The increasing frequency and intensity of extreme weather events, such as typhoons and hurricanes, pose a severe and growing threat to the reliable operation of power systems worldwide. These events often trigger large-scale blackouts, resulting in staggering economic losses and significant societal disruption. While the entire grid is vulnerable, distribution networks are particularly susceptible due to their expansive, exposed infrastructure and traditionally radial, less redundant design. Consequently, research into strategies for enhancing the resilience of distribution grids—defined as their ability to anticipate, withstand, adapt to, and rapidly recover from high-impact, low-probability events—has become a critical focus for utilities and researchers alike.
A cornerstone of resilience enhancement is the flexible mobilization of all available resources to restore critical loads swiftly following an outage. Traditionally, this role has fallen to utility-owned assets like backup diesel generators or, more recently, mobile energy storage systems (MESS). However, the global push towards decarbonization is catalyzing a massive transformation in the transportation sector, marked by the rapid proliferation of battery EV cars. This fleet of millions of mobile batteries represents an unprecedented, distributed energy resource. Through Vehicle-to-Grid (V2G) technology, a parked battery EV car can discharge its stored energy back to the grid, effectively acting as a decentralized generator. This capability positions aggregated fleets of battery EV cars as a potentially game-changing resource for post-disaster load restoration, offering both capacity and geographical flexibility that can complement or even surpass traditional MESS.
Despite this immense potential, a significant gap exists in current research. Most studies on utilizing battery EV cars for grid support treat them as a passive, perfectly obedient resource akin to utility-owned MESS. This perspective is fundamentally flawed. A battery EV car is private property, primarily serving the mobility and economic needs of its owner. Its availability for grid services is not guaranteed but is instead contingent upon the willingness of the owner to participate. This willingness is heavily influenced by economic incentives and the operational constraints of the vehicle (e.g., required state-of-charge for future trips). Ignoring these user-centric factors—the core interests and response willingness of the battery EV car owner—leads to idealized models that are unlikely to succeed in practice. If the compensation is insufficient or the demanded service overly inconvenient, participation will be low, undermining the resilience scheme.
To bridge this gap, we propose a novel bi-level optimization framework for distribution grid resilience enhancement that explicitly models and leverages the economic incentive response of battery EV car owners. The core philosophy is to align the goals of the Distribution System Operator (DSO) with the economic interests of the battery EV car user group, creating a symbiotic relationship. The DSO aims to minimize outage costs, while battery EV car owners seek to maximize their economic benefit from providing grid services. Our model facilitates this by designing two distinct, sequential economic compensation mechanisms for battery EV cars, inspired by demand response programs:
- Incentive-Based Response (IBR): A pre-event monetary incentive paid to battery EV car owners for committing their vehicle to a designated V2G station prior to an anticipated disaster. This compensates for the inconvenience and ensures a pre-positioned fleet of mobile batteries is available at strategic grid nodes for the recovery phase.
- Price-Based Response (PBR): A time-of-use (ToU) tariff for charging and discharging applied during the post-fault restoration period. This dynamic price signal guides the aggregated charge/discharge behavior of the battery EV car fleet to meet the DSO’s real-time load recovery needs while allowing owners to optimize their personal energy costs and ensure their vehicle meets its required departure state-of-charge.
This bi-level structure captures the strategic interaction: the upper-level DSO sets the IBR prices and PBR tariffs, and the lower-level aggregate of battery EV car owners reacts to these prices by deciding their optimal charging/discharging schedules. The equilibrium solution provides a feasible, mutually beneficial strategy for resilience enhancement.
Mathematical Formulation of the Bi-Level Optimization Model
Upper-Level Problem: Distribution System Operator (DSO) Decision Model
The DSO’s objective is to minimize its total economic cost associated with the resilience event, which comprises three components: the cost of unsupplied load, the cost of pre-event incentives, and the cost of real-time energy transactions with battery EV cars.
$$ \min C_{DSO} = C_{PLS} + C_{IBR} + C_{PBR} $$
where:
- $C_{PLS}$ is the cost of load shedding (Power Loss Cost).
- $C_{IBR}$ is the total Incentive-Based Response compensation cost.
- $C_{PBR}$ is the total Price-Based Response compensation cost.
The load shedding cost is calculated as the sum of shed load at each bus, weighted by its priority value $c_{L,i}$:
$$ C_{PLS} = \sum_{t \in T} \sum_{i \in \Omega_B} c_{L,i} P^{sh}_{i,t} $$
Here, $P^{sh}_{i,t}$ is the shed load at bus $i$ in time period $t$, $T$ is the set of restoration time intervals, and $\Omega_B$ is the set of all buses. Loads are typically categorized into priority levels (e.g., critical, important, normal).
The IBR cost is the sum of payments made to all battery EV cars that responded to the pre-event call, dependent on the incentive price $c_{IBR,v}$ set at each V2G station $v$:
$$ C_{IBR} = \sum_{v \in V} c_{IBR,v} N_{E,v} $$
$V$ is the set of V2G stations, and $N_{E,v}$ is the number of battery EV cars that commit to station $v$.
The PBR cost (which can be negative, representing revenue if DSO purchases cheap charging power) is the net payment for the energy exchanged with the battery EV car fleet during restoration:
$$ C_{PBR} = \sum_{t \in T} \sum_{v \in V} c_{PBR,t} (P^d_{V,v,t} – P^c_{V,v,t}) $$
$c_{PBR,t}$ is the ToU price for energy at time $t$, $P^d_{V,v,t}$ is the aggregate discharge power from battery EV cars at station $v$, and $P^c_{V,v,t}$ is the aggregate charge power.
IBR Price-Response Model: The DSO must model how the number of responding battery EV cars $N_{E,v}$ at a station depends on the offered incentive $c_{IBR,v}$. We adopt a linear consumer psychology model:
$$ c_{IBR,v} = k_{IBR,v} \frac{N_{E,v}}{N_{E,max}} + c_{IBR,0}, \quad \forall v \in V $$
Here, $k_{IBR,v}$ is a station-specific response coefficient (slope), $N_{E,max}$ is the total number of battery EV cars in the region, and $c_{IBR,0}$ is a base price below which no vehicle would respond. The response coefficient $k_{IBR,v}$ is uncertain in practice. We model this uncertainty using a chance constraint, ensuring the probability that $k_{IBR,v}$ is above a minimum threshold $k_{IBR,min}$ is greater than a confidence level $\xi$. This can be transformed into a deterministic equivalent using big-M methods:
$$
\begin{aligned}
& k_{IBR,v} \ge k_{IBR,min} – (k_{IBR,min} – k_{IBR,lim}) \cdot (1 – \gamma_v), \quad \forall v \in V \\
& \sum_{v \in V} \gamma_v \ge N_V \cdot \lceil \xi \rceil
\end{aligned}
$$
where $\gamma_v$ is a binary auxiliary variable and $k_{IBR,lim}$ is a lower bound for the coefficient.
Other DSO Constraints:
- Price Bounds: Incentive and ToU prices are bounded by regulatory or market limits.
$$ c_{IBR,min} \le c_{IBR,v} \le c_{IBR,max}, \quad c_{PBR,min} \le c_{PBR,t} \le c_{PBR,max} $$
- EV Commitment Limits: The total number of committed battery EV cars cannot exceed the regional fleet size, and the number per station is limited by physical capacity.
$$ \sum_{v \in V} N_{E,v} \le N_{E,max}, \quad N_{E,v} \le N_{E,max,v} $$
- Network Operational Constraints: The model incorporates standard DistFlow or linearized power flow equations, radiality constraints for network reconfiguration, and operational limits for other resources like MESS and Static VAR Compensators (SVCs). These ensure feasible and stable grid operation during restoration.
$$ \text{Power Flow: } \mathbf{P}_{ij,t}, \mathbf{Q}_{ij,t}, \mathbf{V}_{i,t} $$
$$ \text{Radiality: } \beta_{ij} \in \{0,1\}, \sum \beta_{ij} = N_B – 1 $$
$$ \text{MESS: } E_{m,t+1} = E_{m,t} + \eta^c_m P^c_{m,t} – P^d_{m,t}/\eta^d_m $$
Lower-Level Problem: Aggregate Battery EV Car Owner Decision Model
The collective of battery EV car owners at each V2G station seeks to maximize its total economic benefit during the restoration period, given the prices ($c_{IBR,v}$, $c_{PBR,t}$) set by the DSO.
$$ \max C_{EV} = C_{IBR} + C_{PBR} + C_{SOC} $$
where:
- $C_{IBR}$ and $C_{PBR}$ are the same cost components from the upper level, now representing revenue for the EV group.
- $C_{SOC}$ is a monetized State-of-Charge Satisfaction benefit. It represents the value owners place on having sufficient battery energy at the end of the restoration period for their subsequent travel needs. It is modeled as: $C_{SOC} = \sum_{v \in V} c_{SOC} \cdot E_{V,v,t_{end}}$, where $c_{SOC}$ is a satisfaction coefficient (€/kWh) and $E_{V,v,t_{end}}$ is the aggregate energy stored in the battery EV car fleet at station $v$ at the final time $t_{end}$.
EV Aggregator Operational Constraints: The aggregate behavior of $N_{E,v}$ battery EV cars at station $v$ is constrained by their combined power and energy capacity.
- Power Limits: Aggregate charge/discharge power is bounded by the number of vehicles and individual battery EV car power ratings.
$$ 0 \le P^c_{V,v,t} \le N_{E,v} \cdot P^{max}_{Ec}, \quad 0 \le P^d_{V,v,t} \le N_{E,v} \cdot P^{max}_{Ed}, \quad \forall v, t $$
- Energy Capacity & Dynamics: The aggregate energy state $E_{V,v,t}$ must remain within bounds and follows the storage dynamics equation.
$$ N_{E,v} \cdot E^{min}_E \le E_{V,v,t} \le N_{E,v} \cdot E^{max}_E, \quad \forall v, t $$
$$ E_{V,v,t+1} = E_{V,v,t} + \Delta t (\eta^c_V P^c_{V,v,t} – P^d_{V,v,t} / \eta^d_V), \quad \forall v, t $$
- Departure State-of-Charge Requirement: To guarantee mobility after the grid is restored, the average remaining energy percentage across the fleet at each station must meet a minimum threshold.
$$ r_E = \frac{E_{V,v,t_{end}}}{N_{E,v} \cdot E^{max}_E} \ge r^{min}_E, \quad \forall v \in V $$
Model Solution: Transformation to a Single-Level MILP
The bi-level model, with a linear lower-level problem, can be solved by replacing the lower level with its Karush-Kuhn-Tucker (KKT) optimality conditions, which become additional constraints for the upper level. The complementary slackness conditions introduced by the KKT transformation are linearized using the big-M method. The final reformulation is a single-level Mixed-Integer Linear Program (MILP) that can be efficiently solved using commercial optimization solvers like Gurobi or CPLEX within environments such as MATLAB/YALMIP or Python/Pyomo.
Case Study and Analysis
Simulation Setup
We test the proposed model on a modified IEEE 33-bus distribution system. An extreme event is assumed to have damaged multiple lines, causing isolation of several network sections. The fault repair sequence is known, and the restoration period lasts 16 hours with a 1-hour resolution. The DSO can dispatch MESS units, SVCs, perform network reconfiguration, and utilize the committed battery EV car fleet. The initial post-fault topology and resource locations are shown conceptually (the specific figure from the text is omitted as per instruction). Key parameters are summarized in the table below.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Total Regional Battery EV Cars, $N_{E,max}$ | 100 vehicles | Min/Max IBR Price, $c_{IBR,min/max}$ | 0 / 150 €/vehicle |
| Max Battery EV Cars per Station, $N_{E,max,v}$ | 15 vehicles | Base IBR Price, $c_{IBR,0}$ | 20 €/vehicle |
| Min/Max PBR Price, $c_{PBR,min/max}$ | 0.6 / 2.0 €/kWh | Min Response Coeff., $k_{IBR,min}$ | 1000 |
| SOC Satisfaction Coeff., $c_{SOC}$ | 0.5 €/kWh | Confidence Level, $\xi$ | 0.5 |
| Min Departure SOC, $r^{min}_E$ | 40% | Individual Battery EV Car Capacity, $E^{max}_E$ | 70 kWh |
| Individual Battery EV Car Max Power, $P^{max}_{Ec/Ed}$ | 7 kW | Charging/Discharging Efficiency, $\eta^c_V / \eta^d_V$ | 0.9 / 0.9 |
| Load Shedding Cost, $c_{L,i}$ | Priority 1: 200 €/kWh, Priority 2: 20 €/kWh, Priority 3: 1 €/kWh | ||
Impact of Total Regional Battery EV Car Fleet Size
We first analyze the impact of the total available battery EV car fleet size ($N_{E,max}$) on grid restoration outcomes and EV owner economics. A base case with no battery EV car participation serves as a benchmark.
| Scenario | $N_{E,max}$ (vehicles) | Responding EVs | Total DSO Cost $C_{DSO}$ (€) | Load Shed Cost $C_{PLS}$ (€) | I+PB Cost $C_{IBR}+C_{PBR}$ (€) | Priority 2 Load Rest. Rate |
|---|---|---|---|---|---|---|
| No-EV Benchmark | 0 | 0 | 46,756 | 46,756 | 0 | 86.95% |
| Proposed Model | 50 | 40 | 24,961 | 18,073 | 6,888 | 96.06% |
| Proposed Model | 100 | 58 | 13,898 | 5,894 | 8,004 | 100% |
| Proposed Model | 200 | 58 | 11,287 | 5,893 | 5,394 | 100% |
Analysis: The results demonstrate the profound effectiveness and economic benefit of battery EV car participation. Even with only 50 vehicles, the model reduces the DSO’s total cost by nearly 47% compared to the no-EV case. The load shedding cost $C_{PLS}$ drops significantly as battery EV car discharge supports critical loads. With 100 vehicles, Priority 2 loads are fully restored, causing a major reduction in $C_{PLS}$. Notably, when the fleet size increases to 200, the number of responding vehicles plateaus at 58. This is because the DSO’s primary objective of restoring high-priority loads is already satisfied. It becomes economically suboptimal for the DSO to pay higher IBR incentives to attract more battery EV cars just to restore lower-value (Priority 3) loads. This illustrates the model’s inherent economic efficiency.
| $N_{E,max}$ (vehicles) | Responding EVs | Avg. IBR Price $c_{IBR,v}$ (€/veh.) | Total EV Revenue $C_{EV}$ (€) | Avg. Revenue per EV (€/veh.) |
|---|---|---|---|---|
| 50 | 40 | 144.2 | 7,448 | 186.2 |
| 100 | 58 | 110.0 | 8,816 | 152.0 |
| 200 | 58 | 65.0 | 6,206 | 107.0 |
Analysis: From the battery EV car owner perspective, the model ensures attractive compensation. When the fleet is small (50 vehicles), the DSO must offer a high average IBR price (144.2€) to attract a sufficient number of vehicles, resulting in a high average revenue per vehicle. As the fleet grows, the DSO can achieve its target response with lower IBR prices due to increased competition among vehicle owners, reducing the average IBR revenue. However, the PBR revenue and SOC satisfaction revenue remain substantial, ensuring the battery EV car owners still receive a positive net benefit. This dynamic reflects a realistic market where compensation adjusts based on resource scarcity.
Impact of the State-of-Charge Satisfaction Coefficient
The coefficient $c_{SOC}$ reflects how much battery EV car owners value having a full battery at the end of service. A higher $c_{SOC}$ means owners are more reluctant to discharge deeply. We compare scenarios with $c_{SOC} = 0.5$ €/kWh and $c_{SOC} = 0.8$ €/kWh.
The DSO’s pricing strategy adapts elegantly to this preference shift. With a higher $c_{SOC}$, the ToU price $c_{PBR,t}$ is set at a generally higher level throughout the restoration period to incentivize the now more “reluctant” battery EV car fleet to discharge. Furthermore, the price profile shows a steeper gradient, offering very high prices during early, critical hours to secure essential discharge, and lower (but still elevated) prices later to encourage charging. This results in a different aggregate battery EV car power profile: slightly less discharge in peak hours but significantly more charging overall, ensuring the fleet meets the 100% departure SOC target.
| $c_{SOC}$ (€/kWh) | Final SOC $r_E$ | DSO Cost $C_{DSO}$ (€) | EV PBR Revenue $C_{PBR}$ (€) | EV SOC Benefit $C_{SOC}$ (€) | Avg. EV Revenue (€/veh.) |
|---|---|---|---|---|---|
| 0.5 | 40% | 13,898 | 1,624 | 812 | 152.0 |
| 0.8 | 100% | 12,289 | 0* | 3,248 | 166.0 |
* Net PBR revenue is zero because the higher charge cost cancels out the discharge revenue.
Analysis: This comparison highlights the flexibility of the economic incentive framework. When owners value a full battery more ($c_{SOC}=0.8$), they achieve a 100% departure SOC, translating to a large $C_{SOC}$ benefit. Their net PBR revenue becomes zero because the cost of buying energy at the higher prices to recharge offsets the revenue from discharging. Remarkably, the DSO’s total cost is lower in this case. This is because the DSO pays less in net PBR costs (the high discharge revenue is paid back via the high charging cost), and the slightly increased load shedding cost is minor. The average revenue for the battery EV car owner increases slightly. This demonstrates that the model successfully balances different user preferences—some may prioritize cash revenue, others may prioritize guaranteed range—while still meeting the DSO’s resilience objectives efficiently.
Conclusion
This paper addresses a critical oversight in existing research on distribution grid resilience by formally integrating the economic interests and response behavior of battery EV car owners into the operational planning model. The proposed bi-level optimization framework creates a symbiotic marketplace for resilience services. The DSO minimizes its outage costs by strategically offering two types of economic signals: a pre-event commitment incentive (IBR) and a real-time energy price (PBR). The aggregated battery EV car owners, in turn, maximize their revenue and satisfaction by responding to these signals within their operational constraints.
Simulation results on a standard test network confirm the model’s dual efficacy. From the grid perspective, it enables a significant, cost-effective enhancement in resilience, dramatically reducing load shedding costs and improving the restoration of critical loads compared to scenarios without battery EV car support. From the user perspective, it guarantees that participating battery EV car owners receive fair and attractive compensation, which is adaptive to fleet size and user preferences regarding final battery state-of-charge. The model inherently finds an economically efficient equilibrium, preventing the DSO from over-incentivizing and ensuring battery EV car resources are used where they provide the highest value to the grid.
Future work could extend this framework by incorporating more granular models of individual battery EV car user behavior, uncertainty in disaster location and repair times, and integration with other distributed energy resources like rooftop photovoltaics. Furthermore, the interaction of this resilience-driven mechanism with day-ahead and real-time electricity markets presents an interesting avenue for research. Nonetheless, this study provides a foundational, practical model for unlocking the vast potential of the battery EV car fleet as a key partner in building a more resilient and sustainable power grid.