Stochastic Allocation of Photovoltaic Energy Resources in Distribution Systems Considering Uncertainties Using New Improved Meta-Heuristic Algorithm

Abstract

In this paper, a stochastic-metaheuristic model is performed for multi-objective allocation of photovoltaic (PV) resources in 33-bus and 69-bus distribution systems to minimize power losses of the distribution system lines, improving the voltage profile and voltage stability of the distribution system buses, considering the uncertainty of PV units’ power and network demand. The decision-making variables, including installation location and the size of PVs, are determined optimally via an improved human learning optimization algorithm (IHLOA). The conventional human learning optimization algorithm (IHLOA) is improved based on Gaussian mutation to enhance the exploration capability and avoid getting trapped in local optimal. The methodology is implemented in two cases as deterministic and stochastic without and with uncertainties, respectively. Monte Carol Simulation (MCS) based on probability distribution function (PDF) is used for uncertainties modeling. The deterministic results proved the superiority of the IHLOA compared with conventional HLOA, particle swarm optimization (PSO), to obtain better values of the different objectives and faster convergence speed and accuracy. The results are clear that enhancing the conventional HLOA has increased the algorithm’s ability to explore and achieve the optimal global solution with higher convergence accuracy. Moreover, the stochastic results were clear that considering the uncertainties leads to correct and robust decision-making against existing uncertainties and accurate knowledge of the network operator against the exact values of various objectives compared to the deterministic case.

1. Introduction

The growing trend of electrical energy consumption, the high costs of constructing and commissioning large power plants, and the ending of fossil fuels are caused by installing small power plants with small capacity as distributed generation (DG) resources [1]. The DGs are placed at the lowest level of the system along with the final consumers and have low installation costs and are quick to implement. In addition, the power generation of the DGs has low pollution and their primary fuel cost is cheap or free [2,3]. The critical problem in applying DGs is the optimal placement and the sizing of these sources to take advantage of the distribution system. The lack of optimal location of DGs and illogical power generation by DGs cannot improve the system performance. It may also have adverse effects on the losses of the distribution system line, the voltage deviation, and the voltage stability of the distribution system [4,5]. During recent decades, various studies have been done on the location of DGs, especially renewable DGs. Some of them focused on the objective functions and the use of different criteria, and some others concentrated on the methods which use a new optimization approach [6,7].

1.1. Literature Review

Due to the nonlinear nature and magnitude of the optimal DG location problem, the best methods used to solve this problem are evolutionary or meta-heuristics methods that have been used extensively in recent years. The issue of optimal planning of photovoltaic resources in distribution systems for reduction of losses and voltage profile enhancement is investigated in [8] using the backward, forward sweep (BFS) algorithm. The flower pollination algorithm (FPA) is presented by [9] for a solution allocating renewable resources to reduce losses compared with different algorithms. In [10], the determination of the site and size of DGs is presented using the genetic algorithm (GA) and particle swarm optimization (PSO) to reduce active losses and cost. Mixed-integer nonlinear programming (MINLP) in [11] is applied to find the optimum size of photovoltaic resources for losses minimization. The improved whale optimizer algorithm in [12] is presented for allocating the hybrid photovoltaic/wind energy resources to mitigate the losses and voltage deviation in the distribution system. In [13], allocation of PVs in distribution networks is developed using an Archimedes optimization algorithm (AOA) inspired by physical principles to minimize network dependence and greenhouse gas emissions. PSO is developed in [14] to allocate the photovoltaic resources for active and reactive losses reduction of the distribution systems. Multi-objective PSO is used in [15] to allocate the photovoltaic resources in the distribution systems to minimize the losses and minimize the voltage deviation and voltage stability enhancement. In [16], optimal allocation of PVs in medium-voltage distribution networks is performed using mixed-integer programming (MINLP) to minimize the operating cost. In [17], the optimal location of DGs is evaluated to increase the reliability and power quality of the distribution system using the crow search algorithm (CSA). In [18], the deterministic locating of PVs in the distribution system is performed via the ant lion optimizer (ALO) to minimize the losses, voltage profile, and stability. In [19], a two-stage PSO is proposed to determine the location of new energy resources concerning their maximum size, aimed at loss reduction. In [20], the allocation of DGs to reduce losses and voltage profile enhancement is identified using an improved equilibrium optimization algorithm (IEOA). In [21], optimal allocation DGs are presented to minimize the power losses, voltage profile, and short-circuit level via the GA.

1.2. Research Gap

With unprecedented energy sources penetration, future electricity systems, especially photovoltaic energy sources, are exposed to severe uncertainties that may cause challenges in the distribution systems operation. Not considering the uncertainty of the photovoltaic sources of power leads to a lack of accurate understanding of the distribution system operation based on these sources and changes the characteristics of the electricity system. Therefore, to achieve the actual operation of power distribution systems, it is critical to assess the uncertainty of system performance in these conditions.

A review of previous studies shows that many studies on the distribution systems’ operation using DGs and renewable energy sources have not considered the uncertainty of generation and system load that is constantly changing, which is one of the disadvantages of these studies. Moreover, in [18], locating and determining the optimal size of PVs for losses minimization, enhancement of voltage profile, and stability are presented using the ALO with a weighted coefficients method. In [18], the location of one and two PVs in 33 and 69-bus systems are deterministically investigated and compared with the previous results. Results indicate that the proposed ALO is better at solving the problem for losses minimization and enhancing the voltage conditions [18]. In [18], the uncertainty of load demand and PVs power has not been considered in the problem solution. In this paper, at first, the study carried out in [18] is implemented using a new, improved optimization algorithm. The results are then compared with other algorithms. With application of a new meta-heuristic algorithm, the problem is solved considering the generation and system load uncertainties as stochastic problem.

1.3. Contributions

In this paper, the optimal multi-criteria allocation of PVs in the radial electricity systems is presented for minimizing the losses and enhancing the voltage profile and stability based on the weighting coefficient method as deterministic and stochastic cases without and with uncertainties of PV power and system load, respectively. The Monte Carol Simulation (MCS) [22] is used for modeling the uncertainty of the mentioned parameters based on the probability distribution function. The optimization variable as optimal location and size of PV is found via a new meta-heuristic method named the improved human learning optimization algorithm (IHLOA) as a solver. In the IHLOA, Gaussian mutation has been used to avoid getting trapped in the local optimum of the conventional algorithm in the condition of problem complexity and also to improve the search space exploration and quickly reach the global optimum. At first, the simulation results are presented in a deterministic case without considering uncertainties, and the obtained results are compared with conventional HLOA, PSO solvers, and last studies. Finally, the results of the stochastic case are presented, and the effect of considering uncertainties is evaluated on the problem solving. The performance of the IHLOA is compared with other solvers, such as conventional HLOA and PSO methods.

Highlights of this research are as follows:

• Stochastic allocation of photovoltaic resource in the distribution system
• Multi-objective optimization considering losses, voltage profile, and voltage stability
• Using an improved human learning optimization algorithm
• The superior performance of the IHLOA in comparison with HLOA and PSO
• Increasing losses and weakening the voltage profile and stability with uncertainty

1.4. Paper Structure

In Section 2, the objective function and constraint of the problem are formulated. The structure of the IHLOA is presented in Section 3. The implementation process of IHLOA in the problem solution is expressed considering the uncertainty. Simulation results of PVs application in distribution systems considering the deterministic and stochastic cases are presented in Section 5, and the findings of this study are described in Section 6.

2. Problem Formulation

This paper presents PV panels placement and sizing in distribution systems for minimizing the losses and voltage deviation and voltage stability enhancement in for of a multi-objective model with weight coefficients.

2.1. Objective Function

The optimal location of PVs in the distribution system is presented to reduce the losses and improve the voltage profile and stability. The first objective function that must be minimized in the optimization process is system losses, including the active and reactive losses. The active power losses magnitude ($P_{loss}^k$) at line k is defined as [18,23,24,25,26,27,28,29]:

$$P_{loss}^k=R_k.I_k^2, I_k=\left|\frac{E_j-E_i}{R_k+j{X}_k}\right|$$

(1)

$$P_{loss}^k=R_k.{\left|\frac{E_j-E_i}{R_k+j{X}_k}\right|}^2$$

(2)

where k refers to the lines number between buses i and j, $E_i$ and $E_j$ refer to the voltage of bus i and j, respectively, R and X are the distribution network line’s resistance and the line reactance, respectively.

The total active power losses of the network lines ($P_{loss}$) is obtained by

$$P_{loss}={\displaystyle \sum }_{k=1}^{N_b}{P}_{loss}^k$$

(3)

where N_b is the maximum number of the distribution network lines.

The reactive power losses magnitude ($Q_{loss}^k$) is obtained as follows:

$$Q_{loss}^k=X_k·I_k^2$$

(4)

$$Q_{loss}^k=X_k·{\left|\frac{E_j-E_i}{R_k+j{X}_k}\right|}^2$$

(5)

The total reactive power losses of the network lines ($Q_{loss}$) is obtained by

$$Q_{loss}={\displaystyle \sum }_{k=1}^{N_b}{Q}_{loss}^k$$

(6)

Total apparent power losses magnitude ($S_{loss}$) is obtained by

$$S_{loss}=\left|P_{loss}+j{Q}_{loss}\right|$$

(7)

$$S_{loss}=\sqrt{P_{loss}^2+Q_{loss}^2}$$

(8)

The objective of losses minimization is as follows:

$$F_1=\frac{S_{loss}^{after\_PVs}}{S_{loss}^{before\_PVs}}$$

(9)

In (10), $S_{loss}^{after\_PVs}$ clears the apparent losses with PVs and $S_{loss}^{before\_PVs}$ are the apparent losses without PVs.

Another objective of the study is related to voltage stability index evaluation of the radial distribution network. Considering composite load modeling and power flow analysis, the smallest magnitude of the index at any bus indicates the most sensitive bus to voltage collapse. By using the VSI, one can measure the level of stability of radial distribution networks and thereby appropriate action may be taken if the index indicates a poor level of stability. On the other hand, by using this index, it is possible to calculate the stability index value in each bus. The bus with the minimum voltage stability index value has the highest sensitivity to voltage collapse. The voltage stability index (VSI) according to Ref. [30] is presented as follows [18,23]:

$$VS{I}_{\left(m+1\right)}={\left|E_m\right|}^4-\left[\frac{{\left(P_{m+1}{X}_m-Q_{m+1}{R}_m\right)}^2+\left(P_{m+1}{R}_m-Q_{m+1}{X}_m\right){\left|E_m\right|}^2}{0.25}\right]$$

(10)

where $P_{m+1}$ and $Q_{m+1}$ indicate the active and reactive power of bus m + 1. The objective function related to VSI is defined by

$$F_2=\frac{1}{VS{I}_{total}^{after\_PVs}}$$

(11)

It is necessary to mention that, by optimally allocating photovoltaic resources in the network and determining their size and installation location in the distribution network, power flow is implemented for the set of optimal variables. In other words, network power flow is performed based on the injection of current from photovoltaic sources into the network lines, which has an effect on the voltage drop of the lines and, as a result, the voltage value of the network buses. In this situation, based on Equation (10), the VSI value for each bus of the network and also total network is calculated. Of course, the goal is to maximize this index, which is placed in the denominator of the fraction in Equation (11) to achieve the goal of maximization.

The value of the reference bus voltage is considered relative to the substation voltage of 1 p.u. By injecting power from the substation to the distribution network and by injecting current in the network lines, the voltage drop created in the lines causes the voltage of the network buses to deviate from 1 p.u. The value of the total voltage deviations of the network buses is named the total voltage deviation of the network. In this study, one of the objectives is to reduce voltage deviations or improve the voltage profile by optimal injection of photovoltaic power in the network. Therefore, in Equation (12), the voltage deviations of the network ($V{D}_{total}$) are obtained as the sum of the absolute value of the voltage deviations of each of the network buses relative to 1 unit. The voltage profile enhancement is formulated as minimizing the voltage deviations of the system as follows [8,31,32,33,34,35,36,37,38,39].

$$V{D}_{total}={\displaystyle \sum }_{i=1}^N\left|E_i-1\right|$$

(12)

where N refers to buses number.

Therefore, the objective function of voltage profile enhancement is defined by

$$F_3=\frac{V{D}_{total}^{after\_PVs}}{V{D}_{total}^{before\_PVs}}$$

(13)

where $V{D}_{total}^{after\_PVs}$ and $V{D}_{total}^{before\_PVs}$ refer to the sum of system voltage deviation with and without PVs.

In Equation (13), the voltage deviations value has been normalized after and before the optimal allocation of photovoltaic resources in the network. In other words, the value of Equation (13) is between 0 and 1 and also less than 1, because the voltage deviations value before the optimal allocation of photovoltaic resources is greater than the amount of network voltage deviations after the optimal allocation of resources. Therefore, the objective is to minimize this equation, or in other words, to minimize the voltage deviations after optimizing the photovoltaic sources.

2.2. Constraints

Solving the PVs’ optimal placement is associated with a few technical constraints that the distribution system should meet. The constraints of the optimization problem are illustrated below [18,27,28,29,37,38,39].

• Equality constraints

The load dispatch constraint is defined by

$$P_{post}+{\displaystyle \sum }_{i=1}^{N_{DG}}{P}_{DG}\left(i\right)={\displaystyle \sum }_{i=1}^{N_b}{P}_{loss}\left(i\right)+{\displaystyle \sum }_{m=1}^N{P}_d\left(m\right)$$

(14)

$$Q_{post}+{\displaystyle \sum }_{i=1}^{N_{DG}}{Q}_{DG}\left(i\right)={\displaystyle \sum }_{i=1}^{N_b}{Q}_{loss}\left(i\right)+{\displaystyle \sum }_{m=1}^N{Q}_d\left(m\right)$$

(15)

where $P_{post}$ and $Q_{post}$ are the active and reactive power of post, $P_{DG}$ and $Q_{DG}$ are the active and reactive power of DG, $P_{loss}$ and $Q_{loss}$ are the active and reactive losses, $P_d$ and $Q_d$ refer to active and reactive load, and $N_{DG}$ refers to DG numbers.

• Inequality constraints

Voltage amplitudes of the buses are limited between the allowed lower ($E_i^{\min }$) and upper ($E_i^{\max }$) values as follows:

$$E_i^{\min }\le E_i\le E_i^{\max }$$

(16)

where $E_i^{\min }$ and $E_i^{\max }$ are considered 0.95 p.u and 1.05 p.u, respectively [18].

To prevent reverse power flow, the distributed generations (DGs) installed capacity in distribution network should be less than 75% of the power provided using the post (the power lost in the network lines and also the load demand of the customers). It should be noted here that photovoltaic sources are not able to inject reactive energy into the network.

$${\displaystyle \sum }_{i=1}^{N_{DG}}{P}_{DG}\left(i\right)\le 0.75\times \left({\displaystyle \sum }_{i=1}^{N_b}{P}_{loss}\left(i\right)+{\displaystyle \sum }_{m=1}^N{P}_d\left(m\right)\right)$$

(17)

$${\displaystyle \sum }_{i=1}^{N_{DG}}{Q}_{DG}\left(i\right)\le 0.75\times \left({\displaystyle \sum }_{i=1}^{N_b}{Q}_{loss}\left(i\right)+{\displaystyle \sum }_{m=1}^N{Q}_d\left(m\right)\right)$$

(18)

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

(19)

$$Q_{DG}^{\min }\le Q_{DG}\le Q_{DG}^{max}$$

(20)

Allowed and tolerable current should be obtained from network lines. Therefore, the current passing through the network lines should not exceed the maximum or nominal current. The constraint of the current passing through the network lines is as follows:

$$M{C}_{ij}\le M{C}_{ij-Nominal}$$

(21)

where $Q_{DG}^{\max }$ and $P_{DG}^{\min }$ refer to max and min size of DG generation, $M{C}_{ij-Nominal}$ refers to the allowable current passing through the network lines.

2.3. Weighted Coefficient Method

In this study, multi-objective allocation of PV resources in distribution networks is presented to minimize power losses, improving the voltage profile and voltage stability enhancement. Based on the method of weight coefficients, a compromise is made between the three presented objectives to achieve the optimal solution. The multi-objective function via weight coefficients is defined by

$$OF={\displaystyle \sum }_i{w}_i{F}_i\left(x\right)$$

(22)

where $w_i$ is a positive weight. By minimizing (22), a sufficient condition is created for solving the multi-objective problem. $x$ is vector of decision variables including optimal installation location and size of PV resources in distribution network. Minimizing the losses ($F_1$), minimizing the voltage deviations ($F_2$), and voltage stability maximization ($F_3$), the multi-objective function is defined as below [18]:

$$Min OF=w_1{F}_1\left(x\right)+w_2{F}_2\left(x\right)+w_3{F}_3\left(x\right)$$

(23)

where w₁, w_2, and w₃ refer to the weighted coefficients, and the following equation is established for these coefficients:

$$w_1+w_2+w_3=1$$

(24)

where, in this study, $w_1$, $w_2$, and $w_3$ are considered 0.4, 0.3, and 0.3 based on the trial-and-error method as well as repeated executions and evaluation of the results with the aim of achieving the best result.

2.4. Uncertainty Models of PV Power and Load

The PV power is computed considering irradiance as follows [30,31,32]:

$$P_{PV}=P_{rated}\times \left(\frac{\varphi }{{\varphi }_{ref}}\right)\times {\eta }_{MPPT}$$

(25)

where $P_{rated}$ is the rated power of the PV panel, $\varphi$ is the solar radiation, ${\varphi }_{ref}$ is the reference irradiance (1000 W/m²), ${\eta }_{MPPT}$ is the PV MPPT efficiency (in this study, it is considered as 0.95). Renewable energy based on distributed generation sources such as PVs has an uncontrollable nature, and their generated power is determined based on meteorological conditions. Therefore, the generation of these sources is not deterministic, and the system operation should be done based on predictions. Therefore, the planning of the systems in which these sources are present should be performed based on stochastic PDFs where various conditions that are likely to arise must be considered.

This paper evaluates the uncertainty of PVs power for optimal allocation problem solving in the distribution system. The PV’s power uncertainty modeling is defined based on its power PDF. The most suitable stochastic PDF for the uncertainty of the PV power is beta stochastic PDF, where the stochastic PDF is expressed as follows [33]:

$$f_b\left(\varphi \right)=\{\begin{array}{l}\frac{\Gamma \left(\xi +\psi \right)}{\Gamma \left(\xi \right)\Gamma \left(\psi \right)}\times {\varphi }^{\left(\xi -1\right)}\times {(1-\varphi )}^{\left(\psi -1\right)} for 0\le \varphi \le 1,\xi ,\psi \ge 0\\ 0 otherwise\end{array}$$

(26)

where f_b($\varphi$) is the stochastic PDF of beta for the variable $\varphi$, and $\xi$ and $\psi$ are beta PDF parameters, defined as follows:

$$\psi =\left(1-\mu \right)\times \left(\frac{\mu \times \left(1+\mu \right)}{{\Omega }^2}-1\right)$$

(27)

$$\xi =\frac{\mu \times \psi }{1-\mu }$$

(28)

where μ and $\Omega$ refer to mean and deviation values in the PDF of beta, respectively.

For the load variable, the normal PDF is considered as below:

$$f\left(\mathcal{Q}\right)=\frac{1}{\Omega \sqrt{2\pi }}{e}^{\frac{-\left(\mathcal{Q}-\mu \right)}{2{\sigma }^2}}$$

(29)

where Q is the load demand and $f\left(\mathcal{Q}\right)$ refers to the PDF of normal. The MCS produces the PV power and system load scenarios (samples).

3. Overview of Proposed Algorithm

This study applies the IHLOA for multi-criteria stochastic allocation of PV resources in distribution systems considering uncertainty. In the following, the suggested improved algorithm is presented.

3.1. Overview of HLOA

The human learning optimization algorithm (HLOA) is modeled on four individual learning operators, a social learning operator, a randomized exploratory learning operator, and a learning operator. Imagine a person learning a game based on individual learning by him/her and based on social learning by his/her coach. During learning, one tries new methods without prior knowledge (random learning) to improve one’s skills in that game at random. However, a person has a problem in a difficult and critical situation, and personal learning is not the solution for him/her. He/she benefits from re-learning by the instructor to strengthen his learning in this situation [24,25].

3.1.1. Initialization

In the HLOA, a person (a candidate solution) without prior knowledge based on a binary string whose initial value is assumed to be 0 or 1 is represented as follows [24]:

$$X_{ij}=\left[X_{i1}{X}_{i2}\dots X_{ij}\dots X_{iM}\right],1\le i\le Np,1\le j\le M$$

(30)

where $X_i$ is the ith person, $Np$ refers to the number of populations, and M is the number of components contained in the knowledge, i.e., the dimension of solutions.

The initial population of the HLOA, with initialization for individuals, is generated as follows [24]:

$$X=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_i\\ ⋮\\ x_N\end{array}\right]=\left[\begin{array}{c}{x}_1 x_{12} \dots x_{1j }\dots x_{1M}\\ x_{21} x_{22 } \dots x_{2j }\dots x_{2M}\\ ⋮ ⋮ ⋮ ⋮\\ x_{1i } x_{i2 }\dots x_{ij } \dots x_{iM}\\ ⋮ ⋮ ⋮ ⋮\\ x_{Np1} x_{Np2 } \dots x_{Npj} \dots x_{NpM}\end{array}\right],x_{ij}\in \left\{0,1\right\},1\le i\le Np,1\le j\le M$$

(31)

3.1.2. Learning Operators

(A) Individual learning operator: In the HLOA, the individual learned the problem based on the individual learning operator and based on the experiences gained and stored in the individual knowledge database (IKD) as per the following equations [24]:

$${\mathrm{x}}_{\mathrm{ij}}={\mathrm{ik}}_{\mathrm{ipj}}$$

(32)

$${\mathrm{IKD}}_{\mathrm{i}}=\left[\begin{array}{c}\mathrm{ik}{\mathrm{d}}_{\mathrm{i}1}\\ \mathrm{ik}{\mathrm{d}}_{\mathrm{i}2}\\ ⋮\\ \mathrm{ik}{\mathrm{d}}_{\mathrm{ip}}\\ ⋮\\ \mathrm{ik}{\mathrm{d}}_{\mathrm{iG}}\end{array}\right]=\left[\begin{array}{c}{\mathrm{ik}}_{\mathrm{i}11}{\mathrm{ik}}_{\mathrm{i}12}\dots {\mathrm{ik}}_{\mathrm{i}1\mathrm{j}}\dots {\mathrm{ik}}_{\mathrm{i}}{}_{1\mathrm{M}}\\ {\mathrm{ik}}_{\mathrm{i}21}{\mathrm{ik}}_{\mathrm{i}22}\dots {\mathrm{ik}}_{\mathrm{i}2\mathrm{j}}\dots {\mathrm{ik}}_{\mathrm{i}}{}_{2\mathrm{M}}\\ ⋮ ⋮ ⋮ ⋮\\ {\mathrm{ik}}_{\mathrm{ip}1}{\mathrm{ik}}_{\mathrm{ip}2}\dots {\mathrm{ik}}_{\mathrm{ipj}}\dots {\mathrm{ik}}_{\mathrm{i}}{}_{\mathrm{pM}}\\ ⋮ ⋮ ⋮ ⋮\\ {\mathrm{ik}}_{\mathrm{iG}1}{\mathrm{ik}}_{\mathrm{iG}2}\dots {\mathrm{ik}}_{\mathrm{iGj}}\dots {\mathrm{ik}}_{\mathrm{i}}{}_{\mathrm{GM}}\end{array}\right],1 \le i\le Np,1\le p\le G,1\le j\le M$$

(33)

where IKD_i represents the knowledge database of the ith person, G refers to the size of IKDs, $ik{d}_{ip}$ is the best solution for the ith person. The notation p refers to a random integer that indicates which person is assigned to IKD for individual learning.

(B) Social learning operator: In addition to using the individual learning process to increase their abilities, individuals can also benefit from other people’s experiences to develop their abilities. The HLOA incorporates a social learning process to achieve an effective search. In the HLOA, everyone evaluates the knowledge gained from the collective learning in the social knowledge database (SKD) probabilistically by obtaining a new solution using the following equations [24]:

$$x_{ij}=s{k}_{qj}$$

(34)

$$SKD=\left[\begin{array}{c}\mathrm{ik}{d}_1\\ \mathrm{ik}{d}_2\\ ⋮\\ \mathrm{ik}{d}_q\\ ⋮\\ \mathrm{ik}{d}_H\end{array}\right]=\left[\begin{array}{c}s{k}_{11} s{k}_{12 }\dots s{k}_{1j} \dots s{k}_{1M}\\ s{k}_{21} s{k}_{22 }\dots s{k}_{2j }\dots s{k}_{2M}\\ ⋮ ⋮ ⋮ ⋮\\ s{k}_{q1} s{k}_{q2} \dots s{k}_{qj} \dots s{k}_{qM}\\ ⋮ ⋮ ⋮ ⋮\\ s{k}_{H1} s{k}_{H2} \dots s{k}_{Hj} \dots s{k}_{HM}\end{array}\right],1\le q\le H,1\le j\le M$$

(35)

where H indicates SKD size and skd_q refers to the social knowledge of qth in SKD.

(C) Random exploratory learning operator: Individuals, for various reasons, including a lack of concentration and forgetfulness, cannot always present the learning process based entirely on personal or social experience, and this procedure is a random process. In the HLOA, exploratory learning is presented randomly with the following equation [24]:

$$X_{ij}=RE\left(0,1\right)=\{\begin{array}{l}0,rand<0.5\\ 1,else\end{array}$$

(36)

In (37), rand represents a number in the range (0, 1), randomly.

(D) Re-learning operator: In HLOA, a person is exposed to re-learning to gain new experiences if the competency level does not improve. This operator improves the performance of the HLOA algorithm and escapes local optimization.

3.1.3. IKD and SKD Update

The fitness of the new solutions is calculated after individuals in each generation perform the learning operation. If the qualifications of the new candidates are better than the qualifications of the worst IKDs or the solutions number in the present IKDs is less than G, they will be stored in the IKDs [24].

3.1.4. Implementation of HLOA

The HLOA is implemented as the following equations based on stochastic, individual, and social learning exploratory learning and reaches a new solution [24].

$$x_{ij}=\{\begin{array}{l}RE\left(0,1\right); 0\le rand<pr\\ i{k}_{ipj}; pr\le rand<pi\\ s{k}_{qj}; else\end{array}$$

(37)

where $pr$ indicates the random exploratory learning probability, ($pi$ − $pr$) refers to the rate of individual learning, and ($1-pi$) shows the rate of social learning. If a person does not improve their competence, they are exposed to re-learning to re-experience the learning process without using their prior knowledge. The learning operations and the process of updating the HLOA are repeated until the convergence conditions are met.

3.2. Overview of IHLOA

The conventional HLOA is a powerful algorithm with optimal optimization accuracy. However, optimization algorithms in the case of increasing the complexity of the problem as well as the large size of the system may be poorly trapped in local optimization. In the proposed method, individuals can mutate by making it possible to explore and exploit the search space and achieve a desirable global optimal. Like the conventional HLOA, the fitness function is determined at the end of each iteration. However, the difference is in updating the position of the best individual as R. For example, in the conventional HLOA, the position of the best individual R is only updated when individuals discover new and better situations in subsequent iterations.

On the other hand, if the position of the best individual R is updated in the same current iteration, it is possible to improve the optimization. The Gaussian mutation improves the HLOA performance, and the extracted algorithm is named IHLOA. Therefore, before the end of each iteration, the mutation process explores and exploits the search space as much as possible. Thus, this approach improves and strengthens the algorithm’s performance in identifying and achieving the global optimal. In Figure 1, consider the individual R₁ to R₅ in the search space. These individuals are evaluated according to Figure 1 and updated according to the IHLOA method relative to the particular immune x in the last iteration. The mutation approach is based on a Gaussian distribution [26]. The mathematical expression of the Gaussian mutation is defined as follows:

$$R_g=R+\left(R_{\max }-R_{\min }\right)\ast \mathrm{Gaussian}\left(\mathrm{m},h\right)$$

(38)

where $R_g$ represents the position resulting from the mutation process, $R_{\max }$ and $R_{\min }$ are the lower and upper values of the variables. $\mathrm{Gaussian}\left(\mathrm{m},h\right)$ is a function of the Gaussian distribution where h is the standard deviation of the individual and m represents the mean of all individuals in the current iteration. If the fitness value of $R_g$ is better than $x$, then $R_g$ replaces $R$. The flowchart of the IHLOA is shown in Figure 2.

4. Problem Solving Considering Uncertainties

The steps to implement the problem of PVs allocation in the distribution system via the stochastic–metaheuristic model considering uncertainties are described below and also flowchart of stochastic–metaheuristic model for PVs allocation using the IHLOA is depicted in Figure 3.

• Data initialization

(Step 1) First, the general system data, including the line data and the system bus data, are defined for the IHLOA method, the population, iterations number, and required parameters for the optimization method.

(Step 2) Select the system load PV power, randomly based on their stochastic PDF.

• Problem Solution using the MCS

(Step 3) For Nsamp: i = 1,

(Step 4) A load and PV power sample are selected randomly.

• Deterministic solution

(Step 5) According to the population of the IHLOA, the samples are selected randomly and the objective function is calculated for each variable set. The corresponding variable set to minimum objective function is considered best solution in this step.

(Step 6) The population of the IHLOA is updated and after selecting the samples using the updated population, the objective function is calculated for each set variables. If the obtained value is lower than step 6, the solution is replaced with the old solution.

(Step 7) The obtained optimal variables (optimal installation location and size of PV resources) are stored.

(Step 8) If the total samples are initiated, go to the next step; otherwise, return to step 4.

(Step 9) The stochastic PDF of the optimization variables is extracted, and the numerical results of losses, lowest voltage, and VSI are calculated using these PDFs.

5. Results and Discussion

The results of PVs allocation in the distribution systems are presented in two cases of deterministic and stochastic methods without and with uncertainties of demand and PV power using the IHLOA. The proposed methodology is implemented on two 33 and 69-bus systems. The schematic of 33 and 69-bus systems is shown in Figure 4. The system’s data are extracted using Refs. [34,35,36]. The 33 and 69-bus systems have a 3.716 MW and 3.802 MW active load, and these systems have 2.300 MVAr and 2.696 MVAr, respectively.

The performance of the IHLOA is compared with conventional HLOA and PSO. For conventional HLOA, pr = 0.1 (random exploratory learning probability) and pi = 0.85 ((1 − pi) represent the rate of individual social learning) and for the PSO, the learning factors (C₁ and C₂) are considered 2 and inertia weight is selected between 0 and 1. These parameters are selected based on the settings of the reference paper, the evaluation of the output results, and the trial-and-error method.

5.1. Results of Base Networks

This section presents numerical results related to base systems without PVs in the deterministic case. The losses, minimum voltage, and VSI are obtained at 210.98 kW, 0.9131 p.u, and 0.7004 p.u for the 33-bus system, and also, these values are 224.97 kW, 0.9092 p.u, and 0.6885 p.u, respectively [18].

5.2. Results of Deterministic Allocation of PVs

To compare the IHLOA and evaluate the impact of PVs employed on distribution system operation, one and two panels’ optimal placement and sizing are solved deterministically. The optimization process results cleared the superior capability of the IHLOA in comparison with conventional HLOA and PSO to achieve the better objective function value with the lowest convergence tolerance and in the most inferior iteration in one and two PVs allocation in the 33 and 69-bus systems. For example, the convergence curve of different algorithms in problem-solving considering two PVs for 33 and 69-bus systems is shown in Figure 5 and Figure 6, respectively. In addition, statistical analysis of different algorithms’ performance for allocation of two PVs in the deterministic approach for the systems is given in

Table 1. Statistical Analysis of algorithms performance for allocation of PVs in deterministic approach in 33-bus system.

$\mathbf{Method}$	$\mathbf{Best}$	$\mathbf{Mean}$	$\mathbf{Worst}$	$\mathbf{STD}$
IHLOA	0.4869	0.4872	0.4878	0.08546
HLOA	0.4882	0.4892	0.4897	0.01137
PSO	0.4874	0.4880	0.4886	0.01038

and

Table 2. Statistical Analysis of algorithms performance for allocation of PVs in deterministic approach in 69-bus system.

$\mathbf{Method}$	$\mathbf{Best}$	$\mathbf{Mean}$	$\mathbf{Worst}$	$\mathbf{STD}$
IHLOA	0.4379	0.4385	0.4389	0.09594
HLOA	0.4407	0.4414	0.4421	0.01419
PSO	0.4396	0.4405	0.4409	0.01302

, as the IHLOA achieved fewer Best, Mean, Worst, and STD values than the other algorithms.

The numerical results in two PVs allocation in the systems are presented and compared with previous studies in

Table 3. Comparison of the results of sitting and sizing of two PV in deterministic condition.

System/Item	Size and Location	Losses (kW)	Minimum Voltage (pu)	VSI (pu)	Annual Cost of Losses (USD)	Saving (USD/Year)
33 Bus
IHLOA	846.6 (13), 1160 (30)	82.34	0.9789	29.479	43,277.90	67,613.19
HLOA	822.8 (13), 1143 (29)	82.81	0.978	29.368	43,524.93	67,366.15
PSO	843.5 (13), 1157 (30)	82.55	0.9785	29.457	43,388.28	67,502.80
ALO [18]	850 (13), 1191.1 (30)	82.6	0.9732	29.479	43,414.56	67,476.54
GA [40]	837.5 (13), 1212.2 (29)	82.7	0. 96846	--	--	--
DAPSO [41]	1227 (13), 738 (32)	95.93	0.9651	--	--	--
BSOA [41]	880 (13), 924 (31)	89.34	0.9665	--	--	--
69 Bus
IHLOA	733 (14), 2001 (61)	70.16	0.9876	65.865	36,876.09	81,352.36
HLOA	673 (14), 1975 (62)	70.44	0.9872	65.8312	37,023.26	81,220.96
PSO	731 (14), 1996 (61)	70.29	0.9874	65.8537	36,944.42	81,299.81
ALO [18]	538.77 (17), 1700 (61)	70.75	0.9801	65.8042	37,186.20	81,042.26
GA [40]	1777 (61), 555 (11)	71.79	--	--	--	--
SGA [42]	1000 (17), 2400 (61)	82.9	--	--	--	--
PSO [42]	700 (14), 2100 (62)	78.8	0.9732	29.479	--	--
MTLBO [43]	519.7 (17), 1732(61)	71.77	0.9732	29.479	--	--

. For the 33-bus system, the optimal locations are buses 13 and 30, and for the 69-bus system, buses 14 and 61, where the optimal sizes of the panels are 846.6 kW and 1160 kW for the 33-bus system and 733 kW and 2001 kW for the 69-bus system. It should be said that this is an optimization program that determines the installation location and optimal size of photovoltaic sources, and it must not be the same as other sources. Of course, it is clear that the location of installing photovoltaic resources by IHLOA is the same as the methods in the ALO [18] for the 33-bus distribution system. In addition, for the 69-bus network, many methods have chosen bus 14 or bus 61 to install one of the photovoltaic sources. The loss of the 33-bus system is obtained at 84.81 kW. The minimum voltage is 0.9789; the total VSI is 29.479, the losses cost is USD 43,015.10, and the net saving obtained is USD 67,875.99 per year. The annual cost of losses is obtained from the product of the power losses value in the number of hours of a year (8760 h) in the cost of each kW of losses (USD 0.06) [18]. For the 69-bus system, the system losses are determined as 70.16 kW, the minimum voltage and the VSI index are 0.9876 and 65.8650 p.u, respectively, the cost of losses is USD 36,876.09, and the net saving is determined as USD 81352.36 per year. In addition, comparing the results in two PVs placement with conventional HLOA, PSO, the previous methods in [18,37,38,39,40,41,42,43] showed the optimal performance of the IHLOA. Therefore, IHLOA is a suitable method to optimize the site and size of PV and wind resources in the distribution system. The voltage profile curves of the 33 and 69-bus systems for one and two PVs placement using the IHLOA are depicted in Figure 7 and Figure 8, respectively. As can be seen, by utilizing optimal PVs, the voltage profiles of the test systems are enhanced, and when the two panels are allocated in the distribution systems optimally, the system voltage profile is better compared to the case where only one panel is employed.

5.3. Results of Stochastic Allocation of PVs

The results of PVs allocation with uncertainties of demand and PVs power are presented using IHLOA.

5.3.1. Results of Base Networks

In the worst conditions, peak demand is selected. A normal PDF (Figure 9) with an 85% peak average with a 25% deviation for each bus demand is considered for the load. In buses, the demand can change in 20 to 160%, where most of these changes are around 85%. The PDF for the load is shown in Figure 9.

The power of PVs is generated by following the Beta distribution, which is the most appropriate stochastic for the power of PVs PDF. In this study, the PDF of the power generated by one PV should be such that the maximum capacity is 3000 kW [21], and the power range is close to 3000 kW. The PDF of PV power generation with solar radiation for 25-year-old data is extracted. The PDF of power generated by PVs with two parameters of 50 and 3 are respectively shown in Figure 10. At first, the results without PVs placement are presented. The system load profile, considering the uncertainty shown in Figure 9, is applied to the two systems. The stochastic evaluation for losses, VSI, and the minimum voltage is presented. These evaluations are shown in Figure 11 and Figure 12 for the 33 and 69-bus systems, respectively. The losses for the two systems are around 200 kW, where due to the load uncertainty and the probability of load increasing, they may exceed 300 kW. The system has a minimum voltage of 0.92 p.u, where the voltage occurs in values of less than 0.9. Now, the stochastic placement and sizing of PVs are done considering uncertainties.

Considering the uncertainty, with a low probability, the losses may exceed 300 kW. VSI conditions are so unsuitable, and even the system voltage may reduce to less than 0.88 p.u and 0.87 p.u. in the 33 and 69-bus systems, respectively. These conditions are not observed in the case without considering uncertainty. In other words, considering the uncertainty system operator can be aware of the dangerous situations that are likely to occur. However, the point should also be noted that the probability of losses reaching around 100 kW is also high. However, the operator must prepare the distribution system for the worst possible scenarios; but a critical point is the probability of occurrence.

5.3.2. Results of PVs Placement and Sizing

The optimal locations of two PVs are determined with uncertainties of the load and PV power using IHLOA. So, 1000 samples randomly for uncertain parameters are generated according to their PDFs, and 30 optimization runs are performed. The best location and size of PVs for the 33-bus and 69-bus systems are illustrated in Figure 13, Figure 14, Figure 15 and Figure 16, respectively. The PVs in the 33-bus system have a generation power of nearly 800 kW, with the best locations around the buses 30 and 13, which also matches the results of the deterministic condition. For the 69-bus system, the best size is around 1500 kW, and the best places to install PV are buses 61 and 52.

The stochastic results based on PDF for the 33- and 69-bus systems are depicted in Figure 17 and Figure 18, respectively. As illustrated in Figure 17, for the 33-bus system, the losses are usually around 80 kW, while for the stochastic setting of one PV, it is about 100 kW. In addition, the VSI is in a 1.12 p.u value, and the minimum voltage is close to 0.988, with more occurrence probability. As depicted in Figure 18, for the 69-bus system, the losses in 30% of the values are around 70 kW. The minimum VSI with the value of 1.18 p.u and the minimum voltage around 0.97 p.u are obtained with more occurrence probability.

5.4. Comparison of the Deterministic and Stochastic Results

This section compares the results of the deterministic and stochastic cases using the IHLOA according to

Table 4. Comparison of the results of PVs allocation in deterministic and stochastic conditions for a 33-bus distribution network.

System/Item	Power Loss	Minimum Voltage	Voltage Stability Index
Deterministic	82.34	0.9789	29.479
Stochastic	91.34	0.9665	28.087

and

Table 5. Comparison of the results of sitting and sizing of PVs in deterministic and stochastic conditions for a 33-bus distribution network.

System/Item	Power Loss	Minimum Voltage	Voltage Stability Index
Deterministic	70.16	0.9876	65.865
Stochastic	76.63	0.9752	63.427

. As can be seen in these tables, the power losses of distribution systems in the stochastic case obtained more value compared with the deterministic case. Table 4 and Table 5 illustrate that the minimum voltage and VSI are achieved more in stochastic than in deterministic cases. So, the results indicated that considering uncertainties, the system characteristics can be changed, and the decision-making of the distribution systems operators must be adapted to these changes for a realistic and more accurate view of challenges and worst possible conditions of the distribution systems.

6. Conclusions

In this paper, stochastic–metaheuristic allocation of the PV energy resources is proposed in distribution networks with the objective of minimizing the losses, enhancing the voltage profile, and improving the voltage stability in two stochastic and deterministic cases. The PV generation power and system load are considered uncertain parameters in the stochastic case. The optimization variables, including location and PVs size, are found optimally via IHLOA to be improved based on the Gaussian mutation to explore and exploit better the search space and achieve a desirable global optimal. The capability of IHLOA in the deterministic case solution is proven compared with conventional HLOA and PSO and previous studies to attain fewer power losses and more voltage profile and VSI with more convergence speed and less convergence tolerance. In addition, the simulations cleared that the proposed IHLOA obtained the optimal global solution with the lowest convergence tolerance and highest convergence speed and accuracy compared to conventional HLOA and PSO. Moreover, the problem solution is performed with the stochastic case and considering PDF of uncertainties based on MCS. The results demonstrated that the losses of distribution systems in the stochastic case is obtained more, and minimum voltage and VSI are achieved as smaller than the deterministic case values. Therefore, considering uncertainty as an inherent phenomenon has changed the objectives of losses and voltage profile and stability. Thus, considering uncertainty gives a more realistic and more accurate view to the designers and operators of the distribution systems relative to the challenges and worst possible conditions of the distribution system, which will lead to the implementation of some decisions with more desirable results. The allocation of PV resources in distribution incorporating power quality improvement and uncertainties is suggested for future work.