Is Locational Marginal Price All You Need for Locational Marginal Emission?

Danny H.K. Tsang; Xuan He; Yize Chen

REVIEW 4 major objections 3 minor 1 cited by

Is Locational Marginal Price All You Need for Locational Marginal Emission?

T0 review · 4 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that under the uniqueness condition of Assumption 1, locational marginal emission is a function of locational marginal price, so LME can be looked up from released LMP alone.

desk verdict A useful but unproven shortcut: the LMP-to-LME mapping is new and plausible, but rests on an unverified uniqueness assumption that needs checking. read the letter →

arxiv 2411.12104 v1 pith:EMGE26I5 submitted 2024-11-18 eess.SY cs.SYeess.SP

Xuan He , Danny H.K. Tsang , Yize Chen This is my paper · ORCID

classification eess.SYcs.SYeess.SP

keywords locationalmarginalemissionpricecriticalregionprojectionmulti-parametricprogrammingsecurity-constrainedeconomicdispatchDCoptimalpowerflowcongestionandemissionsLMP-LMEmapping

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that locational marginal emissions (LMEs) at every bus can be recovered from locational marginal prices (LMPs) alone in security-constrained economic dispatch. By partitioning load space into critical regions, each with its own affine generation-sensitivity matrix $G_r$, it writes both LMP and LME as linear images of that matrix, $\alpha_r = c^T G_r$ and $\beta_r = e^T G_r$, so a prerecorded lookup table can map one to the other. The paper's central claim is that, under Assumption 1, there is a well-defined function $\beta = \Phi(\alpha)$ from released LMP vectors to LME vectors, meaning no load vector or optimization solve is needed. If this holds, operators and market participants could publish and consume nodal emission signals as a by-product of existing market clearing, with reported per-sample speedups of roughly 22x over implicit differentiation and much larger gains over finite differences.

What carries the argument

The machinery is critical region projection from multi-parametric programming. Within each polyhedral region of the load space, the first-order optimality system's affine sensitivity lemma (Lemma 1) supplies a constant generation-sensitivity matrix $G_r$; the paper records the pairs $(\alpha_r,\beta_r) = (c^T G_r, e^T G_r)$ for every region. The LMP-to-LME step is then a table lookup: given a released price vector, find the precomputed region with that price and return its emission vector, an operation made well-defined by Assumption 1.

What would settle it

Enumerate all critical regions of any small SCED model with at least two fuel types and multiple congested lines and compare, region by region, the vectors $c^T G$ and $e^T G$; if two regions have identical price vectors but different emission vectors, the map $\beta = \Phi(\alpha)$ is not a function and the lookup fails.

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Extended reading notes

Core claim

The paper asserts that for each critical region of the load space, the same matrix of generation-load sensitivities $G_r$ determines both the locational marginal price $\alpha_r = c^T G_r$ and the locational marginal emission $\beta_r = e^T G_r$, with $c$ the generator cost vector and $e$ the generator emission-rate vector. Because both quantities are linear in the same sensitivity matrix, the authors claim that a unique emission vector is attached to every price vector whenever no two regions share a price vector while disagreeing on emissions (Assumption 1). They verify on standard 14-, 39-, and 118-bus test systems that the resulting critical-region lookup reproduces the values of analytical LME methods and is faster, and they exhibit cases where congestion makes some nodal LMEs negative.

Load-bearing premise

The entire price-only shortcut depends on Assumption 1: no two operating regions may ever show the same locational marginal price vector while having different locational marginal emission vectors, and the paper does not derive this condition from network data or verify it on its test systems.

Editorial extensions

If this is right

System operators could add near-instant emission reporting as a post-processing step on already-released day-ahead or real-time LMP, without solving additional optimization problems.
Nodal emission signals become available to market participants who never see load data, since the price vector itself identifies the operating region.
Congestion directly shapes emission patterns: in the 14-bus case, binding transmission limits create negative LMEs at some nodes, so shifting load toward those nodes would lower total emissions.
The per-sample cost of LME estimation drops by more than an order of magnitude relative to implicit-function methods, and by thousands of times relative to finite differences on larger systems.
Because the mapping is precomputed per operating regime, the same LMP feed can be monitored continuously, with emission rates updating instantly whenever the price vector moves to another critical region.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extending the paper's logic, if Assumption 1 survived tests on large real networks, an ISO could publish an LME lookup table alongside LMP with essentially zero computation per interval; the main remaining task would be proving or systematically checking the uniqueness condition for each network topology.
The LMP-LME table is tied to the active constraint set; a transmission outage, generator outage, or dispatch re-run that changes the active set invalidates the table, so the plug-and-play claim holds only within a fixed topology and parameter regime.
The negative-LME observations imply a concrete operational lever: if regulators priced emissions at the margin, load increases at negatively emitting nodes would be rewarded, turning congestion patterns into carbon-reduction signals rather than just cost signals.
A testable extension is to run the same critical-region projection on markets with piecewise-linear or quadratic costs, where the per-region sensitivity remains affine; if the uniqueness condition still holds there, the price-only route to LME would generalize beyond the linear-cost SCED setting.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper's real contribution is the observation that LMP and LME share the same sensitivity matrix in each critical region, so if LMP uniquely identifies the region, LME is a lookup away. That's a genuinely useful idea and, as far as I know, new. The MPP machinery is standard and the load-based LME computation is sound. The speedups over per-sample implicit differentiation and finite differences are plausible, and the paper is upfront that its central mapping rests on an assumption.

Where it gets shaky is Assumption 1. It says no two critical regions share an LMP vector but differ in LME. That is exactly the uniqueness property Theorem 1 claims to prove, so the theorem is just restating the assumption. The authors never test it on their 14-, 39-, or 118-bus cases. They report only the number of critical regions, not whether the recorded LMP vectors are distinct. If two regions collide, the LMP-to-LME table is multi-valued and the lookup has no principled way to choose. This is a load-bearing flaw, but it's fixable: run the enumeration, check for collisions, and report it. If collisions are rare or absent in practice, the method stands.

The robustness comparison in Section IV-D is unfair. They freeze the IF baseline's LME across ±10% load perturbations, which is not what the IF method does; it recomputes per sample. That stacks the deck and should be redone with a properly updated IF baseline. They also ignore the offline MPP computation time when claiming 22x to thousands of times speedups; for real-time use that's defensible, but it should be stated.

The paper is worth a serious referee. The idea is new, the derivation is clear, and the soft spots are addressable. I'd ask for verification of Assumption 1, a fair robustness comparison, and a report of offline cost. With those, it would be a solid method contribution for anyone working on real-time emission accounting.

Referee Report

4 major / 3 minor

Summary. This paper proposes a critical-region-projection (CRP) method for computing locational marginal emissions (LME) in a DC security-constrained economic dispatch (SCED) model. The load space is partitioned into critical regions using multi-parametric programming; within each region the generation-to-load sensitivity matrix G_r is affine-constant, so the LMP and LME vectors are written as α_r = c^T G_r and β_r = e^T G_r. The main conceptual claim is that, under Assumption 1, there is a unique LME for each LMP, β = Φ(α), so an operator can recover LME from published LMP without knowing the load vector or the network. Case studies on IEEE 14-, 39-, and 118-bus systems report speedups of 22.0x–29.5x over an implicit-function baseline, much larger speedups over finite differences, and faster LMP-based than load-based lookups.

Significance. If the uniqueness claim were established, the paper would provide a practically valuable plug-and-play module that turns publicly available LMP signals into LME estimates without requiring load or network data, which is relevant for ISOs, RTOs, and market participants. The MPP/CRP derivation is standard and self-contained, and the paper does not fit any data to reach its conclusions, which is a strength. However, the central theorem is conditional on an assumption that is neither proved nor numerically verified, and one of the benchmark comparisons is structured in a way that favors the proposed method. These issues must be resolved before the contribution is fully supported.

major comments (4)

[§III-C, Assumption 1 and Theorem 1] The central claim that β = Φ(α) is essentially a restatement of Assumption 1: the proof of Theorem 1 derives a contradiction to the assumption but supplies no independent support for the assumption. Assumption 1 is not checked on any of the test systems; Table I reports only the number of critical regions, and Section IV.F does not verify that all recorded LMP vectors are distinct. The assumption is not innocuous: if two generators have identical marginal cost but different emission rates, then for any two critical regions c^T G_u = c^T G_v, because each column of every sensitivity matrix sums to 1 under the load-balance constraint (1d), while e^T G_u and e^T G_v can differ. Nothing in the model (4) rules out this equal-cost case. The paper should either prove Assumption 1 from the problem data or perform an exhaustive pairwise check across all critical regions in each test system, and also discuss the equal-cost degeneracy explicitly.
[§IV-D, Robustness Analysis] The robustness comparison is not apples-to-apples. The text states that, for the IF method, load samples within a ±10% operating range share the same LME, so when a 1% perturbation is applied, the IF-based LME is not recomputed, whereas the proposed CRP method updates the LME to the new critical region. The reported 86% versus 42% accuracy therefore partly measures update frequency rather than accuracy. A fair comparison should recompute the IF-based LME at the perturbed load using Lemma 1 or the implicit-differentiation method of [20], and then compare the resulting emission estimates.
[§IV-C and §IV-F, Computation Efficiency] The reported speedups are per-sample online costs, but they omit the offline cost of solving the multi-parametric program and constructing all critical regions. The CRP method precomputes the critical regions once, while the IF and FD baselines are evaluated per sample. To substantiate the 'order of magnitude' speedup claim, the paper should report the offline MPP computation time, the number of critical regions, and the breakeven number of online samples at which the CRP method becomes cheaper than repeated IF evaluation.
[§IV-C, Accuracy on the 39-bus system] The paper states that in the 39-bus system the proposed method 'may yield different results for few samples' due to slight deviations in generation sensitivity, but it does not quantify how many samples differ, by how much, or whether those samples lie near critical-region boundaries. Since the abstract and introduction claim accurate LME derivation, this discrepancy should be quantified; if the CRP policy is only approximately optimal for those samples, the LME values are not exact, and the LMP-to-LME mapping inherits the same error.

minor comments (3)

[§III-A, Definition 1 and Eq. (7)] The notation 'CR R' in Eq. (7a) appears to be a typo for 'CR_c', and the redundancy-removal operator ∇ is not defined. Please define the initial region CR_IG and the ∇ operator before using them.
[§II-B and Table I] The unit 'kgCO2/MW' should be 'kgCO2/MWh', and the emission rates should be stated consistently with the energy unit used. In addition, Table I should report the generator cost coefficients and emission rates, since the validity of Assumption 1 depends on the relationship between c and e.
[§III-A, Lemma 1] Lemma 1 describes (5) as a 'first-order approximation' of x and λ, but within a properly defined critical region the affine policy should be exact. Please clarify whether the affine policy is exact in each critical region or only a first-order approximation, since the subsequent LMP/LME formulas rely on this distinction.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 1's unique LMP-to-LME mapping is Assumption 1 restated; the central uniqueness guarantee is imposed by definition rather than derived.

self definitional [Section III.C, Assumption 1 and Theorem 1 (Eqs. 10-13)]
"Assumption 1. There will not be two matrices Gu and Gv, which satisfies cT Gu = cT Gv along with eT Gu ̸= eT Gv. Theorem 1. Based on Assumption 1, given the MPP model (4) representation of SCED, there exists a unique LME for each LMP, that is β = Φ(α). Proof. Assume that one LMP ˜α can correspond to different LME βu and βv. This indicates eT Gu ̸= eT Gv while cT Gu = cT Gv = ˜α, which is a contradiction to Assumption 1. Thus, Theorem 1 can be proved."

With α_r = c^T G_r (Eq. 10) and β_r = e^T G_r (Eq. 11), Assumption 1 is exactly the statement that equal LMP vectors force equal LME vectors, i.e., that β is a single-valued function of α. The theorem's conclusion β = Φ(α) is therefore the assumption restated in functional notation; the proof only checks that a counterexample would violate the assumption and provides no derivation from the SCED or network structure. Because Assumption 1 is never verified in the case studies (Table I reports only critical-region counts, and Section IV.F times lookups without checking whether all recorded LMP vectors are distinct), the paper's central guarantee that LME is uniquely determined by released LMP is not independently established—it is imposed by definition.

full rationale

The core LME-from-load derivation is self-contained: Eqs. (9)-(11) differentiate the SCED KKT system and evaluate α_r = c^T G_r and β_r = e^T G_r inside each critical region, with no fitted parameters and no load-bearing self-citations ([19] is a screening reference only; [23] is an external multi-parametric programming text). The computational benchmarks against implicit-function and finite-difference baselines are honest speed measurements. The only reduction-by-construction is Theorem 1: Assumption 1 states that equal LMP vectors (c^T G) imply equal LME vectors (e^T G), which is precisely the single-valuedness of β = Φ(α); the proof merely restates the assumption as a contradiction. Since Assumption 1 is never verified numerically in Section IV, the paper's headline claim that LME is uniquely recoverable from released LMP is imposed by definition rather than derived from network or cost structure. This is a partial, localized circularity, not a self-citation chain or a fitted-input disguise.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard MPP sensitivity theory, the DC-OPF model, and a fixed linear emission rate model. No parameters are fitted to data; the main free choice is the load variation range omega. The uniqueness assumption (Assumption 1) is external to the derivation and unverified.

free parameters (1)

Load variation range omega = +/-30% for 14-bus; not explicitly stated for 39 and 118-bus
The operating variation range omega in (4d) defines the load polyhedron and therefore the set of critical regions. It is chosen by the authors, not fitted to data, but it affects which regions are enumerated.

assumptions (4)

domain assumption DC power flow approximation with PTDF matrix
The SCED model (1c) uses a DC approximation, which ignores reactive power and losses; LMP and LME are computed on this simplified model.
domain assumption Generators have fixed, known emission rates e
Equation (3) assumes total emissions E = sum_i e_i x_i with constant e_i; in reality emission rates vary with output, so the LME formula is only an approximation.
standard math Multi-parametric programming sensitivity lemma (Lemma 1)
The affine sensitivity (5) is taken from Pistikopoulos et al. [23]; the paper does not prove it, relying on standard MPP theory.
domain assumption The SCED solution is a unique point for each load and the active constraints have linearly independent gradients (non-degeneracy)
The critical region partition and constant sensitivity matrix require that the optimal basis does not change within a region and that the KKT system is non-singular; the paper does not discuss degeneracy.

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Pith. "Pith review of Is Locational Marginal Price All You Need for Locational Marginal Emission?." pith.science (2026). https://pith.science/paper/EMGE26I5

@misc{pith2026241112104,
  author       = {Pith},
  title        = {Pith review of: Is Locational Marginal Price All You Need for Locational Marginal Emission?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EMGE26I5}},
  note         = {Machine review of arXiv:2411.12104}
}

read the original abstract

Growing concerns over climate change call for improved techniques for estimating and quantifying the greenhouse gas emissions associated with electricity generation and transmission. Among the emission metrics designated for power grids, locational marginal emission (LME) can provide system operators and electricity market participants with valuable information on the emissions associated with electricity usage at various locations in the power network. In this paper, by investigating the operating patterns and physical interpretations of marginal emissions and costs in the security-constrained economic dispatch (SCED) problem, we identify and draw the exact connection between locational marginal price (LMP) and LME. Such interpretation helps instantly derive LME given nodal demand vectors or LMP, and also reveals the interplay between network congestion and nodal emission pattern. Our proposed approach helps reduce the computation time of LME by an order of magnitude compared to analytical approaches, while it can also serve as a plug-and-play module accompanied by an off-the-shelf market clearing and LMP calculation process.

Figures

Figures reproduced from arXiv: 2411.12104 by the authors.

**Figure 1.** Illustration of the mappings of Load-LME and LMP-LME via critical region projection, which helps conveniently derive LME for given SCED [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Computation efficiency results. Reported time is averaged over 1000 samples with standard deviation shown as error bars. The y-axis is of logscale. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Robustness Analysis for 118-bus system. A. Benchmark Methods The methods used for our comparison are summarized as follows: 1) Implicit Function (IF)-based LME: This method is utilized in [20], which applies Lemma 1 for each load sample to obtain dx(l) dl to calculate the LME via (3). Note that, in this case, no critical region will be developed. 2) Finite Difference (FD)-based LME: This method uses the finite diffe… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: LME and LMP in different critical regions of the 14-bus system. Negative LME are observed in Critical Region 1. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Calculation time of Load-based and LMP-based LME mappings. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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[1] [20]

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L. F. Valenzuela, A. Degleris, A. El Gamal, M. Pavone, and R. Rajagopal, “Dynamic locational marginal emissions via implicit differentiation,” IEEE Transactions on Power Systems , vol. 39, no. 1, pp. 1138–1147, 2023

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work page 2023

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Z. Liu, Z. Deng, S. Davis, and P. Ciais, “Monitoring global carbon emissions in 2022,” Nature Reviews Earth & Environment , vol. 4, no. 4, pp. 205–206, 2023

work page 2022

[4] [3]

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E. Masanet, A. Shehabi, N. Lei, S. Smith, and J. Koomey, “Recalibrating global data center energy-use estimates,” Science, vol. 367, no. 6481, pp. 984–986, 2020

work page 2020

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S. Mallapaty et al., “How china could be carbon neutral by mid-century,” Nature, vol. 586, no. 7830, pp. 482–483, 2020

work page 2020

[6] [5]

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I. Perissi and A. Jones, “Investigating european union decarbonization strategies: Evaluating the pathway to carbon neutrality by 2050,” Sustainability, vol. 14, no. 8, p. 4728, 2022

work page 2022

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P. Xu, T. S. Adebayo, K. A. Khan, O. ¨Ozkan, and N. K. Shukurullaevich, “United states’ 2050 carbon neutrality: Myth or reality? evaluating the impact of high-tech industries and green electricity,” Journal of Cleaner Production , vol. 440, p. 140855, 2024

work page 2024

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A. Radovanovic, “Our data centers now work harder when the sun shines and wind blows,” 2020, GoogleOfficialBlog, https://bit.ly/3cXPAMu

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