2,\dots$$

(23)

where $x_i\in \left(\mathrm{0,1}\right)$ is the ith value of the state variable of the sine map and $\rho ϵ\left(\mathrm{0,1}\right)$ is its control parameter. Generally, $\rho$ 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].

$$\left\{\begin{array}{c}{x}_{i+1}=1-a{x}_i^2+y_i\\ y_{i+1}=b{x}_i\end{array}\right., i=0,1,2,\dots$$

(24)

where $x_i$ and $y_i$ are the values of the state variables at the ith iteration. a and b are control parameters. Typical values of a and b are 1.4 and 0.3, respectively. For these values of a and b, the Henon map can generate better solutions and reach the system chaotic behavior.

The Henon map can also be used as a one-dimensional map, which can be defined as follows.

$$x_{i+1}=1-a{x}_i^2+{bx}_i, i=0,1,2,\dots$$

(25)

3.3. The Proposed Chaotic Genetic Algorithm

As mentioned above, the proposed chaotic genetic algorithm is based on introducing chaotic functions in the population initialization, crossover and mutation phases of the classical GA. These chaotic maps are used to generate chaotic numbers instead of random numbers to avoid premature convergence and ensure convergence into the global optimum solution. To do this, four chaotic maps, which are the logistic map (Lo), tent map (Te), sine map (Si) and Henon map (He), are used. In order to study the impact of these chaotic maps on the solutions quality, eight combinations of the aforementioned maps are employed and combined with the GA. Therefore, eight chaotic based GAs have been investigated. Acronyms and description of these algorithms are presented in

Table 1. List of studied chaotic-based GAs.

Method	Population Initialization	Crossover	Mutation
LoGA	Logistic map	Logistic map	Logistic map
TeGA	Tent map	Tent map	Tent map
SiGA	Sine map	Sine map	Sine map
HeGA	Henon map	Henon map	Henon map
(Si-He-Lo)GA	Sine map	Henon map	Logistic map
(Te-He-Lo)GA	Tent map	Henon map	Logistic map
(Si-He-Te)GA	Sine map	Henon map	Tent map
(He-Si-Lo)GA	Henon map	Sine map	Logistic map

.

The pseudo-code of the proposed algorithm is given in Algorithm 2.

Algorithm 2 Pseudo-Code of the New Chaotic Based GA

(1) Initialization of GA parameters ($\left|POP\right|$,$G_{max}$,$p_{cr}$, $p_{mu}$, D, $X_i^{min}$ and $X_i^{max}$). (2) Initialization of chaotic maps parameters. (3) Generation of chaotic sequences $\left\{x_i\right\}$ for each chaotic map (Lo, Te, Si, He). (4) Initialization of the population POP.      Select a chaotic map, for initialization, among the suggested maps (Lo, Te, Si, He).      for $k=0$ to $\left|POP\right|$ do          for $j=1$ to D do               $k=k+1$               $\alpha =x_k$               Apply Equation (16)          end for          Evaluate the fitness value of the chromosome $X^i$      end for (5) $g=0$ (6) while $g<G_{max}$ do      $g=g+1$      $PO{P}_{new}=\varnothing$ (6.1)      while $\left|PO{P}_{new}\right|<\left|POP\right|$ do (6.1.1)                Application of crossover operator                            Select a chaotic map, for crossover operator, among the suggested maps.                            Select two chromosomes X and Y from the population.                            $k=k+1$                            $\mu =x_k$                            Generate two new solutions $\tilde{X}$ and $\tilde{Y}$ from X and Y according to Equation (17) (6.1.2)                Application of mutation operator for $\tilde{X}$ and $\tilde{Y}$                            $k=k+1$                            Select a chaotic map, for mutation operator, among the suggested maps.                            $r=x_k$                            Apply mutation operator on $\tilde{X}$ according to Equation (18).                            $k=k+1$                            $r=x_k$                            Apply mutation operator on $\tilde{Y}$ according to Equation (18). (6.1.3)                Greedy selection                            $X_{new}=greedy\_selection\left(X,\tilde{X}\right)$                            $Y_{new}=greedy\_selection\left(Y,\tilde{Y}\right)$                            $PO{P}_{new}=PO{P}_{new}\bigcup \left\{X_{new};Y_{new}\right\}$ (6.2)      end while (7) end while

3.4. Energy Management Strategy (EMS) Based Method

Unpredictable and uncontrollable PVDG-BES system integration on the distribution network leads to a reverse power flow presence and, subsequently, to a grid power loss increase. As a matter of fact, grid power losses minimization depends on not only the allocation and the size of the PVDG-BES system in the distribution network but also on the optimal EMS used to correctly manage the power flow between the different system components (PVDG-BES system, loads and the grid).

For further elucidation, when the power generated by PVDG exceeds the load demand, the BES unit begins to charge and store the excess of power instead of injecting it into the grid. However, when the power consumed by loads exceeds the power produced by the PVDG system in the point of integration, batteries will provide power to loads to satisfy their needs instead of taking power from the electrical grid. Hence, the integration of BES leads to the grid intervention reduction and, subsequently, to a reverse power flow decrease. Therefore, this energy storage unit is an essential element and plays an important role in the best position and capacity determination.

This section explains and highlights the important role of using an EMS on power loss reduction and grid voltage improvement. The flowchart of the proposed power flow management strategy is given in Figure 4. This strategy essentially aims to ensure the continuity of loads satisfaction during all the daily hours. To do this, it is important to set the maximum (${SOC}_{max}$) and minimum (${SOC}_{min}$) bounds of the batteries’ SOC in the first step. In this study, ${SOC}_{min}$ and ${SOC}_{max}$ are chosen to be 10% and 90%, respectively.

The proposed EMS is explained as follows. If the power produced by PVDG system exceeds the load demands and BES units are not completely charged $(10\%\le SOC\le 90\%)$ and three modes can be triggered as follows.

• Mode 1: When the excess of power (dp) is higher than $P_{bat},$ BES system will be fully charged (BES charge mode), and an amount of power will be sent to the bus where the PVDG-BES is connected (injection to the grid).
• Mode 2: In this mode, the excess of power dp is not higher than $P_{bat}$, which means that this excess of power will be stored in the BES system (BES charge mode).
• Mode 3: This mode occurs when all batteries are completely charged $(SOC=90\mathrm{\%})$. As a result, the excess power will be sent directly to the point of connection.

On the other hand, when the power consumed by loads connected to the optimal position exceeds the power generated by the PVDG system (dp < 0). Then, BES units are not completely charged $(10\%\le SOC\le 90\%)$. Three other modes can be triggered (modes 4, 5 and 6) as follows.

• Mode 4: This mode is applied when the power stored in batteries $P_{BES\_i}$ is not enough to compensate the load needs. As a result, the grid intervenes to feed loads (extraction from the grid mode).
• Mode 5: When the power stored in batteries satisfy all load demands, BES units will send power to all the points of consumption connected to the optimal allocations of PVDG-BES systems (BES discharge mode).
• Mode 6: This mode occurs when BES systems are completely discharged $(SOC=10\mathrm{\%})$; the grid will intervene to satisfy all load needs (extraction from the grid).

Consequently, the power stored in batteries $P_{bat\_char}$ and that which is provided by them $P_{bat\_dis}$, as well as the power injected and extracted from the grid $P_{grid}$, will be calculated and provided to satisfy the aforementioned constraints presented in Equations (10) and (11); then, the determination of the optimal placement and size of the PVDG will be reached and realized, as shown in the proposed flowchart in Figure 5.

4. Numerical Validation Using Benchmark Functions

In order to evaluate the efficiencies and performances of the proposed hybrid optimization methods combining GA with various chaotic maps, four benchmark functions, which are detailed in

Table 2. Benchmark functions.

Function	Formula	Search Space	Optimum
Sphere	$F_1\left(x\right)={\displaystyle \sum _1^D}{x}_i^2$	[−5.12, 5.12]	0
Step	$F_2\left(x\right)={\displaystyle \sum _1^D}{\displaystyle {ix}_i^2}$	[−100, 100]	0
Shubert	$F_3\left(x,y\right)={\displaystyle \sum _{i=1}^D}i\times \cos \left(\left(i+1\right)\times x_i+i\right)\times {\displaystyle \sum _{i=1}^D}i\times \cos \left(\left(i+1\right)\times y_i+i\right)$	[−10, 10]	−186.731
SumSquare	$F_4\left(x\right)={\displaystyle \sum _1^D}{(x_i-0.5)}^2$	[−10, 10]	0

, are used.

For a fair comparison between all suggested methods, which were presented in Table 1, the same values of the GA parameters are used for all optimization methods (see

Table 3. GA parameters.

GA Parameters	Value
Maximum Iteration Population size Crossover Percentage (%) Mutation percentage (%) D	500 200 70 20 10

).

The eight hybrid optimization techniques are all implemented using MATLAB software on a personal computer with a CPU Intel Core i7 @ 1.99 GHz and 8 GB of random-access memory (RAM).

In order to investigate the contribution of the employed chaotic maps and select the best optimization method, these algorithms are divided into two groups. The first group (Group 1) includes the algorithms combining GA with one chaotic map, which are LoGA, HeGA, TeGA and SiGA. The second one (Group 2) contains the algorithms combining GA with hybrid chaotic maps, which are (Si-He-Lo)GA, (Te-He-Lo)GA, (Si-He-Te)GA and(He-Si-Lo)GA. The convergence characteristics of the chaotic based algorithms when applied on the above-mentioned benchmark functions are shown in Figure 6, Figure 7, Figure 8 and Figure 9. From these figures, it can be clearly seen that LoGA provides better performance in terms of optimal solution convergence when compared with other optimization techniques of the first group. However, the hybrid (Te-Si-Lo)GA technique shows better performance than all investigated techniques whether from first group or from second group and it provides the lowest minimum values for all studied benchmark functions.

In order to further evaluate and test the effectiveness of each algorithm, statistical results, including the best minimum, mean and standard deviation (SD) values, are calculated through thirty runs of the classical GA, as well as the eight chaotic-based GA techniques. The obtained statistical results are tabulated in

Table 4. Statistical results.

		GA	LoGA	TeGA	SiGA	HeGA	(Te-Si-Lo) GA	(Te-He-Lo) GA	(Si-He-Te) GA	(Si-He-Lo)GA
F1	Best	8.24 × 10−11	1.4 × 10−14	1.63 × 10−14	4.1 × 10−13	4.2 × 10−13	9.89 × 10−15	4.5 × 10−13	1.09 × 10−12	1.2 × 10−11
	Mean	2.42 × 10−9	4.9 × 10−10	3.17 × 10−10	3.7 × 10−9	2.9 × 10−9	2.8 × 10−10	1.63 × 10−9	1.8 × 10−9	2.7 × 10−9
	SD	1.85 × 10−10	1.2 × 10−10	8.58 × 10−15	8.97 × 10−10	9.24 × 10−10	9.76 × 10−11	3.96 × 10−10	4.8 × 10−10	6.7 × 10−10
F2	Best	8.3 × 10−13	5.8 × 10−14	9.6 × 10−13	5.1 × 10−12	8.3 × 10−13	1.1 × 10−15	7.2 × 10−14	2.72 × 10−13	4.9 × 10−12
	Mean	2.54 × 10−9	1.94 × 10−8	1.9 × 10−8	2.9 × 10−8	2.5 × 10−9	2.3 × 10−9	7.2 × 10−10	1.2 × 10−8	2.7 × 10−8
	SD	8.35 × 10−10	6.3 × 10−9	6.3 × 10−9	1.1 × 10−8	8.35 × 10−10	7.23 × 10−10	1.8 × 10−10	4.2 × 10−9	1.08 × 10−8
F3	Best	−186.7291	−186.7305	−186.7302	−186.7291	−186.7291	−186.7307	−186.7305	−186.7301	−186.7304
	Mean	−186.7017	−186.7063	−186.6956	−186.6878	−186.7077	−186.6990	−186.7088	−186.712	−186.6966
	SD	0.0044	0.0035	0.0055	0.0068	0.0036	4.40 × 10−3	0.0035	0.0029	0.0055
F4	Best	2.1 × 10−6	2.5 × 10−13	3.9 × 10−13	8.9 × 10−13	3.7 × 10−12	1.96 × 10−13	5 × 10−13	1.47 × 10−12	1.5 × 10−13
	Mean	2.26 × 10−4	1.8 × 10−8	2.8 × 10−8	2.6 × 10−9	2.03 × 10−8	4.5 × 10−9	7.7 × 10−9	1.3 × 10−9	5.3 × 10−8
	SD	4.17 × 10−5	9.6 × 10−9	1.5 × 10−10	6 × 10−10	9 × 10−9	1.6 × 10−9	2.2 × 10−9	5.2 × 10−10	3 × 10−8

.

According to Table 4, it is obvious that the (Te-Si-Lo)GA technique provides better results and appears the most robust technique compared with the other algorithms. Subsequently, the hybrid (Te-Si-Lo)GA technique will be adopted and applied for solving the optimal placement and sizing of PVDG-BES in a power distribution network.

5. Simulation Results and Discussions

In order to further demonstrate the robustness and applicability of the suggested (Te-Si-Lo)GA based strategy for the optimal placement and sizing of a PVDG-BES system in a power network, the standard IEEE 14-bus distribution network is used. The network buses are numbered from 1 to 14. The selection of this optimization technique is justified by the experimental results found in the previous section. The (Te-Si-Lo)GA is used to find the best location and size of PVDG-BES system considering power loss minimization and several operating constraints such as power flow equalities, bus voltage limits, PVDG capacity, BES capacity and LPSP parameter constraints. The obtained results are compared with those of the classical GA, as well as the LoGA, which provided the best results compared with the other chaos-based algorithms of the first group. The values of the parameters of these optimization algorithms are the same as those used in the previous section.

The single line diagram of the studied system is given in Figure 10. It consists of nine loads, two generators and three synchronous condensers. The initial total active power loss is 13.5 MW; however, the initial total reactive power loss is about 56.9 Kvar [9]. This system is adopted due to the availability of its parameters and data [9].

In this section, two cases are studied to investigate the effect of the integration level of PVDG-BES systems and the power demand variation on the optimal solutions and the total power losses. These two cases are described as follows.

• Case 1: In this case, the problem is solved for one, two and three PVDG-BES systems. However, the power demand is fixed at its rated value during 24 h.
• Case 2: This case is based on taking into consideration the variable atmospheric conditions during 24 h. In fact, the power produced by the PV system depends on daily weather conditions, namely temperature and irradiation. Moreover, the power demand of each load is considered variable and follows a specific profile of power consumption. To do this, an optimal power flow management strategy is used. In this study, the PV-BES characteristics are presented in Table 5.

Table 5. PV and BES parameters, per one system.

PV System	Values	BES System	Value
$P_{pv-min}$	1 MW	$P_{bat-min}$	1 MW
$P_{pv-max}$ $P_{pv-max}$ $N_S$ $N_P$	5 MW 245 W 10 60	$P_{bat-max}$ $P_{bat\_max}$ ${SOC}_{min}$ ${SOC}_{max}$ ${SOC}_0$	4 MW 200 W 10% 90% 50%

Table 5. PV and BES parameters, per one system.

PV System	Values	BES System	Value
$P_{pv-min}$	1 MW	$P_{bat-min}$	1 MW
$P_{pv-max}$ $P_{pv-max}$ $N_S$ $N_P$	5 MW 245 W 10 60	$P_{bat-max}$ $P_{bat\_max}$ ${SOC}_{min}$ ${SOC}_{max}$ ${SOC}_0$	4 MW 200 W 10% 90% 50%

5.1. Case 1: Fixed PV Generated Power and Fixed Load Demand Using Variable PV Penetration Level

Figure 11 presents the convergence characteristics of the objective function corresponding to the total power losses for three PV penetration levels. From Figure 12, it is obvious that the best minimum power loss corresponds to the case when three PVDG systems are integrated into the power grid. However, the worst result corresponds to the case when only one PVDG system is used. In fact, the minimum values of power losses are 0.12981 pu, 0.1252 pu and 0.1208 pu for one PVDG (1 PV), two PVDGs (2 PV) and three PVDGs (3 PV) integration, respectively. Therefore, it can be concluded that the more the number of PVDG systems increases, the more the total grid power loss decreases. The results obtained after the convergence of the proposed optimization algorithm, including optimal location, optimal capacity of PVDGs and minimum power losses in the grid, are tabulated in Table 3. The percentages of loss reduction compared with those which were determined before the integration of PVDGs (0 PVDGs) are also calculated and tabulated in

Table 6. Optimal results for various integration levels of PVDGs.

	0 PV	1 PV	2 PV		3 PV
	-	-	1st PV	2nd PV	1st PV	2nd PV	3rd PV
Optimal position	-	3	6	5	1	2	7
Optimal capacity (MW)	-	4	5	5	3	4	4
Minimum Ploss (pu)	0.1348	0.1289	0.1252		0.1208
Reduction percentage (%)	-	4	7		10.3

.

The integration of PVDG-BES is also studied in this section. To avoid a power loss increase, BES and PVDG are connected at the same node. Figure 12 presents the convergence characteristics of the same objective function after adding various PVDG-BES systems. The aim of this case is to explain the importance of adding the storage system in reducing power losses. This figure shows that the more the number of PVDG-BES systems increases, the more the total grid power loss decreases. In fact, the total power loss is 0.1043 pu for one PVDG-BES (1 PVDG-BES) system, 0.0973 pu for two PVDG-BESs (2PVDG-BES) and 0.0898 pu for three PVDG-BESs (3PVDG-BES). Moreover, it is worth noting that the grid connected PVDG-BES systems presents better loss reduction performance compared with PVDG systems. For instance, the minimum power loss value is 0.0898 pu when three PVDG-BESs are employed; however, it is around 0.1208 pu when three PVDGs are used. The optimal results corresponding to various integration levels of the PVDG-BESs are tabulated in

Table 7. Optimal results for various penetration levels of PVDG-BESs.

	1PVDG-BES			2PVDG-BES					3PVDG-BES
	PV	BES		1st PV	1st BES	2nd PV	2nd BES		1st PV	1st BES	2nd PV	2nd BES	3rd PV	3rd BES
Optimal position	4			1		4			6		5		4
Optimal capacity (MW)	4.5	3		2.5	4.4	1.4	4		1.2	2	1.5	2	1.5	2
Minimum Ploss (pu)	0.1043			0.0973					0.0898
Reduction percentage (%)	22.6			28					33.3

.

Figure 13 shows the voltage profile of all load buses after incorporating the optimal PVDG and PVDG-BES systems under various integration levels. According to this figure, it is clear that the voltage magnitudes at these buses are within their permissible limits given in inequality (11). In this study, the voltage magnitude limits are chosen to be 0.9 pu and 1.1 pu in order to maintain bus voltage stability, avoid under/over voltage and thus enhance grid security.

Figure 14 presents the active power at each system bus after integrating three levels of PVDG-BES systems.

It is worth noting that the integration of PVDG-BES system in the optimal placement with the optimal size can affect the injected active power at each system bus. Taking, for example, the 14th bus, the active power injected in this point is reduced from 2.35 pu to 2.3 pu after the integration of a 1PVDG-BES system. Then, it decreases to 2.25 pu after the integration of 2 PVDG-BES systems. That proves the positive effect of the integration of a PVDG-BES system in the distribution network in terms of grid quality and security amelioration.

5.2. Case 2: Variable Atmospheric Conditions and Intermittent Load Demand with a Fixed PV Penetration Level

In this case, the power consumed by each load follows a specific consumption profile that is given in Figure 15 [8]. The main objective of this subsection is to study and analyze the effects of taking into account the variability of the atmospheric conditions and load consumption profile on the optimal placement and sizing of PVDG-BES systems. To do this, the proposed strategy described in Figure 6 is employed to obtain the optimal size and position of the aforementioned system in such a way that the load demand is covered at each hour. The optimal power flow management strategy presented in Figure 5 is used to correctly manage the power transferred between all system components.

The load consumption profile, temperature (T) and irradiation (G) variations during 24H are presented in Figure 15, Figure 16 and Figure 17, respectively. It is noteworthy that the output power of the PVDG system depends on T and G, and it can be calculated according to Equations (1)–(3).

The convergence characteristics of the proposed energy management strategy based on three optimization algorithms which are the classic GA, the chaotic LoGA and the hybrid (Si-Te-Lo)GA are shown in Figure 18. In this case, a grid-connected PVDG-BES system is investigated.

According to Figure 19, the GA-, LoGA- and (Te-Si-Lo)GA-based optimization methods reached the optimal solutions after 479, 377 and 252 iterations, respectively. The total real power loss is 0.0953 pu for GA, 0.0943 pu for LoGA and 0.0919 pu for (Te-Si-Lo)GA. In conclusion, the hybrid (Te-Si-Lo)GA presents better performance than the two other optimization algorithms in terms of speed convergence and global minimum power losses tracking. These results validate the experimental results performed on the studied benchmark function.

The statistical results for 30 runs using the three optimization algorithms are listed in

Table 8. Statistical results for Case 2.

	GA	LoGA	(Te-Si-Lo)GA
Best minimum Ploss (pu)	0.0953	0.0943	0.0919
Reduction percentage (%)	29.3	30	31.8
Optimal PVDG capacity (MW)	1.5	2.7	5
Optimal BES capacity (MW)	1.5	4.4	4
Optimal location	4	1	1
Mean	0.950	0.094	0.091
SD	3.4 × 10−4	1.1 × 10−3	3.18 × 10−6
Number of iterations	479	377	252
LPSP	0.0899	0.0812	0.03315

.

In Table 8, the standard deviation (SD) is used to further evaluate the convergence performance of the employed optimization algorithms. Indeed, a low value of SD implies that all optimal solutions provided after several runs of the optimization algorithm are close to the best optimal solution. However, the high value of the SD means that all optimal solutions given by the proposed optimization technique are spread out over the search space, and therefore, that technique cannot be efficient for solving such problems. Besides, a LPSP parameter is used to verify loads satisfaction during the day (24 h). In fact, lower LPSP ensures continuous load power supply and proves system reliability. From Table 8, it can be seen clearly that the (Te-Si-Lo)GA technique has the best results, such as the best minimum real power losses and minimum SD and LPSP parameters. Therefore, the combination of the classical GA with tent, sine and logistic maps has improved its robustness and the optimal solution quality.

Figure 19 shows the active powers injected at the grid buses after integrating the optimal PV-BES system obtained using GA, LoGA and (Te-S-Lo)GA for Case 2. From this figure, it can be clearly seen that the active power injected at each system bus is affected by the integration of the PVDG-BES system. However, it is obvious that the values of these powers are more sensitive to the PVDG-BES system optimized using the hybrid (Te-Si-Lo)GA. For instance, the active power injected at bus 14 is reduced to 1.85 pu by using the LoGA; however, it is decreased to 1.8 pu when the (Te-Si-Lo)GA is used. Thus, the hybrid (Te-Si-Lo)GA based PVDG-BES system presents better performance compared with the other techniques in terms of injected active power reduction.

Figure 20 shows the voltage profile of all buses obtained after integrating the PVDG-BES system optimized with GA, LoGA and hybrid (Te-Si-Lo)GA. This figure shows that whatever the algorithm, the voltage magnitudes are within their limits, as given in inequality (11).

In order to assess and analyze the applicability of the power flow management strategy, the daily variations in pu of the PVDG ($P_{pv}$) due to the variability of weather conditions, the power output of the BES ($P_{BES}$), the power injected by the main grid ($P_{grid}$) and the power demand ($P_{load}$) are plotted in Figure 21. It is to be noted that the location and size of the PVDG-BES system are found using the hybrid (Te-Si-Lo)GA algorithm (see Table 8).

As shown in Figure 21, it is clear that the PVDG system cannot produce power from 8 p.m. to 7 a.m. 1, due to the irradiation insufficiency (G = 0 W/m²). During this period, an amount of power is consumed by loads. Hence, the BES unit will compensate this demand and will provide power to the loads ($P_{BES}>0$). This means that loads will take power from batteries instead of extracting it from the electrical grid.

Besides, from 7 a.m. to midday and from 5 p.m. to 8 p.m., the PVDG system generates power ($P_{pv}>0$). However, this production is insufficient to cover the load demand ($P_{pv}<P_{load}$), therefore, BES unit intervenes to compensate this lack of power ($P_{BES}>0$) to the loads. As results, the point of consumption where the PVDG-BES is integrated receives power from the PVDG system and BES unit instead of taking it from the grid. Nevertheless, when the PVDG-BES system cannot satisfy all load needs, the grid intervenes to provide power, especially from 6 p.m. to 9 p.m. After that period, only the BES unit continues to produce the required power until 7 a.m. At midday, the power output of the PVDG system becomes greater than the load demand ($P_{pv}>P_{load}$), and this situation continues until 5 p.m. Therefore, the PVDG system is the only producer of the energy in that time period, and the excess of power will be absorbed by the battery ($P_{BES}<0$) instead of being injected into the grid.

Figure 22 presents the total power loss during 24 h both before adding the PVDG-BES systems (initial state) and then after integrating the DG system coupled with the energy storage system with and without using the EMS. It can be noted that the aim of this step is to analyze the effect of using the EMS on grid power loss.

It is clear that integrating PVDG-BES system after applying the optimal EMS reduces grid power losses. In comparison with the case before integration of PVDG-BESs, the reduction percentage of losses when using only PVDG-BES units is about 10.86%, and it is about 20.14% when EMS is applied. In fact, optimal management of power flows between power grid components such as PVDG, BES and loads can reduce the intervention of the main grid, and consequently, it helps avoiding the reverse power flow, which is one of the main causes of grid power losses and safety problems. That proves the efficiency of power flow management strategy in power loss reduction and voltage profile improvement.

6. Conclusions

Reducing real power losses and improving voltage profile in a distribution network are the main objectives of this study. To deal with these issues, a new strategy for optimally determining the allocation and the size of PVDG-BES systems connected to a distribution network is presented. In fact, the appropriate, optimal placement and sizing of PVDG-BES systems can help prevent reverse power flow by stocking the surplus energy during periods of high production of PV systems or low power demand and then injecting it into the grid during periods of lower production or higher power demand. Since the optimal allocation of the PV-BES system should be close to the consumption point, an optimal power flow management strategy based on LPSP parameter determination is proposed and used in this paper to optimally manage the power flow between all system components, including the PV system, BES unit, load and grid. A new hybrid chaotic GA is established for tackling the problem of power loss reduction and the optimal placement and sizing of PVDG-BES systems. Various combinations of chaotic maps, including logistic, tent, sine and Henon maps, are embedded into the classical GA phases to escape the drawbacks associated with the utilization of random numbers. The performances of these combinations are evaluated and compared using different benchmark functions. The best combination, which corresponds to (Te-Si-Lo)GA, is adopted for solving the studied problem.

To further test the applicability and effectiveness of the proposed strategy, the IEEE 14-bus distribution network under fixed and variable load demands is used. From simulation results, it is found that the suggested optimization algorithm (Te-Si-Lo)GA provided better results compared with the other combinations. Moreover, it is shown that determining the optimal placement and size of PVDG-BES system based on an optimal EMS can manage the power flow between the system components and give better results in terms of power losses reduction and reverse power flow elimination.

As a part of future research, the proposed methodology can be tested on a practical case, which is located in southeast Tunisia in the presence of various energy storage technologies and other types of DG units such as wind turbines. Additionally, as a part of an expansion of this study, other optimization algorithms will be investigated and proposed for optimal placement and size determination of DG systems by considering both economic and technical concerns.