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Differentially Private Communication of Measurement Anomalies in the Smart Grid

Nikhil Ravi, Anna Scaglione, Sean Peisert, Parth Pradhan

TL;DR

The proposed framework provides a robust solution for detecting bad data while preserving the privacy of sensitive power system data and proposes a novel DP chi-square noise mechanism that ensures the test does not reveal private information about power injections or the system matrix.

Abstract

In this paper, we present a framework based on differential privacy (DP) for querying electric power measurements to detect system anomalies or bad data. Our DP approach conceals consumption and system matrix data, while simultaneously enabling an untrusted third party to test hypotheses of anomalies, such as the presence of bad data, by releasing a randomized sufficient statistic for hypothesis-testing. We consider a measurement model corrupted by Gaussian noise and a sparse noise vector representing the attack, and we observe that the optimal test statistic is a chi-square random variable. To detect possible attacks, we propose a novel DP chi-square noise mechanism that ensures the test does not reveal private information about power injections or the system matrix. The proposed framework provides a robust solution for detecting bad data while preserving the privacy of sensitive power system data.

Differentially Private Communication of Measurement Anomalies in the Smart Grid

TL;DR

The proposed framework provides a robust solution for detecting bad data while preserving the privacy of sensitive power system data and proposes a novel DP chi-square noise mechanism that ensures the test does not reveal private information about power injections or the system matrix.

Abstract

In this paper, we present a framework based on differential privacy (DP) for querying electric power measurements to detect system anomalies or bad data. Our DP approach conceals consumption and system matrix data, while simultaneously enabling an untrusted third party to test hypotheses of anomalies, such as the presence of bad data, by releasing a randomized sufficient statistic for hypothesis-testing. We consider a measurement model corrupted by Gaussian noise and a sparse noise vector representing the attack, and we observe that the optimal test statistic is a chi-square random variable. To detect possible attacks, we propose a novel DP chi-square noise mechanism that ensures the test does not reveal private information about power injections or the system matrix. The proposed framework provides a robust solution for detecting bad data while preserving the privacy of sensitive power system data.
Paper Structure (26 sections, 3 theorems, 56 equations, 6 figures, 1 table)

This paper contains 26 sections, 3 theorems, 56 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation: Collective Defense
  3. Literature Review
  4. Anomaly Detection in the Smart Grid
  5. DP for Smart Grids
  6. DP Bad Data Detection in Smart Grids
  7. Contributions
  8. Preliminaries, Threat Model and Problem Statement
  9. Measurement Model and False Data Attack
  10. BDDFDA detection via Weighted Least Squares
  11. Special Case of the Measurement Model
  12. Threat Model
  13. Differentially Private Bad Data Detection
  14. Preliminaries
  15. DP for Stochastic Queries
  16. ...and 11 more sections

Key Result

Theorem 1

[thm]thm:eps-del_chis The mechanism in eq:chi_sq_mech is $(\epsilon,\delta)$-DP for all pairs of neighboring measurement sets $\bm{z}$ and $\bm{z}'$ differing in exactly one measurement, where the guarantee $\delta$ is given by: where $\mathrm{Q}_{s}(a,b)$ is the Marcum Q-function of order $s>0$ with $a>0$ and $b\geq 0$.

Figures (6)

  • Figure 1: Illustration of bad data attacks.
  • Figure 2: Illustration of our proposed DP mechanism for FDA detection.
  • Figure 3: Sensitivity of the test's performance to attack strength.
  • Figure 4: AUROC vs. DP Normalized privacy budget and noise variance scaling ($k$-factor) for input-perturbed measurements $\tilde{\bm{z}} = \bm{z} + \bm{w}$.
  • Figure 5: The AUROC of the hypothesis test for different values of $\sigma_{\nu|z}$ when using our mechanism.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Remark 3: Internal and External Threats
  • Remark 4: Stealth Attacks
  • Definition 1: $(\epsilon,\delta)$-Differential privacy
  • Definition 2: $(\epsilon,\delta)$-Probabilistic Differential privacy
  • Definition 3: Distance one neighborhood
  • Remark 5
  • Theorem 1: Chi-square mechanism is $(\epsilon,\delta)$-DP
  • Theorem 2: Gaussian Approximation of $\mathbbm{q}(\bm{z})$
  • ...and 1 more