## 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]. Figure 1: Resilience curves related to a catastrophe. 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;  
\[\sum_{(e,f)\in A}\chi_{c,ef,ts}^{Trans}=1,\qquad\forall c\in\mathbb{C},ts\in TS \tag{1}\]  
\[\sum_{e\in A^{+}}\chi_{c,ef,ts+1}^{Trans}=\sum_{e\in A^{-}}\chi_{c,ef,ts}^{Trans }\,\qquad\forall c\in\mathbb{C},e\in\mathbb{N},ts\in TS \tag{2}\]  
\[\sum_{e\in A^{+}}\chi_{c,ef,1}^{Trans}=\chi_{c,e,0}^{Trans}\,\qquad\forall c\in \mathbb{C},e\in\mathbb{N} \tag{3}\]  
\[\chi_{c,ef,ts}^{Trans}+\chi_{c,ef,ts+1}^{Trans}\ \leq 1\,\qquad\forall c\in \mathbb{C},(e,f)\in A,e\
eq f,ts\in TS \tag{4}\]  
Figure 2: Virtual nodes in a transportation network. 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.