Evolutionary Approach for DISCO Profit Maximization by Optimal Planning of Distributed Generators and Energy Storage Systems in Active Distribution Networks

Abstract

Distribution companies (DISCOs) aim to maximize their annual profits by performing the optimal planning of distributed generators (DGs) or energy storage systems (ESSs) in the deregulated electricity markets. Some previous studies have focused on the simultaneous planning of DGs and ESSs for DISCO profit maximization but have rarely considered the reactive powers of DGs and ESSs. In addition, the optimization methods used for solving this problem are either traditional or outdated, which may not yield superior results. To address these issues, this paper simultaneously performs the optimal planning of DGs and ESSs in distribution networks for DISCO profit maximization. The utilized model not only takes into account the revenues of trading active and reactive powers but also addresses the active and reactive powers of DGs and ESSs. To solve the optimization problem, a new hybrid evolutionary algorithm (EA) called the oppositional social engineering differential evolution with Lévy flights (OSEDE/LFs) is proposed. The OSEDE/LFs is applied to optimize the planning model using the 30-Bus and IEEE 69-Bus networks as test systems. The results of the two case studies are compared with several other EAs. The results confirm the significance of the planning model in achieving higher profits and demonstrate the effectiveness of the proposed approach when compared with other EAs.

1. Introduction

In recent years, research on active distribution networks (ADNs) has rapidly improved, with a focus on numerous applications that integrate the latest technologies into such systems [1]. Distributed generators (DGs) are among the best technologies to integrate into ADNs due to their high availability, cost-effectiveness, efficiency, and overall advantages [2]. Several benefits can be achieved by installing DGs in ADNs, such as reliability improvement, loss reduction, voltage improvement, etc. [3]. Recent research studies suggest that even more advantages can be gained from the integrated DGs if energy storage systems (ESSs) are planned alongside them [4]. ESSs are becoming more involved in energy and power system planning due to their ability to provide techno-economic advantages, such as improving power quality, peak shaving, and energy management [5]. However, improper planning of energy sources in any power or energy system can result in the loss of desired benefits and create additional performance problems, which can negatively impact the overall functioning of the system [6]. Therefore, research on the effective coordination between DG and ESS units is of vital importance in the power and energy sector [7,8].

1.1. Literature Review

Various objectives, such as overall performance improvement [9], loss reduction [10], voltage profile improvement [11], and cost minimization [12,13,14], have been adopted to achieve optimal planning of DGs or ESSs in ADNs. However, the role of ADNs has evolved significantly compared to that of conventional (passive) networks. ADNs locally contribute to the power generation process through the integration of DG and ESS units. As a result, rather than minimizing total planning and operating costs, distribution companies (DISCOs) tend to increase their profit from selling energy to end users [15]. In deregulated electricity markets, electricity companies at different levels aim to maximize their revenues. Meanwhile, the system operators focus on maintaining the safe and secure operation of the corresponding networks [16]. In this regard, DISCOs can effectively utilize DG and ESS units to maximize their profit. The planning of DGs and ESSs for DISCO profit maximization was performed in [17], where the model was solved using the particle swarm optimization (PSO) method. The bi-level model proposed in [18] included techno-environmental criteria in the developed objective function to maximize the DISCO profit as well as the electric vehicle (EV) parking lot owner’s profit. A dynamic reliability planning model of DGs in ADNs for DISCO profit maximization under load uncertainty was used in the study presented in [19]. However, most of the previous studies on maximizing DISCO profit in deregulated electricity markets have focused on planning either DG or ESS. Moreover, these studies only included the revenue from active power trading in their models, considering the active power of the integrated generation units.

As mentioned before, it is necessary to optimize the utilization of DGs and ESSs in ADNs to obtain the maximum benefits from their installation. This has been achieved in previous works using different optimization methods. Conventional methods, such as the analytical method proposed in [11], have been used. Linear programming (LP) [20], mixed-integer LP (MILP) [21], and mixed-integer nonlinear programming (MINLP) [13,15] have been applied to build the proposed models. The main advantage of LP and NLP methods is their solvability with a variety of commercial solvers such as MOSEK [13] and IPOPT [15]. Nonetheless, the functions and codes of these solvers are masked and cannot be edited or modified. This reduces flexibility, especially for complex systems. Various concepts from game theory [16] and Karush-Kuhn-Tucker (KKT) conditions [18] have been adopted to solve the developed models. However, conventional methods suffer from increased complexity and inaccurate results in most applications, especially for complex systems. On the other hand, evolutionary algorithms (EAs) have been used to achieve the optimal planning of DGs and ESSs in ADNs. EA techniques provide more flexibility than conventional methods even when system nonlinearity increases. PSO has been applied with various objective functions such as cost minimization [12] and DISCO profit maximization [17]. The authors of [14] proposed an equilibrium optimization (EO) method to determine the optimal locations and sizes of PVs and ESSs by total cost minimization. In [22], the locations and sizes of wind turbines and batteries were optimized by GA based on a techno-economic model. Moreover, some hybrid methods have been developed. In [9], the original artificial bee colony (ABC) algorithm was combined with two other methods: fitness scaling and chaotic methods to avoid being trapped in local optima. The harmony search algorithm (HSA) was integrated with the firefly algorithm (FA) in [19]. This provided a more efficient method in terms of accuracy, convergence, and computation time. Other EA methods have been used in previous papers, such as the artificial ecosystem optimization (AEO) [23], the hybrid arithmetic optimization algorithm-sine-cosine approach (AOA-SCA) [24], and the hybrid gradient-based optimizer with moth–flame algorithm [25]. As observed from the literature, most of the methods do not guarantee obtaining global optima. Hence, it is still necessary to develop effective methods that are particularly compatible with the corresponding problem. This can be achieved by advanced hybrid methods. In this context, various state-of-the-art EAs have been recently proposed but not yet used to solve the optimal planning of DGs and ESSs in ADNs. An example of these EAs is the social engineering optimizer (SEO) algorithm proposed by Fathollahi-Fard et al. [26] in 2018. The original SEO and several modified versions have been applied to various optimization problems in many research areas. However, SEO still has significant potential for unique improvements and applications to other engineering problems.

1.2. Research Motivations and Contributions

The following research motivations and contributions can be highlighted based on the literature reviewed in this paper:

• Existing research on maximizing DISCO profit in deregulated electricity markets separately performs the planning of DGs or ESSs, while planning both technologies simultaneously has been rarely addressed. Therefore, this paper simultaneously investigates the optimal planning of DGs and ESSs in ADNs to maximize the DISCO profit;
• Unlike most previous studies, the model presented in this paper includes both active and reactive power of DGs and ESSs. This can greatly increase the reactive power support and enhance their role as effective ancillary services;
• In deregulated electricity markets, not only active power but also reactive power is traded between the upstream grid and customers. Therefore, the revenues from trading both active and reactive power are included in the model, which has not been properly studied in previous relevant papers;
• The optimization techniques used so far for the DISCO profit maximization are either traditional, software-based, or outdated EAs, which may not provide superior solutions. Therefore, developing hybrid methods to be specifically compatible with the studied model is necessary and worth investigation;
• Moreover, although SEO has been applied to solve various optimization problems, it has not yet been applied to solve the optimal planning of DGs and ESSs in ADNs for DISCO profit maximization;
• However, despite the remarkable results obtained by the SEO in solving the above problems, it may require further improvements to specifically solve the optimal planning of DGs and ESSs in ADNs;
• Hence, this paper proposes a new hybrid approach that combines the optimization mechanisms of SEO, differential evolution (DE), Lévy flights (LFs), and quasi-oppositional-based learning (QOBL). With this developed combination, the global best of SEO is improved by distinctively applying the search mechanisms of DE and LFs. In addition, the QOBL technique is applied to improve the initial population of the proposed algorithm;
• The new algorithm called the oppositional social engineering differential evolution with Lévy flights (OSEDE/LFs) is benchmarked and compared to several state-of-the-art EAs;
• Furthermore, the developed OSEDE/LFs is applied to solve the optimal planning problem of DGs and ESSs in ADNs for DISCO profit maximization. The standard 30-Bus and IEEE 69-Bus distribution networks are used as test systems. The results are obtained for two case studies and compared to various algorithms, including the original SEO.

The rest of this paper is structured as follows: Section 2 demonstrates the integration of DGs and ESSs in distribution networks, while Section 3 presents the mathematical model of the DISCO profit maximization problem. The proposed OSEDE/LFs algorithm is introduced in detail in Section 4, and its application to the standard 30-Bus and IEEE 69-Bus networks is provided in Section 5 with several discussions and comparisons. Finally, Section 6 summarizes the conclusions.

2. Integrating DGs and ESSs into Distribution Networks

DG units are integrated into the distribution network to generate electrical power locally near the end user. Although a DG is mainly considered to be an active power source, it can also produce reactive power [13]. As shown in Figure 1a, for a typical DG system, active power (P kW) is generated and supplied to the grid. At the same time, reactive power (Q kVAr) can be produced or absorbed (bidirectional) [17]. Hence, a DG unit could be used as a reactive power compensator by setting the active power output to “zero” and generating only reactive power. A DG can be modeled as a P-Q bus or P-V bus. However, the P-Q model is more appropriate for distribution network applications, where the DG is considered to be a specified load with fixed values of P and Q [22].

Electrical energy can also be generated locally in distribution networks through the integration of ESSs. Nonetheless, the main difference between DG and ESS in this respect is that both P and Q of the ESS can flow bidirectionally, as depicted in Figure 1b.

As a controllable load or generator, an ESS can absorb P from the network and store it for later use. This power can then be injected back into the network on demand [9]. As shown in Figure 1b, a generic battery ESS unit needs to include a storage device, an inverter, and a transformer [12]. A DC voltage $V_{dc}$ is generated by the storage device, which is then converted into a controllable AC voltage $V_{ac}$ by means of the DC–AC inverter. To deliver this AC voltage to the distribution network, it must be raised by a suitable transformer. At the DC terminal, the storage device only absorbs (in charge mode) or injects (in discharge mode) active DC power. The inverter then connects the storage device to the transformer, where the power is converted to AC power at the AC terminal (the inverter output) [12]. By controlling the voltage magnitude and angle ($V_{ac}\angle {\delta }_{ac}$), P and Q delivered to the network by the transformer can be controlled independently. This provides four possible supply/absorption cases for both P and Q [5,22,27].

3. Mathematical Model of the DISCO Profit Maximization Problem

As discussed above, the optimal planning of DGs and ESSs (batteries) in ADNs is performed in this paper to maximize the DISCO profit subject to several constraints and take into account the active and reactive power of DGs and ESSs. Therefore, the decision variables of this problem are the locations and sizes of DG and ESS units (including their active and reactive power).

3.1. Objective Function

The objective function of the problem, represented by the net profit of the DISCO (${PROFIT}_{DIS}$ (USD/year)), is to be maximized. The year is divided into 4 seasons of 91 days each, and each day is divided into 24 h. The objective function, including the active and reactive power of the system, is given as follows [6,17]:

$$\begin{array}{c}\mathrm{Max} \left({PROFIT}_{DIS}\right),\\ {PROFIT}_{DIS}=\left[{PR}_P+{PR}_Q\right]-\left[{Cost}_{DG}+{Cost}_{ESS}\right],\end{array}$$

(1)

where ${PR}_P$ and ${PR}_Q$ denote the DISCO revenues from selling active and reactive power to the customers (loads) (USD/year) as given in Equations (2) and (3), respectively:

$${PR}_P={\sum }_{s=1}^{4}91\times {\sum }_{t=1}^T\left({\alpha }_{DIS}\times P_{t,s}^{sold}\times {\alpha }_{t,s}^P-P_{t,s}^{purch.}\times {\alpha }_{t,s}^P\right),$$

(2)

$${PR}_Q={\sum }_{s=1}^{4}91\times {\sum }_{t=1}^T\left({\alpha }_{DIS}\times Q_{t,s}^{sold}\times {\alpha }_{t,s}^Q-Q_{t,s}^{purch.}\times {\alpha }_{t,s}^Q\right),$$

(3)

where $P_{t,s}^{sold}$ (kW) and $Q_{t,s}^{sold}$ (kVAr) represent the active and reactive power sold to the customers at time $t$ in season $s$, and $P_{t,s}^{purch.}$ (kW) and $Q_{t,s}^{purch.}$ (kVAr) are the active and reactive power purchased by DISCO from the upstream grid at time $t$ in season $s$, respectively. The prices of active and reactive power at time $t$ in season $s$ are denoted by ${\alpha }_{t,s}^P$ (USD/kWh) and ${\alpha }_{t,s}^Q$ (USD/kVArh), while ${\alpha }_{DIS}$ is a percentage that defines the DISCO profit from this process.

The third term of Equation (1) represents the total cost of DGs in the ADN, which is calculated by:

$${Cost}_{DG}={Cost}_{DG}^{inv.}+{Cost}_{DG}^{O\&M},$$

(4)

where ${Cost}_{DG}^{inv.}$ and ${Cost}_{DG}^{O\&M}$ (USD/year) denote the investment and operation and maintenance costs of DGs. The investment cost of a DG is mainly related to its apparent power. The maintenance cost is related to the performance, service fees, and the price of other equipment, and its value is usually fixed. The operation cost depends on the type of DG and its output power. However, for conventional DG units, the operation and maintenance cost is slightly increased on a seasonal basis, which is considered in this paper [6]. The investment and operation and maintenance costs of DGs are given in Equations (5) and (6), respectively:

$${Cost}_{DG}^{inv.}=\left[{\sum }_{n=1}^{N_{DG}}\left(S_n^{DG}\times C_{inv.}^{DG}\right)\right]\times EAC,$$

(5)

$${Cost}_{DG}^{O\&M}={\sum }_{n=1}^{N_{DG}}{\sum }_{s=1}^{4}91\times {\sum }_{t=1}^T\left(P_{n,t}^{DG}\times {CP}_{O\&M,s}^{DG}+Q_{n,t}^{DG}\times {CQ}_{O\&M,s}^{DG}\right),$$

(6)

where $S_n^{DG}$ represents the apparent power (kVA) of the $nth$ DG, $N_{DG}$ is the number of DG units, while $P_{n,t}^{DG}$ (kWh) and $Q_{n,t}^{DG}$ (kVArh) are the active and reactive power of the $nth$ DG at time $t$. The parameters $C_{inv.}^{DG}$ (USD/kVA), ${CP}_{O\&M,s}^{DG}$ (USD/kWh), and ${CQ}_{O\&M,s}^{DG}$ (USD/kVArh) denote the investment unit cost of DG and seasonal operation and maintenance unit costs of active and reactive power, respectively. To convert the investment cost of DGs to the annual value, the equivalent annual cost (EAC) factor is used in Equation (5), which is calculated as follows:

$$EAC=\frac{d\times {(1+d)}^Y}{{(1+d)}^Y-1},$$

(7)

where $d$ represents the discount rate, and $Y$ (years) is the selected lifetime [5,6,17].

Moreover, the total cost of ESSs in the ADN represented by the fourth term of Equation (1) is given by:

$${Cost}_{ESS}={Cost}_{ESS}^{inv.}+{Cost}_{ESS}^{O\&M},$$

(8)

where ${Cost}_{ESS}^{inv.}$ and ${Cost}_{ESS}^{O\&M}$ (USD/year) are the investment and operation and maintenance costs of ESSs as given in Equations (9) and (10), respectively:

$${Cost}_{ESS}^{inv.}=\left[{\sum }_{n=1}^{N_{ESS}}\left(S_n^{ESS}\times {CS}_{inv.}^{ESS}+E_n^{ESS}\times {CE}_{inv.}^{ESS}\right)\right]\times EAC,$$

(9)

$${Cost}_{ESS}^{O\&M}={\sum }_{n=1}^{N_{ESS}}{\sum }_{s=1}^{4}91\times {\sum }_{t=1}^T\left(S_{n,t}^{ESS}\times C_{O\&M,s}^{ESS}\right).$$

(10)

The investment cost of ESSs should be calculated for batteries and inverters, as shown in Equation (9). Hence, for the total number of ESS units $N_{ESS}$, $S_n^{ESS}$ denotes the apparent power (kVA) of the $nth$ ESS’s inverter and $E_n^{ESS}$ is the $nth$ ESS’s capacity (kWh). To calculate the operation and maintenance cost in Equation (10), the apparent power $S_{n,t}^{ESS}$ (kVAh) of the $nth$ ESS’s inverter at time $t$ is used. The parameters ${CS}_{inv.}^{ESS}$ (USD/kVA) and ${CE}_{inv.}^{ESS}$ (USD/kWh) are the investment unit costs of the inverter and ESS capacity, respectively, while $C_{O\&M,s}^{ESS}$ (USD/kVAh) is the seasonal operation and maintenance unit cost [6].

3.2. Constraints

DISCO profit is maximized subject to several constraints on network power flow and DG and ESS operation. Network power balance is maintained for active and reactive power as defined in Equations (11) and (12), respectively [5,6,17]:

$${\sum }_{b=1}^{N_B}{P}_{b,t}^{IN}={\sum }_{b=1}^{N_B}{P}_{b,t}^{OUT},$$

(11)

$${\sum }_{b=1}^{N_B}{Q}_{b,t}^{IN}={\sum }_{b=1}^{N_B}{Q}_{b,t}^{OUT},$$

(12)

where $P_{b,t}^{IN}$ and $Q_{b,t}^{IN}$ represent the active and reactive power entering bus $b$ at time $t$; $P_{b,t}^{OUT}$ and $Q_{b,t}^{OUT}$ denote the active and reactive power leaving bus $b$ at time $t$, and $N_B$ is the set of network buses.

Network voltages are also constrained as follows [18,28]:

$$V_{min}\le V_{b,t}\le V_{max};b=1,\dots ,N_B,$$

(13)

where $V_b$ denotes the voltage on bus $b$ at time $t$, and $V_{min}$ and $V_{max}$ are the maximum and minimum voltage limits.

The apparent power of the $nth$ DG at time $t$ should not exceed the maximum limit $S_{max}^{DG}$ as described in Equation (14):

$$S_{n,t}^{DG}\le S_{max}^{DG};n=1,\dots ,N_{DG},$$

(14)

where the relationship between $S_{n,t}^{DG}$, $P_{n,t}^{DG}$, and $Q_{n,t}^{DG}$ is defined as follows [13,17,22]:

$$S_{n,t}^{DG}=\sqrt{{\left(P_{n,t}^{DG}\right)}^2+{\left(Q_{n,t}^{DG}\right)}^2}.$$

(15)

The operation of ESS units (batteries) is also constrained, where the apparent power of the $nth$ ESS’s inverter at time $t$ must be maintained within the permissible limit as described below:

$$S_{n,t}^{ESS}\le S_{max}^{ESS};n=1,\dots ,N_{ESS},$$

(16)

where $S_{n,t}^{ESS}$ is calculated based on the inverter’s active power $P_{n,t}^{ESS}$ and reactive power $Q_{n,t}^{ESS}$ at time $t$ as follows [22,27]:

$$S_{n,t}^{ESS}=\sqrt{{\left(P_{n,t}^{ESS}\right)}^2+{\left(Q_{n,t}^{ESS}\right)}^2}.$$

(17)

The $nth$ ESS’s capacity is also restricted, where its value at time $t$ should not exceed the maximum limit $E_{max}^{ESS}$ as given below:

$$E_{n,t}^{ESS}\le E_{max}^{ESS};n=1,\dots ,N_{ESS},$$

(18)

where $E_{n,t}^{ESS}$ is calculated as follows [5,12,28]:

$$E_{n,t}^{ESS}=E_{n,t-1}^{ESS}+P_{n,t-1}^{ESS}\times {\eta }_{ch}-\frac{P_{n,t-1}^{ESS}}{{\eta }_{dis}},$$

(19)

where ${\eta }_{ch}$ and ${\eta }_{dis}$ (%) represent the charge and discharge efficiencies of the inverter.

The initial energy stored in ESS at time $t=0$ should be predefined, which is described as follows:

$$E_{n,t}^{ESS}=E_{n,0}^{ESS}; for t=0.$$

(20)

In this paper, $E_{n,0}^{ESS}$ is taken as 10% of the maximum capacity, i.e., the depth of discharge is 0.9.

Moreover, the energy balance of the ESS should be preserved at the end of the day ($t=T=24$). Hence, the following equation is required:

$$E_{n,t}^{ESS}=E_{n,0}^{ESS}; for t=T.$$

(21)

4. The Proposed Algorithm for DISCO Profit Maximization

In this section, the proposed OSEDE/LFs algorithm is presented in detail. First, the mechanisms and steps of the algorithm are explained. Then, the performance analysis is performed by solving benchmark functions and comparing the results with those obtained by other original algorithms. Finally, the proposed approach is applied to solve the DISCO profit maximization problem.

4.1. Mechanisms of the Proposed Algorithm

The proposed algorithm is a unique hybridization of three mechanisms, namely SEO, DE, and LFs. In addition, QOBL is applied to improve the initial population. On this basis, the developed algorithm is called “oppositional social engineering differential evolution with Lévy flights” (OSEDE/LFs).

4.1.1. Oppositional and Quasi-Oppositional-Based Learning

Several EAs suffer from performance-related drawbacks, such as being trapped around local optima or slow convergence. This especially occurs for complex and high-dimensional optimization problems. Recently, the concept of oppositional-based learning (OBL) has been presented to further enhance the performance of EAs in terms of convergence, local optima avoidance, and computational time [29]. The main advantages of OBL are the simplicity and effectiveness when processing EA-based populations, either in the initialization step or within the main loop [30]. The OBL is structured by comparing the current population with its opposite, as the latter could be closer to the global optimum. Furthermore, the quasi-opposite number has been shown to be even closer to the global optimum than the opposite number [31]. Thus, the quasi-opposite population is calculated by a random probabilistic value and compared to the current population, then the best candidate between them is selected, as shown in Figure 2.

The mathematical definition of quasi-oppositional-based learning (QOBL) is presented as follows [30,31]:

$$X_{ij}\left(It+1\right)=\left\{\begin{array}{c}{C}_j+rand\left(\right)\times \left(C_j-X_{ij}\left(It\right)\right),if\left(X_{ij}\left(It\right)<C_j\right),\\ C_j-rand\left(\right)\times \left(X_{ij}\left(It\right)-C_j\right),if\left(X_{ij}\left(It\right)\ge C_j\right),\end{array}\right.$$

(22)

where $X_{ij}\left(It+1\right)$ denotes the quasi-opposition number in dimension j of solution i at iteration $It+1$, rand() represents a random number, and $C_j$ is the midpoint of the distance between the upper bound (UB) and the lower bound (LB) in dimension j, which is calculated as follows:

$$C_j=\frac{(UB+LB)}{2}.$$

(23)

4.1.2. Social Engineering Optimizer (SEO)

The original SEO was developed as a single-solution metaheuristic algorithm by Fathollahi-Fard et al. [26], inspired by the social interrelationship between individuals, i.e., how a person and his counterpart might interact given their conditions and environment. Based on this principle, each potential solution in SEO is a vector containing an individual and its counterpart. The characteristics of each member in this vector symbolize its social abilities to represent the variables of each solution. To initialize the SEO, two random individuals (representing two initial solutions) are generated and compared. After that, the better solution between them is defined as (attacker), while the other is defined as (defender). Then, the search process is led by the attacker trying to evaluate the defender by its merits through a process called training-retraining. During this random pattern process, the attacker replaces some of its variables with the best merits found in the defender, and then the fitness function is tested once again. The training-retraining mechanism continues the search until the best attacker-defender pair is found, which will guide the searching process by defining a set of trait-exchange experiments that are calculated as follows:

$$N_{tr}=round\left\{\propto ,N_{var}\right\},$$

(24)

where $N_{tr}$ denotes the number of tested traits (merits), $\propto$ is the selected trait’s percent, and $N_{var}$ represents the total number of a person’s traits, i.e., decision variables.

The algorithm then proceeds to the process of spotting an attack by performing four unique techniques, as illustrated in Figure 3. During this process, a key parameter $\beta$ is used as input to improve the exploration of the search space. These four techniques are briefly described as follows [26]:

• The first technique is known as “obtaining”, in which the attacker directly mistreats the defender to effectively obtain its desired traits. Based on that, the defender’s new position is updated using the following equation:

$${Def}_n={Def}_c\times \left(1-\sin \beta \right)\times {rand}_1\left(\mathrm{0,1}\right)+\frac{({Def}_c+Att)}{2}\times \sin \beta \times {rand}_2\left(\mathrm{0,1}\right),$$

(25)

where ${Def}_n$ and ${Def}_c$ denote the new and current positions of the defender, $Att$ represents the attacker’s current position, and ${rand}_{\mathrm{1,2}}\left(\mathrm{0,1}\right)$ are randomly generated numbers;

2.

The second technique, known as “phishing”, involves the attacker faking an attack against the defender. The defender then reacts by moving to a safe place. As a result, two new positions of the defender are generated based on the movement of both the attacker and defender, as described in the following equations:

$${Def}_n^1=Att\times \left(1-\sin \beta \right)\times {rand}_1\left(\mathrm{0,1}\right)+\frac{({Def}_c+Att)}{2}\times \left(1-\sin \beta \right)\times {rand}_2\left(\mathrm{0,1}\right),$$

(26)

$${Def}_n^2={Def}_c\times \left(1-\sin \left(\frac{\pi }{2}-\beta \right)\right)\times {rand}_1\left(\mathrm{0,1}\right)+\frac{({Def}_c+Att)}{2}\times \sin \left(\frac{\pi }{2}-\beta \right)\times {rand}_2\left(\mathrm{0,1}\right),$$

(27)

3.

The next technique is called “diversion theft”, in which the attacker deceives the defender by leading the defender to a desired position (set by the attacker). This is achieved using the average distance between the defender and a scaled amount of the attacker. The defender’s new position is then updated by:

$${Def}_n={Def}_c\times \left(1-\sin \beta \right)\times {rand}_1\left(\mathrm{0,1}\right)+\frac{({Def}_c+Att\times \sin \left(\frac{\pi }{2}-\beta \right)\times {rand}_2\left(\mathrm{0,1}\right))}{2}\times \sin \beta \times {rand}_3\left(\mathrm{0,1}\right),$$

(28)

4.

The final technique is defined as “pretext”, in which the attacker uses some of the defender’s favorite traits as bait to completely guide and defeat the defender. By the end of this process, the defender’s new position is re-updated using a scaled amount of the defender’s current position and the average distance between the weighted attacker and defender as follows:

$$\begin{array}{c}{Def}_n=({Def}_c\times \sin \left(\frac{\pi }{2}-\beta \right)\times {rand}_1\left(\mathrm{0,1}\right))\times \left(1-\sin \beta \times {rand}_2\left(\mathrm{0,1}\right)\right)+\\ \frac{(\left({Def}_c\times \sin \left(\frac{\pi }{2}-\beta \right)\times {rand}_3\left(\mathrm{0,1}\right)\right)+Att)}{2}\times \sin \beta \times {rand}_4\left(\mathrm{0,1}\right),\end{array}$$

(29)

Finally, after the completion of the four techniques, the eventual position of the defender is evaluated by comparing it with its old position, where the best position is selected. Moreover, if the selected defender’s position is better than the attacker’s, it will be defined as the new attacker, while another position of the defender will be randomly generated. The whole procedure will be iteratively repeated until the termination condition is met.

4.1.3. Differential Evolution (DE)

The well-known DE is an effective technique for solving optimization problems in various applications. Mutation, crossover, and greedy selection are the three main mechanisms that define the structure of DE [32]. Based on the mutation process, new mutant solutions $S_M$ are generated in each iteration as follows:

$$S_M=S_B+AP·\left(\left(S_{A1}-S_{A2}\right)+\left(S_{A3}-S_{A4}\right)\right),$$

(30)

where $S_B$ represents the best solution in every iteration while $S_{A1}, S_{A2}, S_{A3},$ and $S_{A4}$ denote arbitrary solutions. The amplifying parameter $AP$ is calculated as follows:

$$AP=\overline{AP}-(It-1)·\left(\overline{AP}-\underset{\_}{AP}\right)/({It}_{max}-1),$$

(31)

where $\underset{\_}{AP}=0, \overline{AP}=2$ are the limits of $AP$; $It$ and ${It}_{max}$ denote the current and maximum number of iterations, respectively.

The obtained mutated solution is then improved by applying the crossover process, evolving a trail solution $S_T$ in the next iteration by:

$$S_T^{It+1}=\left\{\begin{array}{c}{S}_M^{It+1} if R\le R_{CO}\\ S^{It} if R\ge R_{CO}\end{array},\right.$$

(32)

where $R$ represents a parameter within the range [0, 1] and $R_{CO}$ represents the crossover rate.

After that, $S_M^{It+1}$ and $S_T^{It+1}$ are compared by greedy selection to keep the best solution in the population [32].

4.1.4. Lévy Flights (LFs)

The LFs is a powerful search mechanism defined as the mathematical representation of the random walks of the creatures as given in Equation (33), where the new solution is obtained by [33]:

$$S^{It+1}=S^{It}+STEP,$$

(33)

where $STEP$ represents the step size given by:

$$STEP=C·\left(S_{A1}^{It}-S_{A2}^{It}\right)⨁Levy\left(\beta \right)\approx 0.01\frac{x_1}{{\left|x_2\right|}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}}\left(S_{A1}^{It}-S_{A2}^{It}\right),$$

(34)

where $C$ denotes a constant, $S_{A1}^{It}$ and $S_{A2}^{It}$ are arbitrary solutions, $⨁$ stands for the entry-wise multiplication, $Levy\left(\beta \right)$ is the Lévy probability distribution function of $\beta$, and $x_1$ and $x_2$ are calculated by the normal distribution function as follows:

$$\begin{array}{c}{x}_1=N(0,{\sigma }_{x_1}^2)\\ x_2=N(0,{\sigma }_{x_2}^2)\end{array},$$

(35)

where ${\sigma }_{x_1}={\left[\frac{\mathsf{\Gamma }(1+\kappa )\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\pi \kappa }{2}\right)}{\mathsf{\Gamma }\left[\frac{(1+\kappa )}{2}\right]{\kappa 2}^{\frac{(\kappa -1)}{2}}}\right]}^{\frac{1}{\eta }}$, $\kappa$ is an index within the range [1,2], $\mathsf{\Gamma }$ represents the gamma function, $\eta =1.5$, and ${\sigma }_{x_2}=1$.

4.1.5. The Proposed OSEDE/LFs Algorithm

To build the proposed algorithm, the above optimization mechanisms are uniquely combined. The flowchart of the OSEDE/LFs algorithm showing its detailed steps is demonstrated in Figure 4. A stepwise variation process is applied to the stochastic parameters of the algorithm to determine their optimal values, which guarantees that the best performance is achieved by the combined mechanisms. As depicted in Figure 4, the algorithm starts by initializing a random population and defining the required operating parameters. At this stage, the initial values of the attacker and the defender are also defined. The randomly generated initial population is then improved by applying the QOBL mechanism. It is worth mentioning that the QOBL technique is used only at the initialization stage to enhance the initial population without its application within the main loop of the algorithm. After that, the main loop begins by applying SEO, where the training-retraining mechanism is performed. Then, a social attack is spotted and responded to through an iterative process until all attacks are over. Subsequently, a new defender is selected, and the global best is updated according to the value of the new attacker. This updated global best is then improved using the DE mechanisms (mutation, crossover, and greedy selection). Furthermore, greedy LFs are performed, where the LF perturbation is improved by reapplying crossover and greedy selection. This ensures achieving the best performance. Then, the global best is updated and set as the new attacker. The main loop is executed iteratively until the termination criteria are met. Finally, global optima are obtained.

4.2. Benchmarking of the OSEDE/LFs Algorithm

In this subsection, the performance of the proposed OSEDE/LFs is verified by solving a set of benchmark functions (BFs). These BFs include unimodal, multimodal, fixed-dimensional, and free-dimensional objective functions. A total of 23 BFs with multiple local minima and different shapes (valley, bowl, and plate shapes) are used. Their detailed mathematical formulations can be found in [34,35]. Hence, by solving these BFs, the performance of OSEDE/LFs is compared to that of 9 well-known state-of-the-art algorithms from the literature: the ant lion optimizer (ALO) [36], dragonfly algorithm (DA) [37], grasshopper optimization algorithm (GOA) [38], grey wolf optimizer (GWO) [39], moth–flame optimizer (MFO) [40], multi-verse optimizer (MVO) [34], sine cosine algorithm (SCA) [35], salp swarm algorithm (SSA) [41], and whale optimization algorithm (WOA) [42]. This comparison is carried out using the original parameters of each algorithm, as recommended by their developers and using the same 23 BFs. For a fair comparison, the population size and maximum iterations are set to 100 and 500, respectively, for all algorithms. The results of 20 individual runs of each algorithm are recorded as given in

Table 1. The benchmarking of the proposed OSEDE/LFs.

Function		ALO	DA	GOA	GWO	MFO	MVO	SCA	SSA	WOA	OSEDE/LFs
$F_1$	Avg	1.03 × 10−8	3.949819	1.42 × 10−8	1.67 × 10−27	1.241253	0.014665	1.58 × 10−12	2.69 × 10−7	1.64 × 10−74	1.94 × 10−229
	STD	8.12 × 10−9	9.015141	9.60 × 10−9	3.28 × 10−27	0.837278	0.005057	3.00 × 10−12	4.86 × 10−7	5.58 × 10−74	0
	Rank	5	10	6	3	9	8	4	7	2	1
$F_2$	Avg	0.468302	1.599196	1.405281	7.46 × 10−17	23.15880	0.037429	5.18 × 10−10	0.000147	6.99 × 10−51	6.33 × 10−116
	STD	1.135663	1.284330	2.105733	5.56 × 10−17	20.41869	0.010797	8.11 × 10−10	0.000341	2.26 × 10−50	2.32 × 10−115
	Rank	7	9	8	3	10	6	4	5	2	1
$F_3$	Avg	0.057536	413.5663	1.15 × 10−5	2.48 × 10−5	17309.75	0.107485	0.002350	3.86 × 10−6	42423.25	2.93 × 10−228
	STD	0.114693	1254.954	3.48 × 10−5	7.41 × 10−5	9753.122	0.082220	0.005775	1.63 × 10−5	15003.66	0
	Rank	6	8	3	4	9	7	5	2	10	1
$F_4$	Avg	0.002035	3.347957	0.000246	7.15 × 10−7	40.44441	0.095011	0.001631	2.34 × 10−5	52.05566	9.89 × 10−116
	STD	0.002150	2.075659	0.000441	8.15 × 10−7	10.34281	0.031363	0.005651	1.03 × 10−5	29.41085	2.27 × 10−115
	Rank	6	8	4	2	9	7	5	3	10	1
$F_5$	Avg	84.75871	1453.581	245.0097	26.88655	23671.32	260.3831	7.531969	268.6981	27.85087	28.689685
	STD	154.9974	1966.364	598.7379	0.731425	39396.17	471.3051	0.561004	641.2532	0.413844	0.0212941
	Rank	5	9	6	2	10	7	1	8	3	4
$F_6$	Avg	6.90 × 10−9	20.02457	2.76 × 10−8	0.713936	1.120453	0.013753	0.374948	0.547076	0.494683	9.06 × 10−10
	STD	2.36 × 10−9	31.59926	5.67 × 10−8	0.354781	0.896267	0.006095	0.133507	0.141783	0.321910	3.99 × 10−10
	Rank	2	10	3	8	9	4	5	7	6	1
$F_7$	Avg	0.027927	0.026891	0.026351	0.001713	2.231993	0.003225	0.003166	0.016537	0.003342	2.17 × 10−5
	STD	0.014574	0.015940	0.038760	0.001462	6.655355	0.001524	0.003221	0.011196	0.002799	1.87 × 10−5
	Rank	9	8	7	2	10	4	3	6	5	1
$F_8$	Avg	−2294.46	−2611.78	−1502.40	−6216.58	−8779.65	−2981.60	−5987.54	−2862.42	−9428.42	−2207.91
	STD	422.8099	312.7750	175.4154	619.3926	919.0249	374.4465	860.8516	318.0720	1492.768	252.9189
	Rank	2	3	4	8	9	6	7	5	10	1
$F_9$	Avg	26.76432	27.10466	9.728792	3.484453	127.0368	18.16475	1.325983	20.34688	1.14 × 10−14	0
	STD	13.63410	13.54998	8.528671	4.477779	46.11280	7.303766	4.098378	8.942808	3.50 × 10−14	0
	Rank	8	9	5	4	10	6	3	7	2	1
$F_{10}$	Avg	0.356285	2.999678	1.789284	1.03 × 10−13	12.71217	0.109276	2.67 × 10−6	0.937622	4.26 × 10−15	8.88 × 10−16
	STD	0.658060	1.104676	3.609173	2.05 × 10−14	8.279653	0.248928	9.34 × 10−6	1.026054	3.15 × 10−15	0
	Rank	6	9	8	3	10	5	4	7	2	1
$F_{11}$	Avg	0.194509	0.665993	0.139938	0.005206	14.26329	0.341601	0.100953	0.235602	0	0
	STD	0.104195	0.365076	0.059471	0.008578	32.85796	0.106914	0.151824	0.140456	0	0
	Rank	6	9	5	3	10	8	4	7	1	1
$F_{12}$	Avg	1.846053	2.098039	0.033959	0.042583	3.188568	0.052179	0.109533	0.354158	0.031228	0.095399
	STD	2.207358	1.551511	0.142477	0.016548	3.829839	0.170313	0.038326	0.538863	0.042258	0.031320
	Rank	8	9	2	3	10	4	6	7	1	5
$F_{13}$	Avg	0.004241	0.840557	2.869110	0.547333	4.282131	0.005557	0.342977	0.002747	0.431719	0.001131
	STD	0.008568	1.215182	0.329120	0.208381	3.569888	0.007426	0.085893	0.004881	0.247168	0.003385
	Rank	3	8	9	7	10	4	5	2	6	1
$F_{14}$	Avg	11.87036	11.51923	289.5171	9.702993	203.7239	9.958775	6.62373	6.336665	7.455311	0.998004
	STD	6.678188	7.662714	227.8556	5.229677	217.4068	6.613166	4.770185	4.018664	4.929486	9.30 × 10−12
	Rank	8	7	10	5	9	6	3	2	4	1
$F_{15}$	Avg	0.011380	0.039980	2.064983	0.001570	0.327337	0.018534	0.001430	0.005387	0.001545	0.001493
	STD	0.017426	0.050395	7.749271	0.004541	1.386298	0.027542	0.000476	0.007532	0.002281	0.004136
	Rank	6	8	10	4	9	7	1	5	3	2
$F_{16}$	Avg	−0.95001	−0.70088	178.4440	−1.01906	77.97936	−1.03159	−1.03038	−1.03163	−1.00860	−1.03163
	STD	0.251210	0.408504	368.3589	0.015822	215.1951	3.51 × 10−5	0.001595	2.36 × 10−13	0.033520	4.73 × 10−9
	Rank	7	8	10	5	9	3	4	1	6	2
$F_{17}$	Avg	0.397887	0.518911	13.61041	0.397961	7.499672	0.397959	1.817831	10.88912	0.724203	0.397887
	STD	1.83 × 10−10	0.344932	17.48689	0.000156	13.10111	0.000120	2.164800	10.22242	1.046828	2.96 × 10−8
	Rank	1	5	10	4	8	3	7	9	6	2
$F_{18}$	Avg	11.1	85.69132	757.7885	11.11088	6094.757	38.10078	7.088386	3	33.92338	3.000001
	STD	19.78277	182.3800	986.2936	19.78185	21684.51	36.22453	18.11531	2.87 × 10−12	51.15743	1.14 × 10−6
	Rank	4	8	9	5	10	7	3	1	6	2
$F_{19}$	Avg	−3.84879	−3.61741	−2.61897	−3.57359	−3.47050	−3.86112	−3.66962	−3.86071	−3.70897	−3.86239
	STD	0.01954	0.347895	0.968000	0.879876	0.669847	0.002963	0.727990	0.004423	0.190813	0.000588
	Rank	4	7	10	8	9	2	6	1	5	3
$F_{20}$	Avg	−3.26794	−2.93263	−2.45848	−3.26838	−3.15473	−3.25628	−2.44660	−3.21957	−2.93217	−3.24402
	STD	0.077511	0.686012	0.672128	0.123730	0.179302	0.067931	0.719194	0.070374	0.403542	0.062463
	Rank	2	7	9	1	6	3	10	5	8	4
$F_{21}$	Avg	−4.47386	−3.77144	−1.65018	−5.64516	−3.95988	−5.73927	−0.79106	−5.51181	−3.86936	−5.05515
	STD	1.771646	1.899046	1.761302	3.554868	3.106963	2.797988	0.968610	3.268046	2.583665	8.35 × 10−5
	Rank	5	8	9	2	6	1	10	3	7	4
$F_{22}$	Avg	−5.45753	−4.56419	−1.03874	−8.58711	−3.88754	−5.97999	−1.38504	−5.71782	−4.41368	−5.61545
	STD	3.449533	3.064162	0.714230	2.95061	1.981279	3.746717	1.161892	3.25645	2.421325	1.62468
	Rank	5	6	10	1	8	2	9	3	7	4
$F_{23}$	Avg	−6.75264	−4.66103	−0.91397	−8.45723	−4.15571	−6.53090	−2.16716	−6.25604	−3.87789	−5.12839
	STD	3.598021	2.814268	0.324818	3.336493	3.298330	4.116066	1.672882	3.715353	2.273931	0.000214
	Rank	2	6	10	1	7	3	9	4	8	5
Total rank		117	179	167	88	206	113	118	107	120	49
Average rank		5.09	7.78	7.26	3.83	8.96	4.91	5.13	4.65	5.22	2.13

for fixed-dimensional BFs ($F_1$ to $F_{13}$) and free-dimensional BFs ($F_{14}$ to $F_{23}$). As shown in this table, the average (Avg) and standard deviation (STD) values are used for a comprehensive analysis of the obtained results. Then, by ranking the performance of each algorithm for all the BFs, it is observed that the best rank among all the compared algorithms is recorded by the proposed OSEDE/LFs (the best total rank of 49 and the best average rank of 2.13). Moreover, Figure 5 depicts the convergence curves of the compared algorithms for some of the BFs, which further validates the performance of the proposed OSEDE/LFs against several powerful original algorithms found in the literature. In addition, the Wilcoxon signed rank test is applied to all the algorithms corresponding to the solved BFs, as shown in

Table 2. The Wilcoxon signed rank test results.

Function	p-Value
	ALO	DA	GOA	GWO	MFO	MVO	SCA	SSA	WOA
$F_1$	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5
$F_2$	8.86 × 10−5	1.03 × 10−4	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5
$F_3$	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5
$F_4$	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5
$F_5$	0.601200	8.86 × 10−5	0.550300	8.86 × 10−5	8.86 × 10−5	0.145400	8.86 × 10−5	0.601200	1.03 × 10−4
$F_6$	8.86 × 10−5	3.90 × 10−4	8.86 × 10−5	0.052200	0.001700	8.86 × 10−5	0.002200	8.86 × 10−5	0.350700
$F_7$	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5
$F_8$	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	0.295900	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5
$F_9$	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.75 × 10−5	8.86 × 10−5	8.86 × 10−5	1.32 × 10−4	8.77 × 10−5	0.500000
$F_{10}$	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	7.69 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.82 × 10−5	0.000488
$F_{11}$	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	0.031250	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	1.00000
$F_{12}$	0.009996	8.86 × 10−5	0.001507	0.000120	0.000103	0.013741	0.390530	0.295880	0.000254
$F_{13}$	8.86 × 10−5	0.000254	8.86 × 10−5	8.86 × 10−5	0.191330	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5
$F_{14}$	0.000130	8.86 × 10−5	8.86 × 10−5	0.000130	8.86 × 10−5	8.86 × 10−5	8.73 × 10−5	0.000282	8.43 × 10−5
$F_{15}$	0.000892	0.000681	8.86 × 10−5	0.217960	8.86 × 10−5	8.86 × 10−5	0.002204	0.001507	0.033340
$F_{16}$	0.226560	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	0.000488	0.000103
$F_{17}$	8.83 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	0.000103	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	0.000103
$F_{18}$	0.247140	0.000103	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5
$F_{19}$	0.004550	0.000120	8.86 × 10−5	0.001713	0.000120	0.040044	8.86 × 10−5	0.278960	8.86 × 10−5
$F_{20}$	0.108430	0.005111	8.86 × 10−5	0.125860	0.191330	0.501590	8.86 × 10−5	0.262720	0.000120
$F_{21}$	8.86 × 10−5	8.86 × 10−5	8.86 × 10−5	0.525650	0.116890	0.601210	8.86 × 10−5	0.708910	0.015240
$F_{22}$	0.550290	0.156000	8.86 × 10−5	0.005111	0.006425	0.708910	8.86 × 10−5	0.851920	0.012374
$F_{23}$	0.092963	0.370260	8.86 × 10−5	0.001507	0.232230	0.061953	0.000140	0.217960	0.003592

. By conducting this nonparametric test, the p-values for the compared algorithms are obtained. These values demonstrate that the proposed OSEDE/LFs is statistically significant compared to the other algorithms since most of the resulting p-values are below the 5% significance level. Therefore, the proposed OSEDE/LFs can be recommended as a powerful method for solving real-world problems and engineering applications.

4.3. Applying OSEDE/LFs Algorithm to Maximize DISCO Profit

In this subsection, the proposed OSEDE/LFs is applied to solve the mathematical model established in Section 2 for DISCO profit maximization. This application is demonstrated in detail below Algorithm 1.

Algorithm 1: OSEDE/LFs for DISCO profit maximization

I—Input the distribution network’s data, define the algorithm’s parameters, and decision variables (number and type). II—Run the power flow and calculate the base-case value of the objective function (before adding DGs or ESSs). III—The algorithm’s initialization: 1. Generate a random population of initial solutions (${IP}_1$) containing locations and sizes of DGs and ESSs. Active and reactive powers of DGs and ESSs are considered, and charging and discharging schedules of ESSs are defined. 2. Initialize random values for the attacker and defender. 3. Run the power flow and evaluate ${IP}_1$ by the objective function (OF) using Equation (1) subject to all constraints using Equations (11)–(21). 4. Regenerate an initial population by QOBL technique (${IP}_2$) using Equation (22) and run the power flow to evaluate it by the OF subject to all constraints. 5. Compare ${IP}_1$ and ${IP}_2$, save the best population, and assign it as the input population to the main loop. IV—Main loop: 6. While stopping criteria are not satisfied: 7. Perform SEO on the current population: ● Train-retrain process. ● Set the 1st social attack. ● While number of attacks < max. number of attacks:    ➢ Spot a social attack by the “obtaining,” Equation (25), “phishing,” Equations (26) and (27), “diversion theft,” Equation (28), and “pretext,” Equation (29).    ➢ Respond to the social attack.    ➢ Number of attacks is increased by 1. End while ● Evaluate the population by the OF subject to all constraints. ● Select a new defender. 8. Update the Global Best based on the new attacker. 9. Apply DE to the current population: ● Mutation using Equation (30) and evaluation of the population using the OF subject to all constraints. ● Crossover using Equation (32) and evaluate the population using the OF subject to all constraints. ● Greedy selection to compare the populations and save the best. 10. Execute LF perturbation on the best population using Equation (33) and evaluate it using the OF subject to all constraints. 11. Crossover using Equation (32) and evaluate the population using the OF subject to all constraints. 12. Greedy selection to compare the populations and save the best. 13. Update the Global Best and set its value as the new attacker. 14. End while. V—Save the global best solutions and display the final results.

5. Results and Discussion

The optimal planning of DGs and ESSs in ADNs for DISCO profit maximization is performed using the proposed OSEDE/LFs. The standard 30-Bus and IEEE 69-Bus distribution networks shown in Figure 6 are used as test systems, and the full line and load data can be obtained from [24,30]. The base power and voltage are 10 MVA and 11 kV for the 30-Bus system and 100 MVA and 12.66 kV for the 69-Bus system. Moreover, all parameters required to perform the simulations are given in

Table 3. Load levels and hourly energy prices during the day (24 h) [6,17].

Hours	1–5	6–8	9–10	11–14	15–16	17–20	21–22	23–24
Load (%)	50	60	70	80	90	100	90	70
Energy prices in autumn and winter
${\alpha }_{t,s}^P$ (USD/kWh)	0.14	0.14	0.22	0.22	0.30	0.30	0.30	0.22
${\alpha }_{t,s}^Q$ (USD/kVArh)	0.028	0.028	0.044	0.044	0.060	0.060	0.060	0.044
Energy prices in spring and summer
${\alpha }_{t,s}^P$ (USD/kWh)	0.18	0.20	0.24	0.24	0.26	0.33	0.33	0.24
${\alpha }_{t,s}^Q$ (USD/kVArh)	0.036	0.04	0.048	0.048	0.05	0.066	0.066	0.048

and

Table 4. Simulation parameters [6,17,18,24].

Parameter	Value	Parameter	Value
${\alpha }_{DIS}$ (%)	15	${CE}_{inv.}^{ESS}$ (USD/kWh)	81
$d$ (%)	20	${\eta }_{ch}$$, {\eta }_{dis}$ (%)	95
$Y$ (years)	5	$C_{O\&M,s}^{ESS}$ (USD/kVAh)	0.02
$V_{min}$ (p.u.)	0.9	${CP}_{O\&M,s}^{DG}$ (USD/kWh), autumn and winter	0.189
$V_{max}$ (p.u.)	1.05	${CQ}_{O\&M,s}^{DG}$ (USD/kVArh), autumn and winter	0.021
$C_{inv.}^{DG}$ (USD/kVA)	1150	${CP}_{O\&M,s}^{DG}$ (USD/kWh), spring and summer	0.2268
${CS}_{inv.}^{ESS}$ (USD/kVA)	805	${CQ}_{O\&M,s}^{DG}$ (USD/kVArh), spring and summer	0.0252

, which are taken from the relevant literature references [6,17,18,24]. The seasonal prices of active and reactive power ${\alpha }_{t,s}^P$ (USD/kWh) and ${\alpha }_{t,s}^Q$ (USD/kVArh) are taken in which the prices in autumn and winter are the same, and the prices in spring and summer are the same. Accordingly, the operation and maintenance unit costs of active and reactive power of DGs ${CP}_{O\&M,s}^{DG}$ (USD/kWh) and ${CQ}_{O\&M,s}^{DG}$ (USD/kVArh) are the same in autumn and winter. Then, these prices are increased by 20% in spring and summer. Since the operation and maintenance cost of ESS is mainly dependent on its apparent power, the operation and maintenance unit cost $C_{O\&M,s}^{ESS}$ (USD/kVAh) is fixed for all seasons [6]. The maximum apparent power of DGs ($S_{max}^{DG}$) and ESSs ($S_{max}^{ESS}$) given in Equations (14) and (16), respectively, should not exceed the sum of total load and network loss without DGs or ESSs [6,17,24].

For each system, the simulation is carried out considering two cases: the planning of DGs only (Case 1) and the planning of DGs and ESSs simultaneously (Case 2). To validate the performance of the proposed OSEDE/LFs, the model is also solved by the original state-of-the-art algorithms: DE, SEO, GWO, MVO, WOA, and PSO for all cases to compare the results. The comparisons are performed based on the original parameters of each algorithm. The parameters of OSEDE/LFs are defined by a stepwise variation to achieve the best performance. For a fair comparison, the population size and the maximum number of iterations are fixed to 50 and 200 for all algorithms, 10 independent runs are executed for each case, and the best solutions are recorded and analyzed using the minimum (Min), maximum (Max), Avg, and STD. Coding and simulations are carried out using Matlab-R2016a on a PC with an Intel Core (TM) i7 processor, 3.2 GHz speed, and 8 GB RAM.

5.1. The 30-Bus Network

5.1.1. Case 1: The Optimal Planning of DGs

In this case, only DG units are considered. Hence, Equations (8)–(10) and (16)–(21) are excluded, where the decision variables are the locations and sizes of DGs (for active and reactive power). The results are obtained for all algorithms and listed in

Table 5. The optimal planning of DGs for the 30-Bus network (Case 1).

Algorithm	Optimal Locations	Optimal Sizes		${\mathit{P}\mathit{R}\mathit{O}\mathit{F}\mathit{I}\mathit{T}}_{\mathit{D}\mathit{I}\mathit{S}}$ (USD/Year)
		P (kW)	Q (kVAr)	Max	Min	Avg	STD
Base case	-	-	-	109,960.54
DE	11	102.64	68.71	167,276.76	167,209.48	167,270.00	21.28
	22	170.92	111.74
	27	77.48	51.46
SEO	11	78.27	52.80	167,354.22	167,278.66	167,346.70	23.89
	21	137.95	94.86
	25	136.29	93.01
GWO	10	112.75	75.18	167,518.23	167,480.62	167,510.70	15.86
	22	154.58	102.27
	26	89.25	60.44
MVO	10	112.45	75.86	167,639.37	167,592.21	167,634.70	14.91
	22	155.04	103.79
	26	88.51	60.17
WOA	11	99.07	66.57	167,308.25	167,214.94	167,298.90	29.51
	21	124.11	81.46
	25	130.36	86.92
PSO	11	102.76	65.84	167,003.63	166,923.33	166,995.60	25.39
	22	164.25	107.38
	26	87.06	55.33
OSEDE/LFs	10	110.58	79.48	168,383.40	168,359.84	168,381.00	7.43
	22	154.02	111.91
	26	86.33	61.60

. As can be seen in this table, the DISCO profit increased from 109,960.54 USD/year, which is the base case before adding DGs to the network, to 167,003.63 USD/year by PSO. This value is further increased to 167,276.76 USD/year by DE, 167,308.25 USD/year by WOA, 167,354.22 USD/year by SEO, 167,518.23 USD/year by GWO, and 167,639.37 USD/year by MVO. However, when using the proposed ODEDE/LFs, the DISCO profit reaches 168,383.40 USD/year, which is obviously the maximum value compared to the other algorithms.

In addition, the STD of the results obtained by ODEDE/LFs (7.43) is smaller than those of the other algorithms. These results verify the robustness of the proposed ODEDE/LFs, further illustrated in Figure 7, which shows the convergence characteristics of all compared algorithms for Case 1. The ODEDE/LFs require a smaller number of iterations to reach the optimal solution.

5.1.2. Case 2: The Optimal Planning of DGs and ESSs Simultaneously

In this case, DG and ESS units are considered. Hence, the decision variables are the locations and sizes of DGs and ESSs (for active and reactive power). The locations and sizes of DGs and ESSs obtained by all the algorithms are listed in

Table 6. The optimal planning of DGs and ESSs for the 30-Bus network (Case 2).

Algorithm	DG Units			ESS Units
	Optimal Locations	Optimal Sizes		Optimal Locations	Optimal Sizes
		P (kW)	Q (kVAr)		P (kW)	Q (kVAr)
DE	12	39.83	24.99	6	56	112
	23	161.02	100.21	9	43	79
	28	27.32	17.22	27	42	87
SEO	12	20.03	12.47	6	101	200
	21	102.79	63.98	10	29	55
	24	99.75	61.82	27	34	67
GWO	11	70.75	45.70	3	41	85
	21	97.16	61.74	8	53	101
	26	84.41	53.20	23	61	114
MVO	10	57.74	38.35	5	48	96
	12	35.63	23.57	18	54	103
	23	157.50	104.61	27	39	74
WOA	11	75.67	51.05	3	51	103
	20	78.64	52.23	8	45	86
	23	113.16	75.75	27	50	96
PSO	21	114.09	71.30	6	59	116
	24	93.72	60.05	10	47	88
	30	21.04	13.26	27	39	76
OSEDE/LFs	11	65.52	40.78	9	40	74
	22	124.63	77.24	18	59	113
	25	54.60	33.98	27	35	68

. Comparing the results of Case 1 (Table 5) to those of Case 2 (Table 6), it can be observed that the locations of DG units obtained by each algorithm are not the same compared to Case 1. This is because the algorithms consider the simultaneous planning of DGs and ESSs in Case 2 rather than only DGs in Case 1. This demonstrates the importance of this strategy as it affects the final results. It is also worth mentioning that all the algorithms are programmed to freely select the locations of ESSs to obtain the maximum profit, even if these locations would be the same as the locations of DGs. However, as can also be seen in Table 6, the optimal locations of ESSs are different from the locations of DGs for all algorithms. This demonstrates that when the DGs and ESSs are planned simultaneously, it is not necessary to allocate DG and ESS units at the same location.

In addition, as can also be seen in Table 6, the reactive power of ESS units is higher than their active power (these active and reactive powers represent the size of the ESS inverter). This explains the high impact of reactive power control on the network. When there is no exchange of active power (P is zero), the ESS unit can still exchange reactive power with the network. In other words, when the inverter is neither charging nor discharging active power, it can still draw or inject reactive power. During these periods, the ESS operates as a capacitor bank, which greatly improves the performance of the network. However, this exchange of reactive power is limited as the apparent power of the inverter must satisfy the technical constraints given in Equations (16) and (17).

Based on the results of the optimal planning of DGs and ESSs shown in Table 6, the DISCO profit is further maximized compared to Case 1, as detailed in

Table 7. DISCO profit for the 30-Bus network, Case 2 (based on the results given in Table 6).

Algorithm	${\mathit{P}\mathit{R}\mathit{O}\mathit{F}\mathit{I}\mathit{T}}_{\mathit{D}\mathit{I}\mathit{S}}$ (USD/Year)
	Max	Min	Avg	STD
Base case	109,960.54
DE	176,975.90	176,907.66	176,969.10	21.58
SEO	177,100.48	177,023.97	177,092.80	24.20
GWO	177,282.74	177,223.70	177,276.80	18.67
MVO	177,359.65	177,322.13	177,352.15	15.82
WOA	177,097.47	177,010.40	177,088.77	27.53
PSO	176,734.33	176,655.29	176,726.40	24.99
OSEDE/LFs	178,314.58	178,295.48	178,308.80	9.23

. When PSO is applied, the DISCO profit is increased to 176,734.33 USD/year. The profit reaches 176,975.90 USD/year, 177,097.47 USD/year, 177,100.48 USD/year, 177,282.74 USD/year, and 177,359.65 USD/year by DE, WOA, SEO, GWO, and MVO, respectively. However, the profit reaches its maximum value when the OSEDE/LFs is applied (178,314.58 USD/year). These results emphasize the usefulness of considering the revenues from active and reactive power trading when calculating the DISCO profit, where higher profits could be achieved. Moreover, the simultaneous planning of DGs and ESSs, considering their active and reactive power in the model, proves to be efficient as the profit is further maximized.

In addition, the effectiveness of the proposed algorithm is validated as the highest profits are obtained when the OSEDE/LFs are applied. As also illustrated in Table 7, the proposed algorithm achieves the optimal solutions with higher robustness as the STD is the lowest among all compared algorithms (9.23). The robustness of ODEDE/LFs is maintained even when the number of decision variables is increased. Accordingly, the proposed algorithm can be applied to larger models with higher complexity. These findings are further illustrated in Figure 8, which depicts the convergence characteristics of all the algorithms compared to Case 2. The ODEDE/LFs require fewer iterations to reach the optimal solution.

To demonstrate the operation of ESS during the optimization process using the OSEDE/LFs, the charging and discharging powers, as well as the stored energy of the ESS unit on Bus 18, are shown in Figure 9. It can be seen that the ESS is charged, discharged, and disconnected in accordance with the load levels given in Table 3. During the light-load hours, the ESS operates in charge mode, where the maximum charge power does not exceed the maximum active power of the ESS. To avoid unnecessary power losses, the ESS is disconnected from the network during the medium-load hours, while it operates in discharge mode during full-load hours. This figure also shows that the applied strategy is sufficient to maintain the energy balance of the ESS, where at the end of the day ($t=T=24$), the residual energy is equal to the initial stored energy ($t=0$).

5.1.3. Technical Impacts of DGs and ESSs

To analyze the technical impacts of DGs and ESSs on the 30-Bus network, the active power loss is calculated when the OSEDE/LFs are applied for Cases 1 and 2 and compared to that of the base case, as illustrated in Figure 10a.

It is obvious that maximizing the DISCO profit also reduces the active power losses in both studied cases, especially during the full-load hours, where the impact is remarkably significant. Nonetheless, when DG and ESS units are considered, the active power losses are still slightly higher during the ESS charging hours. This is reasonable since the ESS units are considered to be loads during this period. However, when the ESS units operate in discharge mode, the losses of Case 2 are further reduced compared to those of Case 1 during the same period. As a result, more benefits can be achieved if the DGs and ESSs are integrated into the network simultaneously, considering their active and reactive power.

To better visualize the impact of the planning strategy on the network voltage, Figure 10b depicts the voltage profile of all buses at Hour 17 (network fully loaded and ESS units in discharge mode). It is validated that the voltages of all buses are greatly improved compared to the base-case voltages. These improvements are clearly seen on the bus with the lowest base-case voltage (the voltage on Bus 27 is improved from 0.8944 p.u. to 0.9631 in Case 1 and to 0.9701 p.u. in Case 2). Hence, the safe and secure operation of the network is maintained in both cases since all voltages are within permissible limits. Nevertheless, the voltages in Case 2 are better improved throughout the network than in Case 1. More precisely, the worse the base-case voltages are, the better Case 2 improves over Case 1.

Finally, it is necessary to demonstrate the active and reactive power that the DISCO exchanges with the upstream network during the day. This will further justify the above results. It will also validate the effectiveness of the planning strategy. Figure 10c shows the active power exchanged with the upstream network using the OSEDE/LFs for Cases 1 and 2. When only DGs are added to the network (Case 1), the received power from the upstream network is well reduced compared to the base case. However, when DGs and ESSs are added (Case 2), the power received is also lower than the base case but higher than that of Case 1 during ESS charging and disconnecting hours. This is because more power is needed to charge the ESS units. Starting from Hour 17, the power received in Case 2 becomes lower than that of Case 1 since the power stored in ESS units is discharged and used. Thus, the DISCO can make more profit when DGs and ESSs are added because the power used to charge the ESS units (during light-load hours) is cheaper than when ESS units are discharged (the load levels are between 100% and 70%).

The reactive power exchanged with the upstream network is also analyzed, as illustrated in Figure 10d. After adding DG units to the network, the DISCO still must receive reactive power but with a lower amount compared to that before adding DGs. Nevertheless, in Case 2, the reactive power is sold to the upstream network during hours 1 to 9 instead of receiving reactive power compared to Case 1 during the same period. Moreover, the DISCO sells more reactive power in the last two hours of the day. Furthermore, during the rest of the day, the reactive power received from the upstream network is clearly lower in Case 2 than in Case 1. This was previously explained since the ESS is still exchanging reactive power with the network when there is no active power being exchanged. Therefore, by adding DGs and ESSs to the network considering their active and reactive power, the DISCO can obtain more income by exchanging higher amounts of reactive power and reducing the amount of power received from the upstream network.

5.2. The 69-Bus Network

5.2.1. Case 1: The Optimal Planning of DGs

Similar to the 30-Bus network, 3 DGs are added. The results for all of the compared algorithms are listed in

Table 8. The optimal planning of DGs for the 69-Bus network (Case 1).

Algorithm	Optimal Locations	Optimal Sizes		${\mathit{P}\mathit{R}\mathit{O}\mathit{F}\mathit{I}\mathit{T}}_{\mathit{D}\mathit{I}\mathit{S}}$ (USD/Year)
		P (kW)	Q (kVAr)	Max	Min	Avg	STD
Base case	-	-	-	729,008.14
DE	22	86.58	54.79	872,448.63	872,360.19	872,439.80	27.97
	61	1003.17	655.84
	65	20.00	12.76
SEO	20	168.26	111.76	871,500.15	871,445.35	871,483.70	26.47
	61	146.47	94.99
	62	853.11	575.52
GWO	24	72.46	48.88	873,201.55	873,145.33	873,195.90	17.78
	61	976.67	646.15
	65	43.91	28.82
MVO	21	90.52	60.60	873,765.87	873,715.50	873,760.80	15.93
	61	786.28	518.14
	64	231.27	151.20
WOA	17	68.23	44.78	871,831.28	871,751.03	871,817.00	30.44
	21	21.56	13.93
	61	1020.37	661.76
PSO	61	147.01	99.18	871,266.96	871,110.72	871,267.00	49.41
	62	244.06	161.47
	63	640.21	440.21
OSEDE/LFs	24	88.03	60.53	875,457.79	875,438.81	875,450.20	9.80
	61	765.34	530.23
	64	240.87	169.38

, which demonstrates that the DISCO profit increases from 729,008.14 USD/year (base case) to 871,266.96 USD/year, 871,500.15 USD/year, 871,831.28 USD/year, 872,448.63 USD/year, 873,201.55 USD/year, 873,765.87 USD/year, and 875,457.79 USD/year by PSO, SEO, WOA, DE, GWO, MVO, and OSEDE/LFs, respectively. The maximum profit is obtained by the proposed algorithm. Moreover, the STD of the results obtained by ODEDE/LFs (9.80) is smaller than those of the other algorithms. Comparing the results of Case 1 for the 30-Bus and 69-Bus networks, it can be noticed that the STD values of all algorithms are higher for the 69-Bus network. This is because the search space is increased, and thus, the complexity of the problem is increased. However, the proposed OSEDE/LFs maintains its robustness, as the optimal solution is obtained with the smallest STD value compared to the other algorithms. These results are further illustrated in Figure 11, showing the convergence characteristics of all compared algorithms for Case 1. The ODEDE/LFs require fewer iterations to reach the optimal solution.

5.2.2. Case 2: The Optimal Planning of DGs and ESSs Simultaneously

In this case, the decision variables are the locations and sizes of 3 DGs and 3 ESSs (for active and reactive power). The locations and sizes of DGs and ESSs obtained by all the algorithms are listed in

Table 9. The optimal planning of DGs and ESSs for the 69-Bus network (Case 2).

Algorithm	DG Units			ESS Units
	Optimal Locations	Optimal Sizes		Optimal Locations	Optimal Sizes
		P (kW)	Q (kVAr)		P (kW)	Q (kVAr)
DE	17	81.14	54.52	19	136	295
	61	398.98	270.19	40	42	141
	64	347.69	230.93	61	181	433
SEO	10	30.78	20.53	8	174	390
	22	180.25	122.07	49	278	747
	61	620.86	410.75	61	289	528
GWO	18	58.618	36.48	61	199	366
	25	41.51	26.27	64	118	221
	61	670.89	429.85	67	212	446
MVO	17	20.27	12.93	42	17	38
	26	20.00	12.61	61	316	584
	61	690.55	449.66	69	159	322
WOA	22	42.90	28.27	62	243	491
	23	62.03	41.84	63	13	24
	62	689.98	460.08	66	114	302
PSO	27	181.65	122.07	56	204	393
	60	155.16	100.23	60	115	262
	62	436.43	289.87	62	176	274
OSEDE/LFs	61	478.16	320.08	12	111	232
	64	178.24	122.09	21	88	178
	65	55.11	36.61	61	281	526

. The results of Case 1 (Table 8) and Case 2 (Table 9) show the difference between planning only DGs and planning DGs and ESSs simultaneously in terms of optimal locations. However, unlike the 30-Bus network, the 69-Bus network requires some DG and ESS units to be placed at the same locations to achieve the maximum profit, as shown in Table 9. These results emphasize that the simultaneous planning of DG and ESS units is also related to the nature and topology of the network under study.

Based on the results of Case 2 shown in

Table 10. DISCO profit for the 69-Bus network, Case 2 (based on the results given in Table 9).

Algorithm	${\mathit{P}\mathit{R}\mathit{O}\mathit{F}\mathit{I}\mathit{T}}_{\mathit{D}\mathit{I}\mathit{S}}$ (USD/Year)
	Max	Min	Avg	STD
Base case	729,008.14
DE	898,119.76	898,054.43	898,100.20	31.56
SEO	891,756.92	891,629.72	891,744.20	40.23
GWO	900,334.02	900,253.20	900,325.90	25.56
MVO	900,813.13	900,767.78	900,799.50	21.91
WOA	897,791.51	897,721.00	897,770.40	34.06
PSO	889,726.09	889,551.00	889,708.60	55.37
OSEDE/LFs	904,013.05	903,979.67	904,006.40	14.07

, the DISCO profit is further maximized to reach 889,726.09 USD/year (by PSO), 891,756.92 USD/year (SEO), 897,791.51 USD/year (WOA), 898,119.76 USD/year (DE), 900,334.02 USD/year (GWO), 900,813.13 USD/year (MVO), and 904,013.05 USD/year (ODEDE/LFs), respectively.

Subsequently, it is proved that the utilized optimization model is effective for DISCO profit maximization. It is also observed that as the complexity of the problem increases (increasing the search space between the 30-Bus and 69-Bus networks), the compared algorithms may perform differently. For example, the results of the SEO algorithm are better than those of PSO, DE, and WOA for the 30-Bus system but worse than those of DE and WOA for the 69-Bus system. Moreover, for both test systems, the SEO performs worse than GWO, MVO, and the proposed OSEDE/LFs, especially for the 69-Bus system. These results confirm that the original SEO needs improvements to handle complex optimization problems like the model used in this paper, especially for large-scale systems. This has been achieved by the proposed OSEDE/LFs algorithm.

It is clear that the highest profits with the lowest STD values are obtained for all cases when the OSEDE/LFs are applied. Using the proposed algorithm, the performance of SEO and DE is remarkably improved to overcome even the powerful state-of-the-art GWO, MVO, and WOA algorithms. To further validate these results, the convergence characteristics of all the algorithms compared to Case 2 are depicted in Figure 12, showing that the ODEDE/LFs reach the optimal solution with fewer iterations.

Furthermore, the charging and discharging powers and the stored energy of the ESS unit on Bus 21 are shown in Figure 13, demonstrating that the energy balance of the ESS is maintained.

5.2.3. Technical Impacts of DGs and ESSs

The technical impacts of DGs and ESSs on the 69-Bus network when the OSEDE/LFs are applied for Cases 1 and 2 are shown in Figure 14, analyzing the (a) active power losses, (b) voltage profile at Hour 17, (c) active power and (d) reactive power exchanged with the upstream network.

Similar observations can be made when compared to the results of the 30-Bus network. As shown in Figure 14a, the power losses are well reduced in both cases, especially during the full-load hours. In Case 2, the active power losses are higher during ESS charging hours, and during the discharging hours, the losses are further reduced, which leads to higher profits. Figure 14b illustrates that the voltages of all buses are significantly improved, especially in Case 2. The lowest base-case voltage (on Bus 65) is improved from 0.9092 p.u. to 0.9621 in Case 1 and to 0.9690 p.u. in Case 2. Figure 14c shows the active power exchange with the upstream network using the OSEDE/LFs for Cases 1 and 2. In Case 1, the power received from the upstream network is well reduced compared to the base case. In Case 2, the received power is also reduced compared to the base case, but it is higher than that of Case 1 during the charging and disconnecting hours of the ESSs. From Hour 17, the received power in Case 2 becomes lower than in Case 1. This leads to higher profits due to the difference in energy prices. The reactive power exchange with the upstream network is illustrated in Figure 14d. In Case 1, the DISCO receives less reactive power than in the base case. In Case 2, the reactive power is sold to the upstream network during hours 1 to 6. During the rest of the day, the reactive power received from the upstream network in Case 2 is clearly lower than in Case 1. This is because the ESS is still exchanging reactive power with the network when there is no active power being exchanged. Thus, the DISCO can generate more income through the exchange of higher amounts of reactive power and the reduction of the power received from the upstream network.

6. Conclusions

This paper has addressed the simultaneous planning of DGs and ESSs in deregulated electricity markets for DISCO profit maximization. The revenues from trading active and reactive power have been considered in the optimization model to further improve the accuracy of the results. Meanwhile, the active and reactive power of DGs and ESSs have also been included in the optimization process, which maximizes their utilization in the reactive power support. Thus, the decision variables have been set to be the locations and sizes (active and reactive power) of DGs and ESSs simultaneously. To solve the designated model, a new hybrid EA called the OSEDE/LFs has been proposed based on the recently developed SEO algorithm. The OSEDE/LFs exploits the advantages of the search mechanisms of DE and LFs to improve the performance of SEO by their distinctive combination within the main loop. Furthermore, the initial population of the algorithm is generated using the QOBL technique. The proposed OSEDE/LFs has been benchmarked and compared with the other nine state-of-the-art EAs using a set of well-known BFs. The results obtained for most of the tested BFs have confirmed the outstanding performance of the OSEDE/LFs over the other algorithms in terms of obtaining the global optima, fast convergence, and robustness with the best total and average ranks achieved. Moreover, the Wilcoxon signed rank test has proved the statistical significance of the OSEDE/LFs. Based on this, the proposed algorithm has been applied to solve the planning model of DISCO profit maximization using the standard 30-Bus and IEEE 69-Bus distribution networks. Two case studies have been considered for each network, namely the optimal planning of DGs and the optimal planning of DGs and ESSs simultaneously. For both networks, the maximum DISCO profits with faster convergence and higher robustness have been obtained by the OSEDE/LFs compared to other original algorithms. In addition, by comparing the results of Case 2 with Case 1 for each network, it has been verified that the proposed algorithm maintains its robustness even when the number of decision variables and the search space are increased, especially for the 69-Bus network. The results have also shown that some algorithms, such as the original SEO, may have worse performance as the complexity of the optimization model increases. This has justified the need to improve its performance for complex optimization problems. Thus, the OSEDE/LFs can be recommended as a robust method for solving more complex and larger-scale problems in different engineering applications, which may be promising for future research. Moreover, the comparisons made between Cases 1 and 2 have highlighted the importance of considering the revenues from active and reactive power trading to achieve higher DISCO profits. This has been done by including the active and reactive power of DGs and ESSs in the optimization model, which has remarkably increased the reactive power support. These results have been validated by observing the reduced power losses, improved voltage profile, and power exchanged with the upstream network.

Finally, several potential directions are worthy of investigation in future research, such as including renewable DGs in the planning model, developing an objective function that considers the environmental revenues, and comparing different types of storage systems. For example, the utilization of pumped hydro storage or gravity energy storage in ADNs for DISCO profit optimization could be an interesting trend for further research.