Congestion Structure and Exceedance Bounds for Locational Marginal Emissions

Motivation

• Prior work uses LMEs to guide flexible demand and places carbon objectives inside dispatch.Examples include shifting data-center load outside operator dispatch and incorporating carbon objectives into dispatch.Source: §1 · p. 1

Contributions

• The paper derives an intrinsic basis for LMEs, proves a rank bound and an exact-recovery observation count, and combines forecast-error distributions with the critical region to bound exceedance probability.Experiments test both contributions across networks up to 1,354 buses.Source: §1 · p. 1

Recovery setting

• Existing LME methods include direct differentiation, LMP mappings within critical regions, and finite differences at every bus.The paper focuses on settings where an external or legacy simulator is available but its equations or solver derivatives are inaccessible.Source: §1 · p. 1

Dispatch model

• The basis applies within a regular active set, where active rows are linearly independent, active inequality multipliers are positive, and dispatch is affine in demand.The model uses strictly convex DC dispatch with line-flow and generator limits.Source: §2 · p. 2

Congestion basis

• The LME vector lies in the span of the uniform vector and binding-line PTDF rows, so its rank is at most one more than the number of binding lines.Cost and generator data determine the coefficients, while demand-dependent constraint rows supply the bus patterns.Source: §2 · p. 2
• With no binding line limits, every bus has the same LME; each independent binding line can add at most one nodal pattern.Binding generator limits can change dispatch response coefficients but add no pattern because their right sides do not depend on demand.Source: §2 · p. 2

Simulation count

• r independent scalar observations recover all LMEs exactly; paired bus perturbations require 2r simulations, or r + 1 with a cached base case.Fewer than r observations leave the coefficients underdetermined; pivoted QR can select independent, well-conditioned bus rows.Source: §2 · p. 2

Active-set changes

• The critical region is a polytope whose facets encode inactive primal constraints and nonnegative multipliers for active inequalities; facet tail bounds yield an exit-probability bound.For Gaussian errors, the exponential facet terms can be replaced with exact facet probabilities.Source: §3 · p. 3

Active-set margin

• κmin summarizes the smallest standardized distance to a critical-region facet, while the complete bound retains every facet.A facet is reached when an inactive primal constraint becomes binding or an active multiplier reaches zero.Source: §3 · p. 3

Exceedance bound

• The bound separates emissions increase while nominal constraints hold from the probability of leaving the critical region, making the limiting term visible.A finite threshold for target probability δ requires region-exit probability εcr < δ.Source: §3 · p. 3

Estimation error

• The estimation allowance combines a variance estimate with the bound, and the total exceedance probability is at most δop + εcr + δest at any stopping time.The variance-informed rule selects buses to reduce emissions-variance uncertainty, assuming future forecast error is independent of simulation noise.Source: §3 · p. 3

Test systems

• Ten of the 14 tested systems have nonzero operating emissions; the other four have zero LMEs under their benchmark rates.A transparent DC dispatch supplies exact derivatives for comparison.Source: §4 · p. 3

Simulation count

• Across 14–1,354 buses, r ranges from 2 to 15; on 300 buses, 24 simulations recover LMEs to 1.2 × 10^-5 relative error versus 600.On the largest system, n/r reaches 90.Source: Figure 1 · p. 4, §4 · p. 4

Emissions exceedance after redispatch

• At 1% nodal forecast error, 23.1% of draws exit the region and nominal LMEs underestimate the redispatch 95th-percentile increase of 3.77 tonnes CO2/h by 8.6%.The bound remains available through 0.4% nodal forecast error; every threshold violation follows an active-set change.Source: §4 · p. 4, Figure 1 · p. 4

When the bound is available

• With 0.5% reserved for estimation, εcr < 0.045 across forecast-error scales from 0.12% to beyond 4% of load, with no relation to network size.Nine of ten systems narrow under correlation; the 588-bus system widens at a⋆ = 0.35.Source: §4 · p. 4

Conclusion

• Small congestion rank can make recovery inexpensive, but proximity to a critical-region boundary can make the recovered vector unreliable.Operational use also requires models of timing, information exchange, and participant response.Source: §4 · p. 4

A.1 Congestion Basis for Marginal Emissions

• Eliminating active-constraint multipliers from stationarity and feasibility yields the demand-response structure used to characterize marginal emissions.The power-balance equality contributes 1, each active line contributes a signed PTDF row, and generator bounds contribute zero.Source: §A.1 · p. 7
• In a connected shuntless network without phase shifters, H = BAL† links PTDFs to the weighted Laplacian and its pseudoinverse.Substitution gives a discrete Poisson equation whose source is supported at no more than 2q binding-line endpoints.Source: §A.1 · p. 7
• Values on binding-line endpoints determine the full congestion-span vector because the remaining buses are harmonic and the discrete Dirichlet problem has a unique solution.When q = 0, the vector is uniform and one bus value suffices.Source: §A.1 · p. 7

A.2 Recovery from Selected Buses

• Choose r buses whose basis submatrix ΦI is nonsingular; their LME observations determine the coefficient vector x.The selected observations satisfy ℓI = ΦIx.Source: §A.2 · p. 7
• The congestion structure permits recovery from r observations, whereas without it fewer than n sparse observations cannot distinguish zero from an unobserved single-bus marginal-emissions vector.Solving for x reconstructs the full vector.Source: §A.2 · p. 7

A.3 Exceedance Bound under Active Set Changes

• Within the nominal critical region, dispatch remains affine and emissions change by ℓ⊤ξ; a union bound over facet violations bounds active-set exit.The proof applies an upper-tail bound to each event F_j⊤ξ > τ_j.Source: §A.3 · p. 8

A.4 Confidence Bound for Emissions Variance

• Cauchy–Schwarz bounds the cross term and a Rayleigh quotient bounds the quadratic term; the simultaneous confidence event also holds at a data-selected stopping time.The confidence event holds simultaneously at every step.Source: §A.4 · p. 8

B Critical Region Construction

• Critical-region construction converts inactive primal inequalities and active multiplier conditions into demand-space facets, then removes zero normals and dependent active rows.Demand directions are converted from per unit to MW before Equation (7) is evaluated.Source: B Critical Region Construction · p. 8

• Among retained systems, congestion sets the rank, and the reduction n/r improves with system size.Source: Table 1 · p. 8
• Four screened systems are excluded because their direct operating emissions are zero, making their LME vectors zero.Source: Table 2 · p. 9, C Full Screening Results · p. 9
• All four selected operating points miss the 10^-8 sensitivity tolerance, with relative disagreements spanning 8 × 10^-8 to 0.42.Source: C Full Screening Results · p. 9
• Three of the four systems have an optimality-system condition number above 109; the well-conditioned 197-bus system has the largest disagreement.Source: C Full Screening Results · p. 9

Experimental setup

• The uncertainty stress test scales the base proxy to demand-error standard deviations of 0.2%–1.0% of load; region-exit estimates use 2,000 draws on systems up to 300 buses.The base proxy uses independent Gaussian nodal errors with standard deviation equal to 1% of nominal demand and a 0.25 MW floor; no empirical exit frequency is reported for larger systems.Source: D Additional Numerical Details · pp. 9, 12

Additional numerical results

• The Gaussian bound tracks empirical exit frequencies within Monte Carlo error, while the distribution-free subgaussian bound overshoots by up to a factor of five.The nearest facet contributes more than 0.93 of the Gaussian bound on eight of ten systems.Source: Table 6 · p. 10
• The usable forecast-error range depends on the nearest facet’s aggregate response ratio, not network size; correlation narrows it when a⋆ > 1 and widens it when a⋆ < 1.Across systems, availability spans from 0.12% to beyond 4% of load; the 588-bus system with a⋆ = 0.35 gains range under correlation.Source: Table 7 · p. 10
• The confidence radius grows with R r log(1/γ) through the log-determinant term and with √γ x̄ through the declared norm bound.The table reports medians over 100 trials after 40 dispatch simulations on the 57-bus system.Source: Table 9 · p. 11
• After 40 simulations, uniform bus selection has median p vt/v = 433, an ill-conditioned diagnostic because it leaves directions relevant to M nearly unobserved.A noisy observation uses a pair of load perturbations and therefore takes two dispatch simulations when the base case is not cached.Source: D Additional Numerical Details · p. 12
• After 20 simulations on the 300-bus system, variance-informed selection reaches a usable bound; leverage selection overestimates its standard deviation by a factor of 218.Once both designs cover the basis, the advantage shrinks to 2.6%.Source: Figure 3 · p. 12
• On the 57-bus system, increasing correlation from ρ = 0 to ρ = 0.5 moves the exceedance-allowance crossing from 0.41% to 0.23% demand error.The nearest facet has aggregate response ratio a⋆ = 5.6.Source: Figure 4 · p. 12
• At 0.4% nodal demand forecast error, the reported threshold is 2.2 times the empirical 95th percentile, with tail-bound, allowance-splitting, and estimation factors of 1.49, 1.25, and 1.15.With a known variance proxy and the exact Gaussian quantile, the threshold is 1.43 tonnes CO2/h versus an empirical 1.40; at 0.7% and above it falls below the true quantile, which the exit term prevents.Source: D Additional Numerical Details · p. 13

Reproducibility

• The repository contains the Julia implementation, benchmark data and provenance, experiment scripts, generated CSV outputs, and automated tests.The numerical results are available with the implementation.Source: E Reproducibility · p. 13
• The experiment manifest records package versions, random seeds, solver settings, exclusion criteria, uncertainty parameters, and sample counts.Source: E Reproducibility · p. 13
• The README explains how to run a short validation case, the test suite, and the complete numerical pipeline.Source: E Reproducibility · p. 13