# Variational Methods for Energy Systems

Due to resource constraints and global climate change, there is an increased need for technological solutions to improve the efficiency and reduce waste of our energy systems. Many of these technological solutions are computationally daunting and therefore require approximate approaches. In this work, we focus on problems on both the demand and generation side of the electrical power system. On the demand side, we investigate the problem of inferring the power consumption of individual loads in a building from aggregate electrical measurements. This problem, also known as Non-Intrusive Load Monitoring or energy disaggregation, involves inference of a latent variable depicting the operational state of individual devices given a set of aggregate observations and so far the existing solutions either require supervised training or make assumptions that limit their applicability and performance in real conditions. On the generation side, we investigate the problem of finding optimal configurations of power generators in a network, also known as the AC-Optimal Power Flow problem.

In this setting, because existing solutions usually cast the problem as constrained optimization, non-linear and non-convex constraints that solutions need to adhere to, cause computational difficulties which results in most solvers lacking robustness and speed. What both problems share is the computational difficulty of inferring an optimal binary vector that describes appliance states or generator configurations, respectively. In order to alleviate the computational cost of this inference step that is otherwise NP-hard, we make use of an approximate technique called Variational Inference which translates statistical inference into an optimization problem by minimizing a divergence measure between the true and an auxiliary but tractable distribution.

Because the choice of the auxiliary distribution determines the goodness of the approximation andconsidering that for both problems the vector of interest is binary, in this thesis, we introduce an auxiliary distribution that can theoretically approximate any distribution over binary states arbitrarily well. Furthermore, in the case of Non-Intrusive Load Monitoring, because the problem requirestracking appliance states over time and modeling temporal dependencies causes the joint distribution required for Variational Inference to become intractable, computationally efficient strategies to approximate this joint distribution are introduced. We ultimately derive an, under some conditions, asymptotically unbiased algorithm for learning and inference in dynamical systems with binary latent states and cast unsupervised Non- Intrusive Load Monitoring as such a problem. The algorithm shows performance comparable to state-of-the-art competitors but overcomes many of their problems because it is truly unsupervised. In the case of AC-Optimal Power Flow, we reformulate the problem as a learning problem. Specifically, we task an agent to produce optimized generator configurations as a function of a demand assignment to the nodes in the network. The application of Variational Inference allows us to efficiently deal with non-convex generator configurations and to ultimately arrive at an algorithm that produces feasible solutions reliably and fast, i.e. it overcomes the robustness and speed issues of existing algorithms, but at the cost of sub-optimality.

## Funding

#### Supported in part by DOE grant DE-EE0007682 and the Pennsylvania Infrastructure Technology Alliance

## History

## Date

24/02/2019## Degree Type

Dissertation## Department

Civil and Environmental Engineering## Degree Name

- Doctor of Philosophy (PhD)