Optimal Integration of Hybrid Energy Systems: A Security-Constrained Network Topology Reconfiguration

Abstract

The integration of distributed energy resources, such as wind farms (WFs) and energy storage systems (ESSs), into distribution networks can lower the economic cost of power generation. However, it is essential to consider operational constraints, including loading margin, which ensures the security line contingency. This study aims to develop a comprehensive hourly distribution network reconfiguration (HDNR) model to minimize the economic cost for the power generation company. The model considers the optimal allocation of WFs and ESSs in terms of capacity and location, as well as the hourly status of the distribution network switches, based on security constraints. The proposed model is applied to an IEEE 33-bus distribution test system, and the capacities and locations of WFs and ESSs are determined. The impacts of security constraints on the optimal capacities and locations of WFs and ESSs, and the hourly configuration of the distribution network, are analyzed based on two case studies. In Case Study I, the model is solved with HDNR conditions, while Case Study II is solved without these restrictions, for comparison purposes. The results show that the optimal allocation of WFs and ESSs is affected by security constraints when HDNR is considered and highlight the crucial role of security constraints under contingency conditions, such as line outages. In the test system, three WFs and two ESSs are optimally allocated, with changes in capacity and location as the loading margin varies.

1. Introduction

As the world confronts the climate change issue, the primary objective is to shift towards a net-zero future by decreasing greenhouse gas emissions to a level that can be compensated by other measures [1,2]. The power industry is a key player in achieving this target since it generates a significant proportion of global emissions. The power sector will have to adopt cleaner and more sustainable energy sources, such as renewable energy, as well as enhance efficiency and minimize losses to achieve the net-zero objective [3]. To achieve this transition, integrating new technologies, such as distributed energy resources (DERs) and fast two-way communication, will be crucial to ensure grid reliability and stability [4,5].

The use of wind energy has seen tremendous growth in recent years due to its environmental and economic advantages. Projected wind total capacity in 2030, even with a steady growth, is more than 1200 GW worldwide, and this trend is expected to continue toward reaching ambitious environmental goals, such as net-zero emissions [6]. However, the delivery of wind power to consumers is challenged by operational and security issues, such as power losses and voltage instability, which are a result of the inherent intermittency of wind power generation. To overcome these issues, it is important to consider the security restrictions associated with wind power integration and investigate the coordination of energy storage systems [7].

Due to technical and economic limitations in expanding transmission systems, utilizing renewable DERs can play a crucial role in delivering power to customers, with lower active power losses and reduced environmental impact. The topology of distribution networks can also have a direct impact on the economic benefits of utilizing distributed generation. Although distribution networks typically have a mesh topology, they should be reconfigured to a radial configuration as much as possible to take advantage of the benefits of radial operation, such as short circuit current limiting and easy protection [8].

To achieve this, distribution network reconfiguration (DNR) is a solution in which the switches in the network are changed to reach a configuration that meets desired goals, while satisfying operational and security constraints, without isolating any network nodes. The optimization of DNR is necessary to ensure the proper operation of future distribution networks. It is also important to consider voltage stability constraints in the formulation of network reconfiguration studies. However, conventional DNR models may not be able to adapt to the increasing trends in network modernization, making an hourly DNR (HDNR) crucial for the proper operation of future distribution networks.

1.1. Literature Review

Recently, there has been a significant number of studies focused on the topic of distribution network reconfiguration, both with and without considering the impact of distributed generation. These studies have defined optimization objective functions based on both technical and economic aspects, as outlined in Table 1.

A review of these studies shows that formulating an objective function for DNR optimization based on economic factors is preferred, as it can encompass technical aspects and also reflect the economic benefits that can be realized from operating a distribution network with wind farms (WFs). In this subsection, we review several methodologies and techniques in the literature for solving the DNR problem. Ref. [6] shows the robustness of the proposed adaptive particle swarm optimization compared to the genetic algorithm for reconfiguration in the presence of DERs. The literature commonly uses power loss reduction as the objective in optimizing DNR in the presence of DERs, as seen in [8,9,10,11]. To address the issue of high-power losses and low reliability in the distribution network, ref. [12] proposed multi-objective reconfiguration models that simultaneously minimize power losses and improve system reliability. The proposed reliability indices include system average interruption frequency index, system average interruption duration index, and average energy not supplied. Amanulla et al. [13] focuses on a probabilistic reliability evaluation method for DNR due to the large computational time needed for reliability. The reconfiguration can also result in economic benefits for the network, as seen in [14], with the reduction of the annual cost after reconfiguration through minimization of power loss and reliability costs.

Load balancing is a technical objective in solving DNR, which is considered with power loss and reliability optimization in [15]. In that study, the non-dominated sorting genetic algorithm II is utilized to find solutions in a multi-objective modeling approach. Joen et al. [16] propose an optimal reconfiguration method for large distribution networks based on simulated annealing, which results in lower computational time compared to other studies. Ref. [17] employs graph theory based on radial DNR to improve computational performance, as well as the Pareto dominance rules of the reconfigured system. The enhanced gravitational search algorithm is used in [18] to improve processing time and the quality of reconfiguration results through special mutation strategy.

The integration of energy storage systems (EESs) is recently considered in DNR optimization. Nick et al. [19] propose a seasonal planning procedure to optimally allocate ESS capacity and location for DNR optimization. Ref. [20] focuses on coordinating power electronic devices, such as smart inverters, to decrease ESS investment costs through a planning schedule for ESSs in conjunction with distributed generators (DGs) and DNR. The application of ESSs to increase renewable DG unit penetration in the grid is studied in [21]. Studies [19,20,21] focus on long-term ESS planning with non-technical objectives, while [22] considers voltage stability constraints in long-term DG planning to ensure power system security. Solving the problem from the DG owner’s viewpoint can result in power losses and security criteria. This can result in exceeding normal values (i.e., normal operation of the system), since the main objective is to increase the penetration of wind energy into the grid for profit and other network characteristics.

While numerous studies have aimed to improve voltage profile through distribution network reconfiguration [5,6,11,19,23,24], none have considered developing an optimal reconfiguration model subject to voltage stability constraints. A sufficient voltage stability margin is crucial for distribution network security and should be considered in the DNR model. While voltage stability constraint models are widely used for power system optimization [4,22,25,26], they have not been applied to DNR studies.

Table 2. Summary of literature review of HNR studies.

Ref.	Objective Function		DER		Line Contingency	HDNR	Security	Optimal Allocation
	Technical	Economic	DG	ESS				DG	ESS
[6]	✓
[8]	✓		✓
[9]	✓		✓
[10]	✓		✓
[11]	✓		✓
[12]	✓	✓
[13]	✓	✓
[14]	✓	✓
[15]	✓
[16]	✓
[17]	✓	✓
[18]	✓	✓
[19]		✓	✓	✓					✓
[20]		✓	✓	✓					✓
[21]		✓	✓	✓				✓	✓
[23]	✓
[24]	✓
This study		✓	✓	✓	✓	✓	✓	✓	✓

summarizes the studies and their methodologies, available in the literature.

Table 1. Application of network reconfiguration in different optimization problems.

Objective	Target	Reference Number
Technical	Active power losses minimization	[6,7,8,9,10,11,12,13,14,15,16,17,18,23,24]
	Voltage profile improvement	[6,8,19,23,24]
	Reliability improvement	[12,13,14,17,18]
	Load balancing	[13,15]
Economic	Switching costs	[9]
	Operation costs	[6,14,18]
	Investment costs	[19,20,21]

Table 1. Application of network reconfiguration in different optimization problems.

Objective	Target	Reference Number
Technical	Active power losses minimization	[6,7,8,9,10,11,12,13,14,15,16,17,18,23,24]
	Voltage profile improvement	[6,8,19,23,24]
	Reliability improvement	[12,13,14,17,18]
	Load balancing	[13,15]
Economic	Switching costs	[9]
	Operation costs	[6,14,18]
	Investment costs	[19,20,21]

1.2. Contribution

The previous studies on DNR have examined various aspects, but there is a lack of research on security-constrained modeling for network reconfiguration involving the presence of wind farms and energy storage systems in order to minimize the economic cost for the generation company. Additionally, there have been no studies that focus on the hourly reconfiguration of the network with the optimal allocation of WFs and ESSs. Line contingency, which significantly affects the objective functions and decision variables, has not been taken into account in any DNR study. The objective of this study is to develop a comprehensive HDNR model which minimizes the economic cost for the generation company (GenCo). The proposed model takes into account the optimal allocation of WFs and ESSs in terms of capacity and location, and specifies the hourly closed/opened switches of the distribution network based on security constraints.

The key features of this study include:

• Consideration of voltage stability as a security constraint for HDNR;
• Development of an HDNR model with a focus on minimizing the cost for the GenCo;
• Analysis of the impact of WFs and ESSs on the results of the proposed model;
• Joint allocation of WFs and ESSs with HDNR;
• Analysis of the sensitivity of HDNR to voltage stability constraints and vice versa;
• Analysis of the security of the reconfiguration model when line contingency occurs in the system.

The study is structured as follows: Section 2 presents the mathematical formulation of the model, while Section 3 provides the simulation parameters. Section 4 discusses the simulation results of the model applied to the IEEE 33-bus distribution test system. Finally, Section 5 concludes the study and proposes recommended actions.

2. Concept of the Network Security Index

The voltage stability margin, which is also known as the loading margin (LM), is an important criterion for measuring the security of the power system. This index is defined as the maximum power of generation units that can be increased to meet the load demand increase prior to violation of operational limits [4,26].

In order to guarantee the secure operation of the system, the network operator defines a specific level of LM. The assigned LM is defined as the difference between the initial operating point (IOP) and the maximum loadability point (MLP) in MW. The model defined in this study takes into account the power flow equations for IOP and MLP, simultaneously [22].

Figure 1 illustrates the concept of system security through the power–voltage (P-V) curve of an arbitrary load bus. In this figure, $P_D^0$ at point A is the load demand at IOP, while the voltage collapse point is point B. The loading parameter (λ_des) represents the desired LM which should be guaranteed during the operation of the system.

3. Problem Formulation

The mathematical formulation of the proposed model, including the objective function, as well as operational and security constraints, are represented in this section. The description of symbols used in this section is given in the Nomenclature.

3.1. Objective Function

As stated in Table 1, the technical and economic aspects are the main objective functions of DNR formulation, where the latter could be inclusive of the former. The objective function defined in this study includes the cost of buying power from the main grid, the WF owners, and the GenCo, as stated below:

$$\min \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}GC={\displaystyle \sum _{t=1}^{N_t}Cos{t}_t^{WF}+Cos{t}_t^{sub}}$$

(1)

$$Cos{t}_t^{WF}={\displaystyle \sum _{i=1}^{N_{WF}}{S}_{base}\times C_t^{pur}}\times P_{i,t}^{WF}$$

(2)

$$Cos{t}_t^{sub}={\displaystyle \sum _{i=1}^{N_s}{S}_{base}\times C_t^{pur}}\times P_{i,t}^G$$

(3)

It is important to note that since the power stored in the ESS is from the main grid or the WFs, it is not included in the costs. The objective function of the proposed model is optimized based on several operational constraints, physical limits, and the radial configuration of the network. The power flow equations are considered simultaneously for both IOP and MLP.

3.1.1. Power Flow Constraints at IOP

The power flow equations at IOP are represented as follows:

$$P_{i,t}^G+P_{i,t}^{WF}+P_{i,t}^{CH}-P_{i,t}^D-P_{i,t}^{DICCH}={\displaystyle \sum _j({\delta }_{ij,t}^L\times P_{ij,t})}$$

(4)

$$Q_{i,t}^G+Q_{i,t}^{WF}-Q_{i,t}^D={\displaystyle \sum _j({\delta }_{ij,t}^L\times Q_{ij,t})}$$

(5)

$$P_{ij,t}=V_{i,t}^2{g}_{ij}-V_{i,t}{V}_{j,t}(g_{ij}\cos {\theta }_{ij,t}+b_{ij}\sin {\theta }_{ij,t})$$

(6)

$$Q_{ij,t}=-V_{i,t}^2{b}_{ij}-V_{i,t}{V}_{j,t}(g_{ij}\sin {\theta }_{ij,t}-b_{ij}\cos {\theta }_{ij,t})$$

(7)

$$\{\begin{array}{l}{P}_{sub}^{\min }\le P_{i,t}^G\le P_{sub}^{\max }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\forall i\in N_s\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\mathrm{otherwise}\end{array}$$

(8)

$$\{\begin{array}{l}{Q}_{sub}^{\min }\le Q_{i,t}^G\le Q_{sub}^{\max }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\forall i\in N_s\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\mathrm{otherwise}\end{array}$$

(9)

$$V_i^{\min }\le V_{i,t}\le V_i^{\max }$$

(10)

$${\delta }_{ij,t}^L(I_{R_{ij,t}}^2+I_{M_{ij,t}}^2)\le I_{MA{X}_{ij}}^2$$

(11)

$$I_{R_{ij,t}}=g_{ij}(V_{i,t}\cos {\theta }_{i,t}-V_{j,t}\cos {\theta }_{j,t})\hspace{0.17em}-b_{ij}(V_{j,t}\sin {\theta }_{i,t}-V_{j,t}\sin {\theta }_{j,t})$$

(12)

$$I_{M_{ij,t}}=g_{ij}(V_{i,t}\sin {\theta }_{i,t}-V_{j,t}\sin {\theta }_{j,t})+b_{ij}(V_{j,t}\cos {\theta }_{i,t}-V_{j,t}\cos {\theta }_{j,t})$$

(13)

where Equations (4) and (5) represent the active and reactive power balance at IOP, respectively, with consideration for different power sources. The active and reactive power flow through the branches at IOP are shown with constraints (6) and (7), respectively. The contribution of the substation for active and reactive power supply is limited by Equations (8) and (9), respectively. The bus voltage limits are represented with Equation (10), while Equations (11)–(13) show the limits on the transmission lines.

3.1.2. Power Flow Constraints at MLP

As stated, this paper considers the power flow constraints at IOP and MLP simultaneously. The power flow constraints at MLP are given below:

$${\widehat{P}}_{i,t}^G+P_{i,t}^{WF}+P_{i,t}^{CH}-{\widehat{P}}_{i,t}^D-P_{i,t}^{DICCH}={\displaystyle \sum _j({\delta }_{ij,t}^L\times {\widehat{P}}_{ij,t})}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(14)

$${\widehat{Q}}_{i,t}^G+Q_{i,t}^{WF}-{\widehat{Q}}_{i,t}^D={\displaystyle \sum _j({\delta }_{ij,t}^L\times {\widehat{Q}}_{ij,t})}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(15)

$${\widehat{P}}_{ij,t}={\widehat{V}}_{i,t}^2{g}_{ij}-{\widehat{V}}_{i,t}{\widehat{V}}_{j,t}(g_{ij}\cos {\widehat{\theta }}_{ij,t}+b_{ij}\sin {\widehat{\theta }}_{ij,t})$$

(16)

$${\widehat{Q}}_{ij,t}=-{\widehat{V}}_{i,t}^2{g}_{ij}-{\widehat{V}}_{i,t}{\widehat{V}}_{j,t}(g_{ij}\sin {\widehat{\theta }}_{ij,t}-b_{ij}\cos {\widehat{\theta }}_{ij,t})$$

(17)

$${\widehat{P}}_{i,t}^D=(1+K_{D,i\hspace{0.17em}}\hspace{0.17em}\lambda )\hspace{0.17em}{P}_{i,t}^D$$

(18)

$${\widehat{Q}}_{i,t}^D=(1+K_{D,i}\lambda )(Q_{i,t}^D)$$

(19)

$${\widehat{P}}_{s,t}^G=\min \left(P_G^{\max },(1+K_{G,s}\lambda )P_{s,t}^G\right)$$

(20)

$$\{\begin{array}{l}{P}_{sub}^{\min }\le {\widehat{P}}_{s,t}^G\le P_{sub}^{\max }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\forall i\in N_s\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\mathrm{otherwise}\end{array}$$

(21)

$$\{\begin{array}{l}{Q}_{sub}^{\min }\le {\widehat{Q}}_{s,t}^G\le Q_{sub}^{\min }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\forall i\in N_s\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\mathrm{otherwise}\end{array}$$

(22)

$$V_i^{\min }\le {\widehat{V}}_{i,t}\le V_i^{\max }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(23)

$${\delta }_{ij,t}^L({\widehat{I}}_{R_{ij,t}}^2+{\widehat{I}}_{M_{ij,t}}^2)\le {\widehat{I}}_{MA{X}_{ij}}^2$$

(24)

$${\widehat{I}}_{R_{ij,t}}=g_{ij}({\widehat{V}}_{i,t}\cos {\widehat{\theta }}_{i,t}-{\widehat{V}}_{j,t}\cos {\widehat{\theta }}_{j,t})\hspace{0.17em}-b_{ij}({\widehat{V}}_{j,t}\sin {\widehat{\theta }}_{i,t}-{\widehat{V}}_{j,t}\sin {\widehat{\theta }}_{j,t})$$

(25)

$${\widehat{I}}_{M_{ij,t}}=g_{ij}({\widehat{V}}_{i,t}\sin {\widehat{\theta }}_{i,t}-{\widehat{V}}_{j,t}\sin {\widehat{\theta }}_{j,t})+b_{ij}({\widehat{V}}_{j,t}\cos {\widehat{\theta }}_{i,t}-{\widehat{V}}_{j,t}\cos {\widehat{\theta }}_{j,t})$$

(26)

$$\lambda \ge {\lambda }_{d\hspace{0.17em}es}>0$$

(27)

where constraints (14)–(17) represent the load balance and power flow through the branches at MLP, respectively. The active and reactive power increase pattern of loads from IOP to MLP are given by Equations (18) and (19), respectively. The substation active power is increased to cover the load increase, as shown in Equation (20). Constraints (21)–(26) represent the physical and operational limits of the substation and transmission lines. The desired loading parameter is defined by Equation (27).

3.1.3. WF’s Capacity Limit

In this study, the optimal locations of WFs are defined based on their effect on the stability of the network. A discrete variable-based formulation is introduced for the WFs allocation, with consideration for the power output of the WFs in IOP and MLP.

$$\hspace{0.17em}\forall i\in N_{WF},\forall t\in N_t$$

$$\{\begin{array}{l}0\le P_{i,t}^{WF}\le {\vartheta }_i^{WF}\times {\eta }_i^{WF}\times P_i^{WF,sd}\hspace{0.17em}\times C{F}_{i,t}^{WF}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\forall i\in N_{WF}\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\mathit{otherwise}\end{array}$$

(28)

$$\{\begin{array}{l}-tg({\phi }_{lead})\times P_{i,t}^{WF}\times {\vartheta }_i^{WF}\le Q_{i,t}^{WF}\le {\vartheta }_i^{WF}\times tg({\phi }_{lag})\times P_{i,t}^{WF}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\forall i\in N_{WF}\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\mathit{otherwise}\end{array}$$

(29)

$${\displaystyle \sum _{i\in N_b}{\vartheta }_i^{WF}\le N_{\max }^{WF}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}$$

(30)

$${\eta }_i^{WF}\le {\eta }_i^{WF,\max }$$

(31)

where constraints (28) and (29) represent the active and reactive power limit of WFs, respectively. The parameter ${CF}_{i,t}^{WF}$ is defined here to specify the expected power output of WF installed in bus i at time t. The binary variable ${\vartheta }_i^{WF}$ is introduced to define the optimal location of WFs based on the security constraint. Constraint (30) limits the number of WFs which could be installed in the network. The uncertainty of wind power generation has not been considered in this study. However, it can be considered as a potential future work. The uncertainty of wind power generation can affect the system security and the network reconfiguration.

3.1.4. ESS Capacity

The charge/discharge power of ESS is an important variable affecting the proposed optimization problem. Due to the effect of ESS on the operational and economic performance of the optimization, their location should be defined for optimal operation of the DN. The ESS model is represented below:

$$\forall i\in N_{ESS},\forall t\in N_t$$

$$\{\begin{array}{l}0\le P_{i,t}^{CH}\hspace{0.17em}\le {\gamma }_{i,t}^{ch}\times P_{i,t}^{ch,\max }\times {\vartheta }_i^{ESS}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\forall i\in N_{ESS}\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\mathit{otherwise}\end{array}$$

(32)

$$\{\begin{array}{l}0\le P_{i,t}^{DISCH}\hspace{0.17em}\le {\gamma }_{i,t}^{disch}\times p_{i,t}^{disch,\max }\hspace{0.17em}\times {\vartheta }_i^{WF}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\forall i\in N_{ESS}\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\mathit{otherwise}\end{array}$$

(33)

$${\gamma }_{i,t}^{ch}+{\gamma }_{i,t}^{disch}\le 1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}({\gamma }_{i,t}^{disch},\hspace{0.17em}{\gamma }_{i,t}^{ch}\hspace{0.17em}\hspace{0.17em}\in \{0,1\})$$

(34)

$$\begin{array}{l}SO{C}_{i,t}^{ESS}=SO{C}_{i,t}^{ESS}+\Delta t.(p_{i,t}^{CH}\hspace{0.17em}{\eta }_{i,t}^{ch}-p_{i,t}^{DISCH}/{\eta }_{i,t}^{dch}\hspace{0.17em})0\\ \end{array}$$

(35)

$$SO{C}_i^{\min }\times {\eta }_i^{ESS}\hspace{0.17em}\le SO{C}_{i,t}^{ESS}\le SO{C}_i^{\max }\times {\eta }_i^{ESS}\hspace{0.17em}$$

(36)

$${\displaystyle \sum _{i\in {\mathsf{\Omega }}_b}{\vartheta }_i^{ESS}\le N_{\max }^{ESS}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}$$

(37)

where the upper and lower bands of the ESS charge and discharge power are shown in Equations (32) and (33). Constraint (34) prevents the simultaneous charge and discharge of ESSs. Constraint (35) shows the state of charge (SOC) of the ESSs, which is limited by Equation (36). Finally, Equation (37) limits the number of ESSs that could be installed in the network.

3.1.5. Topology Reconfiguration

The network reconfiguration can be defined based on the concept of graph theory, which defines a tree as a connected graph without any loop [27]. In radial distribution networks, two important conditions apply: (1) the tree topology which requires the graph theory condition, and (2) connectivity of system buses to the substation. The first condition can be satisfied if the number of closed switches in the network will be equal to the number of nodes minus one. This concept is mathematically expressed as:

$${\displaystyle \sum _{ij\in N_b}{\delta }_{ij,t}^L=2\times (n_b-1)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}$$

(38)

$${\delta }_{ij,t}^L={\delta }_{ji,t}^L\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall i,j\in N_b,t\in N_t$$

(39)

The power balance equations satisfy the second condition, where there is a power flow from the substation to each bus at each hour. Unlike other studies, in this study, the power balance equations are satisfied for both IOP and MLP, simultaneously.

4. Simulation Parameters

The IEEE 33-bus distribution network [28] is used in simulating the proposed model. This system consists of 32 sectionalizing switches and 5 tie switches, and the voltage level of all system buses is 12.66 kV. The active and reactive system load is 3.715 MW and 2.3 MVaR, respectively. Figure 2 depicts the distribution network in its initial configuration. To illustrate the impact of voltage stability on the optimal allocation of WFs, the system load demand is increased by a factor of 1.2. Furthermore, it is assumed that a maximum of 0.5 MW of ESS can be installed in the distribution network. The data of the IEEE 33-bus distribution network used in the simulation is given in [29]. On a personal computer equipped with an Intel i7 processor running at 3.00 GHz with 8 gigabytes of random-access memory, the proposed problem is simulated as a mixed integer non-linear programming (MINLP) problem in the general algebraic modeling system (GAMS) [30] environment using a standard branch and bound solver. The model is solved with SBB in GAMS, which uses a standard branch and bond algorithm to solve the problem. A 24 h operation period is considered for the simulation. It should be noted that the numbers representing the time periods do not represent the time of day. Figure 3 illustrates the pattern of load demand on the system, as well as the price profile of electricity at each hour [31].

Table 3. Capacity factor of WFs in each hour [31].

1	2	3	4	5	6	7	8	9	10	11	12
0.08	0.09	0.12	0.26	0.36	0.59	0.65	0.59	0.49	0.55	0.77	0.72
13	14	15	16	17	18	19	20	21	22	23	24
0.83	0.92	0.97	1.00	0.83	0.65	0.65	0.56	0.57	0.56	0.72	0.84

and

Table 4. Characteristics of ESSs [32].

$P_{i,t}^{ch,\max }(MW)$	$P_{i,t}^{dich,\max }(MW)$	${\eta }_{i,t}^{ch}$	${\eta }_{i,t}^{dich}$	$SO{C}_i^{\max }$	$SO{C}_i^{\min }$
0.20	0.20	0.88	0.88	1.00	0.10

, respectively, provide information about the capacity factor of the WFs [31] and the characteristics of the ESSs [32] used in the simulation. Furthermore, it has been assumed in the simulation that the LM should be 0.15. The maximum number of three WFs, each with eight wind turbines and a capacity of 250 kW and two ESS, each with a capacity of 0.5 MW, are considered for installation in the distribution network.

To demonstrate the link between the constraints of the voltage stability and the network reconfiguration, the following two case studies are considered. In the first case study (Case Study I), the model considers the optimal reconfiguration of the network. However, in the second case study (Case Study II), the proposed model neglects the optimal reconfiguration. Neglecting the reconfiguration can be done by ignoring Equations (36)–(38) from the model. A sensitivity analysis is carried out to investigate how the distribution network security constraints influence the optimal allocation of WFs and ESSs, as well as the hourly reconfiguration of the distribution network. Furthermore, the effects of contingency situations are investigated.

5. Results and Discussion

The simulation findings that describe the linkage between HDNR, network security, and the allocation of WFs and ESSs will be examined in this section. The values of the objective function in Case Study I and Case Study II are USD 2443.80 and USD 2437.54, respectively. Comparing these two values reveals that GenCo’s costs have gone up by a marginal amount. This relatively minor increase in cost is seen as a security cost, which suggests that simultaneous consideration of both distribution network security restrictions and HDNR gives rise to an increase in purchase costs. It is important to note that this increase in the security cost only applies to a day of operation considered in the simulation; hence, it is possible that this cost will drastically increase in the long-term operation of the distribution network.

These two case studies also demonstrate that the optimal allocation of the WTs and ESSs is affected by the optimal HDNR. The simulations for the two case studies were conducted for the initial configuration shown in Figure 4.

Figure 5 shows the optimal HDNR of Case Study I. This figure shows how the network’s configuration can be changed throughout the day. This result demonstrates the importance of optimal HDNR in the future smart grid.

Figure 6 shows the utilized wind power in the distribution network for the two case studies. The evolutions of the SOC of the ESSs for the two case studies are given in Figure 7. From these two figures, it can be seen that the greatest injection of wind power to the distribution network happens during the peak hours (from 3 to 8 p.m.), and the ESSs have been discharged at a high rate during these peak hours. Figure 6 also shows that in Case Study I, the amount of utilized wind power in the distribution network is reduced, and this happens as a result of the simultaneous consideration of security constraints and HDNR.

5.1. Sensitivity Analysis

A sensitivity analysis is performed for different amounts of desired LM, with and without accounting for LM constraints. Neglecting Equations (14)–(27) will lead to ignoring the LM constraints in the problem. The aim of this sensitivity analysis is to demonstrate the effect of LM constraints on the proposed problem.

Table 5. Optimal configuration of network, with and without LM constraints, in 24 h.

Time (h)	Potentially Opened Lines
	Without LM Constraints	With LM Constraints (λdes = 0.10)	With LM Constraints (λdes = 0.15)	With LM Constraints (λdes = 0.20)
1	L4-5, L7-8, L9-10, L13-14, L32-33	L6-26, L9-15, L10-11, L19-20, L32-33	L10-11, L13-14, L14-15, L17-18, L26-27	L2-3, L10-11, L14-15, L16-17, L25-29
2	L9-10, L9-15, L13-14, L14-15, L24-25	L7-8, L10-11, L13-14, L14-15, L16-17	L3-4, L9-10, L12-13, L14-15, L31-32	L4-5, L9-15, L11-12, L12-13, L27-28
3	L6-7, L10-11, L14-15, L21-22, L30-31	L6-26, L9-10, L9-15, L31-32, L32-33	L3-4, L11-12, L13-14, L31-32, L32-33	L4-5, L6-7, L9-15, L10-11, L31-32
4	L6-7, L9-10, L14-15, L21-22, L31-32	L3-4, L9-10, L12-22, L18-33, L31-32	L8-21, L10-11, L27-28, L31-32, L32-33	L8-9, L10-11, L12-22, L31-32, L32-33
5	L8-21, L12-22, L13-14, L15-16, L23-24	L3-23, L8-9, L8-21, L12-13, L17-18	L1-2, L2-19, L3-4, L17-18, L21-22	L4-5, L6-26, L8-21, L9-15, L13-14
6	L3-23, L4-5, L10-11, L15-16, L30-31	L5-6, L6-7, L10-11, L16-17, L24-25	L8-9, L9-15, L21-22, L24-25, L32-33	L1-2, L5-6, L9-15, L12-22, L18-33
7	L12-22, L16-17, L20-21, L26-27, L30-31	L4-5, L10-11, L16-17, L21-22, L30-31	L1-2, L9-10, L9-15, L15-16, L24-25	L1-2, L4-5, L6-7, L11-12, L18-33
8	L1-2, L3-23, L12-22, L16-17, L20-21,	L2-19, L3-23, L4-5, L15-16, L30-31	L3-4, L9-10, L9-15, L15-16, L24-25	L1-2, L2-19, L3-4, L11-12, L32-33
9	L3-23, L6-7, L12-22, L13-14, L16-17	L1-2, L3-4, L6-26, L9-10, L15-16	L3-4, L9-15, L17-18, L21-22, L31-32	L2-3, L4-5, L10-11, L12-13, L18-33
10	L8-21, L9-15, L11-12, L17-18, L21-22	L7-8, L8-21, L10-11, L16-17, L6-7	L1-2, L6-26, L10-11, L20-21, L21-22	L2-3, L6-7, L10-11, L19-20, L32-33
11	L5-6, L7-8, L12-22, L20-21, L23-24	L10-11, L13-14, L14-15, L20-21, L25-29	L3-23, L10-11, L18-33, L21-22, L31-32	L1-2, L2-19, L4-5, L9-10, L18-33
12	L6-26, L8-21, L12-13, L12-22, L16-17	L2-3, L11-12, L12-13, L14-15, L15-16	L2-19, L3-23, L4-5, L9-10, L27-28	L3-23, L6-7, L8-21, L20-21, L32-33
13	L5-6, L9-15, L12-22, L16-17, L23-24	L3-23, L7-8, L8-21, L16-17, L30-31	L3-4, L9-10, L9-15, L13-14, L17-18	L4-5, L6-26, L11-12, L12-13, L23-24
14	L9-15, L10-11, L12-13, L15-16, L23-24	L4-5, L6-7, L9-10, L12-22, L16-17	L1-2, L4-5, L9-10, L19-20, L21-22	L4-5, L6-7, L10-11, L13-14, L25-29
15	L6-7, L6-26, L8-21, L20-21, L23-24	L3-4, L19-20, L21-22, L30-31, L32-33	L4-5, L8-9, L11-12, L31-32, L32-33	L5-6, L6-7, L11-12, L13-14, L23-24
16	L5-6, L6-7, L13-14, L16-17, L20-21	L6-7, L6-26, L10-11, L16-17, L21-22	L3-23, L8-21, L9-15, L21-22, L32-33	L2-19, L3-23, L4-5, L8-9, L18-33
17	L6-26, L8-9, L12-22, L13-14, L16-17	L6-26, L8-21, L9-10, L21-22, L23-24	L4-5, L9-10, L20-21, L25-29, L31-32	L1-2, L4-5, L11-12, L18-33, L19-20
18	L6-26, L11-12, L20-21, L21-22, L23-24	L7-8, L12-22, L18-33, L24-25, L31-32	L3-23, L8-9, L8-21, L11-12, L16-17	L1-2, L2-19, L12-13, L21-22, L32-33
19	L3-4, L9-10, L12-22, L14-15, L15-16	L3-23, L9-10, L17-18, L21-22, L30-31	L5-6, L8-9, L10-11, L18-33, L30-31	L2-3, L10-11, L16-17, L18-33, L26-27
20	L3-23, L9-15, L10-11, L16-17, L21-22	L3-23, L4-5, L11-12, L17-18, L30-31	L7-8, L9-15, L12-22, L16-17, L23-24	L4-5, L7-8, L8-9, L11-12, L13-14
21	L3-23, L7-8, L12-22, L16-17, L30-31	L4-5, L9-10, L24-25, L31-32, L32-33	L3-23, L5-6, L9-10, L9-15, L18-33	L1-2, L3-4, L9-10, L18-33, L31-32
22	L8-9, L12-22, L13-14, L16-17, L20-21	L7-8, L8-21, L10-11, L13-14, L16-17	L8-9, L12-13, L20-21, L24-25, L32-33	L2-19, L3-23, L4-5, L18-33, L21-22
23	L3-23, L6-7, L6-26, L8-9, L16-17	L6-26, L8-9, L11-12, L16-17, L20-21	L2-19, L4-5, L8-9, L24-25, L9-15	L4-5, L18-33, L20-21, L21-22, L30-31
24	L3-23, L4-5, L7-8, L8-21, L13-14	L7-8, L8-9, L11-12, L14-15, L24-25	L1-2, L4-5, L10-11, L17-18, L21-22	L1-2, L9-10, L12-13, L18-33, L31-32

shows the number of switches opened in 24 h for the proposed model, with and without LM constraints for various levels of desired LM, demonstrating that LM limitations have a significant impact on network reconfiguration. This means that as the LM increases, the network is changed every hour to operate the distribution network securely. Note that the numbers in this table represent the bus number.

For example, in the presence of intermittent wind generation, the network configuration for the desired LM of 0.15 is the secured configuration in each hour.

Table 6. Optimal location of WFs and ESS, with and without LM constraints.

Model State	WF (Bus No.)	Number of Wind Turbines	Cumulative Capacity (MW)	ESS (Bus No.)
(a) W/O LM constraints
	14	5	1.25	7, 9
	25	6	1.5
	32	5	1.25
(b) W LM constraints
(λdes = 0.10)	14	6	1.5	5, 30
	30	7	1.6
	32	2	0.5
(c) W LM constraints
(λdes = 0.15)	14	4	1	21, 23
	27	7	1.75
	32	6	1.5
(d) W LM constraints
(λdes = 0.20)	18	8	2	9, 25
	25	6	1.5
	32	2	0.5

gives the optimal location of the WFs and ESSs for the analysis. It is evident from this table that the optimal location and size of WFs and ESSs are affected by security of the system. This means that consideration for security constraints is an important factor in the optimal allocation of DERs. Furthermore, it can be shown in Figure 8 that the values for the objective function of the GenCo’s cost for increasing the desired LM are also increased. This means that the GenCo should spend more money to guarantee the safe operation of the system from the point of view of voltage stability.

5.2. Contingency Analysis

As previously indicated, the accurate determination of LM could ensure the security of the distribution network in terms of voltage stability. To highlight this important role of LM, a simulation has been performed, assuming that a distribution network line is destroyed by a natural disaster or a man-made problem. More precisely, it was assumed in the IEEE 33-bus DN that the line between buses 6 and 26 is affected. The operation of the distribution network is then investigated under the following two conditions, C1: contingency condition, and C2: normal condition. Figure 9 shows the cumulative utilized wind power of WFs under these two conditions. This figure clearly illustrates the critical importance of WFs to cover distribution network loads and prevent load curtailment in contingency situations. On the other hand, the simulation reveals that the values of the cost function in these two conditions are USD 2461.43 and USD 2443.80, respectively. Comparing these two values demonstrates the increase in the GenCo’s costs as a result of the contingency condition. The last operational aspect to investigate under these two conditions is the security of the distribution network under study. For this purpose, the P-V curve of bus 24 at hour 15 is plotted under both operational conditions, C1 and C2 (Figure 10). This figure shows that the LM of the system drops significantly under the contingency condition, providing insight into the security of distribution network in both conditions. The distribution network may undergo a critical operation leading to a failure if at least a minimal value for LM is not evaluated under the contingency condition. Even though the distribution network’s LM drops dramatically under the contingency condition, the value of LM is still appropriate for a secured operation of the distribution network.

6. Conclusions

This study presents a secure reconfiguration method for distribution networks in the presence of wind farms and energy storage systems under normal and emergency conditions. By modeling voltage stability with LM constraints and optimizing the DNR schedule to maintain a desired level of LM, the proposed model ensures the security of the distribution network during the reconfiguration process and emergency situations. Furthermore, the proposed model implements an hourly switch status schedule to optimize the allocation of WFs and ESSs, while ensuring the secure operation of the distribution network. The results of the study, conducted on an IEEE 33-bus distribution system indicate the importance of LM constraints in hourly reconfiguration of distribution networks and in determining the optimal allocation of WFs and ESSs. Additionally, the study concludes that proper LM guarantees the security of the system under emergency conditions and that ESSs can be optimally scheduled to control WF power output and cover system loads. Future research could consider the role of plug-in electric vehicles in supplying system load demand, especially during critical emergencies, and contingencies in long-term planning, as they may impact optimization procedures.