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Locational marginal burden: Quantifying the equity of optimal power flow solutions

Samuel Talkington, Amanda West, Rabab Haider

TL;DR

This work defines Locational Marginal Burden (LMB) as the sensitivity of the energy burden, computed via $\boldsymbol{b}(\boldsymbol{\pi}; \boldsymbol{d}) = \operatorname{diag}(\boldsymbol{d} \oslash \boldsymbol{s}) \boldsymbol{\pi}$, to changes in nodal demand through differentiable optimization of a parameterized DC OPF. By deriving the LMP solution map from OPF dual variables and applying the chain rule, the authors obtain an analytic expression for the LMB matrix $\partial \boldsymbol{b} / \partial \boldsymbol{d}$, incorporating the dual sensitivity $\partial \boldsymbol{\nu}^*/\partial \boldsymbol{d}$ and the PTDF structure $\boldsymbol{F}$. The approach is demonstrated on a synthetic Hawaii network overlaid with ACS income data, revealing that nodal LMB correlates with income (not population density) and that LMB-to-others exhibits a distinct relationship with income and density, offering a tool to guide equity-focused grid investments. The framework is model-agnostic across retail pricing models and holds potential for informing regulatory and investment decisions aimed at reducing energy burden disparities, with extensions to AC power flows and real-world datasets discussed for future work.

Abstract

Fair distribution of benefits in electric power systems is a pertinent energy policymaking problem; however, these efforts cannot be easily quantified in power system engineering studies. Therefore, we propose locational marginal burden (LMB) to provide an interface between a well-studied measure of energy pricing equity, energy burden, with an optimal power flow problem (OPF). This is achieved by investigating the intrinsic link between the dual optimal solution of an OPF problem and the electricity prices, which are used to calculate the energy burden. By applying results from the field of differentiable optimization, locational marginal prices (LMPs) associated with an OPF solution can be differentiated with respect to demand. This enables electricity retail prices, and thereby, energy burden itself, to be differentiated, resulting in the proposed LMB. Simulation of a synthetic Hawaii network interfaced with real-world socioeconomic data shows how the LMB provides new insights into how the operation of the electricity network affects the equity of energy prices.

Locational marginal burden: Quantifying the equity of optimal power flow solutions

TL;DR

This work defines Locational Marginal Burden (LMB) as the sensitivity of the energy burden, computed via , to changes in nodal demand through differentiable optimization of a parameterized DC OPF. By deriving the LMP solution map from OPF dual variables and applying the chain rule, the authors obtain an analytic expression for the LMB matrix , incorporating the dual sensitivity and the PTDF structure . The approach is demonstrated on a synthetic Hawaii network overlaid with ACS income data, revealing that nodal LMB correlates with income (not population density) and that LMB-to-others exhibits a distinct relationship with income and density, offering a tool to guide equity-focused grid investments. The framework is model-agnostic across retail pricing models and holds potential for informing regulatory and investment decisions aimed at reducing energy burden disparities, with extensions to AC power flows and real-world datasets discussed for future work.

Abstract

Fair distribution of benefits in electric power systems is a pertinent energy policymaking problem; however, these efforts cannot be easily quantified in power system engineering studies. Therefore, we propose locational marginal burden (LMB) to provide an interface between a well-studied measure of energy pricing equity, energy burden, with an optimal power flow problem (OPF). This is achieved by investigating the intrinsic link between the dual optimal solution of an OPF problem and the electricity prices, which are used to calculate the energy burden. By applying results from the field of differentiable optimization, locational marginal prices (LMPs) associated with an OPF solution can be differentiated with respect to demand. This enables electricity retail prices, and thereby, energy burden itself, to be differentiated, resulting in the proposed LMB. Simulation of a synthetic Hawaii network interfaced with real-world socioeconomic data shows how the LMB provides new insights into how the operation of the electricity network affects the equity of energy prices.
Paper Structure (37 sections, 1 theorem, 28 equations, 5 figures)

This paper contains 37 sections, 1 theorem, 28 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Equity in electric power systems
  4. Marginal pricing and emissions
  5. Differential Optimization
  6. Contributions
  7. Paper outline
  8. Problem description
  9. Energy burden: an engineering definition
  10. Locational marginal burden
  11. Combined transmission-distribution model
  12. Three retail pricing models
  13. Model 0: Transmission-level approximation
  14. Model 1: Representative model of present-day structures
  15. Model 2: Futuristic distribution-level model
  16. ...and 22 more sections

Key Result

proposition 1

Let $\boldsymbol{s} \in \mathbb{R}^n$ be the net aggregate incomes of each transmission node in an electricity network, and let $\boldsymbol{z}^* := (\boldsymbol{g}^*,\boldsymbol{p}^*,\boldsymbol{\mu}^*,\boldsymbol{\nu}^*) \in \mathbb{R}^{3k + 4m + 1}$ be a solution to the DC OPF program eq:static-d and the Jacobian matrix is non-singular, where $\boldsymbol{Q}$, $\boldsymbol{A}$, $\boldsymbol{G}

Figures (5)

  • Figure 1: Illustration of the flow of information regarding power flow (red) and energy burden (black) between a generating company (GENCO), transmission system operator (TSO), distribution system operator (DSO), and represented census tracts. The outputs intended for the TSO and DSO are in the rectangles. The census tract node is an aggregate of all the census tracts downstream of the transmission node (overseen by the DSO).
  • Figure 2: Mapping of publicly available household income data to the Hawaii synthetic network.
  • Figure 3: Static energy burden vs. mean household income in the Hawaii network, computed using \ref{['eq:burden-in-full-generality']}.
  • Figure 4: Diagonal entries of the LMB matrix vs. regional income in the synthetic Hawaii network with real income and census data.
  • Figure 5: Net marginal burden to others, i.e., off-diagonal column sums of the LMB matrix, vs. regional income in the synthetic Hawaii network with real income and census data.

Theorems & Definitions (3)

  • definition 1: Energy burden
  • definition 2: Locational marginal burden
  • proposition 1: Locational marginal burden in DC OPF