PaperlayerView exact revision on arXiv

Source-linked AI summary

Stochastic Dynamic Pricing for EV Charging Stations with Renewables Integration and Energy Storage

Chao Luo, Yih-Fang Huang, Vijay Gupta

arXiv:1801.02128v1eess.SPecon.EMmath.OC

TL;DR

The paper addresses how EV charging providers can price charging and manage electricity under demand, renewable-generation, and wholesale-price uncertainties while balancing profitability, customer satisfaction, and grid impact. It formulates a multi-objective model with a grid-impact metric and profit safeguard, then applies stochastic dynamic programming and greedy benchmarking to derive policies. In simulation, SDP achieves up to 7% profit gain over greedy, while pricing reshapes spatial-temporal demand to reduce grid impact.

  • Problem

    EV charging providers need pricing and energy-management guidance that balances profitability, customer satisfaction, and grid impact under multiple uncertainties.

  • Method

    The paper formulates a multi-objective optimization model with a fast grid-impact metric and profit safeguard, and uses stochastic dynamic programming alongside a greedy benchmark to derive pricing and procurement policies.

  • Results

    Up to 7% profit gain over the greedy algorithm is achieved by SDP in simulation, while pricing reshapes spatial-temporal charging demand to reduce grid impact.

  • Takeaways & Limitations

    Charging providers can use pricing signals, renewable energy, storage, and forward-looking optimization to balance objectives and manage grid impact under uncertainty.

Abstract

from arXiv · show

This paper studies the problem of stochastic dynamic pricing and energy management policy for electric vehicle (EV) charging service providers. In the presence of renewable energy integration and energy storage system, EV charging service providers must deal with multiple uncertainties --- charging demand volatility, inherent intermittency of renewable energy generation, and wholesale electricity price fluctuation. The motivation behind our work is to offer guidelines for charging service providers to determine proper charging prices and manage electricity to balance the competing objectives of improving profitability, enhancing customer satisfaction, and reducing impact on power grid in spite of these uncertainties. We propose a new metric to assess the impact on power grid without solving complete power flow equations. To protect service providers from severe financial losses, a safeguard of profit is incorporated in the model. Two algorithms --- stochastic dynamic programming (SDP) algorithm and greedy algorithm (benchmark algorithm) --- are applied to derive the pricing and electricity procurement policy. A Pareto front of the multiobjective optimization is derived. Simulation results show that using SDP algorithm can achieve up to 7% profit gain over using greedy algorithm. Additionally, we observe that the charging service provider is able to reshape spatial-temporal charging demands to reduce the impact on power grid via pricing signals.

NOMENCLATURE

The nomenclature defines planning, storage, market, demand, pricing, objective, and power-network variables used throughout the paper.

  • K denotes the total number of planning horizons, while E denotes electricity storage capacity.
  • N and M denote the numbers of buses and PQ buses, η_s is unit storage cost, α and ω are satisfaction-formula shape parameters, and J(I_k, u_k) is maximum expected aggregated utility.
  • p_kj and d_kj denote the charging price and charging demand of station s_j at horizon k, while c_k denotes real-time wholesale electricity price.
  • I_k, W_k, W_min, G_k, F_k, o_k, and u_k represent beginning storage, profit, profit safeguard threshold, customer satisfaction, grid impact, purchased electricity, and renewable energy.
  • Π_k is total utility, λ_1, λ_2, and λ_3 are objective weights, and γ_i,j is the price elasticity coefficient.
  • P_i, Q_i, v_i, and δ_i describe bus active power, reactive power, voltage magnitude, and voltage phase; G_ik and B_ik are admittance-matrix conductance and susceptance.

I. INTRODUCTION

The paper addresses the lack of viable public-charging pricing and energy-management models by jointly optimizing profitability, customer satisfaction, and grid impact under uncertainty. It introduces a multi-objective framework, a fast grid-impact metric, power sensitivities, and a profit safeguard to guide pricing and procurement decisions.

  • Public EV charging lacks a viable and profitable pricing and energy-management model that jointly addresses provider and system objectives under uncertainty.
  • Prior studies addressed revenue, charging-load management, game-theoretic interactions, demand shifting, or service quality, but omitted combinations of customer satisfaction, grid impact, renewables, and demand volatility.
  • The proposed framework jointly optimizes profit, customer satisfaction, and reduced power-grid impact while modeling charging-demand volatility, renewable intermittency, and wholesale-price fluctuation.
  • The multi-objective formulation provides tradeoff insights and guidance for setting charging prices to balance charging demand across the power system.
  • Newton’s method yields a fast grid-impact metric without solving complete nonlinear power-flow equations, and the metric can also analyze other electric loads.
  • Active- and reactive-power sensitivities guide charging-station placement, while a profit safeguard warns when profit may reach a dangerous threshold to help avoid severe financial losses.

II. PROBLEM FORMULATION

The provider mediates between wholesale electricity markets and EV customers while coordinating station pricing, electricity procurement, renewable generation, and storage. The formulation balances profit, customer satisfaction, and grid impact across planning horizons.

  • II. PROBLEM FORMULATION: The provider operates multiple charging stations, procures wholesale electricity, resells it to EVs, and stores harvested renewable energy.The model assumes renewable energy may come from solar or wind and be saved in an energy storage system.
  • II. PROBLEM FORMULATION: The formulation treats profit, customer satisfaction, and grid impact as competing objectives over multiple horizons.The provider must jointly determine pricing and electricity procurement decisions under these objectives.
  • A. Profit of Charging Service Provider: The day is divided into K = 24 hourly planning horizons, with prices and electricity procurement chosen at each horizon.Prices may differ across charging stations, while wholesale real-time prices govern procurement costs.
  • A. Profit of Charging Service Provider: Profit combines charging revenue, wholesale procurement cost, and storage cost, while storage dynamics include charging and discharging efficiencies and process noise.The process noise is Gaussian with zero mean and variance σ2.
  • B. Customer Satisfaction: Customer satisfaction is modeled as a function of aggregate charging demand relative to storage capacity, with α and ω controlling the function’s shape.The satisfaction function is calibrated between 0 and 1 and is non-decreasing with diminishing marginal growth as demand approaches capacity E.
  • B. Customer Satisfaction: Figure 2 illustrates sample customer satisfaction functions for storage capacity E = 200 under different shape-parameter choices.The curves represent satisfaction increasing with aggregate charging demand while becoming saturated near storage capacity.

C. Impact on Power Grid

The paper models grid impact through voltage-related effects and formulates a multiobjective, stochastic optimization problem. It estimates spatial-temporal demand using price-responsive regressions updated recursively with recent observations.

  • C. Impact on Power Grid: The grid-impact objective focuses on voltage magnitude and phase variation, while transmission constraints are handled by a higher-level aggregator or ISO/RTO.The impact metric Fk is designed to increase with charging demand.
  • D. Multi-objective Optimization Framework: The optimization balances maximizing profit and customer satisfaction against minimizing power-grid impact across multiple planning horizons.Decision vector Xk contains station-specific prices and electricity procurement, and expectations account for stochastic outcomes.
  • D. Multi-objective Optimization Framework: An adaptive weighted-sum approach first approximates the Pareto front and then refines its non-convex regions by reducing mesh size recursively.Weight vectors generate convex Pareto optima, while refinement searches for the non-convex portion.
  • D. Multi-objective Optimization Framework: The framework identifies demand estimation, grid-impact measurement, profit safeguarding, and efficient optimization as central challenges.These challenges motivate the methods developed in the following sections.
  • III. CHARGING DEMAND ESTIMATION: Online linear regression models station-level charging demand as a function of current and cross-station prices plus random error.The coefficients include self-price and cross-price elasticities, allowing prices at one station to influence demand at another.
  • III. CHARGING DEMAND ESTIMATION: Recursive least squares continually updates price-elasticity estimates, using a forgetting factor to emphasize recent demand trends.Separate regressions capture geographically distinct stations and changing spatial-temporal demand patterns.

IV. IMPACT ON POWER GRID FROM EV CHARGING

The paper assesses EV charging’s grid impact through network-wide voltage variation and derives a 2-norm sensitivity-based metric using a linearized power-flow model. The metric identifies buses with lower tolerance to active or reactive load variation.

  • Network model: The analysis assumes an N-bus network with one slack bus, M load buses, and N−M−1 voltage-controlled buses, with charging stations connected across load buses.Power-flow analysis determines voltage phases and load-bus voltage magnitudes under three-phase balance and per-unit-system assumptions.
  • Voltage variation model: Newton’s method linearly maps EV-induced active and reactive power increases to voltage-magnitude and phase variations through the inverse power-flow Jacobian.The resulting relation uses ∆P and ∆Q as inputs and ∆V and ∆Φ as voltage-variation outputs.
  • Impact metric: The proposed grid-impact metric is the 2-norm of voltage variation in magnitude and phase.It avoids solving the complete nonlinear power-flow equations for each evaluation.
  • Sensitivity interpretation: A larger active- or reactive-power sensitivity indicates that a load bus has lower tolerance to load variation and is more likely to disturb the network.The sensitivities measure 2-norm voltage variation from a 1 W active-power or 1 var reactive-power injection at a PQ bus.

V. STOCHASTIC DYNAMIC PROGRAMMING (SDP) FOR PRICING AND ELECTRICITY PROCUREMENT

This section develops a stochastic dynamic-pricing and electricity-procurement framework that incorporates renewable generation, wholesale-price uncertainty, system dynamics, and a minimum-profit safeguard. The safeguard constrains the probability of falling below a profit threshold under uncertain charging demand.

  • Policy derivation: The section combines the profit safeguard with renewable-energy, real-time wholesale-price, and system-dynamics modules to derive pricing and electricity-procurement policies using SDP.The procedure is presented as the final step of the section’s policy-derivation framework.
  • Profit safeguard: The model adds a minimum-profit warning mechanism as a constraint so providers maintain a specified profit level under severe demand uncertainty.The safeguard addresses the gap between estimated charging demand used for decisions and uncertain actual demand.
  • Profit safeguard: The safeguard requires the probability that horizon profit W_k falls below threshold W_min to remain below ζ.Here, ζ is a small positive number in (0, 1).
  • Uncertainty model: The uncertainty formulation assumes the demand and process-noise terms represented in the random vector are independent Gaussian random variables.The Gaussian CDF is then used in the probability constraint reformulation.

B. Renewable Energy and Real Time Wholesale Price

The paper represents renewable generation with a Markov-chain forecasting model and treats real-time wholesale-price forecasting as outside its technical scope. Renewable energy is discretized into levels whose transition probabilities can be estimated from historical data.

  • Renewable-energy forecasting: Renewable-energy prediction uses a statistical Markov-chain model, while the framework can also accommodate other forecasting approaches.The paper uses the Markov model to demonstrate how renewable prediction enters the optimization model.
  • Renewable-energy forecasting: Renewable energy is discretized into D levels, and a horizon-specific transition matrix describes movement from current level i to next level j.Each transition probability is denoted t_k,i,j.
  • Renewable-energy forecasting: Each row of the renewable transition matrix sums to one, and its transition probabilities can be estimated from historical data.This supplies the probability model used for renewable-energy evolution.
  • Wholesale-price forecasting: The paper does not study specific real-time wholesale-price forecasting methods because that topic lies beyond its technical scope.The cited forecasting literature includes time-series, machine-learning, big-data, and hybrid approaches.

C. Stochastic Dynamic Programming

The paper formulates pricing and electricity procurement as a finite-horizon stochastic dynamic program whose states evolve with storage, renewable energy, demand, and process noise. SDP exploits overlapping subproblems and optimal substructure to solve the large optimization recursively.

  • SDP solution: SDP partitions the K(L + 1)-variable problem into smaller recursive subproblems by exploiting overlapping subproblems and optimal substructure.Each subproblem can be rewritten in quadratic form, and the SDP engine runs at the beginning of every planning horizon.
  • Dynamic formulation: The planning problem uses K = 24 hourly horizons because wholesale-market electricity is sold hourly.System states evolve under decision variables and random variables across these horizons.
  • Dynamic formulation: The system state includes beginning-of-horizon storage and renewable energy, while procurement, charging efficiency, discharging efficiency, and demand govern state evolution.Storage and renewable-generation process noises are modeled as independent.
  • Objective and constraints: The objective maximizes aggregated expected utility over the planning horizon, including a terminal utility and expectations over demand and process uncertainties.The resulting value function is the maximum aggregated expected utility from the initial state.
  • Objective and constraints: The optimization enforces the profit-probability safeguard together with procurement, pricing, storage, and charging-demand feasibility constraints.The listed constraints include Prob(W_k < W_min) < ζ, procurement bounds, nonnegative prices, storage balance, and nonnegative demand.

VI. SIMULATIONS AND DISCUSSIONS

The simulations compare SDP with a greedy benchmark for pricing and energy management under modeled storage, renewable, and wholesale-price conditions. SDP uses forward-looking information and aggregated utility optimization, achieving higher profit at greater computational complexity.

  • Simulation setup: The simulations use assumed charging and discharging efficiencies of 0.9 and day-ahead PJM wholesale prices to represent real-time price forecasting.The paper notes that other forecasting approaches could also be used.
  • Algorithm comparison: The greedy algorithm optimizes only the current planning horizon, whereas SDP optimizes aggregated utility over multiple horizons using price and renewable-energy predictions.This forward-looking structure explains the algorithms’ different profit outcomes.
  • Algorithm comparison: Up to 7% profit gain is achieved by SDP compared with the greedy benchmark.The comparison is explicitly framed in terms of profitability.
  • Algorithm comparison: O(K) is the greedy algorithm’s time complexity, compared with O(K2) for SDP.SDP’s additional inner backward-recursive calculation increases complexity as the number of planning horizons grows.

B. Aggressive or Conservative Electricity Procurement Strategy

The simulations examine how storage cost changes electricity procurement and how profit safeguards, Pareto analysis, and knee points characterize tradeoffs among objectives. Lower storage costs encourage more aggressive low-price procurement, while higher profit thresholds lead to higher charging prices.

  • Electricity procurement: From 8:00 to 16:00, the provider procures less wholesale electricity because renewable generation is available.Procurement also shifts toward low-price periods and away from high-price periods.
  • Electricity procurement: When ηs is small, the provider procures aggressively during low-price periods; when ηs is large, procurement becomes more conservative.The simulations vary unit storage cost over ηs = 0 to 4.
  • Profit safeguard: Charging prices increase as the profit threshold Wmin increases.The provider raises prices to preserve the probability ζ associated with the safeguard.
  • Pareto analysis: Each Pareto-front point is optimal because improving one objective requires decreasing at least one of the others.The objectives are profit, customer satisfaction, and impact on the power grid.
  • Pareto analysis: Knee points represent the best tradeoffs among profit, customer satisfaction, and power-grid impact by maximizing improvement per unit degradation.The simulations identify several knee regions reflecting different preferences over the three objectives.

E. Interplays between Profit, Customer Satisfaction, and Impact on Power Grid

The simulations show that pricing reshapes charging demand across stations and changes the relationships among profit, customer satisfaction, and grid impact. Higher profit can coincide with lower grid impact but lower customer satisfaction, while reducing grid impact requires shifting demand away from more sensitive buses.

  • Objective interplays: Customer satisfaction decreases as profit increases because higher charging prices reduce total charging demand.The reduced demand lowers satisfaction while increasing provider profit.
  • Objective interplays: Power-grid impact decreases as profit increases, so profit and grid impact are not competing objectives in this projection.Higher prices reduce charging demand and relieve grid stress while improving profit.
  • Objective interplays: Power-grid impact increases as customer satisfaction increases because both are related to total charging demand.Higher demand improves customer satisfaction but increases grid impact.
  • Demand redistribution: As grid impact decreases, demand falls at Charging Station #2 and Charging Station #20 while increasing at other stations.The redistribution shifts load spatially rather than reducing demand uniformly.
  • Demand redistribution: Charging Station #20 has active power sensitivity 1.33, while Charging Station #8 has sensitivity 0.17.Demand decreases faster at the more sensitive station and increases faster at the less sensitive station as grid impact is reduced.
  • Demand redistribution: The provider can reduce grid impact by shifting demand from buses with large active power sensitivity to buses with small sensitivity through charging prices.The conclusion links pricing decisions to spatial demand redistribution.
Loading 1801.02128v1…