Stochastic-Metaheuristic Model for Multi-Criteria Allocation of Wind Energy Resources in Distribution Network Using Improved Equilibrium Optimization Algorithm

Abstract

In this paper, a stochastic-meta-heuristic model (SMM) for multi-criteria allocation of wind turbines (WT) in a distribution network is performed for minimizing the power losses, enhancing voltage profile and stability, and enhancing network reliability defined as energy not-supplied cost (ENSC) incorporating uncertainty of resource production and network demand. The proposed methodology has been implemented using the SMM, considering the uncertainty modeling of WT generation with Weibull probability distribution function (PDF) and load demand based on the normal PDF and using a new meta-heuristic method named the improved equilibrium optimization algorithm (IEOA). The traditional equilibrium optimization algorithm (EOA) is modeled by the simple dynamic equilibrium of the mass with proper composition in a control volume in which the nonlinear inertia weight reduction strategy is applied to improve the global search capability of the algorithm and prevent premature convergence. First, the problem is implemented without considering the uncertainty as a deterministic meta-heuristic model (DMM), and then the SMM is implemented considering the uncertainties. The results of DMM reveal the better capability of the IEOA method in achieving the lowest losses and the better voltage profile and stability and the higher level of the reliability in comparison with conventional EOA, particle swarm optimization (PSO), manta ray foraging optimization (MRFO) and spotted hyena optimization (SHO). The results show that in the DMM solving using the IEOA, traditional EOA, PSO, MRFO, and SHO, the ENSC is reduced from $3223.5 for the base network to $632.05, $636.90, $638.14, $635.67, and $636.18, respectively, and the losses decreased from 202.68 kW to 79.54 kW, 80.32 kW, 80.60 kW, 80.05 kW and 80.22 kW, respectively, while the network minimum voltage increased from 0.91308 p.u to 0.9588 p.u, 0.9585 p.u, 0.9584 p.u, 0.9586 p.u, and 0.9586 p.u, respectively, and the VSI improved from 26.28 p.u to 30.05 p.u, 30.03 p.u, 30.03 p.u, 30.04 p.u and 30.04 p.u; respectively. The results of the SMM showed that incorporating uncertainties increases the losses, weakens the voltage profile and stability and also reduces the network reliability. Compared to the DMM, the SMM-based problem is robust to prediction errors caused by uncertainties. Therefore, SMM based on existing uncertainties can lead to correct decision-making in the conditions of inherent-probabilistic changes in resource generation and load demand by the network operator.

1. Introduction

Today, the application of renewable energy sources is rapidly increasing with the aim of reducing dependence on fossil fuels and reducing environmental pollution due to conventional power generation. On the other hand, due to the extent and variety of the required tools, renewable energy-based generation has been allocated a large amount of power system investment [1]. Therefore, with the correct design of this part of the power system, while improving the condition of the customers, the economic efficiency of the system can be increased. By installing distributed generation (DG) such as clean wind energy sources, which are conventional and widely used types, the costs of the power system production can be reduced in the distribution network [2,3]. The generators in DGs exchange active and reactive power with the network, and this itself causes changes in the basic parameters of the network. Thus, the installation location of these resources should always be investigated in the design of the power system. The design problem with the presence of DGs depends on the definition of several different factors, which include the best technology to be applied, the number and capacity of the DGs, the best installation location and the type of connection to the network of these resources. As mentioned, installing these sources has advantages such as loss reduction and enhancing voltage profile and reliability, which should be evaluated [4,5]. Among the types of DGs based on renewable energy, the use of wind energy sources has received a lot of attention due to its clean and free energy, which has caused sufficient motivation to build more and more wind power plants [6,7]. The energy produced from wind turbines can have a significant effect on the technical and economic improvement of power networks. By identifying the optimal size and location of wind turbines in power networks, the most technical and economic improvements can be achieved. The wind energy is dependent on weather conditions. Thus, accurate forecasting of weather conditions results in accurate prediction of the wind power production. But due to the uncertainty in predicting weather conditions, the prediction of the generated power also exhibits uncertainty. This issue causes many challenges in the power system [8,9]. In addition, production planning plays a fundamental role in obtaining the objectives of the electricity industry because electrical energy cannot be easily stored. Load forecasting is one of the most important inputs to the process, at the end of which the amount of production of existing units in the coming hours, days and months and the necessary capacity for the development of power plants in the coming years are determined. In other words, the main prerequisite for network planning in different time horizons is to estimate and predict the future load of the network [10]. The accurate prediction of the electric load plays an essential role for accurate operation in power networks. Considering that there are always some errors in the forecasts due to the random behavior of the load, the accuracy of load forecasting has been given much attention by the electricity industry. High forecasting causes additional investment to create unused reserves and a lack of optimal productivity. Low forecasting will cause a lack of production and damage to the equipment due to overload. Therefore, in order to prevent the undesirable effects of wind resource production and demand, it is necessary to consider it in the design and network operation with the wind resources allocation.

1.1. Literature Review

Several research efforts have been performed in the allocation of DGs as well as renewable energy resources. In [11], the allocation of DGs in the network is developed through minimizing the losses and voltage deviations considering the uncertainty of load and generation of DGs based on the adaptive genetic algorithm (AGA) and fuzzy logic. In [12], the wind turbines allocation in the network is developed by minimizing the losses and considering the constraint of the maximum allowed wind power capacity via the particle swarm optimization (PSO). In [13], the PSO is employed to find the optimal site and capacity of wind turbines in the networks for minimizing the losses and improving the voltage profile. In [14], the PSO is utilized to identify the optimal site and capacity of wind turbines in the networks to reduce losses and enhance the voltage profile. In [14], the optimal installation location and DG size are determined based on the power loss index and the whale optimization algorithm (WOA). The results show that compared to other methods, the WOA has obtained a greater reduction in network losses. In [15], the loss reduction and voltage fluctuations minimization of the network based on the DGs allocation has been investigated using the improved PSO (IPSO). The results demonstrated that the IPSO has improved the network performance more than other optimization methods. In [16], the DGs allocation has been developed using the PSO to enhance the network voltage profile and stability. The results show that the proposed method has improved the network voltage conditions compared to other methods. In [17], the chaotic search group algorithm (CSGA) is presented for DGs allocation to minimize the losses in the network, and the results showed that the CSGA provided a better performance compared to the traditional version of the algorithm in achieving lower losses. In [18], enhancing the network performance based on the optimal allocation of DGs with the aim of losses reduction and voltage profile and stability enhancement has been studied using the chaotic bat algorithm (CBA). The results show the superiority of the CBA in obtaining less losses and the better enhancement of the network voltage. In [19], the ant colony optimization (ACO) is developed for the DGs allocation in the network with the aim of reducing losses. The results showed that the ACO had a desirable capability in reducing losses and voltage deviations. In [20], the Archimedes optimization algorithm (AOA) is presented to improve voltage stability with wind resource allocation. The results proved the effectiveness of the AOA in stabilizing output power changes. In [21], a modified gravitational search algorithm (MGSA) is used for DGs allocation to minimize the network losses, and this method has achieved the lowest network power losses. In [22], the PSO is used to determine the optimal site and the best capacity of wind resources with the objective of minimizing the losses and voltage fluctuations of the network. The results illustrated the superiority of the PSO in enhancing the voltage profile and improving system reliability. In [23], the allocation of wind energy resources in the network is developed to minimize losses and enhance reliability via the moth-flame optimizer (MFO). The results confirmed that the MFO has enhanced the network reliability. In [24], the PSO is presented for the allocation of a combined heat and power (CHP) system for reducing the losses and voltage fluctuations and enhancing the network reliability. The results showed that through optimal determination of the location and size of the CHP, the best value of each objective can be achieved. In [25], the best installation location and size of the wind turbine in the network is determined to minimize the losses and voltage fluctuations via artificial electric field algorithm (AEFA). Compared to other well-known algorithms, the MOAEFA method results in more loss reduction and better improvement of the network voltage conditions. In [26], the optimal allocation of wind energy resources in the network is developed to minimize losses and improve reliability using hybrid teaching learning optimization-grey wolf optimizer (TLBO-GWO). The results have confirmed the superiority of the combined method in avoiding becoming trapped in the local optimum, as well as achieving lower losses and a higher level of reliability. In [27], an efficient method is presented for allocating wind turbines in radial networks with the aim of improving the voltage stability index and by considering the uncertainties of network load and wind power. In [28], a desirable method for allocating wind turbines in the networks is implemented to enhance the voltage stability index and incorporating the uncertainties of network demand and wind generation. In [27], the allocation of wind resources in the networks considering the uncertainty of wind power is performed with the aim of loss reduction and voltage profile and stability enhancement using the PSO. The literature review is summarized in

Table 1. Literature review summary.

Improved Algorithm	Uncertainty		Device		Objective Function				Year	Ref
	Demand	Generation	WT	DG	ENS	VSI	VD	Loss
-	✓	✓	-	✓	-	-	✓	✓	2015	[11]
-	-	-	✓	✓	-	-	✓	✓	2016	[12]
-	-	-	✓	✓	-	-	✓	✓	2017	[13]
-	-	-	✓	✓	-	-	-	✓	2018	[14]
✓	-	-	-	✓	-	✓	✓	-	2022	[15]
-	-	-	-	✓	-	✓	✓	-	2021	[16]
✓	-	-	-	✓	-	-	-	✓	2022	[17]
✓	-	-	-	✓	-	✓	✓	✓	2022	[18]
-	-	-	-	✓	-	-	-	✓	2021	[19]
-	-	-	✓	✓		✓			2022	[20]
-	-	-	-	✓	✓	✓	✓	✓	2020	[21]
-	-	-	-	✓	✓	-	✓	✓	2020	[22]
-	-	-	✓	✓	✓	-	-	✓	2020	[23]
-	-	-	-	✓	✓	-	-	✓	2020	[24]
-	-	-	✓	✓		-	✓	✓	2021	[25]
✓	-	-	✓	✓	✓	-	-	✓	2019	[26]
	✓	✓	✓	✓	-	✓	-	-	2022	[27]
-	-	✓	✓	✓	-	✓	✓	✓	2020	[28]
✓	✓	✓	✓	✓	✓	✓	✓	✓	2022	This paper

.

1.2. Research Gap, Contributions, and Objectives

It is obvious that in the studies conducted in Table 1, the uncertainty of generated power and load using a multi-criteria stochastic-metaheuristic model (SMM) is not completely and accurately addressed in the network operation based on wind energy sources. The main focus of this paper is to employ the SMM for multi-criteria optimal allocation of wind resources considering uncertainties of wind power generation and network demand via Monte Carlo simulation (MCS) based on the probability distribution function (PDF) and a new meta-heuristic optimization algorithm.

In this paper, the multi-criteria allocation of wind turbines in the distribution network is performed to reduce losses, enhance the voltage profile, stability, and reliability of the network considering uncertainties in wind generation and network load using a new meta-heuristic optimization algorithm based on the SMM. The decision variable including location, capacity, and power factor of the wind turbine in the network is determined optimally using a new improved equilibrium optimization algorithm (IEOA). The traditional equilibrium optimization algorithm (EOA) [29] is inspired by the simple dynamic balance of the mass with proper composition in a control volume, and the global search capability of the algorithm is enhanced based on the nonlinear inertia weight reduction strategy. The proposed methodology is implemented on the IEEE 33-bus radial distribution network. The problem is implemented in two deterministic (DMM) and stochastic (SMM) approaches without and with considering the uncertainties. The superiority of the IEOA in DMM solving is compared with traditional EOA and particle swarm optimization (PSO), manta ray foraging optimization (MRFO) and spotted hyena optimization (SHO). Moreover, the results of DMM and SMM solving based on the IEOA are compared and analyzed in view of losses, network minimum voltage and voltage stability as well as reliability cost.

The main contribution of the paper can be summarized as follows:

• Presenting a stochastic-metaheuristic model for allocation of wind turbines in the distribution network considering uncertainties
• Performing Monte Carlo simulation (MCS) based on the probability distribution function (PDF)
• Using new improved equilibrium optimization algorithm (IEOA) for problem optimizing
• Attesting to the superior capability of the IEOA compared with traditional EOA, PSO, MRFO and SHO
• Correcting decision-making by the network operator against existing risks caused by uncertainties

1.3. Paper Structure

In Section 2, the proposed problem is formulated with the objectives and constraints. In Section 3, the load and wind generation uncertainty model is presented. The proposed IEOA and its implementation process are described in Section 4. In Section 5, the results are presented, and the key conclusion is drawn in Section 6.

2. Formulation of the Problem

The multi-objective allocation of wind resources in the network is formulated to minimize the losses, improve the voltage profile and stability, and minimize the cost of improving the system reliability (the cost of unsupplied energy to customers).

2.1. Objective Function

2.1.1. Power Losses

One of the objectives considered is the minimization of the network power losses in the network lines, which is defined by [15,16]

$${\rho }_{{}_{loss}}^{\varphi }={\mu }_{\varphi }.C_{\varphi }^{{}^2}$$

(1)

$$C_{\varphi }=\frac{{\kappa }_j-{\kappa }_i}{{ƛ}_{\varphi }+j{\hslash }_{\varphi }}$$

(2)

$${\rho }_{{}_{loss}}^k={ƛ}_{\varphi }.{\left|\frac{{\kappa }_j-{\kappa }_i}{{ƛ}_{\varphi }+j{\hslash }_{\varphi }}\right|}^2$$

(3)

$${\rho }_{{}_{loss}}^{}={\displaystyle \sum _{\varphi =1}^{\psi }{\rho }_{{}_{loss}}^{\varphi }}$$

(4)

where, $\varphi$ is the line number between buses i and j, $ƛ$ and $\hslash$ are the resistance and reactance of the line, $C$ is current transferred from the line, ${\kappa }_i$ and ${\kappa }_j$ are voltage of buses i and j, ${\rho }_{{}_{loss}}^{}$ is network line losses and $\psi$ is total number of lines.

2.1.2. Voltage Stability

This objective considers the voltage stability index (VSI) enhancement. The VSI is defined as follows [16].

$$VS{I}_{(z+1)}={\left|{\kappa }_z\right|}^4-4{\left({\rho }_{z+1}{\hslash }_z-{\Phi }_{z+1}{ƛ}_z\right)}^2-4\left({\rho }_{z+1}{ƛ}_z-{\Phi }_{z+1}{\hslash }_z\right){\left|{\kappa }_z\right|}^2$$

(5)

where, ${\rho }_{z+1}$ and ${\Phi }_{z+1}$ are active and reactive power of the bus z + 1. A larger VSI value indicates a better stability of the bus voltage. The objective function of improving voltage stability is considered as minimization of ${\beta }_{VSI}=1/VS{I}_{(z+1)}$, which means maximizing the voltage stability of buses.

2.1.3. Improving the Voltage Profile

Minimizing the network buses voltage deviation is an important objective in the network operation, and is presented as follows [18,19].

$$\kappa d_{}={\displaystyle \sum _{i=1}^{\varsigma }\left|{\kappa }_i-1\right|}$$

(6)

where, $\kappa d$ is the value of network voltage deviations from the value of 1 p.u, $\varsigma$ is number of buses and ${\kappa }_i$ is voltage of bus i.

2.1.4. Reliability Improvement

Due to the outage probability of network lines, the power flow to customers is subject to interruption. Therefore, in the network operation based on wind energy, the cost of energy not-supplied is computed by [30]

$$\mathrm{{\rm E}}n\Im ={\displaystyle \sum _{i=1}^{\psi }{\displaystyle \sum _{j=1}^{\gamma }{\mathrm{\Omega }}_i\times {\ell }_i\times {\Upsilon }_i\times {\sigma }_j}}$$

(7)

$$\mathrm{{\rm E}}n\Im C=\mathrm{{\rm E}}n\Im \times {\chi }_{\mathrm{{\rm E}}n\Im }$$

(8)

where, $\psi$ is total number of lines in the network, $\gamma$ is outages number due to the outage of line i, ${\mathrm{\Omega }}_i$ is outage rate of line i, ${\ell }_i$ is length of line i, ${\Upsilon }_i$ is duration of line repair, ${\sigma }_j$ refers to the amount of interrupted load of bus i, ${\chi }_{\mathrm{{\rm E}}n\Im }$ is cost per kWh of interrupted electricity customers and $\mathrm{{\rm E}}n\Im C$ represents the cost of $\mathrm{{\rm E}}n\Im$.

2.2. Problem Constraints

The objective function should be considered during the problem solving with satisfying the below constraints [15,16,17,18,19].

• WT power

$${\rho }_{Lower}^{WT}\le {\rho }_{WT}\le {\rho }_{Upper}^{WT}$$

(9)

• Maximum installed power of WT

$${\displaystyle \sum _{i=1}^{N_{WT}}{\rho }_{WT}^i}\le Allowable{}_{WT}$$

(10)

• Voltage of buses

$${\kappa }_{Lower}^i\le {\kappa }_i\le {\kappa }_{Upper}^i$$

(11)

• Thermal limit

$${\rho }^{’}{}_{ij}\le Allowabl{e}_{ij}$$

(12)

where, ${\rho }_{Lower}^{WT}$ and ${\rho }_{Upper}^{WT}$ represents the minimum and maximum wind power values, $Allowabl{e}_{WT}$ is the maximum wind power that can be installed in the network, ${\kappa }_{Lower}^i$ and ${\kappa }_{Upper}^i$ the lower and upper voltage values of the bus network, ${\rho }^{’}{}_{ij}$ representing the passing apparent power and $Allowabl{e}_{ij}$ is the thermal limit value.

2.3. Multi-Objective Optimization

The objective function includes losses minimization, voltage deviations and cost of $\mathrm{{\rm E}}n\Im$ and enhancing the voltage stability. Considering that each objective has different dimensions from the others, therefore the multi-objective optimization method of weight coefficients has been applied for normalization of the objective function as below.

$$F={\omega }_1\left(\frac{F_{{\rho }_{loss}}}{F_{{\rho }_{loss},Upper}}\right)+{\omega }_2\left(\frac{F_{{\beta }_{VSI}}}{F_{{\beta }_{VSI},Upper}}\right)+{\omega }_3\left(\frac{F_{{\kappa }_d}}{F_{{\kappa }_d}{}_{,Upper}}\right)+{\omega }_4\left(\frac{F_{E_{n\Im C}}}{F_{E_{n\Im C}}{}_{,Upper}}\right)$$

(13)

where, ${\omega }_1$, ${\omega }_2$, ${\omega }_3$ and ${\omega }_4$ are losses weighted values, voltage stability, voltage deviations and cost of $\mathrm{{\rm E}}n\Im$, respectively, whose absolute value is equal to 1. $F_{{\rho }_{{}_{loss}}^{},Upper}$, $F_{{\beta }_{VSI},Upper}$, $F_{\kappa d,Upper}$ and $F_{\mathrm{{\rm E}}n\Im C,Upper}$ are respectively the maximum value of losses, voltage stability index, voltage deviations and also $\mathrm{{\rm E}}n\Im C$.

3. Uncertainty Model

3.1. Load Demand

Uncertainty in network load estimation is inevitable. Therefore, network design by considering load uncertainty makes the designed network optimal and robust to network load changes. Therefore, the effect of load uncertainty in the operation of the network should be evaluated. In this study, the Monte Carlo simulation method is used to model load uncertainty and it is modeled as a normal PDF as shown below [31].

$$f(Dmd)=\frac{1}{\Gamma \sqrt{2\pi }}{e}^{\frac{-(Dmd-H)}{2{\Gamma }^2}}$$

(14)

where, Dmd represents the network demand, and f(Dmd) represents the PDF of the network demand. In addition, the parameters H and $\Gamma$ refer to the average and deviation from the network demand criterion, respectively.

3.2. Wind Power

Renewable DGs sources such as wind turbines exhibit intermittent characteristics, and their produced power is determined based on meteorological conditions. Therefore, the output of these resources is not certain, and for planning and operating the network, one must act based on predictions. Therefore, the planning of the networks in which these resources are present should be done in a probabilistic manner and should be considered in the different conditions that may occur. The uncertainty of wind power has also been investigated for these resources’ allocation in the network. To model the uncertainty of wind power, the Weibull PDF is used, which is modeled by [32]

$$f_{\upsilon }(\upsilon )=\frac{\delta }{\zeta }{(\frac{\upsilon }{\zeta })}^{\delta -1}{e}^{-{(\frac{\upsilon }{\delta })}^{\delta }}$$

(15)

where, $\nu$ is wind speed, and $\zeta$ and $\delta$ refer to the scale and shape parameter, respectively.

4. The Proposed Algorithm

4.1. Introduction of EOA

The EOA is modeled using the simple dynamic equilibrium of mass with the right combination in a control volume. The first-order differential equation describing the mass equilibrium is presented in (16), based on considering the changes in mass as the sum of the mass entering the system and the mass produced within it minus the amount of mass leaving the system [29]:

$$V\frac{dc}{dt}=Q{C}_{eq}-QC+G$$

(16)

That is, C is the mass inside the control volume (V), VdC/dt is the amount of mass change in the control volume, Q is the amount of inlet and outlet flow rate, C_eq is the equilibrium mass, and G is the amount of mass produced among the control volume. Equation (16) can be rewritten as follows [29]:

$$\frac{dc}{\lambda C_{eq}-\lambda C+\frac{G}{V}}=dt$$

(17)

Integrating (17) results in:

$$C=C_{eq}+\left(C_0-C_{eq}\right)F+\frac{G}{\lambda V}\left(1-F\right)$$

(18)

Therefore, based on (18), F is presented as follows [29]:

$$F=\exp \left[-\lambda \left(t-t_0\right)\right]$$

(19)

where t₀ and C₀ are the initial start time and mass. Equation (18) is used to calculate the mass of the control volume with a certain value.

In EOA, a particle is equivalent to one solution, which is the mass of the same particle position in the PSO.

4.1.1. Initializing and Calculating the Fitness

Raw masses are produced randomly in the search space by the particle number and dimensions with initialization:

$$C_i^{\hspace{0.17em}initial}=C_{\min }+ran{d}_i\left(C_{\max }-C_{\min }\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,n$$

(20)

where C_i^initial is initial mass vector corresponding to the i-th particle, C_min, and C_max the lower and upper edge, rand_i denotes the random vector in the range [1, 0], and n is EOA population particle numbers. The value of the suitability function of each particle is examined and, after storage, equilibrium candidates are identified.

4.1.2. Balance Pool (C_eq)

In EOA, the equilibrium state refers to the convergence condition of the algorithm, which represents the global optimal. Initially, no equilibrium information is available and only equilibrium candidates are selected to achieve a particle search strategy. Four candidates are considered to make the EOA method more explorable and their average strengthens the operation phase. Equilibrium candidates are defined as equilibrium pool vectors based on the following relation [29]:

$${\overrightarrow{C}}_{eq,pool}=\left\{{\overrightarrow{C}}_{eq\left(1\right)},{\overrightarrow{C}}_{eq\left(2\right)},{\overrightarrow{C}}_{eq\left(3\right)},{\overrightarrow{C}}_{eq\left(4\right)},{\overrightarrow{C}}_{eq\left(ave\right)}\right\}$$

(21)

In each iteration of the EOA, each particle randomly updates itself from among the candidates presented with a similar probability.

4.1.3. Exponential Expression (F)

The exponential expression (F) in the EOA strikes a balance among the exploration and exploitation phases. Over time, the rate of return fluctuates in a real control volume so the value of λ is defined as a vector in the range [1, 0], randomly. Equation (19) is rewritten as follows [29].

$$\overrightarrow{F}=e^{-\overrightarrow{\lambda }\left(t-t_0\right)}$$

(22)

The duration t in terms of repetition (Iter) is presented as follows [29]:

$$t={(1-\frac{Iter}{Max-iter})}^{(a_2\hspace{0.17em}\frac{Iter}{Max-iter})}$$

(23)

That is, Iter and Max_iter refer to the number of present iterations and the max number of EOA iterations; respectively. Parameter a₂ is a fixed number and is used for EOA usability. The following equation is presented to strengthen the convergence of the EOA method and to improve the capability of phases related to exploration and exploitation [29]:

$${\overrightarrow{t}}_0=\frac{1}{\overrightarrow{\lambda }}In(-a_{1}sign((\overrightarrow{r}-0.5)\left[1-e^{-\overrightarrow{\lambda }t}\right])+t$$

(24)

where, a₁ is a constant value and is responsible for controlling the discovery of the algorithm. Larger values of a₁ improve exploration and reduce operating phase performance. On the other hand, smaller values of a₁ weaken the exploration and improve the exploitation. The expression sign (r − 0.5) affects the direction of exploration and exploitation, r is a random vector in the range [1, 0]. In this study, a₁ and a₂ were selected as 2 and 1; respectively.

Using (24), Equation (22) can be re-written as [29]:

$$\overrightarrow{F}=a_{1}sign\hspace{0.17em}(\overrightarrow{r}-0.5)\left[e^{-\overrightarrow{\lambda }t}-1\right]$$

(25)

4.1.4. Production Rate (G)

The production rate is one of the important parameters in achieving an accurate solution by improving the operation phase. The final set of production rate equations is expressed as follows [29]:

$$\overrightarrow{F}=a_1\hspace{0.17em}sign(\overrightarrow{r}-0.5)\left[e^{-\overrightarrow{\lambda }t}-1\right]$$

(26)

$$\overrightarrow{G}={\overrightarrow{G}}_0\hspace{0.17em}{e}^{-\overrightarrow{\lambda }\left(t-t_0\right)}={\overrightarrow{G}}_0\hspace{0.17em}\overrightarrow{F}$$

(27)

$$\overrightarrow{G_0}=\overrightarrow{GCP}\left(\overrightarrow{C_{eq}}-\overrightarrow{\lambda }\overrightarrow{C}\right)$$

(28)

$$\overrightarrow{GCP}=\left\{\begin{array}{l}0.5\hspace{0.17em}{r}_1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_2\ge GP\\ \hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_2\prec GP\end{array}\right.$$

(29)

where, G₀ represents the value of initial and k refers to the damping constant, r₁ and r₂ are random numbers in the range [1, 0]. The GCP vector represents the production rate control parameter, and the GP represents the production probability.

Therefore, the EOA update rule is defined based on the following relation [29]:

$$\overrightarrow{C}={\overrightarrow{C}}_{eq}+\left(\overrightarrow{C}-{\overrightarrow{C}}_{eq}\right)\hspace{0.17em}.\hspace{0.17em}\overrightarrow{F}+\frac{\overrightarrow{G}}{\overrightarrow{\lambda \hspace{0.17em}}V}\left(1-\overrightarrow{F}\right)$$

(30)

where, the first term in the left-hand side refers to the equilibrium mass and the second and third terms refer to the mass changes. The role of the second term is global search to achieve the optimal value. The task of the third term is to further strengthen the exploration phase.

4.1.5. Particle Memory Storage

In the current iteration, the fitness value of a particle is evaluated to its previous value, and if the value of the fitness function is better, it is rewritten. Therefore, this strategy strengthens the operation phase and improves the performance of EOA in trapping in local optimization [29].

4.2. Overview of IEOA

In solving the optimization problem, the amount of inertia weight is very effective. In the situation where the inertia weight has a large number, the algorithm has a better capability in a global search. But in a situation with a small number, the algorithm has a better capability in a local search. In the EOA, the value of the inertia weight in the initialization is considered as a fixed number of 1. Therefore, it is necessary to consider the dynamics of inertial weight in strengthening its performance to reach the global optimum faster and avoid premature convergence and getting trapped in the local optimal. In the optimization process of the EOA, to improve the convergence, the nonlinear inertia weight reduction strategy [33] has been used as follows:

$$\varpi (t)={\varpi }_{Lower}+{\left(\frac{1+\cos (\frac{\pi Iter}{Max\_iter})}{2}\right)}^{\epsilon }\times \left({\varpi }_{Upper}-{\varpi }_{Lower}\right)$$

(31)

where, ${\varpi }_{Lower}$ and ${\varpi }_{Upper}$ are lower and upper values of $\varpi$, respectively. here $\epsilon$ = 10 [33]).

By combining Equations (30) and (31), the concentration update is given as follows:

$$\overrightarrow{C}=\varpi (t)\overrightarrow{C_{eq}}+(\overrightarrow{C}-\overrightarrow{C_{eq}})\overrightarrow{F}+\frac{\overrightarrow{G}}{\overrightarrow{\lambda }V}(1-\overrightarrow{F})$$

(32)

$$\overrightarrow{C}=({\varpi }_{Lower}+{(\frac{1+\cos (\frac{\pi Iter}{Max\_iter})}{2})}^{\epsilon }\times \left({\varpi }_{Upper}-{\varpi }_{Lower}\right))\overrightarrow{C_{eq}}+(\overrightarrow{C}-\overrightarrow{C_{eq}})\overrightarrow{F}+\frac{\overrightarrow{G}}{\overrightarrow{\lambda }V}(1-\overrightarrow{F})$$

(33)

4.3. IEOA Implementation

In this section, the implementation steps of solving the problem of allocation of wind turbines in the network optimally are performed based on the SMM model and IEOA. A total of 500 random samples based on the MCS have been considered for load demand as well as wind power for network operation in the presence of uncertainty. In solving the problem, based on the defined scenarios of network load demand and wind power, a PDF for the turbine installation location, a PDF for the capacity, and also a PDF for the power factor of the wind power are presented. The flowchart of the SMM model implementation is shown in Figure 1.

The steps of wind turbine allocation in the network, taking into account the uncertainties, are presented below.

• Step (1) Applying the data of the 33-bus network such as load and network lines data, as well as random generation of the initial population of the variables set, which include the wind turbine location, its power capacity, and its power factor (candidate buses between buses 2 to 33, turbine capacity between 0 and 3 MW and the power factor between 0 and 1) and also determining the parameters of the algorithm including iteration number and algorithm population (number of 200 iteration and 50 populations). Also, at this stage, the set of optimization variables are randomly selected by each member of the IEOA population.
• Step (2) At this stage, wind power and network load demand are considered as uncertain parameters, and the effect of these uncertainties on solving the problem is investigated. According to wind generation and network demand uncertainties, the load PDF is extracted using the MCS with a standard deviation of 20% around the peak load (selection of 500 scenarios (N_sam)). In extracting scenarios, the scenario reduction method is used to achieve better defined patterns and remove scenarios that are very far from deterministic data. At each stage, the problem is solved deterministically by selecting a scenario of wind power and network load demand based on its PDF.
• Step (3) The problem is solved for each load and generation scenario and range of decision variables. Then the objective function (F) value is computed for each variable set and the best set corresponding to the minimum F is considered as the best solution. In other words, in this step, each scenario from the set of scenarios defined as a probability distribution function is applied to the program and the value of the objective function is determined by considering the constraints of the problem.
• Step (4) Each variable set is updated using the IEOA, and if the updated variables achieve a better objective function value, the corresponding set of variables replaces the old set. In other words, in this step, the re-evaluation of the objective function for the set of scenarios generated in the probability distribution function of each variable is presented via IEOA optimization solver.
• Step (5) The conditions for achieving convergence (achieving the lowest value of F, implementing maximum iterations of IEOA and wind production and network demand scenarios) are checked, and if the convergence is achieved, go to step 6, otherwise go to step 2.
• Step (6) The probability distribution function is extracted according to the implementation of different scenarios for the variables of the problem using the IEOA optimization solver.

5. Simulation Results

The simulation results a of multi-objective allocation of a wind turbine in the 33-bus network are obtained based on minimizing the losses, enhancing the voltage profile and stability and also enhancing the reliability based on SMM model using the IEOA. At first, the results without uncertainty based on the deterministic-metaheuristic model (DMM) are presented and then the problem is solved with uncertainty via the SMM model for the 33-bus network. The simulation is implemented on the IEEE 33-bus network shown in Figure 2. The network information, including the lines and buses, is extracted from Ref. [34]. The 33-bus network has 3.716 MW of active power and 2.300 MVAr of reactive power consumption. The base active and reactive losses of this network are 202.67 kW and 135.17 kVAr, and the base minimum voltage of this network is 0.91308 p.u. The network lines outage rate is extracted from [35]. The cost of each kW hour of interruption of the load is also considered to be $0.50 [35].

5.1. Results Based on DMM

The simulation results of the allocation of wind turbine in the 33-bus distribution network are given based on DMM without considering uncertainty. In the analysis of the basic 33-bus network,

Table 2. Basic numerical results of 33-bus distribution network.

Item	Value
${\rho }_{loss}$ (kW)	202.68
Minimum voltage (p.u)	0.91308
VSI (p.u)	26.28
$\mathrm{{\rm E}}n\Im$ (kWh)	6447
$\mathrm{{\rm E}}n\Im C$ ($)	3223.5

shows the numerical results including the values of active power loss, minimum voltage, VSI, $\mathrm{{\rm E}}n\Im$ and $\mathrm{{\rm E}}n\Im C$. The value of the $\mathrm{{\rm E}}n\Im$ is equal to 6447 kWh hours and the $\mathrm{{\rm E}}n\Im C$ is $3223.50.

The simulation results of wind turbine allocation in the 33-bus network based on DMM without considering uncertainties using the IEOA are presented below.

The performance of IEOA in solving the DMM-based problem is evaluated with traditional EOA, PSO, MRFO and SHO methods. The convergence curves of different algorithms are demonstrated in Figure 3. It is clear that compared to other algorithms, the IEOA can achieve the lowest value of the objective function with a higher speed and accuracy of convergence and has converged to the global optimal solution using less iterations.

The allocation results of wind resource considering DMM-based problem using the proposed IEOA and the traditional methods of EOA, PSO, MRFO and SHO in the 33-bus network are given in

Table 3. The results of multi-criteria wind turbine allocation in the 33-bus network via the DMM.

Parameter	Base Network	EOA	IEOA	PSO	MRFO	SHO
Location (Bus)/Size (kW)/Power factor	--	2259.82/30/0.8548	2251.56/30/0.8562	2261.57/30/0.8571	2255.11/30/0.8566	2257.39/30/0.8568
${\rho }_{loss}$ (kW)	202.68	80.32	79.54	80.60	80.05	80.22
Minimum voltage (p.u)	0.91308	0.9585	0.9588	0.9584	0.9586	0.9586
VSI	26.28	30.03	30.05	30.03	30.04	30.04
$\mathrm{{\rm E}}n\Im C$ ($)	3223.5	636.90	632.05	638.14	635.67	636.18

. The IEOA considered 2251.56 kW of wind power with a power factor of 0.8562 in bus 30. The simulation results show that the amount of losses obtained by different algorithms has been significantly reduced compared to the base network. The amount of losses by the proposed IEOA, traditional EOA, PSO, MRFO and SHO are found to be 79.54 kW, 80.32 kW, 80.60 kW, 80.05 kW and 80.22 kW. respectively, and the IEOA has achieved the least losses. The minimum voltage by the proposed IEOA, traditional EOA, PSO, MRFO, and SHO methods are 0.9588 p.u, 0.9585 p.u, 0.9584 p.u, 0.9586 p.u, and 0.9586 p.u; respectively, and the corresponding VSI values are 30.05 p.u, 30.03 p.u, 30.03 p.u, 30.04 p.u and 30.04 p.u; respectively. The results proved the superior capability of the IEOA in obtaining a better voltage profile and stability condition. The $\mathrm{{\rm E}}n\Im C$ for IEOA, traditional EOA, PSO, MRFO, and SHO are found to be $632.05, $636.90, $638.14, $635.67, and $636.18; respectively, and the IEOA method has succeeded in meeting the demand of customers with a higher level of reliability. Also, the capability of the IEOA is evaluated with the PSO in Ref. [36] and the backtracking search optimization algorithm (BSOA) in Ref. [37] in deterministic wind turbine allocation in the 33-bus network. According to

Table 4. Results comparison of the IEOA with the last methods in DMM-based problem solving.

Item	IEOA	BSOA [37]	PSO [36]
Location (Bus)/Size (kW)/Power factor	2251/30/0.8562	2265.24/8/0.82	2567/6/1
${\rho }_{loss}$ (kW)	79.54	82.78	110.90
Minimum voltage (p.u)	0.9588	0.9549	--
VSI	30.05	--	--
$\mathrm{{\rm E}}n\Im C$ ($)	632.05	--	--

, the proposed IEOA achieves the lowest losses and minimum voltage deviation.

5.2. Results Based on SMM

The simulation results of multi-criteria allocation of wind resource in the 33-bus network based on the SMM are presented in this section. The network load is selected with a normal PDF (Figure 4) with a mean of 85% and a std of 20% for the load of each bus. In other words, the demand of each bus should vary between 20% and 160%, and most of these changes are approximately 85%. Based on a Monte Carlo simulation, 500 scenarios have been considered for network load demand. This probability distribution function represents the uncertainty model of the network load demand. Also, the PDF of wind turbine power based on its maximum power of 3000 kW is shown in Figure 5 [24]. This figure shows the wind power uncertainty model with 500 scenarios. The PDF of wind turbine power is defined based on two parameters, scale 10 and shape 1.75 [24].

After performing the probabilistic power flow considering the defined scenarios and the PDFs of load demand and wind generation, the PDF results including losses, minimum voltage, VSI and $\mathrm{{\rm E}}n\Im C$ are presented in Figure 6. By applying the inputs of the problem along with the scenarios of probability distribution functions related to the uncertainty model of wind power and network load demand, the goal of minimizing the objective function has been achieved considering the constraints of the problem. After extracting the probability distribution functions of the decision variables, the probability distribution functions related to each of the objectives, including power loss, minimum and voltage stability, as well as reliability cost, are obtained according to Figure 6. It is clear that the 33-bus network loss is around 200 kW, which can be increased to 300 kW for demand uncertainty. The VSI is near to zero in more than 90% of scenarios. The minimum voltage network is near to 0.92 p.u, and is also lower than the value of 0.9 p.u; this value is not acceptable. Because of demand uncertainty, the $\mathrm{{\rm E}}n\Im C$ is obtained as 4895 dollars.

In the following, the allocation of wind turbine in the 33-bus network via the SMM is implemented using the IEOA, and the results related to the PDF of each decision variable are shown in Figure 7. The probability distribution functions of each of the optimization variables have been extracted according to the defined scenarios based on Monte Carlo simulation and based on the IEOA. As can be seen, bus 30 is the best place to install a wind turbine with 100% probability. More turbine capacity between 1650 kW and 2100 kW is offered with higher probability. The turbine power factor also has a higher probability between the range of 0.80 to 0.87 than the rest of the scenarios.

According to the PDFs related to the location, capacity and power factor of the wind resource in Figure 6 and Figure 7, the PDF of each of the objectives is shown in Figure 8. Based on Figure 8, the results of multi-criteria wind resource allocation via the SMM are given in

Table 5. The results of multi-criteria wind turbine allocation in a 33-bus network based on SMM.

Item	IEOA
Location (Bus)/Size (kW)/Power factor	1928.47/30/0.8543
${\rho }_{loss}$ (kW)	86.12
Minimum voltage (p.u)	0.9485
VSI (p.u)	28.84
$\mathrm{{\rm E}}n\Im C$ ($)	2137.09

. After calculating the product of each of the events in the frequency of that event for different objectives, the value of losses, minimum voltage, VSI and $\mathrm{{\rm E}}n\Im C$ are found to be 86.12 kW, 0.9485 p.u, 28.84 p.u and $2137.09, respectively.

5.3. Comparison of DMM and SMM Results

In this study, the wind turbine allocation is performed on the 33-bus network based on DMM and SMM. To solve the problem considering the DMM, the optimal location, capacity and power factor of the wind turbine are found according to the network demand and wind generation as deterministic parameters. Moreover, to solve the problem via the SMM, with the uncertainties, the location, capacity and power factor of the wind turbine are presented in PDF format. The comparison of DMM and SMM results is presented in

Table 6. Comparison of simulation results of 33-bus network based DMM and SMM.

Item	DMM	SMM
Location (Bus)/Size (kW)/Power factor	2251/30/0.8562	1928.47/30/0.8543
${\rho }_{loss}$ (kW)	79.54	86.12
Minimum voltage (p.u)	0.9588	0.9485
VSI (p.u)	30.05	28.84
$\mathrm{{\rm E}}n\Im C$ ($)	632.05	2137.09

. It is clear that considering the uncertainty based on the SMM, the power loss has increased from 79.54 kW to 86.12 kW and $\mathrm{{\rm E}}n\Im C$ from 632.05 kWh to 2137.09 kWh. Therefore, considering that the uncertainty based on SMM has caused more increase in losses and the network reliability enhancement cost compared to the DMM. In addition, the minimum voltage value has been reduced and weakened from the 0.9588 p.u based on DMM to 0.9485 p.u based on SMM. Also, the VSI value has decreased from 30.05 p.u based on the DMM to 28.84 p.u based on the SMM. Therefore, knowing the uncertainties of power generation as well as load demand can lead to correct and robust decisions by the network operator against existing risks caused by uncertainties.

5.4. Effect of Different Load Cases on SMM Results

In this section, the effect of different load cases (75% and 125% of the nominal load) on solving the SMM is evaluated on the 33-bus network using the IEOA. A comparison of different load cases on solving the SMM is presented in

Table 7. Results of 33-bus network for different load cases on SMM solving.

Item	75%	100%	125%
Location (Bus)/Size (kW)/Power factor	2342/31/0.8880	2251/30/0.8562	2034/30/0.8328
${\rho }_{loss}$ (kW)	60.85	79.54	113.17
Minimum voltage (p.u)	0.9692	0.9588	0.9440
VSI (p.u)	30.53	30.05	29.19
$\mathrm{{\rm E}}n\Im C$ ($)	341.25	632.05	881.87

. It can be seen that with the increase in load demand (25% nominal load), the amount of network losses increases, the minimum value and the voltage stability index are weakened, and the cost of reliability also increases and vice versa. The results show that the proposed methodology in terms of load changes, taking into account the uncertainties of wind power generation and network load demand, based on the IEOA by satisfying the operating constraints, is able to improve all objectives and create a favorable compromise between different objectives.

6. Conclusions

In this study, the multi-criteria allocation of wind energy resource in the 33-bus network is presented to minimize the losses and voltage profile and stability enhancement, as well as enhancing the reliability considering wind generation and network demand uncertainties. First, the problem is implemented based on DMM using the IEOA without considering the uncertainty, and the location, capacity and power factor of wind resource are optimally determined. The results of solving DMM-based problem showed that determining the optimal location, capacity and power factor of wind power using the IEOA method has significantly reduced losses and reliability cost and improved the voltage profile and stability of the network. The superiority of the IEOA is confirmed in solving the DMM-based problem compared to the traditional EOA, PSO, ACO and MRFO methods. In addition, the results of problem solving via the SMM showed that incorporating the uncertainty increases the losses and the cost of reliability enhancement and also weakens the voltage profile and the VSI. The results demonstrated that considering the uncertainties due to the fluctuating nature of wind power as well as the inherent changes in the network load compared to the DMM can lead to robust decisions against forecasting errors caused by uncertainties. The optimal allocation of photovoltaic-wind resources with battery storage to enhance the network reliability considering uncertainty conditions is suggested for future research.