A resilient battery electric bus transit system configuration

The electrification of public transit can significantly reduce greenhouse gas emissions from transportation. Battery electric buses (BEBs) are increasingly viable due to energy efficiency, quiet operation, low maintenance, and zero tailpipe emissions. Designing BEB systems is complex, requiring optimization of charging infrastructure, fleet configuration, energy management, and charging schedules while balancing capital and operational costs, emissions, and grid impacts. En-route fast charging can enable uninterrupted operations but depends on reliable energy supply. BEB operations are vulnerable to disruptions such as power outages and equipment failures; even a single station’s daily disruption can markedly reduce service frequency. Existing models often do not explicitly design for robustness against charging station failures. This study targets the robustness aspect of resilience—ensuring minimal impact on functionality during disruptions—by developing a BEB system configuration that remains fully operational under charging station failure uncertainty.

Prior work on BEB planning under uncertainty has modeled parameters such as energy consumption, travel and charging times, passenger processes, battery degradation, and power supply fluctuation. Energy consumption uncertainty is most examined due to dependence on ridership, traffic, and weather. Recent robust planning studies addressed power supply fluctuations using budgeted uncertainty, modeling charging power delivered to buses as continuous deviations at the trip/line level. In contrast, charging station failures are discrete, network-level events affecting whether a station is operational. The literature has not optimized BEB system design for resilience to charging station disruptions using a network-level uncertainty set on station failures. Broader resilience modeling approaches include survivable network design, fault tolerance, stochastic and robust optimization, interdiction, N−k, and attacker–defender formulations. Two-stage robust optimization is attractive here because it avoids requiring failure probability distributions and is less conservative than single-stage robust models.

The study formulates a two-stage robust optimization model to configure a resilient BEB system under up to k simultaneous charging station failures. The system is represented as a directed graph with candidate charging locations (nodes) and bus route segments (edges). Decisions: first-stage (here-and-now) include selecting charging station locations, charger rated power and number of poles per station, and each bus’s battery capacity; second-stage (wait-and-see) includes charging schedules (whether/when/where to charge, charging power) and failure indicators for buses under realized station failures. Objectives and costs: minimize annual total cost (capital and operational), including station construction, chargers and poles, batteries, fleet (bus without battery), electricity Time-of-Use (ToU) energy charges, demand charges, and time-varying well-to-tank (WTT) GHG emissions costs; annualization is applied to capital costs. Operational constraints include: battery state-of-charge dynamics with minimum/maximum SoC, heterogeneous battery capacities, charging power limits (by pole rating and C-rate factor), continuous charging (plug-in continuity via auxiliary binaries), station-level capacity (poles in use and total charging power), station and network ToU power caps, and demand charge calculation from interval-average peak power. Assumptions: (1) operational timetable satisfied; (2) all buses start day fully charged; (3) heterogeneous BEB battery capacities; (4) heterogeneous numbers of stations, poles, and charger-rated power; (5) disruption is a full-day station failure; (6) no backup buses are used due to high cost. Base Model: a mixed-integer linear program (MILP) without failures determines nominal configuration and schedules. Uncertainty set: budgeted station-failure set allowing up to k failed stations per day (binary failure variables, sum ≤ k). Two-stage robust model: min over first-stage decisions, max over failures in uncertainty set, min over second-stage recourse (operational schedule with penalties for failed buses to ensure relative complete recourse); constraint modifications force zero charging at failed stations. Price of Robustness (PoR) quantifies cost increase relative to deterministic base. Solution algorithm: tri-level problem solved via nested column-and-constraint generation (NC&CG) suited for mixed-integer recourse. Outer level (C&CG) solves the master problem for first-stage variables; inner level iteratively solves an inner master problem to identify worst-case failure scenarios and an inner sub-problem to compute operational response (charging schedule, failures) under those scenarios until convergence. Implementation: models coded in MATLAB; MILPs solved with Gurobi. Data: real-world Oakville Transit network (Ontario, Canada) using GTFS; utility ToU tariffs, demand charges, and WTT GHG intensity considered. Additional computational and convergence details provided in supplementary materials.

Base Model (no station failures): - Optimal system uses 5 en-route charging stations with 8 poles; fleet of 91 BEBs with heterogeneous batteries (majority 100 kWh). - Total annual cost: $6,959,381.19. Cost breakdown: fleet without batteries 76.63%; operations 11.58%; batteries 10.03%; charging infrastructure (stations + chargers) 1.77%. - Robustness assessment of base design: single station failure (r=1) can reduce daily service by up to 34.03% (e.g., failure at Oakville GO Station, ID 1), with up to 30 failed buses at that station; two simultaneous failures (r=2) can reduce service by up to 58.18% (e.g., IDs 1 and 4) with 48 failed buses. Robust Models: - k=1 (robust to any single station failure): 18 charging locations with 19 poles; fleet includes higher-capacity batteries (e.g., 300–600 kWh added). Ensures full operation under any single-station failure by re-optimizing schedules using remaining stations. Daily operational cost varies slightly across single-failure scenarios (e.g., $2,021.47 nominal to $2,069.85 when Station ID 4 fails). Total annual cost: approximately $7.19M; PoR = 3.26%. - k=2 (robust to any two simultaneous station failures): 33 charging locations with 33 poles. Ensures full operation under any two-station failure. Total annual cost: $7,524,798.35; PoR = 8.12%. System behavior and benefits: - Robust designs spread demand across more locations with fewer multi-pole stations, reducing vulnerability concentration. - Peak single-station daily energy demand drops from 5679.78 kWh (base) to 2807.45 kWh (k=1, ~49% of base) and 1805.25 kWh (k=2, ~32% of base). - Operational metrics improve in robust designs: for k=2, on-peak demand is 77% lower than base; total system WTT GHG emissions are 8% lower than base; operational costs are lower than base model. - Price of Robustness prevents large service losses: k=1 avoids up to 34.03% service reduction; k=2 avoids up to 58.18% reduction. Additional summary (Table 7): - Base: Max service reduction 34.03% (one failure) and 58.18% (two failures). - k=1: Max service reduction 0% (one failure), 13.73% (two failures), 22.38% (three failures). - k=2: Max service reduction 0% (one or two failures), 8.92% (three failures).

The results show that BEB networks designed only for nominal conditions are highly sensitive to charging station failures, leading to substantial service reductions. Incorporating robustness through a two-stage robust optimization framework significantly mitigates this risk with modest additional annual cost. The choice of robustness budget k should weigh project budgets, failure risks, and the associated price of robustness against the potential social and economic costs of service loss. Robust configurations diversify charging capacity across more locations, reduce concentration of demand, and, compared to the base design, can lower operational costs, peak demand, and WTT GHG emissions by enabling more flexible, resilient charging schedules. The proposed approach provides actionable planning and operational guidance for agencies to ensure full service under station disruptions, and quantifies cost–risk trade-offs to support decision-making.

This work introduces a two-stage robust optimization model to plan resilient BEB systems that minimize annual capital and operational costs while guaranteeing full operation under up to k simultaneous charging station failures. Contributions include: (1) joint optimization of charging station siting, charger power and poles, and heterogeneous fleet battery capacities under ToU tariffs, demand charges, time-varying WTT GHG emissions, and grid constraints; (2) assurance of full-service operation under any charging station failure scenario via recourse charging schedules; (3) application to a real-world, large-scale network with varying robustness budgets (k=0,1,2); (4) quantification of the price of robustness to balance costs and risks; and (5) demonstration of practical implementability. Case-study results show small PoR (3.26% for k=1; 8.12% for k=2) that avert severe service reductions (up to 34%–58% in the base). The modeling and solution framework can guide transit agencies in resilient electrification planning and operations.

• Disruption modeled as full-day charging station failures (binary on/off), not partial or time-varying outages. - No backup buses are allowed to cover failed operations due to high added cost, emphasizing infrastructure and scheduling recourse. - Annual operational cost in the robust objective uses a worst-case daily scenario for tractability; actual annual operations may involve varying daily realizations. - The approach uses robust optimization and avoids probability distributions; it does not estimate failure likelihoods. - The two-stage robust model with mixed-integer recourse is NP-hard; although solved with NC&CG, computational complexity may limit scalability without additional enhancements. - Energy consumption model and parameters (e.g., C-rate limits, ToU, WTT GHG intensity) are scenario- and context-specific; results depend on these inputs and the chosen robustness budget k.