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Locational Marginal Pricing for Adaptive Robust Look-Ahead Dispatch with Casual Affine Recourse

Aidan Looney, Qian Zhang, Le Xie

arXiv:2609.05156v1eess.SY

TL;DR

Adaptive robust LAED can produce different current-demand marginal values across active worst-case trajectories, complicating single-price settlement. The paper uses causal affine recourse as a tractable price-forming approximation, derives its robust counterpart, and finds that it preserves standard DC LMP structure while supporting dispatch-following settlement. CAR matches deterministic LAED pricing when robustness is inactive and remains a practical middle ground between non-adaptive and fully adaptive robust dispatch, though its gap to FAR can widen under tight ramping and high uncertainty.

  • Problem

    Fully adaptive robust LAED can yield different current-demand marginal values from competing active worst-case trajectories, so it does not always endogenously select one scenario-independent settlement price.

  • Method

    CAR restricts future recourse to a shared causal affine policy and uses a robust counterpart with explicit nominal-balance and network multipliers for price formation.

  • Results

    CAR provides scenario-independent prices with standard DC energy and congestion decomposition, collapses to deterministic LMP when robustness is inactive, and avoids much NAR conservatism while retaining much FAR operational value.

  • Takeaways & Limitations

    CAR provides a practical bridge between deterministic market clearing and fully adaptive robust dispatch while supporting current-period dispatch-following settlement.

  • Takeaways & Limitations

    The analysis is limited to DC economic dispatch, affine recourse decisions, and current-period incentive compatibility; affine recourse is not universally exact.

Abstract

from arXiv · show

This paper develops a marginal pricing mechanism for adaptive robust look-ahead economic dispatch (LAED) under net-load uncertainty. In current market practice, deterministic multi-interval dispatch can misprice flexibility when forecast error is large. Fully adaptive robust (FAR) dispatch captures this uncertainty, but competing worst-case trajectories can imply different marginal values of demand, leaving no single price for market settlement. We propose causal affine recourse (CAR) as a tractable, price-forming approximation. CAR replaces independently optimized trajectory-specific schedules with a causal affine response policy, yielding a market-clearing model that retains the standard energy and congestion decomposition of DC Locational Marginal Pricing. We then demonstrate that CAR-LMP together with a ramp-adjusted current settlement supports dispatch-following and eliminates current-period lost opportunity costs. We solve the CAR problem by deriving a robust counterpart and present computational evidence on exact small DC instances, a ten-generator system, and an IEEE 300-bus public-network case. These simulations show that the proposed prices collapse exactly to deterministic LAED-LMP when robustness is inactive, remain stable in loose-ramp regimes, and produce economically meaningful price shifts when flexibility is scarce. The proposed pricing mechanism provides a practical bridge between adaptive robust dispatch and market-based pricing.

I. INTRODUCTION

The paper motivates adaptive robust LAED pricing by showing that uncertainty and intertemporal constraints can make deterministic LMPs misprice flexibility. It positions FAR as an optimality reference and CAR as a tractable price-forming alternative.

  • Motivation: Uncertain renewable generation and demand increase net-load volatility and the operational value of ramping capability.Rolling-window LAED co-optimizes current and future intervals to account for intertemporal constraints.
  • Motivation: Traditional LMPs may fail to support dispatch-following incentives in look-ahead dispatch without adjusted settlements.This motivates multi-interval pricing methods and explicit ramp or flexibility products.
  • Robust dispatch models: FAR allows future recourse after uncertainty is observed, whereas NAR fixes future decisions and can be overly conservative.FAR is used as the theoretical reference for cost and dispatch.
  • Pricing challenge: FAR can produce set-valued current-demand marginals when simultaneously active worst-case trajectories have different current-dispatch sensitivities.CAR addresses this pricing ambiguity through a shared causal affine policy.
  • Paper contributions: CAR yields a tractable market-clearing model with a single scenario-independent DC LMP, standard energy and congestion components, and ramp-adjusted current settlement.The shared-policy structure prevents competing worst-case realizations from generating different marginal demand values under the stated dual-uniqueness qualification.
  • Dispatch formulation: The models separate the implemented current dispatch from future recourse over a rolling look-ahead window and evaluate future operation over an uncertainty set.The current dispatch is the first-stage decision; future dispatch and shortage decisions are recourse actions.

C. Causal Affine Recourse Approximation

CAR replaces trajectory-specific adaptive recourse with one causal affine policy and converts the robust problem into a finite-dimensional linear program. This preserves tractable price formation while acknowledging that affine policies are not generally exact.

  • Causal affine policy: CAR restricts future recourse to an affine policy whose response matrix is block-lower-triangular.Decisions at each future time depend only on uncertainty revealed by that interval, enforcing non-anticipativity.
  • Robust formulation: The affine robust formulation enforces power balance, operating feasibility, and a worst-case affine-cost epigraph for every uncertainty realization.Operating constraints include capacity, ramping, network, and shortage limits.
  • Price formation: CAR is the causal market-clearing formulation used to price marginal demand, while FAR remains the dispatch optimality reference.Its support-function robust counterpart exposes nominal-balance and network-constraint multipliers.
  • Robust counterpart: For polyhedral uncertainty, support-function inequalities and coefficient matching produce a finite-dimensional robust counterpart used in the computational studies.This reformulation converts the affine robust model into a linear program.
  • Scope of approximation: Affine decision rules are exact only under special structures and can have arbitrarily large optimality gaps in general.Networked LAED combines uncertain balance equations with uncertainty-dependent ramping and transmission constraints.

III. SET-VALUED MARGINAL PRICING UNDER FAR

The paper explains why FAR may fail to select a unique current-period LMP: active worst-case scenarios can transmit different sensitivities through binding intertemporal constraints. Price uniqueness depends on the resulting current-price components, not merely on scenario multiplicity.

  • Source of price multiplicity: Binding intertemporal constraints can transmit different active-scenario sensitivities into distinct current-period marginal prices.The value function is evaluated as current nominal demand changes while future demand, uncertainty, bids, and network data remain fixed.
  • Scenario representation: FAR represents the robust continuation value as the maximum over scenario recourse values associated with extreme uncertainty realizations.Because the recourse value is convex in uncertainty, the maximum is attained at an extreme point.
  • Sensitivity analysis: Each scenario recourse value is convex and piecewise affine in current dispatch and demand, so active worst-case scenarios can have different dispatch sensitivities.These sensitivities generate the subgradients entering current-dispatch KKT conditions.
  • Uniqueness condition: FAR has a unique current-period LMP if and only if the current-price mapping is constant across the dual-optimal set.Other dual multipliers may remain nonunique without changing the economically relevant price.
  • Interpretation: Multiple active worst-case scenarios alone do not imply multiple prices; their current-dispatch sensitivities must differ and affect the full KKT price vectors.Different dual-optimal weightings can otherwise support the same settlement price.
  • Market implication: Price multiplicity is a price-formation problem rather than a failure of dispatch feasibility or cost optimality.Without an exogenous selection rule, the optimization may not determine a single settlement price for the same cleared dispatch and objective value.

IV. MARGINAL PRICING FOR CAUSAL AFFINE RECOURSE

The causal affine restriction makes recourse finite-dimensional through an affine policy and uses coefficient matching to enforce robust balance. In the DC model, the resulting multipliers supply the energy and congestion components of CAR marginal prices.

  • Affine policy representation: CAR replaces the infinite-dimensional recourse-policy space with finite-dimensional affine-policy variables y0 and Y.The policy takes the form y(ξ) := y0 + Yξ.
  • Robust balance: When the uncertainty set is full dimensional, robust balance is equivalent to matching the affine-policy coefficients.This separates nominal-demand effects from uncertainty-response consistency.
  • DC price decomposition: In the DC specialization, nominal-balance and line-flow multipliers provide the energy and congestion components of the CAR marginal price.Nominal demand enters both nominal balance and nominal line-flow constraints.

A. DC-Networked Affine Recourse and Adaptive Robust LMP

The DC robust counterpart enforces balance, network, capacity, ramping, shortage, causality, and worst-case-cost constraints for causal affine policies. Shared-policy structure gives CAR scenario-invariant nominal gradients and standard DC LMPs, while a forward-ramp adder supports dispatch-following.

  • CAR models generation and shortage as affine policies whose response depends only on uncertainty revealed through the corresponding interval.
  • The robust counterpart enforces balance, line-flow, capacity, shortage, ramping, causality, and worst-case cost across uncertain net-load realizations.
  • Multiple active uncertainty realizations can alter response-coefficient subgradients but not direct nominal-policy gradients under CAR.
  • CAR determines a unique DC LMP when the price image of the full dual-optimal set is a singleton, even if individual multipliers are not unique.
  • The CAR bus price equals marginal production cost adjusted by capacity scarcity and adjacent-interval ramp opportunity costs.
  • A unit-specific forward-ramp adder combined with the bus-level CAR LMP restores first-future-interval ramp stationarity and eliminates current-period lost opportunity cost without changing dispatch.

B. ISO Implementation

The ISO settles each rolling-horizon CAR dispatch with the current CAR LMP and, when needed, a forward-ramp side payment, then re-solves after passing the dispatch to the next window.

  • One CAR solve replaces deterministic LAED on the same DC network model and requires no scenario-indexed repricing.
  • The ISO passes the implemented current dispatch to the next rolling window and re-solves with updated conditions.

V. COMPUTATIONAL SIMULATIONS

The computational studies evaluate CAR pricing and operational cost on a toy single-bus example, a ten-generator rolling-horizon system, and an IEEE 300-bus case. The models are linear programs implemented in Pyomo and solved with Gurobi.

  • Three studies test CAR pricing and operational cost across a single-bus example, a ten-generator system, and an IEEE 300-bus case.
  • The ten-generator study compares deterministic LAED, NAR, and CAR across uncertainty budgets and ramp regimes, with selected FAR comparisons.
  • The IEEE 300-bus case tests adaptivity gaps, computation, and ramp-adjusted settlement.
  • All models are linear programs implemented in Pyomo and solved with Gurobi.

A. Toy Example

The toy example contrasts FAR’s scenario-specific recourse and set-valued current prices with CAR’s shared causal policy and unique price. Both formulations select the same current dispatch, but CAR incurs higher total cost to obtain a single market-clearing LMP.

  • Both formulations choose current dispatch (13, 5, 0) with cost $33 before differing in future recourse.
  • FAR recourse: FAR uses separate recourse for each worst-case uncertainty realization, allowing active scenarios to produce different current-dispatch sensitivities.
  • CAR recourse: CAR imposes a common causal affine policy whose t + 1 action is independent of ξ and feasible for every ξ ∈[0, 1].
  • Price comparison: CAR’s objective is uniquely minimized at x = 5, producing total cost 236 and a unique LMP of $4/MW from its shared-policy market-clearing problem.
  • Price comparison: FAR attains total cost 228 with endpoint prices of $3/MW and $6/MW, yielding the full dual-optimal interval [3, 6] USD/MW.

B. Ten-Generator Rolling Market Study

The ten-generator study evaluates CAR across uncertainty budgets and ramp regimes using rolling-horizon simulations. CAR matches deterministic pricing when robustness is inactive, while tighter ramps and larger uncertainty produce larger cost, price, and optimality effects.

  • Uncertainty and ramps: At Γ = 0, CAR collapses exactly to deterministic LMP, while Fig. 1 and Table III show increasing effects as ramping tightens.
  • Study design: The study uses 288 five-minute intervals from a rescaled August 2032 MISO net-load trajectory and evaluates Γ ∈{0, 1, 2, 3} under tight, medium, and loose ramp multipliers.The ramp multipliers are 0.2, 0.5, and 1.0, respectively.
  • Common-state results: At Γ = 3, CAR changes current cost by 0.000%, 0.264%, and 1.587% under loose, medium, and tight ramps, respectively.
  • Common-state results: At Γ = 3, mean signed LMP shifts are 0.000, −0.937, and −0.804 USD/MWh under loose, medium, and tight ramps, respectively.
  • CAR–FAR comparison: The mean CAR–FAR gaps are 0.97%, 1.1%, and 49.54% under loose, medium, and tight ramps, respectively.The larger tight-ramp gap is associated with limited ramping and severe uncertainty making future imbalance costly.
  • Closed-loop results: In closed-loop evaluation, CAR produces no load shedding or ramp slack for Γ = 1, 2, 3, with a 0.12% average current-cost increase and largest mean price shift of 0.43 USD/MWh.

C. Public Network Study

The IEEE 300-bus study evaluates CAR under congested DC conditions and moderate or varying uncertainty. CAR closely tracks FAR operational outcomes, avoids much NAR conservatism, and supports dispatch-following through a ramp-adjusted settlement.

  • 2.139 USD/MWh is the maximum LMP shift, while the robust-objective premium over deterministic LAED is 2.004%.
  • CAR matches FAR’s protected objective at reported precision, whereas the NAR objective gap reaches 0.64% at σrel = 0.012.
  • Under moderate uncertainty, CAR nearly matches FAR’s objective and dispatch, while NAR is more conservative and shifts prices more on average.CAR nevertheless has the larger maximum LMP shift in this comparison.
  • CAR provides a practical middle ground between non-adaptive and fully adaptive robust dispatch while producing scenario-independent prices.The experiments solve FAR by complete vertex enumeration and solve NAR and CAR using finite support-dual robust-counterpart LPs.
  • The IEEE 300-bus validation reduces current lost opportunity cost from 750.84 USD to zero using a forward-ramp adder.The largest absolute adder is 8.62 USD/MWh, and the result verifies local dispatch-following rather than system-wide revenue adequacy.
  • CAR prices bridge deterministic market clearing and fully adaptive robust operation, but the analysis remains limited to DC economic dispatch, affine recourse, and current-period incentive compatibility.

APPENDIX A FULL CAR ROBUST COUNTERPART DERIVATION

The CAR robust counterpart represents future decisions as causal affine policies and converts uncertainty-dependent constraints into a tractable finite-dimensional formulation. Support-function dualization accommodates polyhedral uncertainty sets while preserving the dispatch-side structure.

  • Each support-function occurrence receives a separate dual vector, yielding a finite-dimensional linear program for suitable cost and network models.The support-function representation permits different uncertainty sets without changing the dispatch-side reformulation.
  • CAR uses nominal future decisions and causal affine recourse matrices whose decisions depend only on uncertainty revealed by the corresponding time.
  • Future power balance is imposed as an affine equality in the uncertainty vector.
  • Representing uncertainty in affine-hull coordinates makes robust equality equivalent to coefficient-matching conditions.
  • The nominal balance equation is paired with an affine response condition that tracks uncertainty-induced changes in total net load; Ξ = {0} reduces the model to deterministic LAED.
  • The formulation assumes a nonempty compact uncertainty set and applies componentwise capacity, ramping, network, and balance constraints.
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