Marginal Carbon Intensity

• Marginal Carbon Intensity (MCI) is the measure of additional emissions produced by a marginal increase in electricity demand, offering a consequential metric distinct from average intensity.
• It employs derivative-based and sensitivity analysis methods to capture locational nuances in power systems, supporting demand response and carbon-aware computing.
• MCI informs operational strategies by differentiating between actual emissions reduction and attributional adjustments, while fueling debates on reliability and observability.

Searching arXiv for papers on marginal carbon intensity, locational marginal emissions, and carbon-aware computing. Searching arXiv for "marginal carbon intensity" and closely related terms. Marginal Carbon Intensity (MCI) is an electricity-emissions signal intended to quantify the additional emissions caused by an incremental increase in electricity demand at a particular time and location. Closely related terminology includes the marginal emission factor (MEF), marginal operating emissions rate (MOER), locational marginal carbon emissions (LMCE), and locational marginal emission rate (LME) (Wiesner et al., 15 Jul 2025, Amor et al., 2024, Gorka et al., 2024, Shen et al., 24 Jul 2025). MCI differs from average carbon intensity because it is consequential rather than attributional: average intensity characterizes the emissions content of the prevailing electricity mix, whereas MCI asks how total system emissions change if load is added, shifted, or reduced (Jung, 2024, Dodge et al., 2022). This distinction is foundational in carbon-aware computing, demand response, EV charging, and electricity-market analysis, but it is also the source of persistent disputes over observability, accounting semantics, and operational usefulness (Gorka et al., 2024, Wiesner et al., 15 Jul 2025, Maji et al., 2023).

1. Conceptual meaning and signal semantics

MCI is defined in several papers as the emissions consequence of a marginal change in load. One formulation states that the MEF is “the marginal change in CO$_2$ emissions resulting from a marginal change in electricity demand in a given period, e.g. an hour,” and therefore “the partial derivative of emissions with respect to electricity demand” (Amor et al., 2024). In ElectricityEmissions.jl, the corresponding nodal quantity is LMCE, defined as $$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$, i.e. the short-run emissions sensitivity to a small load perturbation at a specific bus and time (Gorka et al., 2024). A market-clearing formulation defines LMCE as “the incremental change of total system emission in response to a unit load change at a specific location,” formalized as $\partial f(x)/\partial b_i$ through dispatch sensitivity (Lu, 2024).

The primary contrast is with average carbon intensity. One paper defines average carbon intensity as

$$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$

a weighted average over generation sources (Maji et al., 2023). Another emphasizes that average carbon emissions assign the same value to every load at a given time,

$$e_{D,j,t}^{avg} = e_{D,t}^{avg} = \frac{E^{tot}_{G,t}}{\sum_{j\in\mathcal{D}} p_{D,j,t}},$$

whereas LMCE is locational and marginal (Gorka et al., 2024). The practical consequence is that average intensity answers “How carbon-intensive was the electricity mix?” while MCI answers “What emissions change if one more unit of load is served?” (Jung, 2024).

This semantic distinction matters because electricity systems do not provide a unique carbon signal. ElectricityEmissions.jl argues that, in a grid with multiple generation sources and multiple consumers, “the physics of the system do not provide an unambiguous way to trace electricity from source to sink,” which is why multiple carbon metrics coexist (Gorka et al., 2024). The same ambiguity appears in debates over location-based, market-based, residual, average, and marginal formulations (Maji et al., 2023).

2. Mathematical formulations and related metrics

The main MCI formulations in the literature are derivative-based. At system level, a concise form is

$$MEF_t = \frac{\partial \text{emissions}}{\partial \text{load}_t},$$

which the German hourly MEF study uses as its conceptual definition before approximating it by perturbing load by 1 MWh in a selected hour and recomputing total emissions (Amor et al., 2024). At nodal level, ElectricityEmissions.jl computes LMCE as the emissions-weighted redispatch response of generators under a DCOPF active-set sensitivity analysis, while the market-clearing sensitivity paper derives LMCE from the KKT system and then decomposes it into energy-dependent and network-dependent components (Gorka et al., 2024, Lu, 2024). In both treatments, MCI is explicitly short-run, locational, and dispatch-contingent.

Alongside marginal metrics, the literature distinguishes several average or hybrid metrics:

Metric	Definition in the literature	Typical role
ACE	Total system emissions divided by total system load	Carbon accounting
LMCE	$\frac{\delta E}{\delta p_{D,j,t}}$	Carbon-informed load shifting
LACE	Locational average metric based on carbon flow tracing or LMCE path averaging	Locational attribution
ALMCE	LMCE plus a uniform adjustment so assigned emissions match actual total emissions	Hybrid accounting/operational use

This taxonomy is explicit in ElectricityEmissions.jl, which implements ACE, LMCE, LACE, and ALMCE and evaluates their consequences for shifting decisions (Gorka et al., 2024). The market-clearing sensitivity paper adds an important nuance: both LMCE and LACE can be negative, meaning that increasing load at a specific node can reduce total system emissions under the prevailing dispatch and network conditions (Lu, 2024). In its 3-bus example, LMCE at Bus 2 is $-1$ tCO$_2$/MWh, while Bus 3 has $0.8$ tCO$$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$0/MWh, illustrating that nodal marginal effects can diverge sharply even within a small constrained network (Lu, 2024).

A separate but related set of formulas arises in accounting-oriented work. The residual carbon intensity paper defines

$$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$1

for the residual grid mix after contracted carbon-free energy is removed, and

$$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$2

for consumer-specific market-based carbon intensity (Maji et al., 2023). These are not MCI formulas, but they are central to distinguishing consequential operational signals from accounting-adjusted averages.

3. Estimation methodologies

MCI estimation spans dispatch-based, sensitivity-based, perturbation-based, and statistical methods. ElectricityEmissions.jl computes LMCE by solving a piecewise-linear DC optimal power flow, linearizing the active set around the operating point, extracting a generator-redispatch sensitivity matrix $$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$3 from $$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$4, and then evaluating the emissions-weighted marginal response (Gorka et al., 2024). The market-clearing sensitivity model instead differentiates the KKT system of a linear market-clearing problem with PTDF-based constraints, solves the resulting linearized system with an SVD pseudoinverse, and maps dispatch sensitivity into LMCE via the chain rule (Lu, 2024).

A different dispatch-based benchmark appears in the German hourly MEF study. There, EM.POWER Dispatch computes baseline hourly emissions in a rolling 72-hour horizon, then reruns the dispatch with demand increased by 1 MWh in a selected hour and estimates the benchmark MEF as the difference in total emissions divided by the 1 MWh perturbation (Amor et al., 2024). The paper stresses that this benchmark reflects cross-border trade, must-run feed-in, startup decisions, storage optimization, and renewable curtailment, so the resulting MEF is a system marginal response rather than a simple stack lookup (Amor et al., 2024). Because this procedure requires 9,125 runs per normal year, the same paper proposes a Markov Switching Dynamic Regression model as a faster surrogate and reports that it is more accurate than Dynamic Linear Regression in estimating marginal emission factors (Amor et al., 2024).

In European market analysis, hourly MEFs are also approximated with merit-order methods. The study on price-based demand response defines plant-specific emissions intensity as

$$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$5

constructs a merit order using plant or virtual-plant marginal costs, identifies the marginal unit from hourly residual load, and then assigns the hourly MEF as the emissions intensity of that marginal unit, adjusted by transmission efficiency (Fleschutz et al., 2020). The paper proposes both a plant-level method and a piecewise-linear virtual-plant method and applies them to 20 European countries for 2017–2019 (Fleschutz et al., 2020).

At California scale, the carbon emission flow tracing study separates average and marginal computation. Its main algorithm is a linear-time graph-based method for exact quantification of nodal average emission rates under the tracing model, using strongly connected component collapse and topological propagation on an OPF-derived directed graph. Marginal emissions, however, are computed by perturbing nodal load, re-solving OPF, and forming the finite-difference ratio

$$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$6

which the paper interprets as the nodal marginal emission rate (Shen et al., 24 Jul 2025). This reinforces a broad pattern in the literature: average attribution is often flow-traced, whereas MCI is usually redispatch-based.

4. Operational uses in computing and flexible demand

MCI is attractive because it aligns directly with operational decisions about when and where to consume electricity. In the AI cloud-measurement paper, operational emissions are written as $$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$7, with $$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$8 taken to be location-based, time-specific marginal emissions from WattTime, and the practical computation is the time sum of interval energy multiplied by interval MCI at 5-minute resolution (Dodge et al., 2022). Using this signal across 16 Microsoft Azure regions, the paper finds that the same BERT pretraining run can vary from roughly $$e_{D,j,t}^{lmce} = \frac{\delta E}{\delta p_{D,j,t}}$$9 g to $\partial f(x)/\partial b_i$0 g CO$\partial f(x)/\partial b_i$1e depending on region, and projected full 6B Transformer training can vary from 21 to 78 metric tons CO$\partial f(x)/\partial b_i$2e (Dodge et al., 2022). Time shifting matters as well: in Central US, the same BERT finetuning run varied by up to 8% depending on start time (Dodge et al., 2022).

For optimization rather than measurement, the building-level carbon-aware EMS paper integrates day-ahead and real-time MCI directly into a MILP/MPC objective through a monetized carbon term,

$\partial f(x)/\partial b_i$3

so that marginal emissions act as a carbon adder on imports, storage charging, and flexible demand shifts (Cho et al., 9 Mar 2026). Using real-world PJM data, the simulations report a 22.5% reduction in emissions with only a 1.7% increase in cost at a carbon price of $\partial f(x)/\partial b_i$4 (Cho et al., 9 Mar 2026).

ElectricityEmissions.jl offers one of the clearest controlled comparisons between marginal and non-marginal operational signals. In its data-center load-shifting experiments on the RTS-GMLC test system, LMCE is the only tested metric that reduces actual total system emissions after redispatch: actual system emissions fall from 15.828 to 15.669 million tons CO$\partial f(x)/\partial b_i$5, a reduction of 1.00% (Gorka et al., 2024). By contrast, shifting according to ACE reduces the shifter’s assigned emissions but increases actual system emissions to 15.880 million tons, while ALMCE slightly increases actual system emissions and LACE leaves them almost unchanged (Gorka et al., 2024). The paper’s conclusion is explicit: marginal emissions are the appropriate signal for operational shifting, whereas accounting-oriented metrics can reallocate emissions rather than reduce them (Gorka et al., 2024).

The European demand-response study reaches a parallel result with hourly MEFs. Price-based daily 1 kWh shifts from the most expensive hour to the cheapest hour increased operational carbon emissions in 8 of 20 countries and produced an average increase of 2.1% across all countries, even while reducing electricity costs by 10.4% (Fleschutz et al., 2020). Switching from price-based to MEF-based shifting yields an average emissions reduction of 35%, albeit with 56% lower monetary cost savings than price-based shifting (Fleschutz et al., 2020). The practical implication is that low price and low MCI are not equivalent under existing merit orders.

5. Carbon-aware computing beyond strict MCI

A recurrent finding in systems papers is that “carbon-aware” does not imply “MCI-aware.” Several prominent cloud and edge scheduling frameworks use time-varying carbon intensity, but not marginal emissions factors. Google’s Carbon-Intelligent Computing System uses “near-term (48-hour) forecasts for average carbon intensities” from Tomorrow/electricityMap, denoted $\partial f(x)/\partial b_i$6, in a risk-aware optimization that generates daily Virtual Capacity Curves for flexible workloads (Radovanovic et al., 2021). The paper explicitly notes that the deployed system does not use marginal carbon intensity operationally (Radovanovic et al., 2021).

MAIZX is similarly carbon-aware but not MCI-aware. Its central emissions model is

$\partial f(x)/\partial b_i$7

where $\partial f(x)/\partial b_i$8 is energy consumption, $\partial f(x)/\partial b_i$9 is Power Usage Effectiveness, and $$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$0 is carbon intensity (Ruilova et al., 24 Jun 2025). The paper uses 2022 Electricity Maps carbon intensity data, power telemetry every 20 seconds, and hourly carbon intensity across Spain, the Netherlands, and Germany; it never states that the carbon signal is marginal, and its emissions reductions are therefore best interpreted as reductions in estimated operational CO$$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$1 under an average-intensity-style model (Ruilova et al., 24 Jun 2025).

The same pattern appears in adaptive inference and edge scheduling. The DNN inference paper defines carbon intensity as “the amount of CO$$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$2 emissions produced per unit of electricity consumed,” uses historical 30-minute carbon intensity traces from June 2023 in South West England, and adaptively switches among model sizes according to that signal; it does not define or use MCI (Jung, 2024). CarbonEdge uses Electricity Maps hourly traces for 148 carbon zones, maps each edge site to a zone, and optimizes workload placement using $$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$3, “Average carbon intensity of server $$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$4,” subject to latency and capacity constraints (Wu et al., 19 Feb 2025). The paper is explicit that the operative signal is average carbon intensity, not marginal carbon intensity (Wu et al., 19 Feb 2025).

At the container and application level, Carbon Containers also relies on average carbon intensity. Its core enforcement rule is

$$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$5

where $$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$6 is average power over the monitoring interval and $$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$7 is regional average grid carbon intensity from electricityMap (Thiede et al., 2023). The system exposes a per-container carbon-rate cap in g$$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$8CO$$(Carbon\ Intensity)_{avg} = \frac{\sum (E_i \cdot CEF_i)}{\sum E_i},$$9e/hr and enforces it by vertical scaling, migration, and suspend/resume, but it is an average-CI control framework rather than an MCI system (Thiede et al., 2023).

These examples matter because they delimit what many published “carbon reductions” actually mean. A plausible implication is that reductions reported under $$e_{D,j,t}^{avg} = e_{D,t}^{avg} = \frac{E^{tot}_{G,t}}{\sum_{j\in\mathcal{D}} p_{D,j,t}},$$0, $$e_{D,j,t}^{avg} = e_{D,t}^{avg} = \frac{E^{tot}_{G,t}}{\sum_{j\in\mathcal{D}} p_{D,j,t}},$$1, or average-forecast placement objectives are reductions under attributional or accounting-style operational models, not validated reductions in true marginal system emissions. Several of the cited papers state this distinction directly (Ruilova et al., 24 Jun 2025, Wu et al., 19 Feb 2025, Thiede et al., 2023).

6. Accounting conflicts, critiques of MCI, and proposed alternatives

MCI is not only an operational signal; it is also a contested object. One axis of controversy concerns accounting boundaries. The residual-intensity paper shows that location-based, market-based, and residual average carbon intensities can diverge substantially when PPAs and RECs are hidden from public carbon information services (Maji et al., 2023). In its empirical analysis of 123 Electricity Maps regions, up to 66.07% of renewable energy can be double-counted, and carbon intensity estimation errors can reach 194%; in South Australia, the total-mix CI is 125.67 g/kWh while the residual CI rises to 370.22 g/kWh when all solar and wind are treated as contracted (Maji et al., 2023). In a California EV-charging case study, reported emissions intensity is 75.7 g/kWh under the public signal but 194.5 g/kWh under residual accounting, a 156% discrepancy (Maji et al., 2023). The paper does not present MCI itself, but it argues that physical marginal emissions, location-based average CI, and residual/accounting-adjusted signals should not be conflated (Maji et al., 2023).

A second axis of controversy concerns whether MCI is itself an adequate operational metric. The 2025 statement paper argues that MCI is “neither reliable nor actionable” for either carbon accounting or grid-flexibility optimization because it is non-observable, reliant on opaque predictive models, and lacks verifiability (Wiesner et al., 15 Jul 2025). The paper further argues that MCI fails to indicate how much excess power is available and can miss cases where excess power arises from inflexible high-carbon plants such as coal rather than from renewable curtailment (Wiesner et al., 15 Jul 2025). On that basis, it advocates moving beyond MCI toward direct reporting of excess power, explicit modeling of energy storage and grid stability, and integration with emerging granular renewable energy certificate markets (Wiesner et al., 15 Jul 2025).

The debate is therefore not simply “average versus marginal.” ElectricityEmissions.jl explicitly frames the choice of carbon signal as use-case dependent: accounting-oriented signals are appropriate when assigned emissions must sum to actual total emissions, whereas marginal signals are more appropriate when the question is how short-run operations affect actual system emissions (Gorka et al., 2024). The current literature does not resolve this tension into a single canonical metric. Instead, it establishes that operational decarbonization, carbon attribution, and market or contractual accounting are distinct objectives that can require different signals (Gorka et al., 2024, Maji et al., 2023, Wiesner et al., 15 Jul 2025).