Source-linked AI summary
Three-sided mobility-energy market design as a multiperiod stochastic assignment game
Hai Yang, Joseph Y. J. Chow
TL;DR
Existing models do not jointly represent temporal mobility–charging interdependencies and strategic interactions among travelers, mobility providers, and energy providers. The paper proposes a bilevel three-sided assignment game using link-based PURC and decomposed mobility and recharge subnetworks. Experiments on an expanded Nguyen–Dupuis network identify charging-capacity, fleet-sizing, and pricing patterns relevant to platform planning.
Problem
Existing frameworks lack a unified multiperiod equilibrium model that jointly captures mobility assignment, recharge assignment, and strategic interactions among travelers, mobility providers, and energy providers.
Method
The paper develops a bilevel three-sided assignment game with link-based PURC, discrete time intervals, an empirical affine mobility–charging relationship, and chronological decomposition into mobility and recharge subnetworks.
Results
The expanded Nguyen–Dupuis case study identifies localized access and charging bottlenecks, profit maximization near the supply-constrained fleet upper bound, and near-optimal faster solutions from the proposed decomposition heuristic.
Takeaways & Limitations
Platform planning should coordinate fleet sizing, charging capacity, access capacity, and charging-resource distribution rather than treat mobility operations and energy constraints separately.
Takeaways & Limitations
The framework omits nonlinear service–charging relationships, dynamic power-grid constraints, time-varying electricity-price fluctuations, and heterogeneous energy providers.
Abstract
from arXiv · showhide
As mobility service providers (MSPs) and energy providers (EPs) expand electric vehicle ecosystems, models are needed to understand their interactions within a three-sided market. Existing frameworks often overlook the temporal interdependencies between mobility and charging demands. We address this gap by proposing a bilevel problem as an assignment game overseen by a market regulator. The upper level optimizes service pricing to maximize platform profitability. The lower level models a multi-stakeholder equilibrium using a scalable, link-based Perturbed Utility Route Choice (PURC) framework. The evaluation time frame is divided into discrete intervals, capturing the temporal lag between mobility and charging demand via an empirical affine function. We solve the model by chronologically decomposing the lower level into interacting mobility service and recharge subnetworks. Numerical experiments on the expanded Nguyen-Dupuis network reveal several key insights. First, a critical charging capacity threshold exists; operating below it forces a severe reduction in the deployable fleet and creates localized transit deserts. Second, modeling endogenous operating costs reveals a concave profit trajectory, demonstrating that total profit maximizes at a specific fleet size just before market saturation. Third, optimal dynamic pricing operates within a narrow range, where peak pricing acts as a steady revenue driver and off-peak pricing serves as a highly sensitive operational buffer. These findings provide actionable strategies for coordinating fleet sizing and charging infrastructure deployment.
1. Introduction
The paper addresses the missing integration of mobility assignment, recharge assignment, and strategic energy-provider behavior in multiperiod mobility–energy markets. It proposes a bilevel three-sided assignment game with a scalable decomposition method for analyzing equilibrium and platform profit.
- Motivation: 25% of global output is attributed to transportation’s non-renewable energy use, while uncoordinated charging can increase peak electricity demand.The passage also notes that EV environmental benefits depend on energy mix and operating conditions.
- Research gap: Mobility and charging demands are temporally interdependent, while energy costs and capacity affect fares, traveler choices, and provider profitability.These interactions transform the conventional traveler–mobility-provider market into a three-sided market requiring joint modeling.
- Research gap: Existing research treats electrified fleet and charging operations alongside mobility market design, but lacks a unified network-based equilibrium framework.The two streams separately address charging and fleet constraints or platform pricing, assignment, and participation.
- Contribution: The proposed model represents travelers, mobility providers, and energy providers in a bilevel assignment game with multiperiod mobility–recharge coupling.The regulator determines cost transfers and platform profit, while the lower level captures stakeholder equilibrium across periods.
- Solution approach: The framework extends scalable link-based PURC modeling and decomposes the lower-level equilibrium into interconnected mobility-service and recharge subnetworks.The decomposition addresses time-of-day coupling, energy-provider decisions, and fleet-positioning decisions associated with recharging.
2. Proposed model and solution method
The model represents a cyclic, multilayer mobility–energy network in which mobility-service layers connect through recharge subnetworks that assign charging and redistribute fleet capacity across time. Its assumptions and network structure provide a macroscopic steady-state representation of temporal service and charging interactions.
- Model assumptions: The planning model assumes perfect information symmetry and includes traveler and operator individual rationality, while truth-telling mechanisms remain outside scope.The framework is strategic planning rather than mechanism design.
- Model assumptions: The platform includes MOD services, energy providers, a regulator, and out-of-platform options over a cyclic horizon partitioned into discrete intervals.The regulator may be a public agency, private enterprise, or sole mobility service.
- Network structure: The multilayer network couples mobility-service subnetworks with recharge subnetworks to represent spatial and temporal interactions between EV service and charging.It captures time-of-day interactions macroscopically rather than dynamic traffic spillovers or FIFO conditions.
- Cyclic temporal horizon: Mobility layers represent interval-specific demand and costs, with the first interval matching the final interval to model cyclic overnight capacity carryover.This assumption supports long-run consistency in the steady-state day.
- Recharge network: Recharge subnetworks between mobility layers spatially assign charging demand and redistribute fleet capacity between service and charging states.Charging is modeled as a discrete transition phase linking interval t to interval t+1 while preserving fleet-flow conservation.
- Mobility service subnetwork: The mobility service network assigns OD demand through a shared service structure containing MOD subnetworks and a composite out-of-platform option.Each MOD fleet has its own service nodes and links, connected to physical demand through access and egress links.
- Mobility service subnetwork: MOD access links use store-and-forward queues whose chosen capacities limit inflow and represent pickup delays from supply–demand imbalance.Operators choose access-link service capacity in each interval.
- Recharge network: Recharge flows either pass through charging nodes or move directly between service nodes, transferring and conserving average fleet capacity across consecutive intervals.The subnetwork accommodates vehicles that require charging and vehicles that can be redistributed without charging.
Parameters and Input Variables
The model specifies inputs for demand, fleet availability, temporal propagation, network capacities, operating costs, and utility dispersion across mobility and recharge services. These parameters govern travel demand, temporal charging links, physical constraints, and stakeholder utilities.
- Demand and fleet inputs: q^t_s denotes travel demand for OD pair s in time interval t, while V_m denotes the total fleet available to operator m.These inputs define interval-specific demand and operator fleet limits.
- Temporal parameters: π^τ_t is the propagation coefficient connecting service demand in interval τ to interval t.It parameterizes the temporal relationship used to link mobility demand across intervals.
- Network capacities: v_l and h_i scale maximum physical capacities for access link l and node i, respectively.These parameters define capacity limits in the mobility service network.
- Operating costs: d_l and c_l specify link length and operating cost per unit distance, while g_l and c_i specify capacity costs on access links and nodes.Together they represent distance-based and capacity-based operating costs.
- Network representation: δ_il is the node-link incidence indicator, taking values 1 for entering links, −1 for leaving links, and 0 otherwise.It encodes network connectivity for flow relationships.
- Utility weights: w_o, w_sd, w_uc, w_oc, w_rd, w_ocr, and w_rs weight mobility, recharge, traveler, operator, and station utility or dispersion terms.These parameters control the relative contributions of service and recharge utilities in the model.
Decision Variables
The model uses upper-level service prices and lower-level flow, capacity, fleet, charging, and redistribution variables to represent a three-sided mobility–energy assignment game. Its lower level combines mobility and recharge subnetworks, with temporal charging-demand coupling and a decomposition-based solution approach.
- Model variables: Service link pricing p^t_l is an upper-level variable, while flow, service capacity, deployed fleet, charging capacity, and redistribution are lower-level variables.The listed lower-level variables include x^t_s,l, z^t_l, μ^t_i, u^t_i, and r^t,m_ij.
- Integrated lower level: The lower-level model jointly represents mobility assignment, mobility-service capacity allocation, fleet redistribution, charging-capacity allocation, and charging assignment.These decisions are made by three-sided market participants through stochastic coalition choice in the PURC framework.
- Fleet accounting: The total fleet includes cruising, idling, and charging vehicles, while active service fleet reflects demand and their gap measures redundancy from mismatches or charging.Fleet constraints also maintain a buffer for vehicles unavailable because of charging.
- Temporal coupling: An empirical affine function links mobility demand across intervals to charging demand, with propagation and unavailability coefficients governing intertemporal effects.The formulation distinguishes ordinary consecutive intervals from the cyclic transition after the final interval.
- Recharge subproblem: The recharge and redistribution subproblem manages fleet repositioning and charging between consecutive service intervals, with charging demand derived from redistribution requirements.Its operator–energy-provider assignment has a PURC flow-assignment interpretation under fixed mobility-service decisions.
- Solution structure: Simultaneous solution of the recharge variables produces a nonconvex QPQC, so the model uses sequential decomposition whose solutions are ε-optimal under a marginal-utility condition.The decomposition separates service assignment from recharge and redistribution, while the resulting optimality gap can be bounded under the stated condition.
3. Numerical experiments
The numerical experiments validate the lower-level heuristic and evaluate the bilevel framework using synthetic demand and a monopolistic setup with one MOD provider and one charging provider.
- Experimental design: The experiments first benchmark exact solutions against decomposition and AM heuristics using a synthetic network and simulated demand patterns.They assess computational efficiency and accuracy before conducting bilevel framework experiments.
- Experimental design: The experimental setup assumes a monopolistic market with a single MOD service provider and one charging provider.The supplied passage identifies this as the current experimental market structure.
3.1. Evaluation of the proposed heuristic
The proposed decomposition and AM heuristics solve the lower-level model by alternating recharge-routing and redistribution-flow optimization, achieving near-exact results with much shorter computation time. The evaluation compares these methods with a direct exact-solution approach under common parameter settings.
- Experimental design: The evaluation benchmarks the integrated lower-level model against the proposed decomposition and AM heuristics under identical parameter settings.The direct Gurobi solve uses a 10-minute time limit, with the recharge-objective weight set to wo = 0.01.
- Convergence: 10^-4 stopping tolerance is reached within 8 iterations, with most objective improvement occurring during the first three iterations.The stage-2 objective then stabilizes, indicating good numerical convergence behavior.
- Solution quality: 455.80 objective value is obtained by the decomposition and AM heuristics versus 455.91 for the benchmark solution within the 10-minute timeframe.The comparison evaluates the proposed approach against the exact-solution benchmark.
- Computational efficiency: 13.2 seconds replaces 600 seconds of computation, while the proposed approach produces near-exact solutions with substantially improved computational efficiency.The reported runtime reduction rate is 2.2% of the benchmark runtime.
3.2. Case study to illustrative analytical insights from model
The case study shows that charging capacity, fleet size, operating costs, and dynamic pricing jointly determine system efficiency and profitability. Sensitivity analyses identify capacity and fleet thresholds, spatial bottlenecks, and distinct peak- versus off-peak pricing roles.
- Computational validation: The decomposition heuristic reaches an objective within 0.01% of global optimality in 0.3 seconds, compared with 4 seconds for exact Gurobi optimization.The comparison validates the heuristic for the moderate-scale test instances.
- Baseline scenario: The baseline achieves an objective value of 1,463.20 with 1,473 active vehicles out of a 1,600-vehicle fleet.The remaining vehicles are buffered for charging or idle because of demand-supply mismatch.
- Spatial utilization: Access links at nodes 105, 106, and 102 become critical bottlenecks, while node 110 is heavily underused.Node 105 reaches 100% utilization in periods 1 and 2, node 106 reaches 100% in period 1, and node 102 reaches full capacity by the final period.
- Charging capacity sensitivity: Charging capacity has a threshold at 280 units per station: above it, the objective remains near 1,463 and the active fleet near 1,474 vehicles.Below the threshold, charging constraints reduce fleet availability and increase disutility from unmet MOD demand and inefficient OOP routing.
- Charging capacity sensitivity: Lower charging capacity concentrates the remaining fleet at high-demand locations, with node 105 retaining full utilization while other access links decline sharply.In period 3, utilization at nodes 106 and 110 drops near zero as capacity decreases.
- Network dispersion sensitivity: Increasing the dispersion weight raises the objective linearly from 1,301 at wsd = 0.5 to 1,758 at wsd = 2.0.The active fleet remains relatively stable between 1,453 and 1,500 vehicles, while broader route dispersion increases system disutility.
- Fleet-size sensitivity: Endogenous operating costs produce concave profit growth, with total profit peaking at approximately $55,800 before declining as fleet size increases.Profit plateaus between 650 and 750 vehicles, while passenger demand is fully met at 750 vehicles and oversupply begins beyond that level.
3.3. Bilevel optimization: heterogeneous MOD fleets
The heterogeneous-fleet analysis compares operator-specific and platform-wide distance pricing, showing that pooled single-fleet operation yields the highest profit while operator-specific pricing outperforms a unified fare in the two-fleet setting.
- Pricing strategies: Operator-specific pricing decreases profit by 4.16% relative to the single-fleet benchmark, while platform-wide pricing decreases it by 5.14%.The comparison uses the two-fleet configuration against the previous single-fleet case with total fleet size 1000.
- Pricing strategies: $53,404.41 profit under operator-wide pricing exceeds $52,857.06 under platform-wide pricing, while active fleet size falls from 898 to 708 vehicles.The active fleet reduction is 190 vehicles, or 21.2%.
- Pricing strategies: Operator-wide pricing differentiates fees across fleets, with operator 2 charging more than operator 1 in all three periods.Platform-wide pricing compresses cross-operator variation into one average price.
- Fleet structure: Pooling vehicles improves capacity allocation because operator boundaries otherwise require additional capacity in overlapping service areas.The single-fleet scenario uses lower optimal prices and achieves higher total profit, mainly because pooling reduces capacity allocation costs.
4. Discussion and Conclusion
The discussion finds that integrated mobility–energy modeling exposes operational bottlenecks, fleet-size trade-offs, narrow pricing flexibility, and computationally efficient solution strategies. It also identifies missing smart-grid and provider heterogeneity features as important scope boundaries.
- Operational findings: Localized access bottlenecks and charging-resource differences can constrain equilibrium outcomes even when total fleet capacity meets demand.Both charging stations remain heavily used without full saturation, and the cheaper station attracts higher load.
- Fleet sizing: Profitability peaks near the upper bound of the supply-constrained fleet scenario rather than under unlimited fleet expansion.Beyond effective demand absorption, idle and repositioning burdens increase, utilization falls, and total profit declines.
- Dynamic pricing: Optimal prices remain within a narrow band, with high-demand prices declining as fleet size expands and low-demand prices responding more to transitional conditions and charging needs.Sharp price surges would likely suppress ridership more than improve profitability.
- Computational performance: The decomposition and AM heuristic reaches a near-optimal solution with negligible error and substantially shorter runtime for the moderate-size instance.The integrated lower-level problem can also be solved directly to global optimality using commercial solvers.
- Limitations: The model omits nonlinear service–charging relationships, dynamic power-grid constraints, time-based electricity-price fluctuations, and heterogeneous energy providers.Future extensions include localized voltage stability, varying charging capabilities, and more explicit charging-provider decisions.
Appendix A
The appendix lists passenger-demand data by origin–destination pair and time period alongside the integrated eMaaS model’s sets, parameters, and variables.
- Passenger demand: Table A1 organizes passenger demand by OD pair and time period.
- Model notation: Table A2 lists the sets, parameters, and variables used in the integrated eMaaS model.