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Optimal Uniform Pricing for Multi-Interval Dispatch without Make-Whole Uplifts

Valentina Norambuena-Guzman, Cong Chen, Lang Tong, Timothy D. Mount

arXiv:2609.00541v1eess.SYecon.EM

TL;DR

Ramp-constrained dispatch with inaccurate net-demand forecasts can produce LMPs below accepted offers, requiring OOM settlements that create transparency, discrimination, and bidding concerns. The paper derives LMP+U, a closed-form uniform pricing rule that minimizes demand payments while eliminating OOM make-whole uplift and preserving LMP-based congestion charges. Simulations find reduced price volatility and no tested offer-inflation incentive under LMP+U, alongside generally higher generator revenue and demand payments.

  • Problem

    Ramp constraints and net-demand forecast errors can drive LMPs below accepted offers, requiring OOM settlements that create nontransparent price signals, discriminatory compensation, and untruthful bidding incentives.

  • Method

    The paper formulates demand-payment minimization over nodal prices and derives LMP+U, an LMP plus uniform price adder that eliminates OOM make-whole uplift while preserving congestion charges and merchandising surplus.

  • Results

    LMP+U reduces price volatility by as much as 29% and, within the tested bid range, removes the offer-inflation incentive observed under LMP with OOM settlement.

  • Takeaways & Limitations

    In most simulated cases, LMP+U yields higher generator net revenue and demand payment because ramping costs are allocated uniformly in-market while LMP-based congestion charges are preserved.

Abstract

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In a network with ramp-limited generators and inaccurate net-demand forecasts, practical rolling-window dispatch can drive locational marginal prices (LMPs) below generators' bid-in offers. In such cases, out-of-market (OOM) settlements are used to compensate generators and maintain dispatch-following incentives, but OOM can have negative consequences, including nontransparent real-time price signals, discriminatory compensation, and incentives for untruthful bidding. This paper presents an optimal uniform pricing rule that minimizes demand payments, eliminates OOM make-whole payments, preserves LMP-based congestion charges, and ensures revenue adequacy. We derive the proposed pricing rule in closed form and relate it to existing pricing schemes. Numerical comparisons demonstrate favorable generator profits and reduced price volatility. However, higher generator profits are accompanied by increased demand payments, reflecting the in-market, uniform allocation of ramping costs while preserving the LMP-based congestion charges widely used in real-time market settlements. The numerical results also show that, under LMP with OOM settlement, a price-taking generator has an incentive to inflate its offer, whereas this incentive is absent under the proposed pricing rule within the tested bid range.

I. INTRODUCTION

The paper addresses inadequate LMP compensation in ramp-constrained, uncertain real-time dispatch by proposing LMP+U, a uniform pricing rule that removes OOM make-whole uplift while preserving key LMP properties. Simulations examine bidding incentives, generator revenue, demand payments, and price volatility relative to existing schemes.

  • Motivation: Ramp constraints and forecast errors can make LMPs fall below accepted offers, creating revenue shortfalls and violating uniform-price-auction principles.OOM settlements compensate affected generators and support dispatch-following incentives; CAISO’s 2022 real-time OOM bid-cost-recovery payments totaled about $141 million.
  • Motivation: OOM uplifts can create untruthful bidding incentives, discriminatory settlement, and nontransparent ramping price signals.A price-taking generator’s offer may affect uplift revenue without changing dispatch shadow prices or LMPs.
  • Proposed pricing rule: LMP+U minimizes demand payment subject to zero OOM make-whole uplift while preserving LMP-based congestion charges and merchandising surplus.The pricing rule is designed for rolling-window dispatch with ramp-constrained generators and arbitrary net-demand forecast errors.
  • Relation to existing schemes: LMP+U has a closed-form price equal to LMP plus a uniform price adder across locations and includes LMP and MDCP as special cases under stated conditions.Unlike PMP and CMP, the rule has a closed-form solution; MDCP does not naturally generalize to congested networks while extended MDCP may not preserve congestion charges or revenue adequacy.
  • Bidding incentives: Under LMP, PMP, and CMP with OOM uplift, price-taking generators can increase net revenue by inflating offers, whereas this incentive is absent under LMP+U without OOM uplift.The stated LMP+U result applies within the tested bid range and follows from the generator’s inability to influence dispatch shadow prices.
  • Numerical comparisons: In most simulated cases, LMP+U produces higher generator net revenue and demand payment than comparison benchmarks, although negative price adders can reverse the demand-payment ordering.The simulations characterize how network congestion affects the uniform price adder and its effects on revenue and demand payment.

II. DISPATCH MODEL AND BID-COST RECOVERY

The dispatch model uses rolling-window optimization over multiple future intervals, incorporating forecast demand, generator offers and limits, ramp constraints, and transmission-flow limits. Its solution determines the realized dispatch and associated LMP components.

  • Rolling-window dispatch: At each interval t, the operator solves a W-interval rolling-window dispatch over the look-ahead horizon H_t = {t, ..., t + W − 1}.The dispatch session spans intervals H = {1, ..., T}, with buses indexed by [M] and generators grouped by bus.
  • Inputs and variables: The model represents forecast demand, generator dispatch, and aggregate nodal generation and net-demand vectors across the look-ahead window.Forecast demand is indexed by bus and future interval, while generator dispatch is indexed by generator, bus, and interval.
  • Generator constraints: Generator offers represent bid-in marginal costs, while ramp limits and generation limits constrain feasible dispatch.The down- and up-ramp limits and generation limits apply to each generator over the modeled intervals.
  • Network constraints: The multi-interval dispatch optimization uses a shift-factor matrix and line-flow limits to represent transmission constraints.The resulting optimization solution provides the realized dispatch and the rolling-window LMP vector, including a congestion component.

B. Bid-cost Recovery and Make-Whole Payments

Rolling-window LMP can fall below a generator’s bid-in cost, prompting make-whole compensation to recover costs. Although this supports dispatch following, the settlement is discriminatory because only generators with shortfalls receive MWPs.

  • When rolling-window LMP falls below a generator’s bid-in cost, the generator is typically compensated through an OOM make-whole payment.
  • OOM make-whole payments guarantee interval-level bid-cost recovery.
  • The settlement is discriminatory because MWPs are paid only to generators with bid-cost shortfalls.

III. OPTIMAL UNIFORM PRICING WITHOUT MAKE-WHOLE UPLIFT

The proposed pricing optimization minimizes binding-interval demand payments while eliminating OOM make-whole payments and preserving LMP congestion charges. Its constraints imply a uniform price adjustment across locations that preserves LMP merchandising surplus.

  • The pricing optimization minimizes binding-interval demand payments while eliminating OOM make-whole payments and preserving LMP congestion charges.
  • Every feasible price vector differs from LMP by a common, possibly negative, scalar.
  • The uniform price adder preserves the LMP merchandising surplus.

A. Optimal Pricing Formulation

The formulation chooses nodal prices for a rolling-window dispatch to eliminate make-whole payments while preserving LMP congestion charges. The unique optimum uses a scalar adder uniform across locations, recovers dispatched generators’ bid-in costs, and preserves merchandising surplus.

  • A. Optimal Pricing Formulation: The formulation uses binding-interval net demand and the dispatched-generator set to define the pricing problem.
  • A. Optimal Pricing Formulation: The constraints enforce zero OOM make-whole payments and preserve LMP congestion charges.
  • A. Optimal Pricing Formulation: The unique optimal solution is obtained from a scalar price adder uniform across all locations.
  • A. Optimal Pricing Formulation: Every dispatched generator recovers its bid-in cost, and congestion charges under LMP+U equal those under LMP.
  • A. Optimal Pricing Formulation: LMP+U preserves the LMP merchandising surplus and is revenue adequate whenever the stated merchandising-surplus condition holds.
  • A. Optimal Pricing Formulation: LMP+U reduces to MDCP without binding transmission constraints and to LMP when ramping and lower-generation constraints are nonbinding under the stated condition.

IV. NUMERICAL RESULTS

The numerical study compares LMP+U with LMP and two alternative pricing schemes in rolling-window, multi-interval simulations across generator revenues, make-whole payments, surplus, volatility, congestion charges, and bidding incentives.

  • IV. NUMERICAL RESULTS: The evaluation compares LMP+U against LMP and two alternative pricing schemes using rolling-window, multi-interval simulation.
  • IV. NUMERICAL RESULTS: The analysis measures generator net revenue, make-whole payments, adjusted merchandising surplus, bus-price volatility, congestion-charge distortion, and bidding incentives.

A. Parameter Settings

The study simulates rolling-window dispatch in a three-bus system with ramp-constrained generators, CAISO net-demand profiles, and both uncongested and congested transmission.

  • System and demand: The three-bus simulation uses three 500-MW generators with marginal costs of $25/MWh, $30/MWh, and $50/MWh.Demand is located at Bus 3 and follows scaled 100-day CAISO net-demand profiles.
  • Dispatch settings: Dispatch uses a four-interval rolling window, corresponding to a one-hour look-ahead at 15-minute resolution.
  • Experimental cases: The experiment varies common 15-minute ramping capability as a fraction of generator capacity and compares uncongested transmission with congestion at f_13 = 200 MW.
  • Pricing mechanisms: The pricing comparison includes LMP+U, LMP, E-MDCP, and PMP; LMP and PMP use MWPs for remaining bid-cost shortfalls.

B. Generator Net Revenue, Make-Whole Payments, and Merchandising Surplus

The simulations compare generator revenue, MWP requirements, and MWP-adjusted merchandising surplus during morning ramp-down periods across congestion and ramping conditions.

  • Generator revenue: LMP+U generally produces higher aggregate generator net revenue than LMP with MWP because a positive common price adjustment raises every active generator’s net revenue.An exception occurs in the unconstrained 6% ramping scenario, where a negative adjustment produces slightly lower revenue.
  • Make-whole payments: At 6% ramping capability, LMP requires roughly $39k/day in MWPs without congestion and approximately $6.5k/day with congestion.Congestion distributes production across more generators, increasing production cost but providing additional downward flexibility.
  • Merchandising surplus: LMP+U and E-MDCP eliminate MWPs entirely, but only LMP+U is guaranteed to preserve nonnegative LMP merchandising surplus.Under LMP and PMP, MWP-adjusted merchandising surplus becomes negative when required MWPs exceed collected surplus.

C. Price Volatility and Congestion-Charge Distortion

The study evaluates hourly Bus 3 price volatility and Bus 3–Bus 1 congestion-charge distortion under congested transmission with 6% ramping capability.

  • Metrics: Price volatility is measured by the standard deviation of Bus 3 price across 100 simulated days, while congestion-charge distortion is the RMSE from the corresponding LMP spread.
  • Price volatility: Approximately 29%: LMP+U reduces afternoon-ramp peak volatility from approximately $24/MWh to $17/MWh.
  • Congestion signals: LMP+U avoids the tradeoff between reducing volatility and altering nodal price spreads associated with transmission congestion.
  • Congestion signals: LMP+U preserves LMP nodal price spreads and yields zero congestion-charge distortion, whereas E-MDCP and PMP guarantee neither property.

D. Bidding Incentives

The bidding experiment varies G1’s offer while holding its dispatch and prices unchanged over the tested range, isolating offer effects on true net revenue.

  • Experiment design: G1’s offer varies from $25/MWh to $29/MWh while its true marginal cost remains $25/MWh and ramping capability equals 6% of capacity.
  • OOM bidding incentive: Under LMP+MWP, G1’s true net revenue rises from $18.2k to $18.5k as its offer increases, because MWP depends on the submitted offer.
  • Uniform-pricing incentive: True net revenue remains constant at $18.6k under LMP+U and E-MDCP, so G1 cannot increase revenue by overstating its bid within the tested range.
  • Paper-level conclusion: LMP+U eliminates ramp-induced OOM make-whole uplift while minimizing demand payment and preserving LMP-based congestion charges and merchandising surplus.
  • Bidding implications: The conclusion reports that OOM make-whole payments can incentivize price-taking generators to inflate offers, while this incentive is absent under LMP+U within the tested bid range.
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