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Fast Grid Emissions Sensitivities using Parallel Decentralized Implicit Differentiation

Anthony Degleris, Lucas Fuentes Valenzuela, Ram Rajagopal, Marco Pavone, Abbas El Gamal

TL;DR

The paper tackles the expensive computation of dynamic locational marginal emissions (LMEs) in large, sparse power networks with intertemporal constraints. It advances a parallel, reverse-mode decentralized implicit differentiation framework that never forms the full solution-map Jacobian, leveraging vector-Jacobian products to achieve scalability. The key contributions include a dual-decomposition approach to decouple time periods, a method to compute the coupling Jacobian, and a rigorous demonstration that parallelization is essential for achieving speedups in sparse systems, with empirical results showing up to 15× speedups on 500-node networks. The approach generalizes beyond LMEs to arbitrary convex dispatch models, offering substantial practical impact for real-time emissions signaling and emissions-driven operational planning in grids with high renewable and storage penetration.

Abstract

Marginal emissions rates -- the sensitivity of carbon emissions to electricity demand -- are important for evaluating the impact of emissions mitigation measures. Like locational marginal prices, locational marginal emissions rates (LMEs) can vary geographically, even between nearby locations, and may be coupled across time periods because of, for example, storage and ramping constraints. This temporal coupling makes computing LMEs computationally expensive for large electricity networks with high storage and renewable penetrations. Recent work demonstrates that decentralized algorithms can mitigate this problem by decoupling timesteps during differentiation. Unfortunately, we show these potential speedups are negated by the sparse structure inherent in power systems problems. We address these limitations by introducing a parallel, reverse-mode decentralized differentiation scheme that never explicitly instantiates the solution map Jacobian. We show both theoretically and empirically that parallelization is necessary to achieve non-trivial speedups when computing grid emissions sensitivities. Numerical results on a 500 node system indicate that our method can achieve greater than 10x speedups over centralized and serial decentralized approaches.

Fast Grid Emissions Sensitivities using Parallel Decentralized Implicit Differentiation

TL;DR

The paper tackles the expensive computation of dynamic locational marginal emissions (LMEs) in large, sparse power networks with intertemporal constraints. It advances a parallel, reverse-mode decentralized implicit differentiation framework that never forms the full solution-map Jacobian, leveraging vector-Jacobian products to achieve scalability. The key contributions include a dual-decomposition approach to decouple time periods, a method to compute the coupling Jacobian, and a rigorous demonstration that parallelization is essential for achieving speedups in sparse systems, with empirical results showing up to 15× speedups on 500-node networks. The approach generalizes beyond LMEs to arbitrary convex dispatch models, offering substantial practical impact for real-time emissions signaling and emissions-driven operational planning in grids with high renewable and storage penetration.

Abstract

Marginal emissions rates -- the sensitivity of carbon emissions to electricity demand -- are important for evaluating the impact of emissions mitigation measures. Like locational marginal prices, locational marginal emissions rates (LMEs) can vary geographically, even between nearby locations, and may be coupled across time periods because of, for example, storage and ramping constraints. This temporal coupling makes computing LMEs computationally expensive for large electricity networks with high storage and renewable penetrations. Recent work demonstrates that decentralized algorithms can mitigate this problem by decoupling timesteps during differentiation. Unfortunately, we show these potential speedups are negated by the sparse structure inherent in power systems problems. We address these limitations by introducing a parallel, reverse-mode decentralized differentiation scheme that never explicitly instantiates the solution map Jacobian. We show both theoretically and empirically that parallelization is necessary to achieve non-trivial speedups when computing grid emissions sensitivities. Numerical results on a 500 node system indicate that our method can achieve greater than 10x speedups over centralized and serial decentralized approaches.
Paper Structure (17 sections, 1 theorem, 20 equations, 2 figures)

This paper contains 17 sections, 1 theorem, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Locational Marginal Emissions
  4. Dispatch Model
  5. Marginal Emissions
  6. Solution via Implicit Differentiation
  7. Computational Complexity
  8. Efficient LMEs via Dual Decomposition
  9. Dual Decomposition
  10. Computing the Coupling Jacobian
  11. Computational Complexity
  12. Parallel Computation
  13. Numerical Experiments
  14. Data
  15. Computational Performance
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose the solution $z_0$ to eq:dispatch exists uniquely for $d_0 \in \mathbf{R}^{N \times T}$, i.e., $F(z_0, d_0) = 0$ and $F(z, d_0) \neq 0$ for $z \neq z_0$. In addition, suppose $F(z, d)$ is twice continuously differentiable in both its arguments. Then there exists a function $z^*(d)$ such that for all $d \in \Omega$, where $d_0 \in \Omega \subset \mathbf{R}^{N \times T}$. Moreover, the funct

Figures (2)

  • Figure 1: Computation time for both the centralized and decentralized differentiation methods on 50-node and 500-node networks with increasing storage penetration. Forward mode differentiation is much less efficient than reverse mode. Parallel computation enables significant speedup in reverse mode.
  • Figure 2: Speedups achieved for different numbers of allocated threads in the 500-node networks. The speedup is defined as the minimum centralized runtime divided by the minimum decentralized runtime. The speedup curves consists of a linear regime followed by a performance plateau, as predicted. The maximum speedup achieved increases with parallelism, and decreases with the number of batteries in the system.

Theorems & Definitions (1)

  • Theorem 1: Implicit Function Theorem, Dontchev2009-ep