Physics-informed machine learning for Power Systems

Environmental and sustainability concerns are transforming the US electricity sector with aggressive targets to achieve 100% carbon pollution-free electricity by 2035. Achieving this objective while maintaining a safe and reliable power grid in the presence of intermittent renewable generation requires new operating paradigms of computationally fast and accurate decision making in dynamic and service-critical environments. Optimization and machine learning emerge as key approaches, but neither alone are sufficient for future power system control and decision making. Classical approaches for optimization (particularly for large-scale problems and in the presence of mixed integer variables) remain prohibitively slow for dynamic applications where fast and frequent decision making is required. Data driven methods offer a significant speed-up, but out-of-the-box implementations typically cannot enforce hard constraints or address mixed integer variables. To close this gap of accuracy from model-based approaches and speed from data-driven methods, we propose the domain of physics-informed machine learning

Physics-informed ML: This domain aims to bridge the gap between model-based and data-driven methods. We take the knowledge of our system (ex. physical constraints, observed patterns and behaviours) and embed these strategically within ML frameworks. The benefit is that data-driven methods can provide incredible speed-ups in decision making, by transferring some of the computational burden offline. For example, neural networks train offline and try to learn the patterns that exist in the dataset. They then use this information in real time to make decision. 

AI + Optimization: Neural networks can learn patterns that may emerge in repeatedly solving a optimization problem.

Neural networks: Think of these as excellent functional approximators. With different activation functions and number of layers, neural networks can represent complex relationships with highly nonlinear functions. 

End-to-end learning-based optimization framework for dynamic grid reconfiguration

The problem: In this project, we consider a problem in power grid operations: topology optimization. Imagine the power grid as a graph consisting of nodes (electrical buses where loads and generators are connected) and edges (the physical power lines connecting buses to one another). In topology optimization, we can reorganize the graph, to connect different nodes to one another by changing which edges are present. In the physical power grid, this consists of opening or closing switches in lines to either add or remove the electrical connection, sometimes called grid reconfiguration. 

The math in words: To carry out topology optimization, we can write the decision as a mixed integer optimization problem: we want to decide which switches to turn on or off (the integer, or binary, decision) and also decide how much power each generator should produce (the continuous decision). We make these decisions to minimize a system cost (the objective function of the optimization problem). Let's consider the distribution system which distributes power from the transmission grid down to end use customers, like to the room you're sitting in right now. The distribution grid consists of lossy power lines, so a utility company may be interested in delivering power as efficiently as possible -- to maximize efficiency, they can minimize the line losses incurred from transporting power. Our objective function is then to minimize line losses, typically written as a quadratic function. The remainder of the optimization problem consists of the constraints: 

Our method: We propose a physics-informed ML approach. Specifically, we want to embed the discrete decisions of switch on/off status directly within the machine learning loop, so that our framework can be trained end-to-end. Our ML framework consists of a very simple neural network (just 2 layers based on the Universal approximation theorem) which predicts the probability that every switch is open or closed, and the voltages at all nodes in the grid. Then, we design a physics-informed rounding (PhyR) layer which converts the switch open/close probabilities to discrete decisions. We leverage the topological constraints (mainly the radiality) to design the PhyR layer. The radiality constraint explicitly describes the number of switches that can be open or closed in the grid -- let's call this X. We use this information to consider the relative probabilities of the switches: we close X switches with the highest probability of being closed (i.e. the power line is added) and we open the remaining switches (i.e. the power line is removed). This simple approach allows us to effectively learn how to solve the topology optimization problem. Additional details of the framework can be found in our paper.

Some results: We test our framework on distribution grids with different load and generation profiles. Our results show that we can learn to solve the topology optimization problem. To provide some intuitive understanding of why our framework with PhyR works, we summarize some key results here:

GraPhyR -- Embedding more physics: In the framework above we used a very simple neural network architecture, a feedforward neural network with just 2 layers. But can we embed more physics using another architecture? Yes -- let's consider graph neural networks (GNNs)! With a GNN, we can embed the underlying graph describing the power grid into the neural architecture. Rather than trying to learn how nodes are connected to one another, we explicitly tell the GNN these connections. In the framework above, we replace the neural network with a GNN, and continue to use the embedded discrete decisions permitted by PhyR.

To embed the switch decisions into the GNN, we use a few tricks:

You can see more details and simulation results in our paper.