Locational Marginal Emissions (LMEs)

• LMEs are defined as the derivative of total CO₂ emissions with respect to nodal load, quantifying the marginal impact of additional power consumption.
• They are computed using optimal power flow models and implicit differentiation of KKT conditions, reflecting network constraints and generator characteristics.
• LMEs enable carbon-aware operations by guiding real-time dispatch, facilitating strategic resource siting, and promoting dynamic demand response for emission reductions.

Locational Marginal Emissions (LMEs) quantify the rate at which total power-system CO₂ emissions change in response to an incremental increase in electric load at a specific node and time. LMEs, formulated as the carbon analog of Locational Marginal Prices (LMPs), encode the marginal system-wide emission consequence of localized consumption, fully capturing network physics, dispatch constraints, generator characteristics, and, if modeled, temporal couplings such as ramping and storage. Recent advances have extended the computation, application, and integration of LMEs to real-time market operations, carbon-aware dispatch, sectoral carbon-accounting, and carbon-optimized flexible demand response, with rigorous mathematical foundations and algorithmic developments now enabling large-scale, high-fidelity operationalization.

1. Foundational Definitions and Principles

Let $E(d)$ denote the total system CO₂ emissions as a function of the vector of nodal demands $d$. The locational marginal emission at bus $n$ is defined as

$\mathrm{LME}_n = \frac{\partial E}{\partial d_n}$

This quantity has units of (e.g.) kg CO₂/MWh and explicitly measures the instantaneous system-wide emissions impact of an infinitesimal load increase at node $n$. The concept directly parallels the definition of LMP, $\text{LMP}_n = \frac{\partial C}{\partial d_n}$, where $C$ is system production cost.

In the context of optimal power flow (OPF)—whether DC or AC, static or dynamic, transmission or distribution—$E$ is a function of optimal dispatch that itself depends on the load vector: $E(d) = \sum_g e_g(x^*_g(d))$, where $e_g$ is the marginal emission rate of generator $g$ and $x^*_g(d)$ is its dispatch given load $d$ (Valenzuela et al., 2023, Shao et al., 27 Oct 2025, Lu, 24 Jan 2024).

Key contrasts with average emissions intensities:

• LMEs are correctly derived system gradients: they reflect the actual marginal resource that responds to a change, internalizing binding constraints (unit, network, or environmental).
• Regional or mean carbon intensities do not capture marginal displacement, spatial granularity, or dynamic re-dispatch under network congestion (Lindberg et al., 2020, Lindberg et al., 2021, Lindberg et al., 2022, Cote et al., 21 Dec 2025).

2. Mathematical Formulation and Computational Methods

LMEs arise as dual variables or implicit sensitivities in a constrained dispatch optimization.

Consider a DC-OPF for $n$ buses, $g$ generators:

• Objective (cost): $C(x) = \sum_{i=1}^g c_i x_i$
• Constraints: power balance, line flows (via PTDF), generator and flow limits.

Let $e_i$ be the marginal CO₂ emission rate of generator $i$. The total system CO₂ is $E(x) = \sum_{i=1}^g e_i x_i$.

Sensitivity and Dual-Relationship

After solving for cost-optimal dispatch $x^*(d)$: $$\mathrm{LME}_j = \frac{\partial E(x^*(d))}{\partial d_j} = \sum_{i=1}^g \frac{\partial x^*_i}{\partial d_j} e_i$$ By implicit differentiation through the Karush–Kuhn–Tucker (KKT) system, or via market-sensitivity theory (if the solution mapping is affine in a given critical region), one obtains $G = \frac{\partial x^*}{\partial d}$ and thus,

$\beta = e^T G$

where $\beta_j = \text{LME}_j$ (He et al., 18 Nov 2024, Valenzuela et al., 2023). In linear-DC OPF, $G$ can be expressed via basis inversion ($B$ block of $A^{-1}$, where $A$ is the active constraint matrix) (Lindberg et al., 2020, Lindberg et al., 2021, Lindberg et al., 2022).

Dynamic Models and Temporal Coupling

For multi-period OPF with storage and ramping constraints, the total emission function $E(g^*(d))$ is temporally coupled. The full Jacobian $\frac{\partial g^*}{\partial d}$ is computed via implicit differentiation on the (dimension-expanded) KKT system, with adjoint/reverse-mode differentiation enabling tractable computation for large $N, T$ (Valenzuela et al., 2023, Degleris et al., 20 Aug 2024). Parallel-in-time decomposition delivers $O(N/K+1)$ speedup for $K$ storages, $N$ buses, and $T$ periods (Degleris et al., 20 Aug 2024).

Distribution Systems

In radial/meshed distribution systems modeled by SOCP, Distribution Locational Marginal Emission (DLME) extends this framework: $$\mathrm{DLME}_{i,t} = \frac{\partial e_{\mathrm{sum}}}{\partial p^D_{i,t}}$$ where $e_{\mathrm{sum}}$ is total emission, $p^D$ is demand, and the gradient is sequenced through SOCP KKT conditions utilizing the implicit function theorem (Sang et al., 12 Feb 2024).

Computational Acceleration Strategies

• Precomputing critical regions and associated (affine) policies offline (via multiparametric programming) allows real-time LME lookup given either load or LMP vectors, with $O(1)$ or $O(Rn)$ query complexity (He et al., 18 Nov 2024).
• Data-driven generator-to-load mapping (e.g., through regression for $\alpha_{n,g}$ factors) yields closed-form LME expressions and supports fast embedding into DC-OPF as linear constraints (Shao et al., 27 Oct 2025).
• Flow-tracing algorithms using graph-theoretical operations ($O(N+L)$ complexity) allocate average and marginal emissions in large, realistic networks (Shen et al., 24 Jul 2025).

3. LME Versus Alternative Emissions Metrics

Alternative metrics for emission accounting in power systems include:

• Locational Average Emissions (LAE/LACE): Weighted mean emission rate serving a node; fails to reflect marginal displacement or intertemporal redispatch (Shen et al., 24 Jul 2025, Lu, 24 Jan 2024).
• Carbon Emission Flow (CEF): Assigns emissions to network flows/traces based on proportional sharing; always nonnegative and insensitive to economic dispatch basis changes (Lu, 24 Jan 2024).
• LMCE/LME: Derived from market-clearing sensitivities; can be negative (e.g., when new load alleviates a high-emission margin), decomposed into explicit energy and network structural components, ensures allocation fidelity and enables negative or net-zero nodal assignments (Lu, 24 Jan 2024, Cote et al., 21 Dec 2025, He et al., 18 Nov 2024).

Table: Contrasting Emission Metrics

Metric	Physics/Economics	Can be Negative	Captures Network/Dispatch	Use Case
LME/LMCE	Dispatch Sens.	Yes	Yes	Market, Op.
LAE/LACE	Flow-weighted	No	Partial	Reporting
CEF (Flow)	Network only	No	Partial	Intensity Map

See (Lu, 24 Jan 2024, Shen et al., 24 Jul 2025) for details.

4. Integration Into Market Operations and Carbon-Aware Dispatch

LMEs can be directly embedded into dispatch optimization and market design:

• Adding carbon price terms (proportional to LME/LMCEn) into objective functions supports simultaneous optimization of cost and carbon (Shao et al., 27 Oct 2025, Cote et al., 21 Dec 2025).
• Linear LME constraints in DC-OPF admit standard quadratic/linear solution techniques (MATPOWER, Gurobi, CPLEX), with empirically negligible impact on computational time for systems up to 118 or even 1493 buses; real-time or near-real-time capability is routinely demonstrated (Shao et al., 27 Oct 2025, Cote et al., 21 Dec 2025).
• Policy instruments, such as demand-side carbon tariffs, dynamic green tariffs, or LME-based carbon contracts, become straightforward to design and settle hourly or sub-hourly, aligning operational behavior with decarbonization goals (He et al., 18 Nov 2024, Cote et al., 21 Dec 2025).

Carbon Accounting Theorems, as formalized, ensure that total system emissions are exactly decomposable into load, generator, and transmission allocations via LMEs, shadow carbon intensities, and generator differentials, eliminating double-counting and ensuring scope-2/3 compatibility (Cote et al., 21 Dec 2025).

5. Applications in Flexible Demand, Siting, and Resource Planning

LME signals have been deployed in multiple large-scale operational and planning settings:

• Data center operations: Geotemporal load shifting based on LMEs achieves emission reductions up to 1.8–2% (absolute) at negligible cost penalty, with >85% predictive accuracy relative to true system response, outperforming region-average and LMP-based heuristics (Lindberg et al., 2020, Lindberg et al., 2021, Lindberg et al., 2022, Cote et al., 21 Dec 2025).
• Siting: New large loads (e.g., data centers) sited in low-LME regions (Pacific Northwest) can minimize marginal emission impacts and inform zero-carbon procurement strategies (Cote et al., 21 Dec 2025).
• Renewable procurement: Placing new renewable resources at high-LME nodes maximizes the avoided emissions per MW (Cote et al., 21 Dec 2025).
• Distribution demand response: DLME-guided active DR yields emission reductions up to 245% greater than average-factor methods on test feeders (Sang et al., 12 Feb 2024).
• Policy and market design: Publishing LME signals, using LME-based tariffs, and regulating flexible resource participation based on marginal emissions unlocks demand-side flexibility for system-level decarbonization (He et al., 18 Nov 2024, Lindberg et al., 2022, Cote et al., 21 Dec 2025).

6. Empirical and Case Study Results

Key observed characteristics and impacts of LMEs in operational studies include:

• Nodal LMEs span orders of magnitude (e.g., 90–710 kgCO₂/MWh in WECC) and align with resource mix, congestion, and temporal availability (solar/wind) (Cote et al., 21 Dec 2025).
• Spatial and temporal LME clustering reveals region archetypes, with coal-dominated nodes showing consistently high marginal emissions and hydro/renewable regions remaining low and stable (Cote et al., 21 Dec 2025).
• Under grid congestion, LME patterns shift abruptly as marginal generators change, producing piecewise-constant or affine LME profiles over dispatch regimes (Brenner et al., 23 Oct 2025, He et al., 18 Nov 2024).
• Negative LMEs arise where local consumption displaces higher-emitting marginal units—an unambiguous indicator of nontrivial network effects and a diagnostic for carbon-aware flexibility (Lu, 24 Jan 2024, Cote et al., 21 Dec 2025).
• Empirical studies with parallelized implicit differentiation achieve an order-of-magnitude computational speedup for LME calculation in dynamic multi-hour and large-scale systems (Degleris et al., 20 Aug 2024).
• Real-time, algorithmic LME computation for 8,870-bus (CAISO) and 1493-bus (WECC) systems has been demonstrated, supporting practical LME-guided interventions (Shen et al., 24 Jul 2025, Cote et al., 21 Dec 2025).

7. Limitations, Extensions, and Research Directions

Several technical, operational, and policy considerations shape the present and future of LME-based methods:

• Assumptions: Most LME computations rely on the DC approximation, constant emission factors, no losses, and fixed active sets; deviations (transition to AC-OPF, stochastic renewables, nonconvexities) increase complexity but are, in part, tractable via robust/adjoint or graph-based extensions (Shen et al., 24 Jul 2025, Valenzuela et al., 2023).
• Computation: For very large, temporally coupled systems, centralized implicit-differentiation can be prohibitive; decentralized and parallel reverse-mode approaches mitigate bottlenecks (Degleris et al., 20 Aug 2024).
• Physical deliverability: LMEs correctly avoid over-allocation, but practical reporting (contractual, regulatory) must guard against double counting or misalignment with market-based carbon attributes (Cote et al., 21 Dec 2025).
• Policy design: Studies reveal that naïve LMP-based or average-intensity approaches to incentivizing carbon reduction can misfire, increasing emissions under certain network conditions (Cote et al., 21 Dec 2025, Lindberg et al., 2021).
• Transparency and Accessibility: Publishing real-time nodal LMEs, alongside LMPs, is essential for broad market participant engagement and for realizing full emission-reduction potential (Cote et al., 21 Dec 2025).
• Distribution-level modeling: DLME frameworks for AC/detailed distribution systems are emerging, leveraging SOCPs and advanced implicit differentiation (Sang et al., 12 Feb 2024).

Practical deployment requires integration with ISO market-clearing processes, widespread access to LME signals, and harmonization with emissions accounting standards. As network complexity, renewable penetration, and flexible load participation rise, continued algorithmic and methodological enhancements are crucial for scalable, robust, and actionable carbon-aware electricity system management.