# Dynamic locational marginal emissions via implicit differentiation

## TL;DR

This paper introduces a model-agnostic method using implicit differentiation to compute dynamic locational marginal emissions in electricity systems, improving accuracy by accounting for temporal constraints.

## Contribution

The authors develop a novel, model-agnostic approach for calculating dynamic LMEs using implicit differentiation, applicable to complex dispatch models with temporal constraints.

## Key findings

- Incorporating dynamic constraints improves model accuracy by 8.2%.
- Static and dynamic LMEs differ with a normalized RMS deviation of 28.40%.
- The method is validated against state-of-the-art approaches using real U.S. data.

## Abstract

Locational marginal emissions rates (LMEs) estimate the rate of change in emissions due to a small change in demand in a transmission network, and are an important metric for assessing the impact of various energy policies or interventions. In this work, we develop a new method for computing the LMEs of an electricity system via implicit differentiation. The method is model agnostic; it can compute LMEs for any convex optimization-based dispatch model, including some of the complex dispatch models employed by system operators in real electricity systems. In particular, this method lets us derive LMEs for dynamic dispatch models, i.e., models with temporal constraints such as ramping and storage. Using real data from the U.S. electricity system, we validate the proposed method against a state-of-the-art merit-order-based method and show that incorporating dynamic constraints improves model accuracy by 8.2%. Finally, we use simulations on a realistic 240-bus model of WECC to demonstrate the flexibility of the tool and the importance of incorporating dynamic constraints. Namely, static LMEs and dynamic LMEs exhibit a normalized average RMS deviation of 28.40%, implying dynamic constraints are essential to accurately modeling emissions rates.