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Optimal Pricing to Manage Electric Vehicles in Coupled Power and Transportation Networks
Mahnoosh Alizadeh, Hoi-To Wai, Mainak Chowdhury, Andrea Goldsmith, Anna Scaglione, Tara Javidi
TL;DR
Large EV populations couple traffic congestion, charging decisions, and power-grid prices, creating a system-level coordination problem. The paper models individual decisions on an extended graph, studies collective effects, and develops collaborative pricing and reserve-capacity approaches; numerical examples show adverse effects when interdependencies are ignored.
Problem
Large-scale EV adoption couples power and transportation networks, while ignoring this interconnection can destabilize electricity pricing, power delivery, and traffic distribution.
Method
The paper models each driver’s joint charging and routing decision as a shortest path on an extended transportation graph with virtual charging arcs, then studies coordinated operator pricing and reserve capacity.
Results
The study finds that operator collaboration can guide EVs toward socially optimal operation, while disjoint pricing can produce infeasible decisions and unsafe load increases in congested grid locations.
Takeaways & Limitations
Correct pricing requires accounting for power–transportation interdependence; otherwise, large EV populations can create adverse operational effects.
Abstract
from arXiv · showhide
We study the system-level effects of the introduction of large populations of Electric Vehicles on the power and transportation networks. We assume that each EV owner solves a decision problem to pick a cost-minimizing charge and travel plan. This individual decision takes into account traffic congestion in the transportation network, affecting travel times, as well as as congestion in the power grid, resulting in spatial variations in electricity prices for battery charging. We show that this decision problem is equivalent to finding the shortest path on an "extended" transportation graph, with virtual arcs that represent charging options. Using this extended graph, we study the collective effects of a large number of EV owners individually solving this path planning problem. We propose a scheme in which independent power and transportation system operators can collaborate to manage each network towards a socially optimum operating point while keeping the operational data of each system private. We further study the optimal reserve capacity requirements for pricing in the absence of such collaboration. We showcase numerically that a lack of attention to interdependencies between the two infrastructures can have adverse operational effects.
I. INTRODUCTION: A TALE OF TWO NETWORKS
Large-scale EV adoption couples transportation and power networks because charging and travel decisions jointly affect congestion, electricity prices, and system operation. The paper models these decisions and develops coordinated pricing approaches for reliable, socially optimal outcomes.
- I. INTRODUCTION: A TALE OF TWO NETWORKS: Large-scale EV adoption couples transportation and power networks, so ignoring their interconnection can destabilize electricity pricing, power delivery, and traffic distribution.The paper therefore proposes controls that acknowledge interdependence between the infrastructures.
- I. INTRODUCTION: A TALE OF TWO NETWORKS: The paper represents each driver’s joint charge-and-path decision as a shortest-path problem on an extended transportation graph containing virtual charging arcs.The extended graph integrates individual route and charge decisions into system-level analysis.
- I. INTRODUCTION: A TALE OF TWO NETWORKS: Independent power and transportation operators can collaborate on electricity prices, charging-station mark-ups, and road tolls while keeping each system’s operational data private.The collaboration is presented as necessary for correct price design.
- I. INTRODUCTION: A TALE OF TWO NETWORKS: Related EV transportation studies model energy-constrained routing, but generally omit power-grid interactions or treat electricity prices as fixed rather than designing them.The paper positions its contribution around electricity price design for coupled infrastructures.
- I. INTRODUCTION: A TALE OF TWO NETWORKS: An individual EV driver chooses both a travel path and charging locations and amounts, with costs shaped by transportation congestion and charging conditions.The decision problem includes route selection, charging choices, and the associated travel and charging costs.
B. Charging Costs
The paper models EV charging and travel as a joint cost-minimization problem on an extended transportation graph. Virtual arcs encode charging choices, while energy-feasibility constraints ensure the battery remains within operational limits.
- Charging costs: Charging costs combine location-specific electricity prices, one-time station plug-in fees, and inconvenience from charging time and congestion.The inconvenience cost reflects fast-charging-station waiting and charging time, with a separate treatment for charging at the trip origin.
- Extended graph: Virtual arcs represent discrete charging amounts and convert node charging decisions into shortest-path choices.Each virtual arc corresponds to a permitted charge amount and changes the vehicle’s energy level accordingly.
- Path optimization: The transformed EV problem is an Energy-aware Shortest Path Problem over energy-feasible paths connecting the origin and destination.Arc costs include travel-time costs and monetary charges such as electricity bills and tolls.
- Path optimization: Energy-feasibility requires that battery charge stays between zero and capacity throughout the path, and feasible paths can be computed offline.This feasibility test is independent of network congestion represented through arc flows.
- Computational scope: The paper uses dynamic programming in principle, while its small numerical experiment enumerates loop-free energy-feasible paths by brute force.The authors note that efficient ESPP solution methods are beyond the paper’s scope and may matter in more realistic models.
IV. SYSTEM LEVEL MODEL
At the system level, EV route and charging decisions generate coupled traffic and power flows. The extended graph supports joint optimization of vehicle flows, electricity prices, and road tolls by separate system operators.
- System representation: Adding virtual arcs at potential origins and fast-charging stations enables aggregate EV traffic and energy loads to be represented as network flows.The extended graph connects individual driver decisions to system-level control of both infrastructures.
- System representation: Vehicle arc flows, electricity prices, and tolls become jointly optimized system variables rather than externally imposed quantities.This formulation treats the aggregate effects of individual charging and route choices as part of the control problem.
- Social optimization: A centralized controller could minimize total transportation congestion and generation costs while enforcing transportation and power-system constraints.The power constraints include supply-demand balance and transmission-line limits.
- Operator coordination: In practice, the IPSO and ITSO optimize their respective networks separately, keep operational data private, and influence individuals through prices.This institutional separation motivates studying coordination and pricing strategies across the coupled systems.
A. The ITSO’s Charge and Traffic Assignment Problem
The ITSO models EV route and charging choices as flows over feasible paths in the extended graph. Its charge and traffic assignment problem selects path flows subject to demand conservation and network-flow relationships.
- Flow formulation: Drivers are grouped into classes sharing origins and destinations, with each class assigned a set of feasible extended-graph paths.A class contains either EVs or internal-combustion vehicles, and EV paths are energy-feasible.
- Flow formulation: Path-flow variables specify how much demand from each class chooses each path, while conservation requires total path flow to equal class demand.The demand rate is treated as deterministic and given.
- Flow formulation: Arc flows are obtained by aggregating class path flows through arc-path incidence matrices.The incidence indicator is one when an arc belongs to a path and zero otherwise.
- ITSO optimization: Virtual-arc flows induce power-grid charging demand, linking transportation assignment variables to electricity-system load.The charging-demand vector is formed from flow on virtual arcs.
- ITSO optimization: The ITSO can solve a modified static traffic assignment problem, called the charge and traffic assignment problem, on the extended graph.This formulation determines aggregate route and charging patterns through path-flow decisions.
B. The IPSO’s Economic Dispatch Problem
The IPSO selects generation dispatch to serve EV charging and baseload while respecting generator, balance, and transmission constraints. Its objective is the cheapest feasible generation mix, but social alignment requires pricing mechanisms.
- Power-system feasibility: Generation at each grid node must respect capacity limits, while total generation must balance EV charging demand and baseload.The model uses one merged generator at each node for brevity.
- Power-system feasibility: Transmission-line flows are constrained under the DC approximation by power-transfer relationships and bidirectional line limits.The vector of line limits specifies the allowable flow on each transmission line.
- Economic dispatch: The IPSO solves an economic dispatch problem that minimizes strongly convex generation costs over feasible dispatches.The formulation assumes at least one feasible generation mix exists for every possible load profile.
- Pricing motivation: Socially optimal traffic and generation schedules need not minimize the costs faced by individual drivers or generators.The paper therefore motivates pricing as a distributed, incentive-compatible alignment mechanism.
C. Pricing Mechanism for Electric Power
The power-pricing mechanism uses locational marginal prices derived from generator costs and network constraints, while charging demand feeds back into those prices. Tolls on the extended graph align individual EV decisions with the social optimum.
- Power pricing: Locational marginal prices arise from the KKT stationarity condition for power balance and line-flow constraints.The price vector is p = γ1 + H^Tµ, with γ and µ associated with balance and line-flow constraints.
- Power–transport coupling: Electricity prices affect charging demand, which changes grid loading and feeds back into the price vector.The same price vector p enters the transportation optimization and influences charging demand d in the power problem.
- User equilibrium: Without transportation tolls, aggregate EV flows follow user equilibrium and generally need not equal the socially optimal flow.The no-toll case sets θa = 0 on all arcs of the extended graph.
- Marginal congestion pricing: Marginal congestion tolls make selfish EV route-and-charge decisions equivalent to the optimal social decisions when each toll equals the externality imposed on other users.The theorem applies this condition to every arc of the extended graph.
- Charging-station pricing: Charging-station entrance tolls act as congestion mark-ups that are higher where station capacity or route advantages make charging locations more contested.These plug-in fees capture externalities from limited charging-station capacity.
V. INTERACTIVE NETWORK OPERATION
Separate operation of the power and transportation networks ignores price-responsive charging demand and can produce unstable prices. The paper therefore motivates collaborative schemes that account for their interdependence.
- Interactive network operation: The status quo operates the power and transportation networks separately, despite their operational interconnection.The studied interaction schemes are summarized in Fig. 5.
- A. Greedy pricing: The IPSO designs LMPs from historical charging patterns, while the ITSO treats electricity prices as fixed when setting transportation tolls.The disjoint model does not allow charging demand to shift between grid buses in response to posted prices.
- A. Greedy pricing: Under greedy pricing, congestion and electricity prices could oscillate indefinitely.The paper substantiates this claim with a numerical example and links it to welfare loss from suboptimal operation.
B. Collaborative pricing
Collaborative pricing uses dual decomposition to coordinate the IPSO and ITSO while separating their optimization subproblems. With a sufficiently small step size, the method converges to efficient market-clearing prices and operating decisions.
- B. Collaborative pricing: An ex-ante IPSO–ITSO collaboration can post an efficient market-clearing LMP using a dual decomposition algorithm.The approach separates the coupled optimization through Lagrange multipliers for power balance and line-flow constraints.
- B. Collaborative pricing: The coupling constraints jointly involve transportation flow and generation, so dual decomposition lets the IPSO and ITSO solve separate subproblems.The ITSO optimizes extended-graph flow using iteration-specific electricity prices, while the IPSO solves its own power subproblem.
- B. Collaborative pricing: With a small enough step size, dual decomposition converges to the solution of the coupled optimization problem.The iterative scheme updates the balance and congestion components of the LMP through the dual variables.
- B. Collaborative pricing: At convergence, the price p = γ⋆1 + H^Tµ⋆ clears the market, with generator outputs and system flows equal to g⋆ and λ⋆.These values are the optimal generation and flow variables of the coupled problem.
C. Optimal reserve capacity for trial-and-error pricing
Without ex-ante collaboration, trial-and-error price updates may require reserve capacity because Lagrangian relaxation can temporarily violate power-system feasibility. The paper formulates reserve procurement as a robust optimization problem and notes that practical approximation remains necessary.
- C. Optimal reserve capacity for trial-and-error pricing: Trial-and-error pricing updates prices from observed charging demand rather than requiring ex-ante transportation-demand forecasts.This approach assumes network flow reaches a new equilibrium faster than electricity costs or travel demands change.
- C. Optimal reserve capacity for trial-and-error pricing: Lagrangian relaxation most likely violates primal balance and line-flow feasibility during convergence, so reserve generation is used to handle these contingencies.The resulting violations are treated as threats to reliable grid operation.
- C. Optimal reserve capacity for trial-and-error pricing: The IPSO must procure reserve capacity in advance so dispatched generation can correct demand–supply imbalances after price updates.Reserve y adjusts generator output subject to −r ⪯ y ⪯ r.
- C. Optimal reserve capacity for trial-and-error pricing: Finite dual Lipschitz constants and bounded optimal dual variables support the feasibility bounds used to size reserve capacity.The paper attributes Lipschitz finiteness to strong convexity and bounded dual variables to convexity, linear constraints, and finite optimal value.
- C. Optimal reserve capacity for trial-and-error pricing: Optimal reserve procurement is the cheapest nodal reserve combination that restores balance and flow constraints under possible feasibility violations.The formulation is a robust optimization problem over allowable violation sets and reserve prices.
- C. Optimal reserve capacity for trial-and-error pricing: The reserve constraint is convex piecewise linear in reserve capacity because its worst case occurs at extreme points of the relevant polyhedra.The paper uses this structure to characterize the reserve-sizing problem.
- C. Optimal reserve capacity for trial-and-error pricing: Computing the exact worst-case reserve requirement is non-trivial because the extreme points are generally unknown, so the numerical study uses sample/scenario approximation.The paper identifies approximation algorithms as outside its scope.
VI. NUMERICAL EXAMPLES
The numerical examples model coupled power and transportation networks and compare cooperative optimization with myopic pricing as EV demand increases. Cooperation converges toward feasibility, whereas disjoint pricing can oscillate and produce infeasible grid decisions.
- Numerical setup: The study assumes publicly owned charging stations selling electricity to EV drivers at wholesale prices.The transportation case uses the network from Davis to San Jose, while the power case extends the IEEE 9-bus model with additional EV charging load buses.
- Numerical setup: Each fast-charging station supplies 1 kWh every 5 minutes, with charging options of 0, 1, 2, or 3 kWh.EVs consume 1 kWh per 25 miles, have 6 kWh batteries, and start in Davis with 4 kWh.
- Cooperative optimization: As EV demand increases, the socially optimal traffic pattern reroutes vehicles through Winters and other paths as the coupled networks become more congested.The example compares electricity consumed at sites and traffic leaving sites across EV totals per epoch.
- Myopic pricing: Under myopic pricing, equal electricity prices make the transportation operator minimize travel time, creating uneven power-network energy consumption before prices trigger reassignment.The IPSO subsequently lowers the Winters price and raises prices at Fairfield, Fremont, and Mountain View.
- Myopic pricing: Large EV populations can make the disjoint optimization infeasible because greedy pricing misrepresents drivers’ responses and increases load at congested grid locations.The resulting load may require shedding to keep transmission lines within limits.
- Cooperative optimization: The dual decomposition algorithm converges in approximately 100 iterations to an approximately feasible solution, with actual infeasibility following an O(1/k) decay.The experiment fixes the EV population at 2.5 × 10^4 per epoch.
VII. CONCLUSIONS AND FUTURE WORK
The paper concludes that EV adoption creates interdependence between power and transportation networks, while IPSO–ITSO collaboration can achieve socially optimal traffic and energy outcomes. It also identifies static, wholesale-market assumptions and privately owned charging facilities as important boundaries for future work.
- Large-scale EV integration creates interdependence between power and transportation networks in the studied static setting.
- IPSO–ITSO collaboration can guide EVs toward a socially optimal traffic pattern and energy footprint.
- Without direct operator collaboration, the paper analyzes the reserve capacity requirements needed to operate the grid.
- The results rely on an ideal static setting without retail markets, while private charging ownership and hourly dynamics remain for future analysis.Private ownership may limit the IPSO’s ability to impose taxes and maximize social welfare; dynamic traffic assignment also introduces non-convexities.
APPENDIX A MPEC FOR FINDING (γ⋆, µ⋆) IN (30)
The appendix uses an MPEC to bound optimal dual variables across possible EV demand profiles. Its lower-level problem computes optimal power dispatch and the associated dual variables for each demand profile.
- An MPEC enumerates possible EV demand values and their corresponding optimal dual variables (γ*, µ*).
- The demand profile is constrained between dmin and dmax, with regularization parameter δ applied to power-generation constraints.
- The lower-level minimization finds optimal dispatch g and the associated dual variables for each admissible demand profile.The dual variables correspond to the power-balance and network-constraint conditions shown in the formulation.