Electric Autonomous Mobility-on-Demand (EAMoD)

• EAMoD is a coupled mobility–energy system that integrates electric vehicles with routing, dispatch, rebalancing, and battery charging under state-of-charge constraints.
• It employs multi-layer network models and canonical optimization frameworks (e.g., MILP, LP, DAG formulations) to jointly address traffic flow, charging logistics, and infrastructure siting.
• Real-time control strategies leverage hybrid optimization–learning methods and robustness techniques to enhance profit, reduce delays, and ensure service fairness.

Electric Autonomous Mobility-on-Demand (EAMoD, also written E-AMoD) denotes an autonomous mobility-on-demand system in which electric autonomous vehicles provide on-demand mobility, so that fleet operation must jointly manage passenger pickup and drop-off, vehicle rebalancing, and battery charging under state-of-charge (SoC) constraints. In the recent literature, EAMoD is treated not as a simple electrified variant of AMoD, but as a coupled mobility–energy system whose decision space includes routing, dispatch, rebalancing, charging, charger siting, vehicle design, ride-pooling, and, in some formulations, vehicle-to-grid (V2G) or backup-power service (He et al., 2023, Tan et al., 2018, Singhal et al., 2023).

1. System architecture and operational scope

A recurrent systems view models EAMoD within a three-layer architecture. The upper layer is the electric power distribution network, including distributed energy resources and grid-edge assets; the middle layer is the EV network, consisting of battery-electric vehicle fleets, geographically distributed charging and/or battery-swapping stations, communication backhaul, and the digital platform orchestrating dispatch and charging; the bottom layer is the physical transportation network of roads, curbs, parking facilities, and traffic management systems. In this formulation, the middle layer is the interface between traffic flow and power flow, and the charging loads created by the fleet can also be treated as demand-side flexibility for the power system (Tan et al., 2018).

At the operational level, early zone-based EAMoD work places a local fog controller in each service zone. The fog controller receives customer requests, free-vehicle arrivals, SoC reports, and charger availability; classifies vehicles and customers into SoC or distance classes; solves a dispatching and charging optimization in real time; and communicates either dispatch orders or charging-slot assignments. This architecture was introduced precisely to address the two bottlenecks emphasized in that line of work, namely the computational and charging delays, and it supports partial charging, full charging, and direct service decisions within each zone (Belakaria et al., 2017, Belakaria et al., 2018).

Region-based balancing models use a closely related decomposition. A city is partitioned into $N$ regions, and the local state of region $i$ can include the number of vacant EAVs $V_i$, the number of low-battery EAVs $L_i$, predicted passenger demand $d_i$, the number of EAVs currently charging $ST_i$, the number of empty charging spots $ES_i$, and the total charging spots $SP_i$. This state is then used by control policies that dispatch vacant vehicles and low-battery vehicles across neighboring regions while accounting for charging utilization and supply–demand balance (He et al., 2023).

2. Mathematical abstractions and canonical optimization models

The dominant mathematical abstractions are multi-layer network flows, time-expanded graphs, and directed acyclic graphs (DAGs). In one formulation, the E-AMoD control problem is posed as a linear program over a space-charge-time graph, but the resulting optimization problem is too large for real-time implementation in realistic settings (Singhal et al., 2023). Mesoscopic planning models instead build time-invariant multi-layer networks in which each physical road node is replicated across discretized SoC layers, road arcs decrease SoC by a fixed quantum, charging arcs raise SoC, and geo-arcs model pickup and drop-off. This produces mixed-integer linear programs (MILPs) or linear programs (LPs) with explicit passenger-flow, rebalancing-flow, and charging-flow variables, together with charger-siting variables such as $c_i\in\{0,1\}$ or $\kappa_c\in\{0,1\}$ (Paparella et al., 2022, Paparella et al., 2024).

DAG-based formulations condense feasible request-to-request transitions. In the request-DAG and energy-DAG framework, decision variables include binary assignment arcs $i$0, charging-detour indicators $i$1, charging amounts $i$2, battery states $i$3, request-selection variables $i$4, and fleet-design variables such as battery capacity $i$5. Solar-electric variants use the same DAG idea but add solar energy harvested on each arc, $i$6, and allow $i$7 to be positive or negative so that vehicles can either draw energy from the grid or inject it through V2G operations (Paparella et al., 2023, Paparella et al., 2023).

Objective functions differ by planning horizon and operator criterion. Typical objectives include minimizing total vehicle-hours-traveled, minimizing total net cost, or maximizing operator profit. Representative formulations minimize

$i$8

or minimize a profit-equivalent cost containing amortized vehicle and battery cost, charging cost, and served-request revenue, or minimize charging and rebalancing cost subject to service constraints (Paparella et al., 2024, Paparella et al., 2023, Paparella et al., 2022).

Alternative state representations also exist. For outage-response settings, aggregate vehicle densities are modeled continuously over normalized state-of-energy $i$9 and time $V_i$0, with PDE state variables $V_i$1 for charging vehicles, $V_i$2 for idle vehicles, and $V_i$3 for discharging vehicles. This PDE system is then discretized and embedded in a large LP that co-optimizes mobility revenue, charging cost, and power-delivery revenue during outages (Sheppard et al., 2020).

Congestion-aware formulations add another layer of realism by embedding traffic congestion through a Bureau of Public Roads volume–delay function,

$V_i$4

Bang and Malikopoulos study routing and rebalancing with this macroscopic congestion model, then apply heuristic loop recovery and charging-aware rerouting to restore battery feasibility. In their reported small-network study, the congestion-aware quadratic program spreads flows so that $V_i$5, cutting average customer travel times by $V_i$6, while the charging-scheduling reroutes keep overall travel times $V_i$7 below the naïve baseline (Bang et al., 2021).

3. Real-time control, uncertainty, and learning-based methods

Real-time EAMoD control has increasingly shifted toward hybrid optimization–learning methods. A notable example presents a bi-level formulation in which a graph network-based reinforcement learning agent specifies a desired next state in the space-charge graph, while tractable linear programs are solved to best achieve that desired state while ensuring feasibility. On real-world data from San Francisco and New York City, this approach achieves up to $V_i$8 of the profits of the theoretically-optimal solution while achieving more than a $V_i$9 speedup in computational time, outperforms the best domain-specific heuristics with an increase in profits by up to $L_i$0, and exhibits zero-shot transfer on inter-city generalization and service area expansion (Singhal et al., 2023).

Robustness to demand and supply uncertainty is handled in two principal ways. The first is adversarial multi-agent reinforcement learning. In the REBAMA framework, adversarial agents perturb local states to model both EAV supply uncertainty and mobility-demand uncertainty, while region agents choose balancing actions. The shared reward combines charging utilization fairness and supply-demand fairness, and the policy is trained under a centralized-training, decentralized-execution actor-critic scheme. Relative to a non-robust MARL baseline, the reported improvements are $L_i$1 in reward, $L_i$2 in charging utilization fairness, and $L_i$3 in supply-demand fairness; relative to a robust-optimization baseline, the gains are $L_i$4, $L_i$5, and $L_i$6, respectively (He et al., 2023).

The second is distributionally robust optimization. In a data-driven DRO formulation, the operator minimizes the worst-case expected balancing cost under demand and EV-supply ambiguity sets characterized by moment conditions. The original infinite-dimensional problem is reformulated as a computationally tractable convex semidefinite/quadratic program. On Shenzhen e-taxi data, this solution reduces average total balancing cost by $L_i$7, and improves average mobility fairness and charging fairness by $L_i$8 and $L_i$9, respectively, compared with solutions that do not consider uncertainties; the reported runtime is approximately $d_i$0 s per hourly decision (He et al., 2022).

Large-scale predictive control now combines forecasting with stochastic and robust optimization. A recent combined stochastic and robust MPC framework uses spatio-temporal Bayesian neural network forecasts, a multi-stage stochastic MILP, robust constraints on energy consumption and travel times, and a tailored Nested Benders Decomposition. In San Francisco and Chicago, the combined approach reduces median passenger waiting times by up to $d_i$1, 95th-percentile delays by nearly $d_i$2, rebalancing distance by $d_i$3, and electricity costs by more than $d_i$4 relative to deterministic, reactive, and robust baselines (Jacobsen et al., 27 Aug 2025). A related real-time framework that explicitly incorporates power-system charging regulation models EAMoD operation as an MDP and solves it with piecewise linear approximate dynamic programming combined with MPC; on the reported Manhattan study, the proposed ADP+MPC achieves daily profit $d_i$5 k\$, QoS $d_i$6, and CPU $d_i$7 s per decision, while avoiding undervoltage with only $d_i$8 profit loss and $d_i$9 QoS loss under power coordination (Pan et al., 1 Oct 2025).

A consistent implication of these studies is that published “learning-based” EAMoD controllers are often not purely end-to-end; they typically retain explicit feasibility layers, convex subproblems, or MPC structure around the learned component (Singhal et al., 2023, Jacobsen et al., 27 Aug 2025, Pan et al., 1 Oct 2025).

4. Charging management, queueing structure, and fleet dimensioning

Charging management was first formalized through multi-class queueing models. In the multi-class charging and dispatching scheme, vehicles and customers are divided into $ST_i$0 classes, customer class $ST_i$1 must be served by a vehicle whose SoC class $ST_i$2, and an arriving vehicle of class $ST_i$3 either serves immediately with probability $ST_i$4 or partially charges to class $ST_i$5 with probability $ST_i$6. The effective service-ready arrival rate into class $ST_i$7 is

$ST_i$8

and, with sub-class service, the service arrival rate into the $ST_i$9-th customer queue becomes

$ES_i$0

Under M/M/1 stability, the expected response time is $ES_i$1, and the optimization objective is to minimize the maximum response time, equivalently maximize a common response-rate lower bound $ES_i$2 (Belakaria et al., 2018).

The empirical motivation for this queueing treatment is strong in the reported studies. The optimized multi-class scheme with sub-class service yields up to $ES_i$3 reduction in max response time versus the naïve “always partial-charge + same-class dispatch” policy, and $ES_i$4, $ES_i$5, $ES_i$6, and $ES_i$7 improvements over other benchmarks; when charging points $ES_i$8 drop, the sub-class dispatch model sustains up to $ES_i$9 better max response time (Belakaria et al., 2018). In the earlier stochastic linear program without sub-class service, the optimized policy reduces average response time by $SP_i$0 and maximum response time by $SP_i$1 compared to “always-charge,” and the stability analysis yields the class-design rule

$SP_i$2

for the number of classes (Belakaria et al., 2017).

Fleet-dimensioning work then asks for the smallest zone inflow $SP_i$3 needed to guarantee a target response-time bound $SP_i$4. A key lower bound is

$SP_i$5

together with a lower bound on the number of classes,

$SP_i$6

The optimization jointly selects $SP_i$7 and the dispatch/charging proportions $SP_i$8 so as to minimize the required vehicle inflow while preserving stability of the service and charging queues. Reported simulations indicate up to $SP_i$9–$c_i\in\{0,1\}$0 reduction in the required $c_i\in\{0,1\}$1 relative to “always-charge” and “equal-split” baselines under high-demand or tight-$c_i\in\{0,1\}$2 scenarios (Belakaria et al., 2018).

A distinct queueing-theoretic branch compares plug-in charging and battery swapping through a bi-level planning model. Plug-in stations are modeled as M/M/$c_i\in\{0,1\}$3/$c_i\in\{0,1\}$4 queues, whereas battery swapping is represented as a mixed open/closed network coupling EV waiting with battery circulation. The reported planning results differ sharply by technology regime: when choosing plug-in charging, increased charging speed transforms infrastructure from sparsely distributed large stations to densely distributed small stations, while enlarged battery capacity transforms the infrastructure from densely distributed small stations to sparsely distributed large stations; when choosing battery swapping, both increased charging speed and enlarged battery capacity lead to a smaller number of battery swapping stations (Gao et al., 2023).

5. Infrastructure siting, ride-pooling, and vehicle design

Joint charger siting and routing is a central planning problem because charging infrastructure changes the geometry of feasible empty flows. A mesoscopic MILP that optimizes routing jointly with charging infrastructure siting on an iso-energy graph can be solved in less than $c_i\in\{0,1\}$5 min, and in the Manhattan case study it yields up to a $c_i\in\{0,1\}$6 reduction in rebalancing energy relative to a highest-betweenness-centrality heuristic. The same study reports a clear diminishing-returns threshold: with optimal siting, benefits saturate at approximately $c_i\in\{0,1\}$7 stations/km$c_i\in\{0,1\}$8, and under the heuristic one needs at least $c_i\in\{0,1\}$9 as many stations to reach the same empty-mile energy cost (Paparella et al., 2022).

The same line of work also links infrastructure design to vehicle design. In the Manhattan battery-design comparison, the smallest, lightest vehicle uses approximately $\kappa_c\in\{0,1\}$0 less total energy than the largest-battery vehicle, even though its rebalancing energy alone is $\kappa_c\in\{0,1\}$1 higher because it must charge more frequently; optimal fleet sizes differ negligibly across the tested vehicle designs (Paparella et al., 2022). A related joint design-and-operation MILP for Manhattan finds that a fleet of small and light vehicles with battery capacity of $\kappa_c\in\{0,1\}$2 kWh only can strike the best trade-off in terms of battery degradation, fixed costs, and operational efficiency, with sampled subproblems yielding $\kappa_c\in\{0,1\}$3 kWh and $\kappa_c\in\{0,1\}$4 requests per vehicle (Paparella et al., 2023).

Ride-pooling introduces a different design trade-off. In the ride-pooling E-AMoD framework, requests are first matched offline into pooled requests subject to maximum waiting time $\kappa_c\in\{0,1\}$5, maximum detour $\kappa_c\in\{0,1\}$6, and seat-capacity $\kappa_c\in\{0,1\}$7, then routed in the same multi-commodity flow MILP. In the Manhattan taxi case study, jointly optimizing charging infrastructure placement decreases overall energy consumption and vehicle-hours-traveled by approximately $\kappa_c\in\{0,1\}$8 compared to a heuristic placement, but ride-pooling reduces those costs by up to $\kappa_c\in\{0,1\}$9, which is the dominant effect. Vehicle choice also becomes demand-regime dependent: the lightweight two-seater is most convenient for low-demand areas, while the heavier four-seater is most convenient for high-demand areas; in dense regimes above $i$00 req/hr with pooling up to vehicle capacity, the four-seater out-performs the two-seater despite lower per-km efficiency (Paparella et al., 2024).

These results collectively show that battery size, vehicle mass, seat capacity, and charger geography are not separable design knobs. Several case studies explicitly report that larger batteries are not automatically preferable from an operator perspective, because they raise mass, charging time, or fixed cost even when they reduce charging detours (Paparella et al., 2022, Paparella et al., 2023, Paparella et al., 2024).

6. Power-system coupling, economic operation, and market structure

EAMoD becomes a power-system problem as soon as charging loads are large enough to affect distribution or transmission constraints. In Orange County, a joint electric-AMoD/PDN optimization that couples time-expanded vehicle flows with a convex surrogate of unbalanced three-phase power flow shows that coordination between the fleet and the PDNs eliminates $i$01 of the overloads and $i$02 of the voltage drops that the electric AMoD fleet would cause in an uncoordinated setting; the detailed case study reports a $i$03 reduction in serious transformer overload events and a $i$04 reduction in total voltage-violation hours at a $i$05 increase in total social-planner cost (Estandia et al., 2019). In a transmission-level setting for Dallas–Fort Worth, lack of coordination between the AMoD system and the power network can cause a $i$06 increase in the price of electricity in Dallas–Fort Worth, whereas coordination could reduce electricity expenditure compared to the case where no cars are present and yield savings of up to \$147M/year; that work also shows that the social optimum can be enforced as a general equilibrium through locational marginal prices and a dual-decomposition coordination algorithm (Rossi et al., 2017).

The energy-service role of EAMoD extends beyond cost minimization. The AFoD concept treats autonomous electric vehicles as providers of charging flexibilities as a service on demand in energy systems, with four decisions requiring coordination: vehicle assignment, rebalancing, real-time charging/discharging schedules, and flexibility-service allocation (Tan et al., 2018). Revenue-stacking examples support this broader view. During large-scale power outages, a PDE-based dispatch model finds that supplying power is negligible in a moderate outage scenario but accounts for between \$i$072 million, amounting to between $i$08 and $i$09 more revenue than is earned serving mobility only, in an extreme scenario (Sheppard et al., 2020). Solar-electric EAMoD, in which vehicles also harvest solar energy through vehicle-mounted roofs and may perform V2G, is reported to reduce the total cost of operation by $i$10–$i$11 compared to an electric-only counterpart, and larger batteries can even accrue profits through larger V2G trades (Paparella et al., 2023).

Economic analyses also address internal operator trade-offs and market structure. Smart-charging analysis under a random-price model shows that investing in larger battery capacities and operating more vehicles for rebalancing reduces the charging costs while increasing the fleet operational costs, so the operator faces an explicit battery-size versus operating-cost trade-off (Turan et al., 2019). Competition models show that, in a duopoly, equilibrium ride prices have to be symmetric between firms; on Manhattan data at $i$12, the empirical average price ratio $i$13 is $i$14, the aggregate-demand ratio $i$15 is $i$16, the profit-per-firm ratio $i$17 is $i$18, and the consumer-surplus ratio is $i$19, quantifying how competition lowers prices and raises consumer surplus relative to monopoly (Turan et al., 2020).

Taken together, the literature characterizes EAMoD as a cyber-physical optimization domain in which mobility service, battery dynamics, charging infrastructure, congestion, uncertainty, and electric-power constraints are mathematically inseparable. The strongest common result across formulations is not a single preferred algorithmic paradigm, but the consistent need for co-optimization across transportation and energy layers (Tan et al., 2018, Estandia et al., 2019, Singhal et al., 2023).