[Paper Review] Differential Privacy of Aggregated DC Optimal Power Flow Data

Fengyu Zhou, James Anderson+1 more|arXiv (Cornell University)|Mar 27, 2019

Privacy-Preserving Technologies in Data17 references4 citations

TL;DR

This paper proposes a differential privacy framework for aggregated DC optimal power flow (OPF) data by analyzing the sensitivity of generator outputs to load changes through a novel concept of $(\delta,\varepsilon)$-monotonicity. It shows that noise requirements for differential privacy depend critically on network topology, with tree-like networks requiring less noise than cyclic or bottlenecked topologies, where sensitivity can be amplified significantly.

ABSTRACT

We consider the problem of privately releasing aggregated network statistics obtained from solving a DC optimal power flow (OPF) problem. It is shown that the mechanism that determines the noise distribution parameters are linked to the topology of the power system and the monotonicity of the network. We derive a measure of "almost" monotonicity and show how it can be used in conjunction with a linear program in order to release aggregated OPF data using the differential privacy framework.

Motivation & Objective

• To address the challenge of privately releasing aggregated OPF data while protecting sensitive load information from grid operators.
• To analyze how network topology influences the sensitivity of optimal generation to load changes, which directly impacts differential privacy guarantees.
• To quantify the noise required for differential privacy using the Laplace mechanism, linking it to topological properties like monotonicity and bottleneck structures.
• To demonstrate that privacy-utility trade-offs vary significantly across different system topologies, even under the same privacy budget.
• To provide theoretical and empirical evidence that privacy guarantees depend on the structure of the power network, particularly the presence of binding constraints and generator-load connectivity.

Proposed method

• Introduces the concept of $(\delta,\varepsilon)$-monotonicity as a measure of how changes in load propagate through the network to affect generator outputs.
• Uses the Laplace mechanism to add noise to aggregated OPF results, with noise scale determined by the $L_1$-sensitivity of the OPF solution map.
• Derives the $L_1$-sensitivity of the OPF operator based on network topology, particularly identifying bottlenecks that amplify load changes.
• Applies a linear program formulation to compute the required noise parameters for a given privacy budget $(\Delta, \varrho)$, where $\Delta$ is the maximum load change and $\varrho$ is the privacy parameter.
• Analyzes three network topologies—tree, IEEE 9-bus, and ring—using the same privacy budget to compare noise requirements and distribution shapes.
• Employs the signed incidence matrix $\mathbf{C}$ and branch susceptance matrix $\mathbf{B}$ to model the DC power flow and compute power flows via $\mathbf{p} = \mathbf{C}^T \mathbf{B}^{-1} \mathbf{C} \mathbf{d}$, where $\mathbf{d}$ is the load vector.

Experimental results

Research questions

• RQ1 How does network topology affect the sensitivity of optimal generation to load changes in a DC OPF problem?
• RQ2 What is the relationship between the structure of a power network (e.g., presence of bottlenecks or cycles) and the amount of noise required for differential privacy?
• RQ3 Can the concept of $(\delta,\varepsilon)$-monotonicity be used to quantify privacy leakage in OPF data releases?
• RQ4 How does the required noise scale for differential privacy vary across different topologies when the privacy budget is fixed?
• RQ5 To what extent can the Laplace mechanism be effectively applied to OPF data, and what are the limitations imposed by network structure?

Key findings

• The required noise magnitude for differential privacy is highly dependent on network topology, with tree-like networks requiring significantly less noise than cyclic or bottlenecked networks.
• In a monotone tree network with a single bottleneck, the $L_1$-sensitivity is bounded by $\Delta$, leading to a noise scale of $\Delta / \varrho$, which is minimal for the given privacy budget.
• For the IEEE 9-bus network, which is $(\Delta, 2.01\Delta)$-monotone, the sensitivity is higher than in the monotone case, requiring more noise to achieve the same privacy level.
• In a ring network with $N=40$ buses, the system is $(\Delta, (N-4)\Delta)$-monotone, leading to a sensitivity of $36\Delta$, which results in a nearly uniform noise distribution and severely degraded utility.
• The probability density functions of the mechanism outputs become flatter as network topology increases sensitivity, indicating that privacy is harder to achieve in non-monotone or cyclic networks.
• The study confirms that differential privacy for OPF data is independent of the number of aggregation regions and depends only on the underlying network’s structural properties and sensitivity.