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Carbon-Aware Optimal Power Flow

Xin Chen, Andy Sun, Wenbo Shi, Na Li

TL;DR

This work introduces Carbon-aware Optimal Power Flow (C-OPF), a framework that jointly optimizes power and carbon flows by integrating carbon emission flow equations with traditional OPF constraints. It establishes theoretical guarantees for the carbon flow model, including feasibility and uniqueness, and tackles unknown branch power-flow directions via a dual-flow reformulation that remains solvable with standard nonlinear optimizers. Two energy-storage carbon-footprint models are developed—water tank and load/carbon-free generator—each with corresponding accounting rules, enabling demand-side carbon management and decarbonization-oriented decisions. Numerical experiments on a modified 39-bus system demonstrate that C-OPF can reduce total grid emissions and enforce nodal carbon-intensity caps, albeit at higher computational cost than classic OPF. The results highlight C-OPF’s potential to inform carbon-aware dispatch, demand response, and policy design for grid decarbonization, while pointing to avenues for efficiency improvements and broader applications.

Abstract

To facilitate effective decarbonization of the electric power sector, this paper introduces the generic Carbon-aware Optimal Power Flow (C-OPF) method for power system decision-making that considers demand-side carbon accounting and emission management. Built upon the classic optimal power flow (OPF) model, the C-OPF method incorporates carbon emission flow equations and constraints, as well as carbon-related objectives, to jointly optimize power flow and carbon flow. In particular, this paper establishes the feasibility and solution uniqueness of the carbon emission flow equations, and proposes modeling and linearization techniques to address the issues of undetermined power flow directions and bilinear terms in the C-OPF model. Additionally, two novel carbon emission models, together with the carbon accounting schemes, for energy storage systems are developed and integrated into the C-OPF model. Numerical simulations demonstrate the characteristics and effectiveness of the C-OPF method, in comparison with OPF solutions.

Carbon-Aware Optimal Power Flow

TL;DR

This work introduces Carbon-aware Optimal Power Flow (C-OPF), a framework that jointly optimizes power and carbon flows by integrating carbon emission flow equations with traditional OPF constraints. It establishes theoretical guarantees for the carbon flow model, including feasibility and uniqueness, and tackles unknown branch power-flow directions via a dual-flow reformulation that remains solvable with standard nonlinear optimizers. Two energy-storage carbon-footprint models are developed—water tank and load/carbon-free generator—each with corresponding accounting rules, enabling demand-side carbon management and decarbonization-oriented decisions. Numerical experiments on a modified 39-bus system demonstrate that C-OPF can reduce total grid emissions and enforce nodal carbon-intensity caps, albeit at higher computational cost than classic OPF. The results highlight C-OPF’s potential to inform carbon-aware dispatch, demand response, and policy design for grid decarbonization, while pointing to avenues for efficiency improvements and broader applications.

Abstract

To facilitate effective decarbonization of the electric power sector, this paper introduces the generic Carbon-aware Optimal Power Flow (C-OPF) method for power system decision-making that considers demand-side carbon accounting and emission management. Built upon the classic optimal power flow (OPF) model, the C-OPF method incorporates carbon emission flow equations and constraints, as well as carbon-related objectives, to jointly optimize power flow and carbon flow. In particular, this paper establishes the feasibility and solution uniqueness of the carbon emission flow equations, and proposes modeling and linearization techniques to address the issues of undetermined power flow directions and bilinear terms in the C-OPF model. Additionally, two novel carbon emission models, together with the carbon accounting schemes, for energy storage systems are developed and integrated into the C-OPF model. Numerical simulations demonstrate the characteristics and effectiveness of the C-OPF method, in comparison with OPF solutions.
Paper Structure (31 sections, 1 theorem, 27 equations, 11 figures, 3 tables)

This paper contains 31 sections, 1 theorem, 27 equations, 11 figures, 3 tables.

Table of Contents

  1. Sets and Parameters
  2. Variables
  3. Introduction
  4. Related Work and Key Issues
  5. Our Contributions
  6. Carbon Emission Flow and Carbon Accounting
  7. Concept of Carbon Emission Flow
  8. Carbon Emission Flow Model for Lossy Power Networks
  9. Feasibility and Solution Uniqueness of Carbon Flow Model
  10. Use of Carbon Emission Flow for Carbon Accounting
  11. Carbon-Aware Optimal Power Flow Method
  12. Generic C-OPF Model
  13. Objective Function
  14. Power Flow Equations and Constraints
  15. Carbon Flow Equations and Constraints
  16. ...and 16 more sections

Key Result

Theorem 1

Suppose that for each $i\notin \mathcal{J}$ of the matrix $\bm{P}_{\mathrm{C}}$, there is a sequence of nonzero elements of $\bm{P}_{\mathrm{C}}$ of the form $\bm{P}_{\mathrm{C}}[i,i_1], \bm{P}_{\mathrm{C}}[i_1,i_2],\cdots, \bm{P}_{\mathrm{C}}[i_r, j]$ with $j\in \mathcal{J}$. Then, $\bm{P}_{\mathrm

Figures (11)

  • Figure 1: Comparison between carbon emission pool and carbon emission flow. (In sub-figure (a), all end-users in a large area adopt the same grid average emission factor (AEF) to calculate their attributed carbon footprints. In sub-figure (b), each pipeline represents a power line, with the width indicating the magnitude of power flow. Darker colors indicate higher carbon emission intensities. Power in-flows with different carbon emission intensities are mixed at each bus and distributed downstream.)
  • Figure 2: Illustration of virtual carbon flow and physical power flow.
  • Figure 3: A simple 3-node network case example.
  • Figure 4: Power flow and carbon flow under the "water tank" ES model.
  • Figure 5: The modified New England 39-bus test system.
  • ...and 6 more figures

Theorems & Definitions (7)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5