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Admittance Matrix Concentration Inequalities for Understanding Uncertain Power Networks

Samuel Talkington, Cameron Khanpour, Rahul K. Gupta, Sergio A. Dorado-Rojas, Daniel Turizo, Hyeongon Park, Dmitrii M. Ostrovskii, Daniel K. Molzahn

TL;DR

This work derives spectral concentration bounds for random admittance matrices in power networks, enabling principled control of uncertainty in topology and line parameters. By modeling Y as $Y = A^{\top} W A$ and applying matrix Bernstein-type inequalities, the authors obtain bounds on $\mathbb{E}[\|Y\|]$ and tail probabilities, and extend these to random contingencies and the LinDistFlow/LCPF frameworks. They decompose the linear power-flow operator $F$ into Kronecker-structured elementary Jacobians, derive per-edge norm bounds and matrix-variance statistics, and establish concrete error bounds for a family of linearizations under uncertainty. The results support probabilistic guarantees for contingency analysis, network reconfiguration, and the evaluation of linearizations in specialized problems, enabling robust screening and decision-making in uncertain power networks.

Abstract

This paper presents probabilistic bounds for the spectrum of the admittance matrix and classical linear power flow models under uncertain network parameters; for example, probabilistic line contingencies. Our proposed approach imports tools from probability theory, such as concentration inequalities for random matrices with independent entries. It yields error bounds for common approximations of the AC power flow equations under parameter uncertainty, including the DC and LinDistFlow approximations.

Admittance Matrix Concentration Inequalities for Understanding Uncertain Power Networks

TL;DR

This work derives spectral concentration bounds for random admittance matrices in power networks, enabling principled control of uncertainty in topology and line parameters. By modeling Y as and applying matrix Bernstein-type inequalities, the authors obtain bounds on and tail probabilities, and extend these to random contingencies and the LinDistFlow/LCPF frameworks. They decompose the linear power-flow operator into Kronecker-structured elementary Jacobians, derive per-edge norm bounds and matrix-variance statistics, and establish concrete error bounds for a family of linearizations under uncertainty. The results support probabilistic guarantees for contingency analysis, network reconfiguration, and the evaluation of linearizations in specialized problems, enabling robust screening and decision-making in uncertain power networks.

Abstract

This paper presents probabilistic bounds for the spectrum of the admittance matrix and classical linear power flow models under uncertain network parameters; for example, probabilistic line contingencies. Our proposed approach imports tools from probability theory, such as concentration inequalities for random matrices with independent entries. It yields error bounds for common approximations of the AC power flow equations under parameter uncertainty, including the DC and LinDistFlow approximations.
Paper Structure (20 sections, 8 theorems, 88 equations, 1 figure)

This paper contains 20 sections, 8 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions
  4. Proposed Approach
  5. Plan of the paper
  6. Bounding the Spectrum of Admittance Matrices
  7. An illustrative example: Admittances bounded by 1 per-unit
  8. Uncertain contingencies
  9. The Linear Coupled Power Flow Model: Formulation and bounds
  10. Spectral properties of elementary power flow Jacobians
  11. Boundedness
  12. Matrix variance of the LCPF model under uncertainty
  13. Matrix variance statistic
  14. Application: Bounding the Error of a Family of Power Flow Linearizations
  15. Conclusions
  16. ...and 5 more sections

Key Result

Theorem 1

Consider a power system with $n$ nodes and $m$ lines. Let $\Delta = \max_{i} \deg(i)$ be the maximum degree of any node in the network. Suppose that the admittances are distributed according to any bounded distribution $w_l \sim \mathcal{D}$ that satisfies $\left|w_l\right|\leq 1$. Then, we have

Figures (1)

  • Figure 1: Comparison between the analytical bound for expected operator norm $\operatorname{ E}[\left|\left|\boldsymbol{Y}\right|\right|]$ of the admittance matrix and 200 experimental samples, plotted against the number of lines in the network. In this simple numerical experiment, the networks were generated using the homogeneous Erdős-Rényi model, i.e. by switching all possible lines independently with some probability $p$, and changing $p$ to increase the number of switched lines. Minor discontinuities in the theoretical curve are due to randomness in the number of switched lines.

Theorems & Definitions (21)

  • Definition 2.1: Flat start Jacobian
  • Definition 2.2: Elementary Laplacian Matrix
  • Theorem 1: Concentration of the admittance matrix with fixed connectivity and bounded admittances
  • proof
  • Definition 2.3: Contingency factors and nodal criticality
  • Theorem 2: Concentration with fixed admittances and uncertain contingencies
  • proof
  • Definition 2.4: Intrinsic dimension
  • Remark
  • Remark
  • ...and 11 more