Privacy-Utility Tradeoff for Applications Using Energy Disaggregation of Smart-Meter Data

Abstract

Privacy-preserving data mining technologies have been studied extensively, and as a general approach, du Pin Calmon and Fawaz have proposed a data distortion mechanism based on a statistical inference attack framework. This theory has been extended by Erdogdu et al. to time-series data and been applied to energy disaggregation of smart-meter data. However, their theory assumes both smart-meter data and sensitive appliance state information are available when applying the privacy-preserving mechanism, which is impractical in typical smart-meter systems where only the total power usage is available. In this paper, we extend their approach to enable the application of a privacy-utility tradeoff mechanism to such practical applications. Firstly, we define a system model which captures both the architecture of the smart-meter system and the practical constraints that the power usage of each appliance cannot be measured individually. This enables us to formalize the tradeoff problem more rigorously. Secondly, we propose a privacy-utility tradeoff mechanism for that system. We apply a linear Gaussian model assumption to the system and thereby reduce the problem of obtaining unobservable information to that of learning the system parameters. Finally, we conduct experiments of applying the proposed mechanism to the power usage data of an actual household. The experimental results show that the proposed mechanism works partly effectively; i.e., it prevents usage analysis of certain types of sensitive appliances while at the same time preserving that of non-sensitive appliances.

Appendices

A Computability and Convexity of the Optimization Problem

1.1 A.1 Computability

We show that both the objective function (1) and constraint function (2) can be computed respectively and therefore the optimization problem (3) is solvable, assuming $p_{{\varvec{X}}^*, Y}$ is known.

We start with (1). First note that ${\varvec{X}} \rightarrow Y \rightarrow Z$ forms the Markov chain because the smart-meter data Y depends on the appliance states ${\varvec{X}}$ and the distorted data Z depends on the smart-meter data Y. Then,

$$\begin{aligned} p_{Z | {\varvec{X}}^*} (z | {\varvec{x}}^*) = \int _{\mathcal {Y}} p_{Z|Y}(z|y) p_{Y|{\varvec{X}}^*} (y|{\varvec{x}}^*) dy. \end{aligned}$$

(11)

Here,

$$\begin{aligned} p_{Y|{\varvec{X}}^*} (y|{\varvec{x}}^*) = \frac{p_{{\varvec{X}}^*, Y}({\varvec{x}}^*, y)}{P_{{\varvec{X}}^*}({\varvec{x}}^*)}, \end{aligned}$$

(12)

$$\begin{aligned} P_{{\varvec{X}}^*}({\varvec{x}}^*) = \int _{\mathcal {Y}} p_{{\varvec{X}}^*, Y} ({\varvec{x}}^*, y) dy. \end{aligned}$$

(13)

Also,

$$\begin{aligned} p_{Z} (z) = \sum _{{\varvec{x}}^* \in {\varvec{\mathcal {X}}}^*} P_{{\varvec{X}}^*} ({\varvec{x}}^*) p_{Z | {\varvec{X}}^*} (z | {\varvec{x}}^*). \end{aligned}$$

(14)

Therefore, we can confirm that all the members in (1) can be computed from $p_{{\varvec{X}}^*, Y}$ and $p_{Z|Y}$.

As for (2), we can easily confirm the computability by seeing that

$$\begin{aligned} p_{Y}(y) = \sum _{{\varvec{x}}^* \in {\varvec{\mathcal {X}}}^*} p_{{\varvec{X}}^*, Y}({\varvec{x}}^*, y). \end{aligned}$$

(15)

1.2 A.2 Convexity

We additionally note here that (3) is a convex optimization problem. This is because, as with [25], the objective function and the constraint function are convex functions of the optimization variable $p_{Z|Y}$.

Convex optimization has several desirable properties. From an analytical viewpoint, it is assured that any local minimum is a global minimum and finding a global minimum is therefore reduced to finding a local minimum [5]. From a practical viewpoint, efficient algorithms such as interior-point methods have been proposed, and software libraries are available [6].

B Modification to Discrete Power Data

In Sect. 3.2 we considered the case where the smart-meter data and distorted data are continuous. In practical situations, however, it is possible that the smart-meter data is quantized to discrete levels. Indeed, as we describe in detail in Sect. 4, we use discrete power data in our experiment that has been quantized to a resolution of 7 W. It is therefore required to modify the optimization problem (3) to accommodate such cases. We describe here the discretized version of the optimization problem.

Let $\tilde{Y} \in \tilde{\mathcal {Y}}$ be a discrete random variable representing the quantized smart-meter data and $\tilde{Z} \in \tilde{\mathcal {Z}}$ represent the distorted data, where $\tilde{\mathcal {Y}}$ and $\tilde{\mathcal {Z}}$ are finite sets. Let $d: \tilde{Y} \times \tilde{Z} \rightarrow \mathbb {R}^+$ be some distortion function. Then the optimization problem in (3) becomes

$$\begin{aligned} \min _{p_{\tilde{Z}|\tilde{Y}}} I({\varvec{X}}^* ; \tilde{Z}) \quad \text {subject to } E_{\tilde{Y}, \tilde{Z}} [d(\tilde{Y}, \tilde{Z})] \le \delta , \end{aligned}$$

(16)

where

$$\begin{aligned} I({\varvec{X}}^*; \tilde{Z}) = \sum _{{\varvec{x}}^* \in {\varvec{\mathcal {X}}}^*} \sum _{\tilde{z} \in \tilde{\mathcal {Z}}} P_{{\varvec{\mathcal {X}}}} ({\varvec{x}}^*) P_{\tilde{Z}|{\varvec{X}}^*} (\tilde{z}|{\varvec{x}}^*) \log \frac{P_{\tilde{Z}|{\varvec{X}}^*} (\tilde{z}|{\varvec{x}}^*)}{P_{\tilde{Z}} (\tilde{z})}, \end{aligned}$$

(17)

$$\begin{aligned} E_{\tilde{Y}, \tilde{Z}} [d(\tilde{Y}, \tilde{Z})] = \sum _{\tilde{y} \in \tilde{\mathcal {Y}}} \sum _{\tilde{z} \in \tilde{\mathcal {Z}}} P_{\tilde{Z} | \tilde{Y}} (\tilde{z} | \tilde{y}) P_{\tilde{Y}} (\tilde{y}) d(\tilde{y}, \tilde{z}). \end{aligned}$$

(18)