⚡ electrical engineering

LACE-S: Toward Sensitivity-consistent Locational Average Carbon Emissions via Neural Representation

This paper proposes LACE-S, a neural representation-based metric for locational average carbon emissions that ensures physical validity and sensitivity consistency across the entire loading region, thereby enabling spatial load shifting strategies that reliably reduce system-wide emissions rather than paradoxically increasing them.

Original authors: Young-ho Cho, Min-Seung Ko, Hao Zhu

Published 2026-04-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Young-ho Cho, Min-Seung Ko, Hao Zhu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Carbon GPS" Problem

Imagine the electrical grid as a massive, complex city. In this city, electricity is the water flowing through pipes, and power plants are the water treatment facilities. Some facilities use clean rainwater (wind/solar), while others use dirty, smoggy swamp water (coal/gas).

When you turn on a light in your house, you are "drinking" water from this mix. The big question is: How much "smog" is your specific lightbulb responsible for?

Currently, we have a few ways to answer this, but they are like broken GPS systems:

The "Average" GPS: It tells you the average smog of the whole city. It doesn't care if you live next to a clean factory or a dirty one.
The "Marginal" GPS: It tells you the smog of the next drop of water you drink. It's very precise for a tiny sip, but if you try to drink a whole glass, the map gets confusing and wrong.
The "Path" GPS: It calculates smog based on a specific route you took yesterday. If you take a different route today, the map is useless.

The Problem: When we try to use these broken maps to tell people to move their electricity usage to "cleaner" times or places (a process called Spatial Load Shifting), the maps often lie. They tell people to move their load to a "clean" spot, but because the map is wrong, the power grid actually ends up burning more coal. It's like a GPS telling you to take a shortcut that actually leads you into a traffic jam.

The Solution: LACE-S (The "Smart, Sensitivity-Aware" Map)

The authors of this paper built a new, super-smart GPS called LACE-S. Instead of using a static map, they used a Neural Network (a type of AI) to learn the entire city's layout at once.

Here is how they made it work, using three simple rules:

1. The "Total Bill" Check (Emission Balance)

Imagine a dinner party where the total bill is $100. If you ask everyone to split the cost, the sum of their individual shares must equal $100.

Old Maps: Sometimes, when you add up everyone's share, the total is $90 or $110. The math is broken.
LACE-S: The AI has a special "projection layer" (a mathematical safety net) that forces the numbers to always add up perfectly to the total bill. No matter how the load shifts, the total carbon count remains accurate.

2. The "Sensitivity" Check (The Ripple Effect)

Imagine dropping a pebble in a pond. The ripples spread out.

Old Maps: They might tell you that dropping a pebble in one spot creates ripples everywhere, or nowhere. They miss the nuance.
LACE-S: The AI learns exactly how a tiny change in demand at one house ripples through the grid to change the emissions at other houses. It understands that if you turn on a heater in Zone A, it might force a dirty generator in Zone B to kick in, but turning it on in Zone C might not. It captures these "ripples" (sensitivities) perfectly.

3. The "Neighborhood" Check (Jacobian Regularization)

In a real city, if you change your water usage, it mostly affects your immediate neighborhood, not the whole country.

Old Maps: They sometimes think a change in your house affects a generator 500 miles away, which is unrealistic.
LACE-S: The AI is trained with a special rule (regularization) that says, "Hey, keep it local." It learns that your house is part of a specific "cluster" or neighborhood. It ensures the math reflects that your actions mostly impact your local group of power plants, making the map much more realistic.

The "Zone" Shortcut: ZACE-S

Calculating the exact carbon cost for every single house in a huge city is computationally heavy (like trying to count every grain of sand on a beach).

To fix this, the authors created ZACE-S. Instead of mapping every house, they map Zones (like neighborhoods or districts).

Imagine grouping 100 houses into one "Super-House."
The AI calculates the carbon cost for the whole neighborhood.
Result: It's much faster and uses less computer power, but it's still accurate enough to guide people to cleaner energy.

The Grand Test: Did it Work?

The authors tested this new system on a standard electrical grid model (the IEEE 30-bus system).

The Old Way: When they told people to shift their electricity usage based on old maps, the total pollution went up. The "GPS" led them into a trap.
The LACE-S Way: When they used the new AI map, the total pollution went down.
The Result: LACE-S was the only metric that consistently guided the grid toward cleaner energy without accidentally making things worse. It was almost as good as a "perfect" theoretical solution (which requires knowing the future, something humans can't do).

The Takeaway

Think of LACE-S as a smart, self-correcting carbon GPS.

Old Metrics: Like a paper map that is only accurate for one street. If you drive two blocks away, you get lost and drive into a swamp.
LACE-S: Like a live, AI-driven navigation system that knows the whole city, understands how traffic flows, and guarantees that if you follow its directions, you will actually arrive at a cleaner destination.

This technology is a huge step forward because it allows us to finally trust the signals we send to consumers to help them reduce carbon emissions, turning "green energy" from a good idea into a reliable reality.

1. Problem Statement

The paper addresses the critical challenge of designing accurate locational carbon emission metrics for power grids to guide demand-side decarbonization, specifically Spatial Load Shifting (SLS).

The Gap: Existing metrics fail to provide globally consistent emission allocations across varying operating conditions.
- Flow-based metrics (e.g., CEF): Rely on proportional sharing principles that often yield counterintuitive allocations due to complex grid topologies (loop flows) and fail to capture marginal sensitivities under congestion.
- Point-wise metrics (e.g., LMCE): Calculate marginal sensitivities at a single operating point. They cannot allocate total emissions or capture patterns beyond that specific point.
- Trajectory-based metrics (e.g., LACE-R): Integrate marginal sensitivities along a specific loading path. While they balance total emissions, they are limited to that specific path and cannot generalize to other operating regions.
The Consequence: When these flawed metrics are used to signal carbon-sensitive resources (like flexible loads), they often paradoxically increase system-wide emissions after load shifting, rather than reducing them.

2. Methodology

The authors propose LACE-S (Locational Average Carbon Emissions - Sensitivity-consistent), a metric derived via a Neural Representation approach. The core idea is to use a neural network (NN) to learn a continuous mapping from nodal load profiles to locational emission factors that is valid across the entire loading region.

A. Neural Architecture

Input/Output: The NN takes nodal load profiles ( $d$ ) as input and outputs locational emission factors ( $\lambda$ ).
Structure: A feedforward network with $L$ $L$ hidden layers.
- Activation: ReLU for hidden layers; Sigmoid for the output layer to bound values within the range of generator emission factors.
- Sparsity: The first and last layers are sparsified based on $k$ -means clustering of buses with similar marginal generator responses, reflecting the modularity of power systems.

B. Key Constraints and Loss Functions

To ensure physical validity and accuracy, the model is trained with a composite loss function:

Total Emission Balance (Projection Layer):
- An explicit projection layer maps the raw NN output ( $\hat{\lambda}$ ) to a final output ( $\tilde{\lambda}$ ) that strictly satisfies the balance condition: $\sum d_i \tilde{\lambda}_i = E_{total}$ .
- A Balance Loss ( $L_\lambda$ ) minimizes the deviation between the raw output and the projected output.
Sensitivity Consistency (Gradient Loss):
- Since ground-truth LACE values are unknown, the model uses known Locational Marginal Carbon Emissions (LMCE) as gradient labels.
- A Sensitivity Loss ( $L_\mu$ ) minimizes the difference between the NN's estimated marginal sensitivities ( $\partial(d^\top \hat{\lambda})/\partial d$ ) and the true LMCE values.
Jacobian-based Regularization:
- To capture the underlying partition of load buses (where buses in the same cluster share similar generator responses), the authors regularize the Jacobian matrix ( $J = \partial \hat{\lambda} / \partial d$ ).
- Block-Diagonal Regularization ( $L_{bd}$ ): Encourages zero coupling between different clusters.
- Diagonal-Dominance Regularization ( $L_{d}$ ): Encourages self-influence to be stronger than cross-influence.

C. Scalability: ZACE-S

For large-scale systems, the authors introduce ZACE-S (Zonal Average Carbon Emissions).

Strategy: Maps nodal inputs to predefined market zones ( $K \ll D$ ).
Benefit: Reduces model complexity from $O(D^2)$ to $O(D \cdot K)$ , significantly lowering memory and computational requirements while maintaining accuracy at the zonal level.

D. Training Procedure

A multi-stage training strategy is employed to ensure convergence and stability:

Initialize parameters to predict the Average Carbon Emission (ACE) baseline.
Train with the primary loss (Balance + Sensitivity).
Add block-diagonal regularization.
Add diagonal-dominance regularization.

3. Key Contributions

LACE-S Metric: Development of a sensitivity-consistent metric that generalizes over the entire loading region, overcoming the path-dependency of LACE-R and the locality of LMCE.
Neural Representation Framework: A novel approach using NNs with explicit projection layers and Jacobian regularization to enforce physical laws (emission balance) and market structures (sensitivity consistency).
Scalable Zonal Metric (ZACE-S): A strategy to reduce computational complexity for large grids by aggregating nodes into market zones without sacrificing modeling fidelity.
Effective SLS Signaling: Demonstration that LACE-S acts as a reliable signal for spatial load shifting, consistently reducing emissions where existing metrics fail.

4. Numerical Results

The models were validated on the IEEE 30-bus system with 20 loads and 6 generators (reassigned fuel types to create diverse emission profiles).

Modeling Accuracy:
- Balance: LACE-S achieved near-perfect total emission balance (average deviation < 0.005), comparable to a fully connected baseline but with fewer extreme corrections.
- Sensitivity: LACE-S significantly reduced the maximum deviation in approximating LMCE (from 8% in the baseline to <4%) and improved consistency across load scenarios.
- Structure: The learned Jacobian matrix correctly exhibited the expected block-diagonal structure, confirming the model captured the correct market partitioning.
Scalability (ZACE-S):
- Reducing the output dimension from 20 nodes to 5 zones decreased trainable parameters by 51.56% (3,200 to 1,650) and reduced training time per epoch by roughly 64%, while maintaining high accuracy.
Spatial Load Shifting (SLS) Performance:
- Scenario: Shifting flexible loads to reduce emissions.
- Result:
  - LACE-S: Successfully reduced total emissions by 0.10% (from 176.062 to 175.887 tCO2), approaching the theoretical optimal limit (Opt-shift).
  - Existing Metrics (LMCE, LACE-R, CEF): All increased system-wide emissions (by 0.05% to 0.13%) when used as signals.
- Robustness: Across 1,000 distinct load profiles, LACE-S never increased emissions, whereas the other metrics frequently led to higher emissions.

5. Significance

This work represents a significant advancement in carbon-aware grid operations.

Paradox Resolution: It solves the critical issue where current carbon signals inadvertently increase emissions by providing a metric that is consistent with global emission patterns, not just local or path-dependent ones.
Practical Implementation: By introducing ZACE-S and a scalable training procedure, the method is viable for large-scale power systems and aligns with existing zonal market structures.
Demand-Side Potential: It enables reliable demand-side management (SLS), allowing flexible loads to effectively participate in decarbonization without requiring perfect knowledge of the grid's internal state, bridging the gap between theoretical optimization and market-based implementation.