Optimal Configuration of Mobile–Stationary Hybrid Energy Storage Considering Seismic Hazards

Abstract

The occurrence of extreme disasters, such as seismic hazards, can significantly disrupt transportation and distribution networks (DNs), consequently impacting the post-disaster recovery process. Restoring load using distributed generation represents an important approach to improving the resilience of DNs. However, using these resources to provide resilience is not enough to justify having them installed economically. Therefore, this paper proposes a two-stage stochastic mixed-integer programming (SMIP) model for the configuration of stationary energy storage systems (SESSs) and mobile energy storage systems (MESSs) during earthquakes. The proposed model comprehensively considers both normal and disaster operation scenarios of DNs, maximizing the grid’s economic efficiency and security. The first stage is to make decisions about the location and size of energy storage, using a hybrid configuration scheme of second-life batteries (SLBs) for SESSs and fresh batteries for MESSs. In the second stage, the operating costs of DNs are evaluated by minimizing normal operating costs and reducing load loss during seismic events. Additionally, this paper proposes a scenario reduction method based on hierarchical sampling and distance reduction to generate representative fault scenarios under varying earthquake magnitudes. Finally, the progressive hedging algorithm (PHA) is employed to solve the model. The case studies of the IEEE 33-bus and 12-node transportation network are conducted to validate the effectiveness of the proposed method.

1. Introduction

In recent years, the frequent occurrence of high-impact low-probability (HILP) events has inflicted significant damage on national energy infrastructures. Among various HILP disasters, earthquakes are among the most unpredictable, frequently resulting in widespread power outages within distribution networks (DNs) [1]. In 2023, Türkiye was struck by a magnitude 7.8 earthquake, with a subsequent magnitude 7.5 earthquake occurring nine hours later, leading to over 90% power loss [2]. According to the China Earthquake Networks Center, in 2022, China recorded a total of 726 earthquakes with magnitude 3.0 or above. This total included 526 earthquakes with magnitudes between 3.0 and 3.9 and 10 earthquakes with magnitudes between 6.0 and 6.9, with none above magnitude 7.0 [3]. Given that large earthquakes are less frequent than other natural disasters and unpredictable in both timing and location, analyzing their impact on DNs presents a substantial challenge [4]. Therefore, identifying vulnerabilities that impact DNs’ reliability during the planning phase is critical to enhancing the safety and resilience of the entire power distribution system, with the ultimate goal of minimizing economic losses.

Resilience refers to the ability of a system to withstand and recover quickly from extreme natural hazards [5]. Currently, two main aspects are considered for DN resilience enhancement strategies: pre-disaster planning of resilience resources to improve the system’s capacity to endure disasters and post-disaster scheduling of these resources for rapid power restoration. In pre-disaster planning, most strategies involve reinforcing existing distribution lines [6], deploying battery energy storage systems (BESSs) [7], and installing line switches [8] to bolster system resilience. Reference [9] proposes a three-layer robust optimization model based on the N-K security criterion, employing a “defense-attack-defense” approach to determine the best planning scheme under the worst disaster scenario. Reference [10] establishes a two-stage stochastic programming model for extreme events, where the first stage involves reinforcing the line and the second stage minimizes the operating costs of failure scenarios. Reference [11] proposes a risk-averse two-stage stochastic bi-level programming problem. The upper level minimizes the total investment costs, and the lower level minimizes the expected load loss by mobile energy storage systems (MESSs). Concerning post-disaster resilience resource scheduling, reference [12] examines strategies for post-disaster network reconfiguration and mobile energy storage deployment, identifying the optimal recovery path. Reference [13] utilizes various flexible resources to ensure power supply to critical loads. Reference [14] coordinates repair crews for equipment restoration. However, they primarily consider the impact of disasters on DNs, whereas in reality, extreme disasters also damage transportation infrastructure, and they do not consider the impact of road damage on the dispatch of mobile emergency resources. In addition, given the low probability of extreme events, planning only for extreme disasters will result in inefficient utilization of energy storage resources.

BESS has been widely used to provide services such as voltage regulation [15], energy arbitrage [16], and improving power supply reliability [17] in DNs. Beyond the commonly used stationary energy storage systems (SESSs), MESSs present a greater potential for grid support due to their mobility [18]. Reference [19] establishes the line fault model under seismic hazards by peak ground acceleration (PGA) and enhances the earthquake resilience of DNs through SESSs. Reference [20] compares MESSs and SESSs, concluding that after 10 h of power outage, an MESS outperforms an SESS. Reference [21] demonstrates that a BESS composed of an SESS and an MESS met the demand for sudden load increases and emergency power source for essential equipment during the Winter Olympics. However, they only considered a BESS composed of fresh batteries, which remain costly despite recent declines in energy storage prices, while overlooking the potential application of second-life batteries (SLBs) in energy storage systems, resulting in high initial investment costs.

In recent years, SLBs have attracted attention for their environmental and economic benefits [22]. Reference [23] describes the application scenarios and technical details of SLBs as SESSs. Reference [24] proposes a cascading utilization strategy for SLBs considering battery health to achieve the recycling of fresh batteries and SLBs throughout the planning period. Reference [25] presents an economic operation model for BESSs comprising SLBs under varying states of health, taking into account the costs associated with battery aging. However, they only focused on the application of retired batteries as SESSs under normal scenarios and did not explore the potential value of SLBs in multi-scenarios such as disaster scenarios.

With the increasing frequency of extreme disasters and the impending retirement of a large number of electric vehicles, determining how to rationally configure different types of energy storage to meet the demands of various application scenarios is of great significance for enhancing the resilience and economic efficiency of DNs. This paper proposes a two-stage SMIP model that considers both stationary retired batteries and MESSs, categorizing DNs’ operational scenarios into normal and earthquake disaster scenarios. The model comprehensively accounts for the dual damages caused to DNs and TNs, balancing the resilience level and economic efficiency of DNs. In the first stage, planning decisions are made about the location and size of the BESS. The second stage evaluates the operating costs of the BESS in each operating scenario to ensure the optimal utilization and load supply of the BESS. Furthermore, typical fault scenarios under different seismic levels in DNs and TNs are generated by PGA based on the scenario reduction method of hierarchical sampling and distance reduction. The progressive hedging algorithm (PHA) is utilized to solve the proposed model. The contributions of this paper are as follows:

• There are optimal access points for hybrid energy storage under different operating scenarios, and the operational scenarios of DNs are divided into normal and disaster scenarios to maximize their value. The model adopts the minimization of annual operating costs as the objective for normal operation and the minimization of load reduction as the optimization goal for extreme scenarios, thereby enhancing the overall benefits of energy storage applications. Compared to existing energy storage planning models, this model considers the comprehensive performance of energy storage under different scenarios, balancing the economics and resilience of the DNs.
• We propose an energy storage planning model that leverages the low costs associated with SESSs derived from SLBs as well as the mobility of fresh batteries as MESSs. It alleviates the initial economic burden on energy storage investors while significantly enhancing grid resilience, providing a new idea for realizing the economics and sustainability of energy storage systems.
• The roulette algorithm is utilized to generate a vast array of power distribution equipment and road traffic fault scenarios, and then stratified sampling and a distance reduction method are employed, grounded in load loss considerations, to generate representative scenarios for each magnitude. The proposed method effectively captures the relationship between earthquake magnitude and system load loss, thereby reflecting actual disaster scenarios, incorporating HILP events.

The rest of the paper is organized as follows: Section 2 presents the hybrid energy storage planning framework and classifies the operational scenarios of DNs. Section 3 models the earthquake disaster scenarios that integrate TNs and DNs. Section 4 illustrates the two-stage SMIP model and the solution algorithm. Section 5 presents case studies. Section 6 concludes the paper.

2. Modeling Framework for BESS Planning

The overall structure of this paper is shown in Figure 1. The two-stage framework aims to determine the optimal planning decisions in the first stage and minimize expected operating costs in the second stage for both normal and disaster scenarios. These two stages are not separate but are solved as a single optimization problem. In this study, the siting and sizing of the SESS and MESS are considered planning decisions, while the objective of the second stage is to minimize the operating costs of each scenario, thereby reducing the overall objective function.

For second-stage operational scenarios, this paper determines the impact of seismic hazards on the TNs and DNs through a roulette algorithm and employs stratified sampling and distance reduction to determine representative scenarios for each magnitude. Finally, the planning problem is solved in a two-stage stochastic optimization model based on the number of scenarios obtained from the reduction.

2.1. Modeling of Mobile–Stationary Hybrid Energy Storage

SLBs have the potential to break the constraints of high investment costs associated with BESSs due to their low price. As shown in Figure 2, although both SLBs and fresh batteries are electrochemical storage devices, they differ significantly in parameters such as cost, charge–discharge efficiency, the initial state of health (SOH), and lifespan. After testing, classification, and reassembly, SLBs retain substantial residual value despite their relatively lower charge–discharge efficiency and rated capacity, which makes them suitable for SESSs, microgrids, and short-term backup power applications with lower performance requirements. This effectively reduces the initial economic burden on energy storage investors. In this study, we assume that all SLBs used in the system have an initial SOH of 80%.

In the event of a large-scale power outage, an MESS can provide rapid emergency power, helping the system restore critical loads quickly, thus reducing the economic impact of power interruptions. This capability significantly enhances the grid’s flexibility and emergency response during extreme events.

Due to the security risks associated with SLBs, in this paper, SLBs are used as SESSs, while fresh batteries are configured as MESSs, forming a hybrid configuration. This approach helps mitigate the limitations of SLBs in low-power scenarios. Such a hybrid energy storage system not only reduces overall investment costs but also ensures system reliability while enhancing flexibility and emergency response capabilities during extreme events.

2.2. Operation Scenarios of DNs

Given the unpredictability and short duration of earthquake disasters, research on pre-disaster planning and defense strategies is especially critical. However, because the probability of HILP events is low, focusing solely on planning for such events may lead to excessive redundancy in energy storage equipment. Therefore, we categorize the operational scenarios of DNs into normal and disaster conditions, balancing grid resilience with economic efficiency. The optimal operating strategies for BESSs are developed for both normal and disaster scenarios, defined by sets $S^N$ and $S^E$, which are indexed by s, and together, they compose set $S\in [S^N\cup S^E]$ which is considered in the proposed optimization. During normal operation, MESSs are stationed at designated stations. In the event of a fault, these units are dispatched through TNs to reach connection points in the DNs, thereby ensuring power supply to critical loads. Each scenario is allocated a probability weight $w_s$ to satisfy ${\displaystyle {\sum }_{s\in [S^N\cup S^E]}}{w}_s=1$, and we consider the various operational states of the DNs throughout the year for comprehensive planning.

2.2.1. Normal Operation

In the normal operational phase, we primarily focus on implementing peak shaving and valley filling strategies within mixed energy storage systems, as well as the benefits derived from accommodating PV. The optimal number of clusters for normal scenarios is identified using the elbow method, which is applied to historical data through k-means clustering, resulting in the generation of output characteristic curves for PV and load. To simplify the model, the MESS is treated as a fixed resource during normal operations, allowing for its relocation during natural disasters. The objective function for normal operating scenarios is to minimize the annualized operational cost of the DNs, as shown in Equation (1):

$$\min \hspace{0.33em}{\displaystyle \sum _{s\in S^N}{w}_s}{f}_s^N$$

(1)

$$f_s^N=f_s^{pur}+f_s^{op}+f_s^{pv}$$

(2)

$$f_s^{pur}={\displaystyle \sum _{t\in T}{P}_{s,t}^{sub}}{c}_{pur}$$

(3)

$$f_s^{op}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in N}(P_{s,t,i}^{dis}-}{P}_{s,t,i}^{ch})c_{pur}}$$

(4)

$$f_s^{pv}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{g\in G}{c}_g}}(P_{s,t,g}^{pre}-P_{s,t,i}^{pv})$$

(5)

where $f_s^{pur}$ is the power purchase cost, $f_s^{op}$ is the BESS’s operating cost, $f_s^{pv}$ is the PV power abandonment penalty, $P_{s,t}^{sub}$ is the electricity purchased from the grid, $P_{s,t,i}^{ch}$ and $P_{s,t,i}^{dis}$ are the charge and discharge power of the BESS, $P_{s,t,g}^{pre}$ and $P_{s,t,i}^{pv}$ are the PV-predicted output and actual output, and $c_{pur}$ and $c_g$ are the time-of-use price and the penalty price of power abandonment. Equation (2) calculates the operational cost of DNs for scenario s, which includes the cost of electricity procurement, the annual operation maintenance cost, and the penalty for the curtailment of PV.

2.2.2. Disaster Operation

To analyze the impact of seismic hazards on power systems, we utilize regional earthquake historical data to determine the locations and magnitudes of earthquakes. The fault model is constructed for distribution equipment and road traffic based on PGA, considering the dual damage caused to both the transportation network and the power grid. When seismic hazards occur, the DNs develop a post-disaster recovery strategy based on available flexible resources, coordinating multiple resources, including energy storage, to minimize load loss. An MESS can be deployed to supply power to loads from fixed locations depending on the damage to the transportation network. Thus, the post-disaster operation strategy for an MESS depends on the fixed setting determined during normal operations. This paper establishes a post-disaster optimization model that considers the fault conditions of both the DNs and the TNs based on candidate nodes identified under normal conditions and disaster scenarios reduced by different magnitudes, as shown in Equation (6):

$$\min \hspace{0.33em}{\displaystyle \sum _{s\in S^E}{w}_s}{f}_S^E$$

(6)

$$f_S^E={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in N}{c}_i^d}}{P}_{s,i,t}^{lr}+{\displaystyle \sum _{t\in T}{P}_{s,t}^{sub}}{c}_{pur}$$

(7)

where $f_S^E$ is the disaster operation cost, $P_{s,i,t}^{lr}$ is the active load shedding on node i, and $c_i^d$ is the cost of the lost load. Unlike Equation (2), Equation (7) accounts for load shedding, which is penalized based on the value of lost load.

3. Scenario Sampling and Reduction Under Seismic Hazards

The key characteristic parameters of seismic events include the epicenter location, magnitude, and geological parameters. For the same earthquake, there is only one magnitude value, and as the distance from the epicenter increases, the seismic intensity gradually decreases. Common seismic parameters include PGA, peak ground velocity, and peak ground displacement. Among these, PGA serves as the primary basis for engineering seismic design and has been widely used to assess damage to power distribution equipment and road traffic systems [26,27]. The PGA attenuation characteristic is formulated in Equation (8) [26]:

$$\ln Y_{PGA}=0.583+0.651M-1.6521\lg (D+0.182e^{0.707M})$$

(8)

where M is the earthquake magnitude, and D is the source distance.

3.1. Vulnerability of Power Distribution Equipment

The necessary condition for the normal operation of power lines is that none of the utility poles along the line segment have collapsed. The failure rate of power distribution lines during earthquake disasters can be obtained as shown in Equation (9):

$$p_l=1-{\displaystyle \prod _{o=1}^k{(1-p_o)}^k}$$

(9)

where $p_l$ is the fault probability of line l, $p_o$ is the collapse probability of the pole in the line segment, and k is the number of poles.

Distributed generation systems (DGs) play a critical role in power systems, and their failure directly reduces the power generation capacity of the distribution system. Many studies focus on line failures and overlook failure scenarios involving DGs. We use a lognormal cumulative distribution function to represent the damage function for DGs, indicating the probability that a generator reaches or exceeds a given damage state, as shown in Equation (10):

$$P(G_c|Y_{PGA})={\displaystyle {\int }_0^{Y_{PGA}}\frac{1}{\sqrt{2\pi }{\xi }_{c}x}\exp [\frac{{(\ln x-\ln {\epsilon }_c)}^2}{-2{\xi }_c^2}]\mathrm{d}x}$$

(10)

where c is the different damage states, ${\epsilon }_c$ is the logarithmic mean under the limit failure state c, and ${\epsilon }_c$ is the logarithmic standard deviation in the limit failure state c.

Vulnerability curves for various damage levels can be used to directly assess the cumulative probabilities of each damage state. These cumulative probabilities are then used to derive the individual probabilities of equipment damage states, as shown in Equations (11)–(15):

$$P_{G_4}=P(G_4|Y_{PGA})$$

(11)

$$P_{G_3}=P[G_3|Y_{PGA}]-P(G_4|Y_{PGA})$$

(12)

$$P_{G_2}=P[G_2|Y_{PGA}]-P(G_3|Y_{PGA})$$

(13)

$$P_{G_1}=P[G_1|Y_{PGA}]-P(G_2|Y_{PGA})$$

(14)

$$P_{G_0}=1-P(G_1|Y_{PGA})$$

(15)

where $G_0$, $G_1$, $G_2$, $G_3$, and $G_4$ represent none, slight, moderate, extensive, and complete damage states, respectively. Following an earthquake, DGs may operate under different availability modes. The reduction coefficients for PV output at different damage levels can be determined [28] as shown in

Table 1. PV output and road post-earthquake attenuation factor.

Damage State	PV Output	Road Attenuation
none	1.0	1.0
slight	1.0	0.8
moderate	0.5	0.5
extensive	0	0
complete	0	0

.

3.2. TNs’ Impedance Under Seismic Hazards

To analyze road traffic conditions during seismic hazards, we apply the Bureau of Public Roads (BPR) function to calculate movement time between nodes [29]. Using Dijkstra’s shortest path algorithm, it then calculates the shortest time to travel between the nodes and uses that as the road weight.

Seismic hazards primarily disrupt road traffic by reducing road capacity. Based on the extent of earthquake-induced damage and its impact on transportation capacity, road damage is typically classified into five categories: basically unscathed, slightly damaged, moderately damaged, severely damaged, and completely ruined. Table 1 presents the traffic capacities for roads at each damage level [30]. The modified BPR travel time function is accordingly shown in Equation (16):

$$t{r}_d=t_{d,0}[1+\alpha {({\displaystyle \frac{Q_{d,e}}{C_{d,e}{C}_d}})}^{\beta }]$$

(16)

where $t{r}_d$ is the road impedance travel time of road d after an earthquake, $t_{d,0}$ is the free flow travel time, $Q_{d,e}$ is the traffic flow on toad d after an earthquake, $C_{d,e}$ is the capacity reduction factor after an earthquake of road d, $C_d$ is the capacity of road d under normal conditions, and $\alpha$ and $\beta$ are BPR parameters. In this paper, extensive damage and complete damage are grouped into a single category. Parameters for vulnerability curves at different levels of road damage are obtained from the literature [31]. The probabilities for each level of road damage are then calculated using Equations (11)–(15).

3.3. Seismic Hazards Scenario Generation and Reduction

After calculating the probabilities of faults in distribution equipment and road traffic under seismic disasters, we utilize a roulette wheel algorithm to simulate the damage states of all distribution equipment and urban traffic roads. The roulette wheel algorithm produces a significant number of scenarios, complicating the computation of all scenarios in stochastic planning problems. Managing multiple scenarios within the planning framework poses a substantial challenge in stochastic planning. Traditional clustering methods, such as K-means clustering and information entropy, can generate typical scenarios but may neglect some relatively isolated extreme scenarios.

In regions with frequent seismic activity, low-magnitude earthquakes are common but have limited impact. Conversely, high-magnitude earthquakes, although less frequent, can cause catastrophic damage. Focusing solely on high-magnitude events often disregards economic considerations in systematic planning, leading to excessive investment. To appropriately consider the influence of high-impact low-probability (HILP) events, it is crucial to simulate system operations under varying earthquake magnitudes. By generating scenarios for different magnitudes, a comprehensive assessment of potential situations, including HILP events, can be ensured. This approach provides a better understanding of the impact on DNs and TNs under varying disaster intensities, enabling the integration of these scenarios into subsequent optimization decisions.

In this paper, the characteristic variables of random scenarios consist of binary variables (fault locations) and continuous variables (load distribution at nodes). Scenarios generated by the K-means clustering method may include different fault locations with identical total cut load amounts. Therefore, this paper introduces a scenario reduction method based on stratified sampling and distance reduction. This method samples scenarios for various magnitudes through stratification and selects a simplified scenario that closely matches the original scenario for each magnitude, based on average load loss and average road transit time, to reduce the total number of scenarios. This intelligent scenario selection strategy ensures that the realities of fault conditions during seismic hazards are accurately represented while also including HILP events in the generated scenarios. The specific steps are detailed in Algorithm 1.

Algorithm 1: Fault Scenario Generation and Reduction

1. Available information and initialization: 2. Power grid and transportation network model, equipment fragility curve, seismic data 3. Scenario Generation: 4. For m = 1, 2, …, M 5.  For n = 1, 2, …, N $6. \mathrm{Calculate} p_l$ for each line (i, j) by Equation (9) 7.    Calculate the cumulative probability of PV and road by Equation (10) $8. \mathrm{If} p_l>rand[u(0,1)]$: $9. z_{ij}=0$ $10.\mathrm{If} P(G_4|Y_{PGA})>rand[u(0,1)]$ $11. \mathrm{PV} \mathrm{assigned} \mathrm{to} \mathrm{the} G_4$ damage stage 12.    Else $13. \mathrm{Find} \mathrm{c} \mathrm{that} \mathrm{satisfies} P(G_c|Y_{PGA})>rand[u(0,1)]\ge P(G_{c-1}|Y_{PGA})$$, \mathrm{PV} \mathrm{assigned} \mathrm{to} \mathrm{the} G_c$ damage stage 14.    Go to step 10 calculate the road damage stage $15. \mathrm{Calculate} \mathrm{power} \mathrm{loss} P_{m,n}$$\mathrm{and} \mathrm{total} \mathrm{transportation} \mathrm{time} T_{m,n}$ $16. \mathrm{Calculate} P_M^{avg}={\rm E}(P_{m,n})$$\mathrm{and} T_M^{avg}={\rm E}(T_{m,n})$ 17. Scenario Reduction: 18. For m = 1, 2, …, M 19.  For n = 1, 2, …, N $20. \mathrm{If} \left|P_{m,n}-P_M^{avg}\right|\approx \min (\left|P_{m,n}-P_M^{avg}\right|)$ 21.     Selecting scenario as typical for DNs $22. \mathrm{If} \left|T_{m,n}-T_M^{avg}\right|\approx \min (\left|T_{m,n}-T_M^{avg}\right|)$ 23.     Selecting scenario as typical for TNs

4. Energy Storage Planning Considering Seismic Hazards

4.1. Two-Stage SIMP Model

Utilizing the established models for both normal and disaster operations, we establish a two-stage SIMP model based on a mobile–stationary hybrid BESS. Grounded in probability theory, this model describes uncertain information through a scenario-based approach or by utilizing random variables with specific probability distributions. It establishes a stochastic framework aimed at minimizing expected costs, with probabilistic representations of uncertainty to enhance realism. In the first stage, the cost-effectiveness of planning investments is considered, and planning decisions are made based on existing energy storage resources, exploring configurations for both fixed and mobile storage. The second stage accounts for both normal and emergency scenarios, coordinating the scheduling of flexible resources, including a BESS to achieve peak shaving and valley filling while reducing load losses under disaster conditions. This approach supports resilience-oriented energy storage planning. The objective function of the model aims to minimize the investment and operational costs of the DNs, as shown in Equation (17):

$$\min ( f^{inv}+{\displaystyle \sum _{s\in S^N}{w}_s}{f}_s^N+{\displaystyle \sum _{s\in S^E}{w}_s}{f}_s^E)$$

(17)

$$f^{inv}={\displaystyle \sum _{i\in N}\kappa (1+C_{om}^e)(C_e^e{E}_i^e+C_p^e{P}_i^e)}{x}_i^e$$

(18)

$$\kappa ={\displaystyle \frac{r{(1+r)}^y}{{(1+r)}^y-1}}$$

(19)

where $f^{inv}$ is the BESS investment cost; $x_i^e$ is the position of BESS; $\kappa$ is the annual investment equivalence coefficient; $C_{om}^e$, $C_e^e$, and $C_p^e$ are the BESS’s operation and maintenance costs, unit capacity cost, and unit power cost; $E_i^e$ and $P_i^e$ are the rated capacity and rated power of the BESS; $r$ is the discount rate; and $y$ is the storage life.

Equation (17) minimizes the sum of the three components. The first component represents the investment cost of the BESS, including energy and power capacity, as shown in Equation (18). The second and third components represent normal and disaster operational costs, as shown in Equations (11) and (16).

4.2. Constraints

4.2.1. Operational Constraints of BESS

$$N_e^{\min }\le {\displaystyle \sum _{i\in N}{x}_i^e}\hspace{0.33em}\le \hspace{0.33em}{N}_e^{max}$$

(20)

$$0\le P_{s,t,i}^{ch}\le u_{s,t,i}^{ch}{P}_{s,i}^e$$

(21)

$$0\le P_{s,t,i}^{dis}\le u_{s,t,i}^{dis}{P}_{s,i}^e$$

(22)

$$u_{s,t,i}^{ch}+u_{s,t,i}^{dis}\le x_i^e\hspace{1em}{u}_{s,t,i}^{ch},u_{s,t,i}^{ch}\in \{0,1\}$$

(23)

$$E_{s,t,i}=SO{C}_{ini}\cdot E_{s,i}^e$$

(24)

$$SO{C}_{\min }{E}_{s,i}^e\le E_{s,t,i}\le SO{C}_{\max }{E}_{s,i}^e$$

(25)

$$E_{s,t,i} = E_{s,t-\Delta t,i}+\left[{\eta }_i^{ch}{P}_{s,t,i}^{ch},-,P_{s,t,i}^{dis},/,{\eta }_i^{dis}\right]\Delta t$$

(26)

where $N_e^{\min }$ and $N_e^{max}$ are the minimum and maximum numbers of BESSs installed; $u_{s,t,i}^{ch}$ and $u_{s,t,i}^{dis}$ are the charge and discharge identifiers of the BESS; $SO{C}_{ini}$, $SO{C}_{\min }$, and $SO{C}_{\max }$ are the initial value and minimum and maximum charged states of the BESS; $E_{s,t,i}$ is the charged state of the BESS; and ${\eta }_i^{ch}$ and ${\eta }_i^{dis}$ are the charge and discharge efficiencies of the BESS.

4.2.2. Distribution System Operating Constraints

$$\begin{array}{l}{\displaystyle \sum _{l(i,:)\in {\Omega }_L}{P}_{s,t,l}}-{\displaystyle \sum _{l(:,i)\in {\Omega }_L}{P}_{s,t,l}}=P_{s,t}^{sub}\\ +P_{s,t,i}^{pv}+P_{s,t,i}^{dis}-P_{s,t,i}^{ch}-P_{s,t,i}^{load}+P_{s,t,i}^{lr}\end{array}$$

(27)

$$\begin{array}{l}{\displaystyle \sum _{l(i,:)\in {\Omega }_L}{Q}_{s,t,l}}-{\displaystyle \sum _{l(:,i)\in {\Omega }_L}{Q}_{s,t,l}}=Q_{s,t}^{sub}\\ +Q_{s,t,i}^{pv}+Q_{s,t,i}^{dis}-Q_{s,t,i}^{ch}-Q_{s,t,i}^{load}+Q_{s,t,i}^{lr}\end{array}$$

(28)

$$U_{s,t,i}-U_{s,t,j}\le M_0\left(1-z_{s,t,l}\right)+{\displaystyle \frac{r_l{P}_{s,t,l}+x_l{Q}_{s,t,l}}{U_0}}$$

(29)

$$U_{s,t,i}-U_{s,t,i}\le -M_0\left(1-z_{s,t,l}\right)+{\displaystyle \frac{r_l{P}_{s,t,l}+x_l{Q}_{s,t,l}}{U_0}}$$

(30)

$$U_{\min }\le U_{s,t,i}\le U_{\max }$$

(31)

$$-z_{s_0,t,l}{S}_l^{\max }\le P_{s,t,l}\le z_{s,t,l}{S}_l^{\max }$$

(32)

$$-z_{s_0,t,l}{S}_l^{\max }\le Q_{s,t,l}\le z_{s,t,l}{S}_l^{\max }$$

(33)

$$0\le P_{s,t}^{sub}\le P_{sub}^{max}$$

(34)

$$0\le Q_{s,t}^{sub}\le Q_{sub}^{max}$$

(35)

$$0\le P_{s,t,i}^{lr}\le P_{s,t,i}^{load}$$

(36)

$$0\le Q_{s,t,i}^{lr}\le Q_{s,t,i}^{load}$$

(37)

$$0\le P_{s,t,i}^{pv}\le G_c{P}_{s,t,i}^{pre}$$

(38)

$$0\le Q_{s,t,i}^{pv}\le G_c{Q}_{s,t,i}^{pre}$$

(39)

where $P_{s,t,l}$ and $Q_{s,t,l}$ are the active and reactive power flowing on line l; $P_{s,t,i}^{load}$ and $Q_{s,t,i}^{load}$ are the active and reactive power of the load on node I; $U_{s,t,i}$ is the square of the voltage; $M_0$ is a large constant; $r_l$ and $x_l$ are the resistance and reactance of line l; $z_{s,t,l}$ is a binary variable, which is equal to 0 if line l is faulty and otherwise; $U_{\min }$ and $U_{\max }$ are the upper and lower limits of the voltage at node i; $S_l^{\max }$ is the transmission capacity of line l; and $P_{sub}^{max}$ and $Q_{sub}^{max}$ are the upper limits on the active and reactive power of purchased power.

The linearized DistFlow model is employed to relax the constraint of line faults on voltage using the Big-M method. Equations (27)–(33) represent the power flow model of DNs. Equations (34) and (35) establish limits on the active and reactive power that DNs can purchase from the power grid. Equations (36) and (37) establish limitations on the active and reactive loads during DN restoration. Equations (38) and (39) establish limitations on the active and reactive power outputs of PV.

4.2.3. MESS Routing Constraints

At any given moment, each mobile energy storage unit is allowed to connect to only one node. The spatio-temporal transfer model for mobile energy storage across nodes can be considered a point-to-point model, as outlined below:

$${\displaystyle \sum _{i\in N^{\prime }}{\alpha }_{s,i,t}^e\le }{x}_i^e$$

(40)

$${\displaystyle \sum _{e\in V}{\alpha }_{s,i,t}^e}\le N_i^e$$

(41)

$${\alpha }_{s,i,t+\tau }^e+{\alpha }_{s,j,t}^e\le 1 , i\ne j, \forall \tau \le t{r}_{s,ij}, \forall t\le T-\tau$$

(42)

$${\alpha }_{s,i,1}^e=lo{c}_i$$

(43)

where ${\alpha }_{s,i,t}^e$ is a binary variable, which is equal to 1 if the MESS is connected to node i and 0 otherwise, $N_i^e$ is the maximum number of MESS devices connected to node i, $\tau$ is the transfer time of mobile energy storage at node i and node j, and $lo{c}_i$ is the initial storage position.

Equation (40) is the candidate node to which the MESS is connected. Equation (41) establishes a limit on the number of MESSs that can be connected to each candidate node based on the capacity of the corresponding station. Equation (42) ensures that the transportation of the MESS among distinct nodes meets the travel time requirement. Equation (43) is the initial position of the MESS.

4.3. Solution Technology

Planning problems for a single stage or specific scenario typically involves small-scale SMIP models that can be readily addressed using optimization solvers. In contrast, multi-stage planning problems typically comprise large-scale SMIP models, making them challenging to solve directly with standard solvers. The PHA is frequently employed to tackle SMIP issues. The PHA decomposes the original problem into scenario-based subproblems, which are subsequently solved in parallel, resulting in a significant reduction in computational complexity. The two-stage SMIP model can be represented as shown in Equations (44) and (45):

$$\underset{x}{\min \hspace{0.33em}\hspace{0.33em}}{\mathit{c}}^{\top }\mathit{x}+\sum _{s\in S}{w}_{s}g{(s)}^T\mathit{y}(s).$$

(44)

$$\mathrm{s}.\mathrm{t}.\hspace{0.33em}\hspace{0.33em}(\mathit{x},\mathit{y}(s))\in K_s.$$

(45)

where $\mathit{x}$ represents decision variables that do not include scenarios, $\mathit{y}(s)$ represents decision variables that incorporate scenarios, and the set of constraints for the problem is denoted as $K_s$. The non-scenario variable $\mathit{x}$ can be equivalently expressed as $\mathit{x}=\{{\mathit{x}}_1, {\mathit{x}}_2, \cdots , {\mathit{x}}_s\}$, where ${\mathit{x}}_1={\mathit{x}}_2=\cdot \cdot \cdot ={\mathit{x}}_s$, ensuring that the decision vector in the first stage remains independent of the scenarios. Finally, the model can be rewritten equivalently as shown in (46), and Algorithm 2 outlines the specific steps of the PHA.

$$\min \left\{\sum _{\left|S\right|}^{s=1}{w}_s\left({\mathit{c}}^{\top }\mathit{x}+\mathit{g}{(s)}^{\top }\mathit{y}(s)\right):(\mathit{x},\mathit{y}(s))\right.\in K(s),s=1,\dots ,|S|,{\mathit{x}}_1=\cdots ={\mathit{x}}_s\}.$$

(46)

The size of $\rho$ is the penalty factor, which directly influences the solution speed and convergence of the model and is usually set in direct proportion to the cost of the decision variable. The convergence precision $\varsigma$ is set as 0.001.

Algorithm 2: PH Algorithm

$1. \mathbf{Initialize}\mathbf{:} k=0, \mathrm{for} \mathrm{all} s\in S, x_s^k=\arg \min \{c^{T}x+g^T(s)y(s)\}$ $2. \mathbf{Aggregation}\mathbf{:} {\overline{x}}^k={\displaystyle \sum w_s}{x}_s^k$ $3. \mathbf{Multiplier} \mathbf{Calculation}\mathbf{:} {\varpi }_s^k=\rho (x_s^k-{\overline{x}}^k)$ $4. \mathbf{Iteration} \mathbf{Update}\mathbf{:} k=k+1$ $5. \mathbf{Iteration} \mathbf{k}\mathbf{:} s\in S, x_s^k=\arg \min \{c^{T}x+{\varpi }_s^{k-1}x+{\displaystyle \frac{\rho }{2}}‖x-{\overline{x}}^{k-1}‖+g^T(s)y(s)\}$ $6. \mathbf{Aggregation}\mathbf{:} {\overline{x}}^k={\displaystyle \sum w_s}{x}_s^k$ $7. \mathbf{Multiplier} \mathbf{Update}\mathbf{:} {\varpi }_s^k={\varpi }_s^{k-1}+\rho (x_s^k-{\overline{x}}^k)$ $8. \mathbf{Gradient} \mathbf{Calculation}\mathbf{:} g^k={\displaystyle \sum w_s(x_s^k-{\overline{x}}^k)}$ $9. \mathbf{Convergence} \mathbf{Check}\mathbf{:} \mathrm{if} g^k<\varsigma$, terminate the algorithm, otherwise go to Step 4

5. Case Studies

This section performs a case study utilizing the system shown in Figure 3, which combines the improved IEEE 33-bus distribution network with the 12 traffic network nodes from [32], and the coupling relationships between the power and transportation nodes are shown in

Table 2. Coupling of power and transportation networks.

Traffic Node	Distribution Node	Traffic Node	Distribution Node
1	21	7	14
2	2	8	15
3	12	9	27
4	8	10	29
5	3	11	18
6	1	12	32

. Photovoltaic power stations are configured at nodes 7, 22, and 32, each with a rated power of 500 kW. The penalty for curtailing solar power is 2 CNY/kW. The cost of load shedding for critical loads is 100 CNY/kW, while the cost for non-critical loads is 10 CNY/kW. The discount rate is 0.06. The distance between power poles is 50 m, and the MESS operates at a speed of 30 km per hour. The specific parameters for the hybrid BESS are provided in

Table 3. Hybrid energy storage system model parameters.

Hybrid Energy Storage	Fresh	SLBs
$N_e^{max}$	3	3
$C_e^e$	2150 CNY/kWh	1000 CNY/kWh
$C_p^e$	1100 CNY/kW	400 CNY/kW
$y$	10	5
${\eta }_i^{ch},{\eta }_i^{dis}$	0.95	0.85
$SO{C}_{ini}$	0.2	0.2
$SO{C}_{\min }$$–SO{C}_{\max }$	0.1–0.9	0.1–0.9

, and the time-of-use tariff of DNs is shown in

Table 4. Time-of-use tariff.

Time Period	Time	Tariff (CNY/kWh)
Trough	23:00–7:00	0.35
Flat	13:00–18:00	0.7
Peak	8:00–12:00; 19:00–22:00	1.4

. The proposed model is implemented using the YALMIP in MATLAB R2019b, with the GUROBI 10.0 solver used for optimization.

5.1. Scenario Generation and Reduction

In the normal scenario planning model, this paper employs the elbow method of k-means clustering based on historical data to determine the number of normal scenarios as 5. Clustering results in the generation of PV and load output characteristic curves, as shown in Figure 4. The probabilities of normal scenarios derived from K-means clustering are shown in

Table 5. Normal operation scenario probabilities.

Normal Scenario	Probability	Normal Scenario	Probability
1	0.135	4	0.176
2	0.218	5	0.262
3	0.209

.

When analyzing the impact of earthquakes on DNs and TNs, the first step is to establish models for the location and intensity of the earthquake. For the seismic source location, this paper uses historical data to select locations that are more likely to be affected by an earthquake. The study area is divided into circular regions, with the nearest distance from the roads to the epicenter set as 5 km, and the seismic source locations are randomly generated. For seismic intensity, the PGA at different locations is calculated using Equation (1) based on the distance from the seismic source and the earthquake magnitude. Next, the impact of the earthquake on the DNs and TNs is quantified by PGA. The damage probability of the DNs and TNs varies with the PGA, as shown in [21,32]. For simplicity, in the case of power poles along the same line, the midpoint of the line is taken as the reference point for calculating the PGA. Subsequently, the method discussed in Section 1 is applied to generate and reduce earthquake fault scenarios. Stratified sampling is performed for earthquakes of magnitudes 3 to 7 using the roulette wheel algorithm.

This generates fault scenario sets for the DNs and TNs. For each magnitude, an initial set of 1000 fault scenarios is generated. Figure 5 shows the average weighted load loss and road passage time for 1000 scenarios generated for a magnitude 5 earthquake. It can be observed that, after 1000 trials, the values of load loss and road passage time converge. Finally, typical scenarios for each magnitude are selected based on the average load shedding and average road passage time. Historical earthquake data from the past decade are utilized to ascertain the probability distribution of each magnitude. Figure 6 depicts the occurrence probabilities of various magnitudes and their respective load shedding amounts for the representative scenarios. The overall trend is that the larger the magnitude, the greater the system load loss. Additionally, the method accounts for HILP events, making it highly suitable for resilience planning studies. The reduced fault scenarios are then used as the stochastic planning scenarios in Section 4.

5.2. Planning Result

The simplified scenarios described above represent various potential situations in DNs, with each scenario corresponding to the damage caused by a specific earthquake magnitude to the distribution equipment and road traffic. In solving the problem in Equation (17), the probability weights of normal and disaster scenarios are artificially generated as 0.8 and 0.2. with ${\displaystyle \sum _{s\in S^N}{w}_s}$ = 0.8 and ${\displaystyle \sum _{s\in S^E}{w}_s}$ = 0.2.

In the planning problem, this paper first selects candidate nodes for the MESS based on normal operating conditions. BESS devices are installed at locations where they generate the greatest economic benefit under normal operational conditions. The primary role of the hybrid BESS in normal scenarios is facilitating energy arbitrage through peak shaving and valley filling while enhancing the absorption of PV energy to mitigate carbon emissions. According to the proposed optimization framework, six MESS access stations are identified, located at nodes 2, 3, 8, 18, 21, and 29. In all fault scenarios, we assume that the earthquake occurs at 9:00, coinciding with the time when the load is about to reach its peak. It is assumed that a repair time of two hours is needed per line and that only one line can be repaired at a time, while repairs for motors and roads are temporarily ignored. For the selection of the initial positions of the MESS, we apply the principle of choosing the node with the shortest distance from the initial position to each candidate node. Based on this principle, the initial positions of the MESS are determined to be nodes 3, 21, and 29. The final allocation results for the hybrid BESS are presented in Figure 3.

The optimization strategy proposed in this paper favors the installation of the SESS at nodes 1, 12, and 19, with an annual investment cost of CNY 1,557,000 and maintenance costs of CNY 70,000. The annual investment cost for deploying an MESS is CNY 2,089,000, with maintenance costs of CNY 62,000. Due to the high probability of earthquake disasters and the large number of damaged devices, energy storage investment costs are relatively high. Under normal conditions, the hybrid BESS implements a peak shaving and valley filling strategy, yielding a maximum revenue of CNY 1,962,000 for the SESS and CNY 2,910,000 for the MESS. The results show that SLBs, due to their lower unit cost, require less initial investment. However, because of their lower charge–discharge efficiency and shorter battery life compared to new batteries, the net profit and return on investment for an MESS are higher, making it more valuable for investment. Using retired batteries, however, can reduce the initial economic pressure on energy storage investors.

As an example, consider the fault scenario with a magnitude of 5. In this scenario, the following power lines are damaged: L5, L7, L15, L16, and L17. Roads R9 and R10 experience slight damage, while R8 sustains moderate damage. The PV systems are also moderately damaged, with their output reduced to 50% of their original capacity. To restore normal power supply to the DN more quickly, three MESSs are deployed to support different fault areas within the damaged transportation network.

Table 6. The location of mobile energy stored during the recovery period.

Magnitude 5 Earthquake		Time Period
		9	10	11	12	13	14	15	16–19
Dispatch	MESS1	21	→	8		→		2
	MESS2	3			→	8
	MESS3	29					→		18

lists the node locations of the MESS during each time period in the recovery phase, where “→” indicates movement between different nodes. Figure 7 illustrates the output status of the MESS during the recovery period. At the moment of disaster, the mobile energy storage systems are located at nodes 3, 21, and 29.

During the 9:00–11:00 period, MESS1 discharges at node 21 and then moves to node 8. MESS2 and MESS3 discharge at nodes 3 and 29, respectively, with a maximum power output of 600 kW. Meanwhile, SESS2 and SESS3 also begin discharging. However, SESS1, located at node 1, is unable to provide power to the faulted nodes and remains inactive. From 11:00 to 13:00, line L15 is repaired, and MESS1 moves to node 8 to discharge. MESS2 moves from node 3 to node 8, while MESS3, SESS2, and SESS3 continue discharging at their original locations. Between 13:00 and 16:00, lines L5 and L7 are repaired. Due to capacity limitations, MESS1 cannot provide sufficient power at node 8. Affected by the damaged transportation network, MESS1 moves to node 2 to begin charging. MESS2 replaces MESS1 to discharge at node 8 and then starts charging. MESS3 first charges at its original location and then moves to node 18. SESS2 and SESS3 begin charging after completing their discharge. From 16:00 to 19:00, MESS1 and MESS2 discharge after completing charging, while MESS3 continues discharging at node 18. At 19:00, all faulted lines are repaired, and the DNs can be maintained through external power supply. The recovery ratio of the load after the fault is high, reaching over 83% by t = 17, with full load recovery achieved. In summary, the planning results ensure that the DN operates normally during the earthquake disaster and that the majority of the load is restored under disaster conditions, effectively minimizing further economic losses.

5.3. Comparison of Different Cases

To verify the effectiveness of the proposed mobile–stationary hybrid BESS model, three scenarios are established for analysis:

Case 1: The energy storage system consists solely of conventional fresh batteries, considering only disaster scenarios.

Case 2: The MESS is not considered. The energy storage system comprises both conventional fresh batteries and SLBs, while considering both normal and disaster scenarios.

Case 3: The approach proposed in this paper, where the energy storage system consists of both conventional fresh batteries and SLBs, and considers both normal and disaster scenarios.

Based on the load recovery power in each time period, the load recovery ratio curves for each strategy are derived, as shown in Figure 8. The optimal planning results for the three strategies are presented in

Table 7. Optimal planning results of different strategies.

Case	Type	Capacity (kWh)	Investment Cost (CNY 103)	Benefit of BESS (CNY 103)	Load Shedding Cost (CNY 103)
1	SESS	1200, 2000, 2000	1955	2328	456
	MESS	1900, 2000, 2000	2125	2914
2	SLBs	2000, 2000, 2000	1642	2061	1400
	Fresh	1700, 2000, 1800	1981	2780
3	SESS	2000, 2000, 1700	1557	1962	825
	MESS	2000, 1800, 2000	2089	2910

and Figure 2, Figure 9 and Figure 10. From Table 7, it is evident that, in comparison to Case 1, Cases 2 and 3 reduce the annual investment costs by CNY 457,000 and CNY 434,000, respectively, by incorporating retired batteries. Additionally, since the planned lifespan of retired batteries is only 5 years, their initial investment cost is significantly lower than that of new batteries. The annual net profits from the energy storage systems for each strategy are CNY 1,040,000, CNY 1,085,000, and CNY 1,094,000, respectively, and the loss of load costs are CNY 456,000, CNY 1,400,000, and CNY 825,000, respectively. Since Case 1 only considers disaster scenarios, its benefits during normal operation are relatively low, but it has the lowest loss of load costs. As it considers only the deployment of new batteries, Case 1 results in higher overall investment costs. Compared to Case 2, Cases 1 and 3 deploy mobile energy storage, which allows for optimal energy distribution across time and space. By fully utilizing the mobility of mobile energy storage, the load recovery ratio is higher after faults, reaching over 80% in the scenario of a 6.8-magnitude earthquake. Case 3 incurs a higher loss-of-load cost compared to Case 1, but it achieves greater energy storage benefits, indicating that while emphasizing economic efficiency and resilience, the reliability of power supply may be partially compromised.

6. Conclusions

This paper comprehensively considers both normal and disaster operation scenarios of DNs, aiming to maximize the grid’s economic efficiency and security. The following is a summary of the key points drawn from the analysis above:

• The impact of seismic hazards on distribution equipment and road traffic is quantified using the PGA, and failure probability models for both are established. Typical scenarios are created for each magnitude by employing the roulette algorithm and considering distance reduction based on load loss. The proposed method captures the relationship between earthquake magnitude and system load loss, ensuring that disaster scenarios realistically represent actual conditions and include HILP events.
• A two-stage stochastic optimization model is established, with the second stage dividing operational scenarios into normal and disaster conditions. During normal operations, BESSs generate profit through peak shaving and valley filling, while in disaster operations, BESSs ensure a continuous power supply to loads. Compared to models that consider only a single operational scenario, the efficient utilization of energy storage resources is achieved based on meeting the dual goals of good economy and resilience of DNs.
• This paper proposes a novel hybrid energy storage planning model that utilizes low-cost SLBs to construct SESSs and fresh batteries as MESSs. The proposed planning scheme can effectively reduce the initial economic pressure of the energy storage investor while improving the grid resilience, providing a new idea for realizing the economy and sustainability of the energy storage system.

Future work will investigate the cyclic replacement mechanism of fresh batteries and SLBs during the planning cycle and assess the impact of secondary disasters, such as aftershocks, following strong earthquakes.