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Threshold Pricing for Distributed Scheduling of Flexible Demands in Energy Communities

Minjae Jeon, Lang Tong, Qing Zhao

arXiv:2608.27174v1eess.SY

TL;DR

The paper addresses distributed scheduling of renewable-backed flexible demand when household responses create an intractable bilevel stochastic dynamic program. It derives a two-threshold centralized policy and uses it to construct the Threshold Pricing Rule, which achieves individual rationality, revenue adequacy, and asymptotic welfare optimality under light traffic.

  • Problem

    Jointly scheduling deferrable EV charging and price-elastic thermostatic loads with stochastic renewables creates a generally intractable bilevel stochastic dynamic program.

  • Method

    The paper characterizes a two-threshold centralized policy and derives the Threshold Pricing Rule, which broadcasts NEM-based prices so households respond in their own interest.

  • Results

    The Threshold Pricing Rule is individually rational, revenue adequate, and asymptotically optimal in community welfare under a light-traffic condition.

  • Takeaways & Limitations

    The threshold structure supports distributed scheduling with closed-form household TCL and EV decisions under the broadcast price.

Abstract

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This paper develops a price-based distributed scheduling in an energy community whose members own behind-the-meter renewable generation with deferrable EV charging and price-elastic thermostatic loads. A coordinator transacts with the distribution utility under a Net Energy Metering tariff and broadcasts a community price to which each household responds in its own interest, giving a bilevel stochastic dynamic program that is intractable in general. Our main result characterizes that the joint optimal centralized policy is a two-threshold policy on the community's aggregate renewable generation. Building on this structure, we adopt the Threshold Pricing Rule, which is uniform, individually rational, revenue adequate, and asymptotically optimal in terms of community welfare under a light-traffic condition. Simulations using synthetic and real world data confirm asymptotic optimality and individual surplus gains.

I. INTRODUCTION

The paper studies price-based distributed scheduling for energy communities with renewable generation, deferrable EV charging, and price-elastic thermostatic loads. It characterizes a threshold-based centralized policy and builds the Threshold Pricing Rule to coordinate household responses.

  • I. INTRODUCTION: The coordinator sets internal prices so self-interested households collectively maximize community social welfare under a Net Energy Metering tariff.The coordinator transacts with the utility on households’ behalf.
  • I. INTRODUCTION: EV charging is deferrable until a deadline, while thermostatic load consumption is non-deferrable but responds to price through private comfort utility.These demands are jointly scheduled with local renewable generation.
  • I. INTRODUCTION: Joint scheduling creates a bilevel stochastic dynamic program that is generally intractable, because each household solves a dynamic program beneath the coordinator’s pricing problem.Pricing must also satisfy individual rationality and revenue adequacy.
  • I. INTRODUCTION: The centralized optimum is a two-threshold policy on aggregate renewable generation, with closed-form decisions outside the net-zero zone.The net-zero zone is characterized by a Bellman equation.
  • I. INTRODUCTION: The Threshold Pricing Rule uses centralized-policy thresholds, broadcasts NEM-based prices by generation zone, and is individually rational, revenue adequate, and asymptotically welfare-optimal under light traffic.Numerical studies use synthetic and real-world EV charging and residential solar data.

3) Renewable generation:

The model represents household and community renewable states and defines net consumption under a Net Energy Metering tariff. Internal payments use a uniform, time-varying price function based on household net consumption.

  • 3) Renewable generation:: Household renewable generation evolves independently across households as a discrete-time Markov process on an uncountable state space.The household state combines EV status, remaining demand, time to deadline, and renewable generation.
  • 3) Renewable generation:: The community state is the collection of all household states, while community and household net consumption are defined from their aggregate energy balances.The community is defined analogously to an individual household.
  • 3) Renewable generation:: The utility’s NEM tariff charges retail rate π+ for net imports and credits compensation rate π− for net exports, with π+ > π−.π+ is the retail import rate and π− is the export compensation rate.
  • 3) Renewable generation:: The coordinator maps aggregate renewable generation to a time-varying internal price parameter through policy χ.Each household’s payment applies the same NEM-like form using internal price parameters.
  • 3) Renewable generation:: A household with negative net consumption is credited for its net production under the internal payment rule.This mirrors the treatment of net exports under the NEM tariff.

1) Lower level, individual household surplus maximization:

The lower level models each household’s surplus-maximizing response to the coordinator’s broadcast price. TCL consumption is scheduled first, and EV charging is then optimized against the remaining renewable generation.

  • 1) Lower level, individual household surplus maximization:: Household decisions respond to the broadcast price and realized renewable generation, with households treating the induced price process as exogenous.The coordinator’s pricing policy determines the stochastic process of broadcast prices.
  • 1) Lower level, individual household surplus maximization:: Within each interval, the non-deferrable TCL is scheduled first and receives priority over household renewable generation.The TCL is treated as the household’s only load during this step.
  • 1) Lower level, individual household surplus maximization:: The EV charging policy is chosen after TCL scheduling to maximize expected cumulative household surplus over the scheduling horizon.The EV can claim only the renewable generation remaining after TCL consumption.
  • 1) Lower level, individual household surplus maximization:: The coordinator’s pricing policy must satisfy constraints for coalition stability and avoidance of coordinator deficits.These constraints are formalized through revenue adequacy and individual rationality.
  • 1) Lower level, individual household surplus maximization:: The coordinator maximizes total community surplus using the household policies induced by its pricing policy.The coordinator is assumed to know EV deadlines and remaining demands and to measure household renewable generation.

III. OPTIMAL CENTRALIZED SCHEDULING

The centralized scheduling problem is formulated as a finite-horizon Markov decision process whose optimal policy partitions aggregate renewable generation into net-consuming, net-zero, and net-producing zones. Decisions are closed form in the outer zones, while the net-zero allocation requires solving a Bellman equation.

  • Problem formulation: The coordinator solves a finite-horizon Markov decision process over household states and aggregate EV-charging and TCL decisions.The centralized problem provides an upper bound on welfare achievable under pricing policies.
  • Two-threshold structure: The optimal centralized policy is a two-threshold policy on aggregate renewable generation.The thresholds define the boundaries between the three operating zones.
  • Two-threshold structure: The thresholds partition renewable generation into net-consuming, net-zero, and net-producing zones.The community imports shortfalls below the lower threshold and exports surplus above the upper threshold.
  • Outer-zone decisions: In the net-consuming zone, TCL demand is limited and each EV charges only its deadline-required minimum, deferring remaining charging.The marginal energy is purchased at π+ in this zone.
  • Outer-zone decisions: In the net-producing zone, TCL demand expands and EVs charge to their maximum because consumption is preferable to selling at π−.These outer-zone decisions and thresholds have closed-form expressions with computational cost linear in N.
  • Net-zero allocation: The net-zero allocation ρt requires solving a Bellman equation and suffers from the curse of dimensionality.This is the source of complexity in the otherwise closed-form centralized policy.

IV. THRESHOLD PRICING RULE AND PROPERTIES

The Threshold Pricing Rule uses the centralized policy’s thresholds to post one of three prices, enabling households to respond independently. Household decisions are myopic and have closed-form responses to the broadcast price.

  • Pricing rule: The rule replaces direct bilevel optimization with prices that induce centralized decisions in the two outer zones and uses NEM in the net-zero zone.Optimality is traded for simplicity only where no price can implement the centralized allocation directly.
  • Pricing rule: TPR uses the centralized thresholds and posts π+ in the net-consuming zone, π− in the net-producing zone, and the NEM tariff between them.The coordinator computes the thresholds from N reported household terms.
  • Household response: Under χTPR, each household’s optimal decision is myopic and depends only on the current broadcast price.The response is characterized in closed form for TCL and EV decisions.
  • Household response: At π−, TCL consumption reaches p−i,t and EVs charge to Mi,t rather than sell at the low rate.The lower compensation rate makes additional consumption preferable to exporting.
  • Household response: Under the NEM tariff, households behave as stand-alone customers, with TCLs absorbing local generation within [p+i,t, p−i,t] and EVs claiming the residual.This is the response in the net-zero zone.

C. Properties of TPR

The paper establishes that TPR satisfies individual rationality and revenue adequacy, and that its welfare loss vanishes under a light-traffic condition as community size grows. Numerical results support these properties and show individual surplus gains.

  • Pricing properties: TPR is individually rational and revenue adequate in every interval.These are the two pricing constraints required by the upper-level optimization.
  • Pricing properties: Individual rationality follows because TPR offers households a price at least as favorable as NEM in every zone.Revenue adequacy follows because internal prices are never more generous than the coordinator’s utility-facing prices.
  • Asymptotic optimality: TPR is suboptimal only in the net-zero zone, whose probability vanishes as community size grows under the homogeneous-community assumptions.Those assumptions include Bernoulli EV arrivals, bounded durations, i.i.d. stationary renewables, and common utility.
  • Asymptotic optimality: If α < (θr − p−)/(c̄ T̄), TPR is asymptotically optimal in expected community welfare.The condition ensures surplus generation can serve TCLs at p− and fully charge present EVs.
  • Numerical evaluation: Numerical experiments using synthetic and real-world data demonstrate asymptotic optimality and quantify individual surplus gains.The evaluation covers EV charging and residential solar data.

A. Simulation setting

The simulations model 24-hour energy-community scheduling with EVs, TCLs, and renewable generation using synthetic and real-world data. They compare TPR with centralized and alternative pricing or scheduling policies, finding asymptotic optimality and individual surplus gains.

  • Each of the N members has a behind-the-meter solar generator, an EV charger, and a TCL scheduled over 24 one-hour intervals.
  • Experiments use ACN-Data EV demand, stationary lognormal renewable draws, and measured Pecan Street solar output.The renewable models differ by experiment: i.i.d. stationary draws support asymptotic-optimality analysis, while measured solar supports individual-surplus analysis.
  • The asymptotic-optimality study compares TPR with an Oracle upper bound, a threshold policy using LLF in the net-zero zone, and MPC.The Oracle has perfect knowledge of all random variables and solves an open-loop optimization.
  • B. Asymptotic optimality of distributed scheduling: TPR and the threshold policy with LLF show exponentially decaying per-household optimality gaps as community size grows, whereas MPC’s gap persists.The gap is measured against the Oracle benchmark on a logarithmic scale; the LLF threshold policy converges faster than TPR.
  • C. Comparisons of individual surplus gains: In a community of N = 14, TPR produces 10.08%–83.13% larger surplus gains than the stand-alone NEM decision with the comparison pricing rule.Members with smaller renewable-generation capacity obtain larger gains by purchasing shared renewables at π− and scheduling TCLs with higher consumption surplus.
  • The paper concludes that TPR is individually rational, revenue adequate, and asymptotically optimal in community welfare.The conclusion also identifies truthful reporting and finite-community net-zero-zone welfare bounds as open directions.

APPENDIX A PROOF OF THEOREM 1 AND PROPOSITION 1

The appendix formulates the finite-horizon scheduling problem as an MDP with community net consumption determined by aggregate demand minus renewable generation. It defines feasible actions and the NEM payment’s piecewise-linear structure.

  • The feasible action at time t is represented using EV demand and remaining-time vectors, together with charging and TCL decisions.
  • The community net consumption is z_t = y(p,c) − r_t, and the finite-horizon MDP has terminal value V_T+1 ≡ 0.
  • A charger coordinate is feasible when d_i,t ≤ τ_i,t c̄, ensuring that its deadline can still be met; every arrival is feasible by assumption.
  • Because π+ > π−, the NEM payment is the upper envelope of two linear-price objectives associated with π+ and π−.

B. Sandwich bound on the value of deferrable demand

This section establishes a sandwich bound for the value of deferrable demand. Concavity and marginal-cost bounds constrain how value changes with residual demand and support the later scheduling characterization.

  • One kWh of EV demand can be served later at marginal cost at most π+, while relieving one kWh saves at least π−.
  • The value function V_t(d,τ_t,r_t) is concave in residual demand, proved by backward induction from the zero terminal value.The stage reward is jointly concave, the successor demand is affine, expectations preserve concavity, and the feasible action set is convex.
  • For d ≤ d′, the value difference is bounded by π−1_T(d′ − d) below and π+1_T(d′ − d) above when d′ is feasible.
  • Truncating charging above the coordinatewise threshold M_t preserves feasibility while combining payment and continuation-value bounds.
  • The proof handles residual-demand changes through induction, including coordinates near or beyond the charging-capacity boundary.
  • Within the restricted set C_t, the continuation value Φ_t is concave and actions outside C_t are strictly dominated.Increasing charging from c_i < m_i,t to m_i,t strictly improves the objective when q′(0) > π+.

C. Proof of Theorem 1

The proof splits the feasible actions into net-consuming and net-producing regions, where the NEM objective becomes linear-price optimization. Concavity then places the optimum at the corresponding threshold actions or on the net-zero boundary.

  • Net-consuming zone: In the net-consuming region, the optimizer uses p_i,t satisfying ∂U_i,t(p_i,t) = π+ and the charging action m_t.
  • Net-producing zone: The analogous net-producing optimization uses M_t and establishes v− as optimal under the π− linear-price objective.
  • Net-consuming zone: The threshold action v+ attains the upper bound and is optimal in the net-consuming region.
  • The feasible set is divided into A+ with y(p,c) ≥ r_t and A− with y(p,c) ≤ r_t, where the NEM objective equals the corresponding linear-price objective.
  • Net-zero zone: When an action produces more than r_t, concavity moves the maximum to the face y = r_t, identifying the net-zero boundary.
  • Net-zero zone: The net-zero-zone policy ρ_t satisfies y(ρ_t(x_t)) = r_t, so the community is net-zero.

D. Proof of Proposition 1

The proof reduces household surplus maximization to a price-inelastic scheduling problem with net renewable generation, then uses threshold-based local adjustments to establish path-wise optimality. The resulting action depends only on the current price state and renewable generation.

  • Step 1 (TCL): The TCL’s static concave program is characterized by one-sided derivatives, with linear payments yielding first-order conditions at prices π− and π+.The reduction identifies the relevant marginal-price thresholds from the signs of ∂U_i,t(p)−π±.
  • Step 2 (Reduction): Because the TCL is scheduled first and claims renewable generation with priority, residual renewable generation is g_i,t = r_i,t − p_i,t(ψ_t).The residual depends on the price state and TCL renewable use, not on EV charging or the EV state.
  • Step 2 (Reduction): With an exogenous price process, TCL utility is constant on each realization, so surplus maximization reduces to cost minimization with net renewable g_i,t.The resulting problem is price-inelastic, and each payment function is convex piecewise linear with slopes in [π−, π+].
  • Step 3 (EV, interchange): Successive adapted exchanges move each completing schedule toward the asserted action without increasing realized cost, preserving completion throughout the process.The local slope remains π+ or π− until the target is reached, so repeated modifications establish the claim path by path.
  • Step 3 (EV, interchange): For ψ_t = π+, moving excess current charging into the future does not increase cost, while for ψ_t = π−, borrowing charging from the future does not increase cost.The two exchanges use the payment slopes: current savings are exactly π+ϵ or π−ϵ, while the offsetting future change is bounded in the favorable direction.
  • Conclusion: Every completing sequence is dominated on every realization by the asserted action, which depends only on ψ_t and the current state.This establishes the proposition’s optimality conclusion without requiring dependence on the full EV history.
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