Multi Microgrid Framework for Resilience Enhancement Considering Mobile Energy Storage Systems and Parking Lots

Abstract

This paper proposes a practical and effective planning approach that takes advantage of the mobility and flexibility of mobile energy storage systems (MESSs) to increase distribution system resilience against complete area blackouts. MESSs will be very useful for boosting the system’s resilience in places affected by disasters when the transmission lines are damaged. A joint post-disaster restoration strategy for MESSs and PEV-PLs is proposed, along with distributed generation and network reconfigurations, to reduce total system costs, which include customer interruption costs, generation costs, and MESS operation and transportation costs. The integrated strategy accounts for the uncertainty of the production of wind- and solar-powered microgrids (MGs) and different forms of load demand. Therefore, this paper assesses the effect of MESSs on distribution system (DS) resilience in respect of MG cost reduction and flexibility. Due to the multiple networks, PEV-PLs, DGs, and MESS limitations, the suggested restoration problem is stated as a mixed-integer linear programming problem. The suggested framework is complemented using a benchmark testing system (i.e., 33-bus DS). To assess the effectiveness of the proposed model, the model’s output is contrasted with results from typical planning and a traditional model. The comparison of the data shows that the suggested model, in addition to MESSs, effectively achieves a large decrease in cost and enhances the DS resilience level.

1. Introduction

Recent significant blackouts triggered by extreme weather occurrences have had disastrous economic and social impacts [1,2]. As a result, taking necessary measures to improve a DS’s resilience is vital to avoiding significant load shedding [3,4]. Electric service restoration may be easily handled in a more resilient DS, by integrating wind turbines (WT-DGs), photovoltaic (PV-DGs) systems, and plug-in electric vehicles, and an innovative option is offered for enhancing the operation of smart grid technologies. The notion of mobile energy storage devices is used as an alternate option for transportation electrification. Following a catastrophe, fixed sources, such as DERs and MGs, may be utilized to power one island of the DS [5,6]. However, coupling with PEV parking lots (PEV-PLs) to share energy with the electric power grid and transportable sources, such as MESSs, has the potential to greatly enhance resilience through effective scheduling across numerous islands within the DS.

Because of the critical significance of DGs in supplying system loads in today’s power systems, multiple studies on stationary sources in DS network restoration in various conditions have indeed been published [2,7]. When the DS is severed from the main network, DGs are commonly used in restoration [8]. When DGs are insufficient to satisfy energy demands, MGs can assemble a diverse set of distributed energy resources (DERs) to support the restoration of DS networks [9,10]. Furthermore, the DS may be divided into many islands, each with its own MG [11]. These studies show that a DS may gain a lot from stationary DERs, particularly MGs, in terms of improving resilience against widespread blackouts. Additionally, MGs offer chances to combine MESSs [12,13] and PEV-PLs [14] for DS service restoration due to the rising prevalence of public charging/discharging infrastructure.

Numerous articles have been published up to this point regarding the operation of the smart grid under both normal and emergency circumstances. The literature for this article has been divided into categories based on the types of components: MESSs, PEV-PLs, and distribution network reconfiguration (DNR).

MESSs: For the interconnection of MESSs, several studies have been conducted to use MESSs in the management of power systems under both normal [15,16] and catastrophic events [17]. For example, the authors of [18] developed a multi-agent paradigm for service restoration in DSs with MESSs. This architecture had three layers: a physical layer connected to the DS power network, a cyber layer connected to the communication system, and a MESS transportation layer. Its goal was to restore DS functionality in disaster situations. In [19], the author suggests an integrated power, gas, and transportation energy system using two transportation networks for MESSs. In the first instance, liquid hydrogen is transported and stored, while in the second, electrical storage systems are transported. As a result, RESs were stored in MESS facilities, or injected, transported, and reutilized in the natural gas network. Additionally, the redundant power generated by RESs may be immediately stored by MESSs, and the railway transportation system transports both to the load point of electrical and gas networks. Additionally, a stochastic platform was employed to capture the inherent uncertainties in the predicted values of the load and generation of RESs. In [20], two proposed approaches were created to find the optimal distribution of SESSs and MESSs in resilient DSs when operating under extreme events. In [21], a strategy for enhancing DS resilience with MESSs was presented. To enable DS performance in isolated operation, MESSs were routed in this framework during catastrophes. The authors of [22,23] have suggested service-restoration solutions that employ MESSs to boost DS resilience against catastrophic events and extreme weather disasters. Additionally, to increase DS resilience after extreme events, the studies in [24,25] introduced restoration strategies that also integrated network reconfiguration with MESSs transportation. In [15,26], MESSs were implemented to perform load shifting for the distribution network. An integrated unit commitment problem for a battery-based energy transportation system was addressed in [27] in an attempt to decrease transmission line congestion and lower operating costs. In addition to increasing system efficiency, battery-based energy transportation technology was utilized to decrease wind power curtailment [28]. In [22], a combined post strategy for mobile energy transportation was presented, and a two-stage methodology was used in this article to enable resilient route discovery and management of mobile power sources (MPSs). MPSs were pre-positioned in the DS during the first stage, i.e., before the event, to allow rapid pre-restoration, hence increasing the endurance of the electrical supply to critical loads. In the second stage, which occurs after the disaster, MPSs were dynamically transmitted to the DS to cooperate with traditional restoration attempts to improve system recovery. For load restoration, authors of [29] proposed a post-disaster joint approach integrating numerous MGs, MESSs, network reconfiguration, and generation scheduling. In [30], the authors developed a co-optimization model with the scheduling and routing of repair teams and mobile power stations to restore distribution networks and improve grid resilience. To integrate the routing decisions made by MESSs with the reconfiguration of distribution networks, a load-restoring technique was proposed in [31] based on rolling optimization. However, the multi-period dynamic dispatch was neglected because the assigned MEGs remained stationary in the same location.

PEV-PLs:

Despite their limitations, PEVs have exceptional economic and operational gains for distribution networks. As a result, several research papers have focused on examining the various elements of PEVs in power grids. The artificial bee colony optimization technique was used in [32] to present a multi-objective optimization model for the efficient allocation of PLs throughout the distribution grid. In [33], DGs and PEV-PLs were assigned in the network at the same time using a two-level optimization technique that investigated the cooperative behavior of PEV-PLs and renewable-based DGs. In [34], a comprehensive management model for the efficient scheduling of electric vehicles’ active and reactive power while dealing with multiple uncertainties was developed. The authors applied the Benders decomposition to speed up execution. From the viewpoint of parking lot owners, a noteworthy innovative planning model was presented in [35] for the long-term planning of PEV-PLs. The research discussed shifting energy usage from on-peak to off-peak times using a coordinated charging mechanism. When the connection between home loads and the upstream grid is damaged, an approach was suggested in [36] that leveraged PEV energy as a source of power for such loads. As described in [37], a parking-lot-equipped renewable energy sources based micro grid (RMG), was proposed to manage and aggregate PEVs. Although PEVs were present, the stochastic framework was used to describe the optimal energy management of the renewable energy-based microgrid. The hybrid information gap decision technique (IGDT) incorporated with the stochastic method was utilized in [38] to handle a transmission-constrained AC-unit commitment model integrated with electric vehicles (EV) and wind energy. A scenario-based approach was used to predict the behavioral uncertainty associated with owners of electric vehicles, while IGDT was used to control the uncertainty associated with wind energy.

Distribution network reconfiguration (DNR): Lots of studies on the subject of DNR have recently been published. In general, researchers of these publications have used a variety of strategies and techniques to address DNR challenges. In [39], a planning model was proposed that integrated expansion planning for the active distribution network and the DNR, while taking the demand response program into account to serve critical load demand as virtual distributed resources. The findings obtained by [40] illustrated the resilience of the adaptive control particle swarm optimization in contrast to other strategies, such as the genetic algorithm used by [41] for network reconfiguration in the presence of DGs. In [42], the idea of DNR was integrated with microgrid creation to restore system loads after a catastrophic event. A tri-level defender–attacker–defender paradigm was used in [43] as a primary component of power system resiliency. As a third stage or defense plan, the authors implemented reconfiguration and microgrid islanding methods to assess grid performance from the standpoint of a system operator. Table 1 provides the taxonomy of the research papers listed above. This table shows that some key details were overlooked in earlier research projects.

From the resilient operation perspective especially, the collaborative planning of MESSs, PEV-PLs, and network reconfiguration with inner uncertainties during post-disaster restoration has not previously been taken into consideration. The gap in the application of MESSs to DS critical load profile restoration must be filled, nevertheless, by a proper planning model. To reduce overall costs in the after-effects of disasters, a collaborative post-disaster stochastic restoration scheme is presented in which MESSs and PEV-PLs are dynamically designed in collaboration with DS reconfiguration through MGs. Additionally, all system buses are possible nodes for MGs that are paired with PLs to be installed optimally. More specifically, MGs leverage the DGs’ generating capabilities to their fullest potential and serve as root buses to dynamically generate islands for the pickup of various load kinds. Meanwhile, the vehicle scheduling problem (VSP) uses the time-space network (TSN) and is designed to represent mathematically the scheduling challenge of MESSs.

Because internal uncertainties and external events are taken into account, stochastic programming may be the appropriate method for addressing the outlined issue. Therefore, this paper develops a comprehensive resilience-driven model for the optimal planning and joint post-disaster restoration problems of networked MGs, capturing both internal uncertainties and external contingencies. The optimization problem is formulated as an MIP model.

According to a comparison of the essential aspects of the related literature in Table 1, there is a gap in the research relating to the uncertainty modelling techniques of a joint scheduling of power electrical vehicles parking lots PEV-PLs, MESSs, and DNR in post-disaster restoration is still not considered, especially from the perspective of resilience operation. In this regard, the point of the study is to close the gap between the application PEV-PLs and MESSs for distribution system restoration. Therefore, a cooperative post-disaster restoration strategy has been designed for MESS, PEV-PL, and DS interaction with multiple MGs to provide more resilient DSs scheduled in cooperation with DNR to increase the flexibility and reduce the overall cost of the system. Nevertheless, through scenario-based stochastic programming, internal uncertainties related to the DG generation sources and various load profiles are considered in the operating framework, while external damages are used to emphasize the importance of meeting the load requirements of critical customers under contingency situations in conformance with their priorities. The main contributions are concluded as follows.

A joint post-disaster restoration scheme for the coordination of MESSs, PLs, and DSs with multiple MGs is proposed to bridge the gap in the application of MESSs and PLs to achieve more resilient DSs. A temporal-spatial MESS model, which connects the transportation networks and DSs, is proposed in terms of flexibility and total cost reduction.

• The total cost, considering both the benefits (mitigating the customer interruption cost) and operation costs (Wind-DG and PV-DG generation cost, MESS transportation cost, PEV-PL operation cost, and battery maintenance cost), is proposed to evaluate the feasibility of using MESSs and PEV-PLs in DS restoration.
• Internal uncertainties with DG generation sources and different load profiles are considered in the operational model via scenario-based stochastic programming, while external contingencies, including multiple line outages, are considered.
• The key role of different load types is demonstrated in the model in addition to the reconfiguration of the network to show the importance of serving critical customer loads in contingency conditions according to their priorities.

Table 1. A comparison of the relevant works in the literature.

Ref. No.	DNR	MESS	PEL-PLs	DGs	Considering MGs	Uncertain Parameters	Contingency	Test System
[1,2,3,4,5,6,7] General descriptions of the resiliency enhancement methods and recommendations for MG and DS network restoration.
[8]	✕	✕	✕	✓	✓	DGs, load	✓	123-bus
[9]	✕	✕	✕	✓	✓	✕	✓	-
[10,11]	✕	✕	✕	✓	✓	✕	✓	37-bus 123-bus
[12,13]	✕	✕	✓	✓	✓	✕	✕	33-bus 37-bus
[14]	✓	✕	✓	✓	✕	✕	✓	33-bus
[15]	✕	✕	✓	✓	✕	✕	✕
[16]	✕	✓	✕	✓	✕	PV, wind, load	✕	33-bus 69-bus
[17]	✕	✓	✕	✓	✕	Load, wind speed, solar radiation	✕
[18,19,20,21]	✕	✓	✕	✓	✕	✕	✓	33-bus 15-bus
[19]	✕	✓	✕	✓	✕	Load and wind	✕	118-bus 14-bus gas network
[20]	✕	✓	✕	✓	✕	PV	✓	33-bus
[22]	✕	✓	✕	✕	✕	✕	✓	33–123 bus
[23]	✕	✓	✕	✕	✓	Line failure	✓	15-bus
[24]	✕	✓	✕	✕	✕	✕	✓	33-bus
[25]	✕	✓	✕	✓	✕	Wind, PV	✓	33-bus
[26]	✕	✕	✓	✓	✕	✕	✕	6-bus
[27]	✕	✓	✕	✕	✕	✕	✕	118-bus
[28]	✕	✓	✕	✓	✕	TSN	✕	118-bus
[29]	✓	✓	✕	✓	✓	✕	✓	33-bus
[30]	✓	✓	✕	✕	✕	Line failure	✓	33-bus
[31]	✓	✓	✕	✕	✓	✕	✓	33-bus
[32]	✕	✕	✓	✕	✕	✕	✕	33-bus
[33]	✕	✕	✓	✓	✕	Wind, PV	✕	15-bus
[34]	✕	✕	✓	✕	✕	Load, price	✕	33-bus
[35]	✕	✕	✓	✓	✕	✕	✕	Real system in Toronto
[36]	✕	✕	✓	✓	✓	Wind, market prices	✓	33-bus
[37]	✕	✕	✓	✓	✓	Market price, wind and load demand	✕	RMG owner plans
[38]	✕	✕	✓	✓	✕	wind	✕	6-bus
[39]	✓	✕	✕	✕	✕	✕	✕	33-bus
[40]	✓	✕	✕	✓	✕	Wind, loads	✕	Taiwan Power system
[41]	✓	✕	✕	✕	✕	✕	✓	33-bus
[42,43]	✓	✕	✕	✓	✓	✕	✓	33-bus
This Paper	✓	✓	✓	✓	✓	(C, R, IN) Load profiles, wind, and PV	✓	33-bus

Table 1. A comparison of the relevant works in the literature.

Ref. No.	DNR	MESS	PEL-PLs	DGs	Considering MGs	Uncertain Parameters	Contingency	Test System
[1,2,3,4,5,6,7] General descriptions of the resiliency enhancement methods and recommendations for MG and DS network restoration.
[8]	✕	✕	✕	✓	✓	DGs, load	✓	123-bus
[9]	✕	✕	✕	✓	✓	✕	✓	-
[10,11]	✕	✕	✕	✓	✓	✕	✓	37-bus 123-bus
[12,13]	✕	✕	✓	✓	✓	✕	✕	33-bus 37-bus
[14]	✓	✕	✓	✓	✕	✕	✓	33-bus
[15]	✕	✕	✓	✓	✕	✕	✕
[16]	✕	✓	✕	✓	✕	PV, wind, load	✕	33-bus 69-bus
[17]	✕	✓	✕	✓	✕	Load, wind speed, solar radiation	✕
[18,19,20,21]	✕	✓	✕	✓	✕	✕	✓	33-bus 15-bus
[19]	✕	✓	✕	✓	✕	Load and wind	✕	118-bus 14-bus gas network
[20]	✕	✓	✕	✓	✕	PV	✓	33-bus
[22]	✕	✓	✕	✕	✕	✕	✓	33–123 bus
[23]	✕	✓	✕	✕	✓	Line failure	✓	15-bus
[24]	✕	✓	✕	✕	✕	✕	✓	33-bus
[25]	✕	✓	✕	✓	✕	Wind, PV	✓	33-bus
[26]	✕	✕	✓	✓	✕	✕	✕	6-bus
[27]	✕	✓	✕	✕	✕	✕	✕	118-bus
[28]	✕	✓	✕	✓	✕	TSN	✕	118-bus
[29]	✓	✓	✕	✓	✓	✕	✓	33-bus
[30]	✓	✓	✕	✕	✕	Line failure	✓	33-bus
[31]	✓	✓	✕	✕	✓	✕	✓	33-bus
[32]	✕	✕	✓	✕	✕	✕	✕	33-bus
[33]	✕	✕	✓	✓	✕	Wind, PV	✕	15-bus
[34]	✕	✕	✓	✕	✕	Load, price	✕	33-bus
[35]	✕	✕	✓	✓	✕	✕	✕	Real system in Toronto
[36]	✕	✕	✓	✓	✓	Wind, market prices	✓	33-bus
[37]	✕	✕	✓	✓	✓	Market price, wind and load demand	✕	RMG owner plans
[38]	✕	✕	✓	✓	✕	wind	✕	6-bus
[39]	✓	✕	✕	✕	✕	✕	✕	33-bus
[40]	✓	✕	✕	✓	✕	Wind, loads	✕	Taiwan Power system
[41]	✓	✕	✕	✕	✕	✕	✓	33-bus
[42,43]	✓	✕	✕	✓	✓	✕	✓	33-bus
This Paper	✓	✓	✓	✓	✓	(C, R, IN) Load profiles, wind, and PV	✓	33-bus

2. Integration Service Restoration

This section presents a successful service restoration strategy that coordinates MESS, PEV-PLs, microgrid resource dispatching, and distribution network reconfiguration. Resilience is referred to as the ability to anticipate and adapt to changing situations, as well as to resist and recover quickly from disturbances [44]. For demonstration purposes, a hypothetical resilience curve related to an event in [45] is used in this work, as illustrated in Figure 1 (R), as a measure of the level of system resilience. The set of points is as follows: pre-event robust state (t₀, t₁), event progress (t₁, t₂), post-event damaged state (t₂, t₃), restoration state (t₃, t₄), post-restoration state (t₄, t₅), and infrastructure recovery (t₅, t₆). In this article, it was considered that catastrophic occurrences create transmission grid failures at (t₁) and that distribution systems can no longer be powered by transmission grids. As a result of the faults and their effects being noted in (t₂, t₃), microgrids were used to coordinate PEV-PLs and MESSs to restore service at the distribution level. To increase the system resilience level to (R₂), the suggested joint restoration strategy was applied from (t₃). Until the main transmission systems resumed operation, the DS continued in the post-restoration condition. Therefore, the period (t₃, t₅) in this study refers to the restorative state and the post-restorative period. The joint restoration technique for the DS was implemented throughout the optimization horizon (T), as illustrated in Figure 1, and it was considered that the upstream grid was repaired at (t₅). As a result, during the time (T), the employment of MESSs and PEV-PLs in the DS was suggested for a cooperative post-disaster restoration strategy that utilizes MESSs and PEV-PLs in MGs in cooperation with renewable DGs for post-disaster recovery. Utility firms were expected to own and control MESSs. At the same time, PEV parking lots (PEV-PLs) could be represented by the DER [14]. Furthermore, it was expected that a transportation network connected all MGs inside the DS. As a result, PEVs could play an important role in sourcing system loads. Even though PEVs absorb electric power and operate as consumers, the deployment of vehicle-to-grid (V2G) technology enables PEVs to share energy with the power network.

A resilient service restoration in a DS after natural disasters was made possible by the scheduling of MESSs, PEV-PLs, and the reconfiguration of the distribution network using MGs. By adjusting the ON/OFF status of remote-controlled switches, the DS was divided into a few islands and reconfigured. An MG on each island had a different load profile (residential, commercial, and industrial) and served as a root bus to deliver power to critical loads while also acting as an interface between DSs and transportation systems.

3. Mobile Energy Storage System Modeling

A dynamic TSN model was used to simulate the trip network of MESSs to more effectively manage scheduling issues in both the temporal and space domains. To better understand the driving dynamics and charging/discharging behaviors of MESSs, as well as their effects on DS restoration, this section presents a temporal-spatial MESS model [31,46].

A realistic transportation network is considered in Figure 2 with three stations {1, 2, and 3}, where the number on each arc connecting two stations is associated with the trip time, expressed in time spans. Thus, the traveling time between every two MGs is 1 h. However, there are two-time spans, or twice as long as a distance, between MG 1 and 3. To make the problem easier to understand, a fictitious station 4 was created between MGs (stations) 1 and 3. Therefore, the trip durations of each arc between any two MGs were equivalent to one time period [27,47].

The TSN depicted in Figure 3 represents the possible hourly connections for a transportation network with four MGs (stations) that were already employed in public transportation scheduling tasks [27,28,47]. As shown in Figure 3, multiple types of arcs were used to represent the traveling and parking behavior of MESSs to show their dynamics. The TSN consisted of MG terminals stations and arcs connecting different MGs, which show the allowable transport paths. Two different forms of TSN arcs were addressed in this work. The first, which consists of horizontal solid arcs, denotes the MESS stopping to transfer power at any station tied to the MG. Grid connecting arc is the name given to an arc of this type. Secondly, the slanted dotted arcs depict MESS transit between stations at any given time span. It is known as a transferring arc when an arc of this type exists. Additionally, since the MESS cannot be connected to the MG stations without charging or discharging devices, these MG terminals were classified as fictional stations. The TSN-Based MESS Model may be formulated as follows;

$${\displaystyle \sum _{(e,f)\in A}{ƛ}_{c,ef,ts}^{Trans}}=1,\forall c\in ℂ,ts\in TS$$

(1)

$${\displaystyle \sum _{ef\in A^{+}}{ƛ}_{c,ef,ts+1}^{Trans}=}{\displaystyle \sum _{ef\in A^{-}}{ƛ}_{c,ef,ts}^{Trans}},\forall c\in ℂ,e\in \aleph ,ts\in TS$$

(2)

$${\displaystyle \sum _{ef\in A^{+}}{ƛ}_{c,ef,1}^{Trans}=}{ƛ}_{c,e,0}^{Trans},\forall c\in ℂ,e\in \aleph$$

(3)

$${ƛ}_{c,ef,ts}^{Trans}+{ƛ}_{c,fe,ts+1}^{Trans}\le 1,\forall c\in ℂ,(e,f)\in A,e\ne f,ts\in TS$$

(4)

Equation (1) implies that MESS’s position can be located on any arc in time span ts. In (2), any MESS c at the end of the time span (ts) is in MG (e, ts) in the TSN, which indicates that in the following time span (ts + 1), MESS k should indeed be in one of the arcs that start from MG (e, ts). Specifically, Constraint (3) specifies that the outflow at each station equals the starting state of MESS k in the time span (1). Constraint (4) ensures that TESSs will never be able to perform an instantaneous round trip, meaning MESSs going from one MG to another are not allowed to return to the first MG immediately. When the mobile energy storage units arrive at any MG, the following constraints must be met.

$$0\le P_{c,e,t,s}^{MESS-C}\le {ƛ}_{c,ee,t}^{Trans}\times P_k^{MESS-MAX}\forall c\in ℂ,\forall e\in \aleph ,t\in T,s\in \omega$$

(5)

$$0\le P_{c,e,t,s}^{MESS-D}\le {ƛ}_{c,ee,t}^{Trans}\times P_c^{MESS-MAX}\forall c\in ℂ,\forall e\in \aleph ,t\in T,s\in \omega$$

(6)

$${\displaystyle \sum _{e\in \aleph }{P}_{c,e,t,s}^{MESS-C}\le {\mathsf{\Gamma }}_{c,t,s}^C\times P_c^{MESS-MAX}}\forall c\in ℂ,t\in T,s\in \omega$$

(7)

$${\displaystyle \sum _{e\in \aleph }{P}_{c,e,t,s}^{MESS-D}\le {\mathsf{\Gamma }}_{c,t,s}^D\times P_c^{MESS-MAX}}\forall c\in ℂ,t\in T,s\in \omega$$

(8)

$${\mathsf{\Gamma }}_{c,t,s}^C+{\mathsf{\Gamma }}_{c,t,s}^D\le {\displaystyle \sum _{e\in \aleph }{ƛ}_{c,ee,t}^{Trans}}\forall c\in ℂ,t\in T,s\in \omega$$

(9)

$$E_{c,ts+1,s}^{MESS}=E_{c,ts,s}^{MESS}-[{\displaystyle \sum _{e\in \aleph }\frac{P_{c,e,ts,s}^{MESS-Dis}}{{\eta }_c^{MESS-Dis}}}-{\eta }_c^{MESS-Ch}\times {\displaystyle \sum _{e\in \aleph }{P}_{c,e,ts,s}^{MESS-Ch}]}\forall c\in ℂ,ts\in TS,s\in \omega$$

(10)

$$E_c^{MESS-MIN}\le E_{c,ts,s}^{MESS}\le E_c^{MESS-MAX}\forall c\in ℂ,ts\in TS,s\in \omega$$

(11)

The limitations on the charge/discharge variables of MESSs are stated in Constraints (5) and (6). When connected to the MG, the MESS can operate in one of two states, charge or discharge, which is controlled by Constraints (7) and (8), and the operation mode is restricted by (9). By the end of period t, Constraint (10) defines the energy storage (SOC) in the batteries of the MESS. Last but not least, the variable of SOC is constrained by upper and lower boundaries as indicated in (11).

4. Formulation of the Proposed Modeling Framework with MESSs

In this article, the modeling of PEV-PLs and MESSs into DSs allowed for the coordination of MESSs and DSs via mixed-integer programming (MIP) with stochastic scenarios to produce a more resilient restoration strategy. The GAMS program with the Gurobi solver was used to handle the code using a stochastic optimization strategy [48]. Indeed, Gurobi solved a sequence of LP subproblems for MILP while solving problems comprising discrete variables using a branch and cut algorithm. Gurobi also ensured the validity of the inputs as an integer-feasible solution. If this procedure was successful, the response would be accepted as an integer solution to the current problem.

A main grid is disrupted by unexpected natural catastrophes in the resilient operation of MGs, which forces an MG to keep operating off-grid and curtail all the connected loads. The mathematical formulation of the integrated post-disaster restoration scheme is highlighted in Figure 4, which also provides notable details of the proposed model framework of the most effective restoration. As seen in Figure 4, with consideration of operational and topological constraints in addition to the input data, the objective aimed to minimize the overall cost. The implementation of continuous variables for each type of load recognizes partial load restoration.

According to Figure 4, it can be shown that the uncertainty of DGs (wind and PVs) and load profiles (residential, commercial, and industrial) were modeled using stochastic optimization for a set of stochastic scenarios attached to their probabilities. Furthermore, a dynamic TSN equation was employed to model the MESS trip network to manage the scheduling challenges in both the temporal and spatial domains and achieve the charging and discharging patterns of MESSs. In contrast, several limitations were taken into account while modeling the number and position of PEV-PLs within each parking lot station at MGs. Finally, to maximize load restoration and minimize overall operating costs, the proposed system’s outcomes for post-disaster restoration were achieved by taking into account the corresponding constraints and equations in addition to the input data.

4.1. Objective Function

As previously stated, the goal of this work is to minimize an objective function that includes in the first section: the operation cost of the MESS, interruption cost for both critical and non-critical loads, PEV operation cost, MG generation cost (costs of supplied power from PV-DGs and Wind-DGs), and MESS transportation cost in the second section, as in Equation (12).

$$\begin{array}{l}\min {\displaystyle \sum _{S\in \omega }{\pi }_w{\displaystyle \sum _{t\in T}\left[\begin{array}{l}{\displaystyle \sum _{c\in ℂ}{C}_c{\displaystyle \sum _{e\in \aleph }(P_{c,e,t,s}^{MESS-Ch}+P_{c,e,t,s}^{MESS-Dis})+}}\\ +{\displaystyle \sum _{i\in {\mathsf{\Phi }}_{C\&IN}}{C}_i^{C/IN}(P{D}_{i,t,s}^{(C/IN)}-P_{i,t,s}^{D-restored})+{\displaystyle \sum _{i\in {\mathsf{\Phi }}_R}{C}_i^R(}P{D}_{i,t,s}^R-P_{i,t,s}^{D-restored})}\\ +{\displaystyle \sum _{e\in {\aleph }_{pev}}{C}_e^{PEV}(P_{e,t,s}^{PEV.Ch}+P_{e,t,s}^{PEV.Dis})}\\ +{\displaystyle \sum _{e\in {\aleph }_w}{C}_e^{Wind}{P}_{e,t,s}^{Wind}}+{\displaystyle \sum _{e\in {\aleph }_{pv}}{C}_{\aleph }^{PV}{P}_{e,t,s}^{PV}}\end{array}\right]}}\\ \\ +{\displaystyle \sum _{c\in ℂ}{C}_c^{Trans}{\displaystyle \sum _{ts\in TS}{\displaystyle \sum _{\begin{array}{l}(f,e)\in A\\ f\ne e\end{array}}{ƛ}_{c,fe,ts}^{Trans}}}}\end{array}$$

(12)

4.2. Operation Constraints of Distribution Systems and Network Topology

In this research, the power flow analysis of DS was performed using a linearized power flow model [31] as described by the following constraints.

$${\displaystyle \sum _{(i,j)\in l}{P}_{ij,t,s}}-{\displaystyle \sum _{(k,i)\in l}{P}_{ik,t,s}}=P_{i,t,s}^{MG-MESS}+P_{i,t,s}^{MG-PL}-P_{i,t,s}^{D-restored},\forall i\in \mathsf{\Phi },t\in T,s\in \omega$$

(13)

$${\displaystyle \sum _{(i,j)\in l}{Q}_{ij,t,s}}-{\displaystyle \sum _{(k,i)\in l}{Q}_{ik,t,s}}=Q_{i,t,s}^{MG-MESS}+Q_{i,t,s}^{MG-PL}-Q_{i,t,s}^{D-restored},\forall i\in \mathsf{\Phi },t\in T,s\in \omega$$

(14)

$$V_{i,t,s}-V_{j,t,s}\le MF(1-{\vartheta }_{ij,t,s})+\frac{R_{ij}{P}_{ij,t,s}+X_{ij}{Q}_{ij,t,s}}{V_0},\forall (i,j)\in l,t\in T,s\in \omega$$

(15)

$$V_{i,t,s}-V_{j,t,s}\ge MF(1-{\vartheta }_{ij,t,s})+\frac{R_{ij}{P}_{ij,t,s}+X_{ij}{Q}_{ij,t,s}}{V_0},\forall (i,j)\in l,t\in T,s\in \omega$$

(16)

$$-{\vartheta }_{ij,t,s}\times P_{ij}^{Max}\le P_{ij,t,s}\le P_{ij}^{Max}\times {\vartheta }_{ij,t,s}\forall (i,j)\in l,t\in T,s\in \omega$$

(17)

$$-{\vartheta }_{ij,t,s}\times Q_{ij}^{Max}\le Q_{ij,t,s}\le Q_{ij}^{Max}\times {\vartheta }_{ij,t,s}\forall (i,j)\in l,t\in T,s\in \omega$$

(18)

$$V_i^{Min}\le V_{i,t,s}\le V_i^{Max}\forall i\in \mathsf{\Phi },t\in T,s\in \omega$$

(19)

$$V_{i,t,s}=V_0\forall i\in \mathsf{\Phi },t\in T,s\in \omega$$

(20)

$$0\le P_{i,t,s}^{C\&IN-restored}\le P{D}_{i,t,s}^{}\forall i\in {\mathsf{\Phi }}_{C\&IN},t\in T,s\in \omega$$

(21)

$$0\le P_{i,t,s}^{R-restored}\le P{D}_{i,t,s}^{}\forall i\in {\mathsf{\Phi }}_{\mathrm{R}},t\in T,s\in \omega$$

(22)

$$0\le Q_{i,t,s}^{C\&IN-RES}\le Q{D}_{i,t,s}^{}\forall i\in {\mathsf{\Phi }}_{C\&IN},t\in T,s\in \omega$$

(23)

$$0\le Q_{i,t,s}^{R-RES}\le Q{D}_{i,t,s}^{}\forall i\in {\mathsf{\Phi }}_{\mathrm{R}},t\in T,s\in \omega$$

(24)

Constraints (13) and (14) define the AC linearized active and reactive power flow, which must be evaluated at each bus (i), time interval (t), and scenario (S). Equation (15) use the big-M method to impose constraints on line voltage drop [31]. As in Equations (17) and (18), the active and reactive line capacity constraints should be limited to allowable levels. The lower and upper bounds on voltage magnitude are enforced by (19). The initial voltage of MG buses is set to V₀ by (20). The active and reactive restored load restriction for critical (commercial and industrial) and non-critical (residential) loads is represented by Constraints (21)–(24), respectively.

$$\{\begin{array}{l}{\vartheta }_{ij,t,s}=(1-S{T}_{ij,t})\mathrm{if}{\vartheta }_{ij,t,s}\in \mathsf{\Phi },t\in T,s\in \omega \\ {\vartheta }_{ij,t,s}=\left\{0,1\right\},\mathrm{if}{\vartheta }_{ij,t,s}\in {\mathsf{\Phi }}_{Flex},t\in T,s\in \omega \end{array}$$

(25)

$${\vartheta }_{ij,t,s}={\vartheta }_{ji,t,s}\forall (i,j)\in {\mathsf{\Phi }}_{Fixed},t\in T,s\in \omega$$

(26)

$${\displaystyle \sum _{(i,j)\in l}{\vartheta }_{ij,t,s}}=\left|\mathsf{\Phi }\right|-|\aleph |\forall t\in T,s\in \omega$$

(27)

The constraints on the network topology can be separated into two categories: the integer Constraint (25) for the binary variables of the connection matrix (ST), and the radiality Constraints (26) and (27) to ensure the radial operation of the DS. As a result of the flexibility of the operation and protection coordination, radial architecture was employed with various MGs throughout the distribution network. As a result, the radial DS configuration was employed in this study and should be kept during the reconfiguration procedure. Then, the cardinality of the set in the radial DS arrangement was indicated, as seen in (27), which is a condition for redial DS configuration.

4.3. Operation Constraints of Microgrids

MGs were represented in this study as a single bus with distributed generation PV-DGs and Wind-DGs, combining the entire production resources, an equivalent local load, and charging/discharging services. Hence, the following constraints can be used to represent the operation of the MG when the MESS is integrated.

$$P_{e,t,s}^{MG-MESS}=P_{e,t,s}^{Wind}-{\displaystyle \sum _{c\in ℂ}{P}_{c,e,t,s}^{MESS-Ch}+{\displaystyle \sum _{c\in ℂ}{P}_{c,e,t,s}^{MESS-Dis}}}-P{D}_{e,t,s}^{R,C,IN}\forall e\in \aleph ,t\in T,s\in \omega$$

(28)

$$Q_{e,t,s}^{MG-MESS}=Q_{e,t,s}^{Wind}-Q{D}_{e,t,s}^{R,C,IN}\forall e\in \aleph ,t\in T,s\in \omega$$

(29)

$$P_{e,t,s}^{MG-PL}=P_{e,t,s}^{PV}-P_{e,t,s}^{PEV-Ch}+P_{e,t,s}^{PEV-Dis}-P{D}_{e,t,s}^{R,C,IN}\forall e\in \aleph ,t\in T,s\in \omega$$

(30)

$$Q_{e,t,s}^{MG-PL}=Q_{e,t,s}^{PV}-Q{D}_{e,t,s}^{R,C,IN}\forall e\in \aleph ,t\in T,s\in \omega$$

(31)

Equations (28) and (29) represent the cumulative active and reactive power supply from MGs to distribution systems through bus e, taking the MESS charging from and discharging to MGs, as well as Wind-DGs deployed in the same MG, into consideration. When considering PEVs-PLs charging from and discharging to MGs, as well as PV-DGs located in the same MG, Equations (30) and (31) gather the active and reactive power supply from MGs to distribution systems through bus e. In addition, the loads that were located at each microgrid were considered.

4.4. The PEV-PLs’ Constraints

In this work, many constraints were considered to model the number and location of PEV-PLs among MGs, the capacity of electric vehicles, and the number of electric vehicles. The initial energy of PEVs represents the energy within a PEVs battery when it enters an MG’s parking lot station for the charging-discharging process. As a result, the initial energy at PEVs in the first time interval was considered in the planning to estimate the energy of PEVs, as anticipated in (32) and (33). As seen in the (34), the energy stored inside the battery in the final period equals the energy stored in the first time intervals. The energy storage capacity of PEVs was limited by the upper and lower energy stored and the number of PEVs that enter the PLs for charging and discharging as denoted in (35). The limitations of charging and discharging power are represented in (36) and (37), respectively, whereas Equation (38) limits the BESS to choose just one action of the charging or discharging process by using the (M) method for linearization purpose. Constraints (39) and (40), specify the charging and discharging powers in terms of installed maximum power, which should not be greater than the considered maximum power. In addition, constrain (42) is defined as the number of PEVs that should not exceed the maximum permitted number. The deployment of PLs in the MGs is important due to the obvious benefits of PEVs and rising trends in growing PEV penetration into the grid via MGs. In this regard, the binary variable is introduced in Constraint (43) to ensure the PL charging stations support all electric vehicles in the network. For that purpose, (44) restricts the maximum capacity of ith PLs for each MG.

$$E_{e,t,s}^{Pev}={\Upsilon }_{e,t,s}^{PEV}\times E_e^{Initial}+E_{e,t-1,s}^{Pev}+\Delta E_{e,t,s}^{Pev}\forall e\in \aleph ,t\in T,s\in \omega$$

(32)

$$\Delta E_{e,t,s}^{pev}={\eta }_{pev}^{Ch}{P}_{e,t,s}^{Ch.Pev}-\frac{P_{e,t,s}^{Dis.Pev}}{{\eta }_{pev}^{dis}}\forall e\in \aleph ,t\in T,s\in \omega$$

(33)

$$E_{e,t=24,s}^{pev}=E_{e,t=1,s}^{pev}\forall e\in \aleph ,s\in \omega$$

(34)

$$E_{e,t,s}^{Pev}\le {\Upsilon }_{e,t,s}^{PEV}\times E_e^{Max.Pev}\forall e\in \aleph ,t\in T,s\in \omega$$

(35)

$$P_{e,t,s}^{PEV-Ch}\le {\upsilon }_{e,t,s}^{Ch}\times M\forall e\in \aleph ,t\in T,s\in \omega$$

(36)

$$P_{e,t,s}^{PEV-Dis}\le {\upsilon }_{e,t,s}^{Dis}\times M\forall e\in \aleph ,t\in T,s\in \omega$$

(37)

$${\upsilon }_{e,t,s}^{Ch}+{\upsilon }_{e,t,s}^{Dis}\le 1\forall e\in \aleph ,t\in T,s\in \omega$$

(38)

$$P_{e,t,s}^{PEV-Ch}\le P_e^{Max.pev}\times {\Upsilon }_{e,t,s}^{PEV}\forall e\in \aleph ,t\in T,s\in \omega$$

(39)

$$P_{e,t,s}^{PEV-Dis}\le P_e^{PEV-Max}\times {\Upsilon }_{e,t,s}^{PEV}\forall e\in \aleph ,t\in T,s\in \omega$$

(40)

$${\displaystyle \sum _{e\in \aleph }{\displaystyle \sum _{t\in T}{\displaystyle \sum _{s\in \omega }{P}_{e,t,s}^{PEV-Ch}}}}\ge {\displaystyle \sum _{e\in \aleph }{\displaystyle \sum _{t\in T}{\displaystyle \sum _{s\in \omega }{P}_{e,t,s}^{PEV-Dis}}}}$$

(41)

$${\Upsilon }_{e,t,s}^{PEV}\in \left\{0,1,2,3,\dots \dots {\Upsilon }_{PEV}^{Max}\right\}\forall e\in \aleph ,\mathrm{t}\in \mathrm{T},s\in \omega$$

(42)

$$\left\{\begin{array}{l}{\displaystyle \sum _{e\in {\mathsf{\Phi }}_{PEV-PLs}}{\xi }_e^{PLs}\in \left\{0,1\right\}}\\ {\xi }_e^{PLs}=0\mathrm{otherwise}\end{array}\right\}$$

(43)

$${\displaystyle \sum _{e\in {\mathsf{\Phi }}_{PEV-PL}}{\xi }_e^{PLs}={\mathbb{N}}_{PEV-PLs}^{Max}}$$

(44)

To achieve the appropriate size and position of PEV-PLs, both binary and integer variables must be satisfied at the same time, which resulted in a nonlinear constraint. Therefore, a linearization factor (ζ) was used to indicate the optimal location of PLs as well as the capacity of each PEV-PL that is installed in MG (e), as expressed in (45) and (46). Constraints (47) and (48) express the limitation of an integer variable by the linearization factor that was considered in the model.

$$0\le Ca{p}_{e,t,s}^{Pev.Pls}\le Ca{p}_{MAX}^{Pev.PLs}\forall e\in \aleph ,\mathrm{t}\in \mathrm{T},s\in \omega$$

(45)

$$0\le {\zeta }_{e,t,s}\le {\xi }_e^{PLs}\times Ca{p}_{MAX}^{Pev.PLs}\forall e\in \aleph ,\mathrm{t}\in \mathrm{T},s\in \omega$$

(46)

$$Ca{p}_{e,t,s}^{Pev.Pls}-(1-{\xi }_e^{PLs})\times Ca{p}_{MAX}^{Pev.PLs}\le {\zeta }_{e,t,s}\le Ca{p}_{e,t,s}^{Pev.Pls},\forall e\in \aleph ,\mathrm{t}\in \mathrm{T},s\in \omega$$

(47)

$$0\le {\Upsilon }_{e,t,s}^{PEV}\le {\zeta }_{e,t,s}\forall e\in \aleph ,\mathrm{t}\in \mathrm{T},s\in \omega$$

(48)

4.5. Distribution Generation (DGs) Constraints

The following is a limitation of DGs (Wind and PVs) and their capacity, which is expressed in the following constraints.

$$0\le P_{e,t,s}^{Wind}\le P_e^{(W)unit}\times {\gamma }_{t,s}^{Wunit}\forall e\in {\aleph }_w,t\in T,s\in \omega$$

(49)

$$0\le Q_{e,t,s}^{Wind}\le P_{e,t,s}^{Wind}\times \tan ({\cos }^{-1}(P{F}_W))\forall e\in {\aleph }_w,\mathrm{t}\in \mathrm{T},\mathrm{s}\in \omega$$

(50)

$$0\le P_{e,t,s}^{PV}\le P_e^{(PV)unit}\times {\hslash }_e^{PV}\times {\gamma }_{t,s}^{PV}\forall e\in {\aleph }_{pv},\mathrm{t}\in \mathrm{T},s\in \omega$$

(51)

$$0\le Q_{e,t,s}^{PV}\le P_{e,t,s}^{PV}\times \tan ({\cos }^{-1}(P{F}_{PV}))\forall e\in {\aleph }_{pv},\mathrm{t}\in \mathrm{T},s\in \omega$$

(52)

$$\left\{\begin{array}{l}{\displaystyle \sum _{e\in E_{PV}}{\hslash }_e^{PV}=}1,\mathrm{if}{\xi }_e^{PLs}=1\\ {\hslash }_e^{PV}=0,\mathrm{otherwise}\end{array}\right\}$$

(53)

where (49) and (50) define, respectively, the active and reactive power restrictions of wind-DGs installed at selected MGs with MESSs, while PV-DG active and reactive power restrictions are denoted in (51) and (52). Constraint (53) was implemented to ensure that the installed PV-DGs are at any MGs that include PLs.

5. Case Study and Input Data

The planning model [49] was tested using an IEEE 33-bus test system with 8 MVA and 12.66 KV base values, as shown in Figure 5. In this work, the planning model problem was a mixed-integer quadratically constrained programming (MIQCP) task with stochastic scenarios. As a result, the planning model was solved utilizing a stochastic optimization approach using GAMS software, as well as the Gurobi solver. The provided planning model was evaluated using a historical dataset of different types of loads and daily variation levels of (wind and PV) DGs. A significant number of 1000 stochastic profiles for each sort of load demand, and (PV-DGs and wind-DGs) were generated using a normal distribution function. However, a large number of stochastic profiles increase the complexity of the computation burden. To reduce the number of scenarios, a strong scenario reduction technique was necessary. In this work, the SCENTED2 tool by (GAMS) was used for scenario reduction [50]. As a result, five scenarios with joint probability were applied to illustrate stochastic variations in the system.

The whole DS was isolated from the main grid by disconnecting line (1, 2), as well as seven lines (3, 4), (12, 13), (21, 22), (24, 25), (6, 26), (8, 21), and (32, 33) were at failure as a result of disastrous occurrences. Loads were divided into three profiles: industrial, commercial, and residential, each type with five scenarios, with joint probabilities as listed in

Table 2. Probabilities of different load profiles and DG profiles.

Scenario	PDC	PDR	PDI	Wind-DGs	PV-DGs	Merage ALL
S1	0.34	0.25	0.13	0.140	0.19	0.128
S2	0.24	0.25	0.21	0.200	0.17	0.191
S3	0.01	0.18	0.24	0.330	0.13	0.068
S4	0.18	0.16	0.16	0.140	0.27	0.355
S5	0.23	0.16	0.26	0.190	0.24	0.258

. However, all of the load demand types were classified into critical (commercial and industrial) and non-critical (residential) loads and supplied according to their priority levels [51]. These daily profiles were considered based on historical data obtained from [52] and the percentage of their maximum values are depicted in Figure 6, Figure 7 and Figure 8 below.

Five MGs were connected to the distribution network through buses, three of which were used for MESSs stations and were linked to MGs 14, 21, and 25, interconnected to each other via a transportation arc based on the (1 h) travel time between any two microgrids. Whereas the remaining two MGs were coupled with PLs and could be optimally located at any MG (e), three MESSs were taken into consideration and are assumed to have the same features as listed in

Table 3. MESSs Parameters.

MESS Parameters	1	2	3
Initial position (MG-Bus)	14	21	25
Charging/discharging power (KW)	200
Energy capacity (KWh)	1000
Initial SOC (KWh)	200
SOCmax/SOCmin (KWh)	0.90/0.10%
Charging/discharging Efficiency	0.95/0.95%

. This paper assessed the maximum number of PEV-PLs that may be deployed at candidate MG as two and the number of PEVs per hour as 40. As a result, 1500 PEVs were deemed to be in the network, and they should provide different types of loads to minimize power curtailment in the case of all the DS being separated from the main grid. It is important to emphasize that the drivers’ travel patterns were assumed to be identical, and the focus was on the impact of PEVs in post-disaster emergency conditions. All the PEV parameters are listed in

Table 4. PEVs Parameters.

Charging/discharging power (KW)	2.3
Nominal Energy capacity (KWh)	16
Initial SOC (KWh)	1.6
SOCmax/SOCmin (KWh)	0.90/0.10%
Charging/discharging Efficiency	0.95/0.95%

[14].

In addition, the maximum allowable limitations related to the rated power for each unit (Wind-DGs and PV-DGs) were 1 MW and 0.8 MW. The daily real dataset was considered for Wind and PV-DGs based on the historical dataset from [53], and the percentage of their maximum values are shown in Figure 9 and Figure 10, with associated joint probabilities tabulated in the above Table 2.

The costs of power from wind-DGs and PV-DGs were 1.9 and 2.3 USD/MW, respectively [16]. The transportation cost was considered to be USD 80 per transit [31]. For critical and non-critical loads, unit interrupting costs were chosen to be USD 10 per kWh and USD 2 per kWh, respectively [31]. For MESS and PEV, the unit battery maintenance costs were 0.2 and 0.15 USD/KWh, respectively [14,31]. The maximum and minimum voltages limitation was defined as being 1.05 p.u. and 0.95 p.u., respectively.

6. Numerical Result of the Proposed Model

In this article, the suggested planning model was used to determine the 33-bus DS optimal operation in both the restorative and post-restoration states. The test results were divided into three groups as a consequence. In Case I, the DS had MGs with wind-DGs and PV-DGs and did not consider PEV-PLs or MESSs; in Case II, the DS had MGs and considered PEV-PLs and DNR; and in Case III, the DS had MGs and considered PEV-PLs, DNR, and MESSs. According to objective value and load recovery, the optimization findings of three cases are reported and compared in

Table 5. Simulation results for distribution system restoration.

Results			CASE 1	CASE 2	CASE 3
Objective ($)	Interruption cost	PDR	17,581.044	18,854.366	18,981.718
		PDC andPDI	80,093.427	70,179.259	34,620.604
	Transportation cost		0	0	720
	Battery maintenance cost		0	0	1325.001
	PEV-PLs operation cost		0	313.491	333.031
	Wind-DGs cost		49.184	49.665	54.971
	PV-DGs Cost		12.546	13.026	13.026
	Total cost		97,736.202	89,409.807	56,048.351
Load Restoration (%)	Priority 1(C andIN-Loads)		78	80.94	90.59
	Priority 2(R-Loads)		38	36.5	36.11
	Total		66.5	68.17	74.93

.

In Case 1, microgrids depended entirely on local distributed generation for service restoration in distribution systems in the case absence of PEV-PLs and MESSs. As a result of applying the default faults, the network reconfiguration result was that switches (9, 15), (12, 22), (18, 33), and (25, 29) were opened. According to the table above, the estimated total cost was USD 97736.202, and the recovery percentage of critical (Priority-1), non-critical (Priority-2), and overall loads were, 78%, 38%, and 66.5%, respectively.

In the second case, it was taken into account that local PV-DG resources coordinated with PLs at the microgrids. For the restoration of various load demands, the PEV-PLs were dispatched to microgrids at #15, and #20, respectively. The findings collected show that the open switches were (9, 15), (12, 22), (18, 33), and (25, 29). As listed in Table 5, the whole cost was USD 89409.807, which is 8.52% less than Case 1 (The base cost). The critical load (Priority-1) restoration percentage was 80.94%, the non-critical (Priority-2) load restoration percentage was 36.5%, and the overall load restoration percentage was 68.17%. The results demonstrate the advantages of appropriately utilizing PEV-PLs to pair up with microgrids for efficient service recovery.

Case 3 is achieved when PEVs-PLs cooperate with the scheduling of MESSs, and DGs at MGs and DNR. Regarding the DS reconfiguration, the same line switches became energetic as mentioned in Case 1 and Case 2, as depicted in Figure 11, due to the location of the same events that were considered for all test cases. By following Table 5, the total cost decreased by 37.3% when compared with Case 2 and by 42.65% when compared with Case (1) to reach USD 56048.351. Critical, non-critical, and total load restorations improved to 90.59%, decreased to 36.11%, and increased to 74.93%, respectively. It should be noticed that the restoration of the non-critical load was slightly less than in Case 2; this is because MESSs transmit energy across microgrids to energize the critical loads, which results in increased losses during charging and discharging. These findings indicate that MESSs can greatly increase the resilience of DSs.

The hourly position and status of the MESSs are described in

Table 6. Status and location of MESSs.

	MESS#1			MESS#2			MESS#3
Time	Location	Status	Time	Location	Status	Time	Location	Status
1–3	14	C	1–6	21	C	1–8	25	C
4	14	D	7	21–14	T	9	25	D
5	14	C	8	14	C	10	25	C
6	14–25	T	9	14	D	11–12	25	D
7–8	25	C	10	14	C	13	25–14	T
9	25	D	11	14–25	T	14–15	14	C
10	25	C	12–13	25	D	16	14	D
11	25	D	14	25	C	17	14	C
12	25–14	T	15–18	25	D	18	14–25	T
13–14	14	C	19	25–14	T	19–22	25	D
15	14–25	T	20–21	14	C	23	25	C
16–19	25	D	22	14	hD	24	25	D
20–21	25	C	23	14–25	T
22	25	D	24	25	D
23	25	C
24	25	D

. As specified in Table 5, MESS2 remained at the starting position, MG21, for 6 h (00:00–06:00) for charging purposes. Then, it transferred from MG21 to MG14 at 7:00 and stayed there for 3 h (8:00–10:00) for charging and discharging. Then, it transferred to MG25 at 11:00 to stay for 7 h until being discharged between 12:00 and 13:00 and charged from 15:00 to 18:00. Following that, travel was performed from MG25 to MG14 at 19:00, and then charging began for 2 h (20:00–21:00) before discharging started at 22:00 for 1 hour, until 23:00. The final trip was back to MG25 at 22:00 to be discharged during the last hour (24:00).

The SOC of MESSs is shown in Figure 12 for all scenarios. Using MESS2 as an example to show the timeline, MESS2 remained in its starting position for charging purposes for 6 h (00:00 to 06:00), but as can be seen, the SOC gradually rose until it reached 80%. Then, as shown in Figure 11, MESS2 departed to MG14 at 07:00 and stayed there for 3 h to charge, discharge, and recharge. As a result, the SOC at the period from 7:00 to 10:00 was slightly raised at 08:00 and decreased at 09:00 to achieve 70% of the SOC level according to the discharging process, then it increased until it reached 90% of the SOC at 10:00. The MESS2 then went from MG14 to MG25 at 11:00 and stayed there for 6 h, discharging from 12:00 to 13:00, charging at 14:00, and discharging from 15:00 to 18:00, reducing the SOC to 10% of its original level, as shown in Figure 12. At 19:00, MESS2 traveled from MG25 to MG14 for charging purposes between 20:00 and 21:00, thus the SOC level increased to 30%, while at 22:00, MESS2 began discharging for 1 h at the same MG. Last but not least, MESS2 returned to MG25 at 23:00 for discharge purposes during the last hour. Other MESSs have shown a similar pattern of behavior. An MG was modeled as a single bus that pooled all generating resources, an equivalent local load, and charging and discharging services, as was already described. Figure 13 shows the power balance at MG, which further includes the MESS station, Wind-DG generation, and equivalent loads. Indeed, during the time horizon, the power produced by wind turbines was adequate. Therefore, for restoration purposes, these MGs offered the extra energy to the DS and transferred it by MESSs to any other MG at any hour. Since there was a lot of power discharged from different MESSs at MG25, the power balance rose, as seen in Figure 13.

It is clear in Figure 11 that the position of PEV-PLs in the initial configuration of the DS placed MG15 and MG20 buses to charge during system off-peak hours and injected energy into the system during system peak hours in the event of an emergency.

Table 7. The number of PEVs In Each MG.

Time	MG-15					Time			MG-20
#	s1	s2	s3	s4	s5	#	s1	s2	s3	s4	s5
T1	40	40	40	40	40	T1	40	40	40	40	40
T2	40	40	40	40	40	T2	40	31	40	40	24
T3	40	40	40	40	40	T3	40	40	40	40	39
T4	40	40	40	40	40	T4	40	33	40	40	27
T5	40	40	40	40	40	T5	40	40	40	40	37
T6	40	40	40	40	40	T6	40	37	40	40	27
T7	40	40	40	40	40	T7	40	37	40	40	29
T8	40	40	40	40	40	T8	40	37	40	40	29
T9	40	40	40	40	40	T9	40	40	40	40	40
T10	40	40	40	40	40	T10	40	40	37	32	31
T11	40	40	40	32	40	T11	33	40	40	33	15
T12	40	40	40	21	19	T12	40	40	40	33	11
T13	40	40	19	40	10	T13	40	40	40	40	19
T14	40	40	9	40	5	T14	40	40	40	40	37
T15	40	40	40	40	40	T15	40	40	40	40	40
T16	40	40	40	39	24	T16	40	40	40	40	40
T17	40	35	40	40	12	T17	40	40	40	40	16
T18	40	40	40	40	6	T18	40	28	40	20	8
T19	40	40	12	16	3	T19	40	14	40	7	4
T20	40	17	4	8	2	T20	31	7	38	4	2
T21	40	9	4	5	1	T21	15	3	13	2	1
T22	40	5	7	2	1	T22	6	2	7	1	1
T23	19	2	13	1	1	T23	3	1	4	1	1
T24	10	1	7	1	1	T24	2	1	2	1	1

provides a summary of the number of electric vehicles that should be in each PL across the time horizon under each scenario.

In this research, an MG was represented as a single bus pooling the entire generating resources, an equivalent local load, and charging/discharging services. Figure 14 shows the power balance at each MG as an outcome of PV-DG power generation as well as PEV-PLs services. As can be seen, the power generated during the first and last hours was insufficient because of a lack of PV-DG power generation and the low initial state of PEVs, which began charging throughout these times by importing power from the DS. While the power from 07:00 to 17:00 was sufficient for PL services, and PV-DG power was available, MGs began to support the DS by providing additional power to assist in post-disaster restoration. As seen in Figure 15, the SOC of PEVs for all scenarios began to rise between the hours of 01:00 and 06:00, indicating that the charging process continued until the SOC reached 100%. Then, between 07:00 and 11:00, PEVs began to discharge, bringing the SOC down to 10%. Following that, due to the charging and discharging procedure, SOC raised to 0.25% of the SOC level and decreased until that last time span.

7. Comparative Analysis and Discussions

The numerical results indicate that PEVs and MESSs are discharged to the MGs whenever power from DGs (wind and PVs) is or is not adequate, and charged where and whenever power from DGs (wind and PVs) is noticeably higher than the amount to critical loads. Furthermore, exchanging energy and power between MGs to provide critical loads in the DS, PEV-PLs, and MESSs can reduce the overall cost. Additionally, PEVs discharge and function as storage systems at times of high demand. Table 5 clearly shows that in comparison to Case 2, the overall load restoration grew by 6.76%, while the load restored for Priority 1 has grown by 9.65% and dropped for Priority 2 by 0.39%.

Throughout this case, MESS1 charged to the SOC of 70% in MG25 from 01:00 to 03:00 (see Figure 12 and Table 6). Then, at the same MG14, MESS1 discharged and charged, respectively, from 03:00 to 05:00; in this case, the MESS operated as an SBESSs. Then, from 05:00 to 06:00, MESS1 travelled to MG25 and got charged for two hours (06:00–08:00) to a SOC of 90% before being discharged there for one hour from 08:00 to 09:00 while the SOC fell to 70%. After that, it was charged and discharged, respectively, from 9:00 to 11:00. Then, from 11:00 to 12:00, MESS1 moved to MG14 to get charged for two hours (12:00–14:00) to a SOC of 80%. Next, MESS1 brought that charged energy from MG14 and moved to MG25 between 14:00–15:00, before being discharged until 20:00, and so forth. Therefore, MESS1 was used to transmit energy from MG14 to MG25. Likewise, it is possible to see similar behavior in the other MESSs. Furthermore, Figure 14 indicates the power balance at each MG as a combination of the services offered by PEVs-PLs with PV-DG power generation. As can be noticed, the first and last hours of power generation were grossly inadequate due to the lack of PV-DG power generation and the low initial condition of PEVs, which started charging during these hours by consuming power from the DS, while there was enough power from 07:00 to 17:00 for PLs services. The reason for this is that PV-DG power generation was widely available.

The comparison of three cases reveals the importance of MESS storage and transportation when applied in combination with PEV-PLs for post-disaster restoration. The effectiveness of PEV-PLs would dramatically reduce the costs associated with supporting the MGs by permitting energy to be transmitted between MGs by using MESSs to serve critical loads and attain an effective restoration scheme. Moreover, the PEV-PLs with MESSs might occasionally operate as stationary battery energy storage systems, SBESSs, to adopt the load-shifting mechanism in the DS.

The difference in the overall cost savings achieved by the unit interruption cost and unit transportation cost were USD 10/kWh for Priority 1 (commercial and industrial load) and USD 2/kWh for Priority 2 (residential load), respectively. Comparing the total costs in Cases 2 and 3 yields the percentage of the overall cost saving. Due to the use of PEV-PLs in cooperation with MESSs to transfer energy throughout other MGs for the restoration of critical loads, in this case compared to Case 2, the overall cost dropped by 49.33% and 0.99% for Priorities 1 and 2, respectively. The strengths of the PEV-PLs integrated with MESS and vice versa will therefore be felt as a result of the reduced unit interruption cost and higher unit transportation cost. Furthermore, if the unit transit cost is too high, MESSs will remain at their reference MGs to function as SBESSs. It should be highlighted that despite the cost of unit interruptions continually rising, the overall reduction in cost will remain stable. The main reason is that the scheduling of MESSs might end in the same trip cycle when the unit interruption costs are excessively high and the transportation cost is therefore ignored.

This comparative analysis can therefore be used as a decision support tool to demonstrate the adaptability of MESSs units and how well they cooperate with PEV-PL operations across various MGs post disasters, as well as to highlight the effectively suggested model to bring up the distribution system’s level of resilience.

8. Conclusions

This paper studied an innovative integration of MESSs with PEV-PLs into a DS to assess a post-disaster joint restoration scheme, which attempts to minimize overall cost by scheduling MESSs and planning PLs in collaboration with distribution network reconfiguration and MG operations for recovery purposes. By managing the ON/OFF state of remotely controlled switches, MGs act as supply buses to construct islands dynamically when charging or discharging services are available. The uncertainties in various forms of load consumption, PV-DGs, and Wind-DGs are evaluated by generating a large number of 1000 scenarios using the normal distribution function technique. To minimize the computational burden, the SCENRED2 tool was employed to capture five desert scenarios with joint probabilities. To transport MESSs across MGs, a simplified TSN-based MESS scheduling model is provided as a MILP for both space decisions and continuous time intervals. During the operating horizon, the size and optimum position of MG-integrated plug-in electric parking lots are also specified, along with the ON/OFF state of the remote-control switches of the lines. Three case studies on a modified 33-bus test system were used to prove the efficacy of the suggested restoration approach. To show the effects of MESSs and PEV-PLs on DS restoration, comparison simulations have been conducted. Thus, PEVs are perfectly accommodated within MGs in proper size and positions for load service restoration according to priorities. Meanwhile, MESS transportability enables an effective transfer schedule among several MGs to support a DS. Moreover, the overall cost reduced by 37.3% compared to Case 2 to reach USD 56048.351, while including MESSs and PEV-PLs. Nevertheless, employing MESSs raise critical load restoration to 90.59%, anddecreased non-critical to 36.11%, while the overall load restoration is increased to 74.93%. The simulation results show the flexibility of MESSs and how well PEV-PLs work across different MGs during disasters, as well as highlight the efficiently suggested model to enhance the resilience of the distribution system.