Identification of Power Systems with Droop-Controlled Units Using Neural Ordinary Differential Equations

TL;DR

The results demonstrate that even though SINDy yields more accurate models, NODEs achieve good prediction performance without prior knowledge about the system’s nonlinearities which SINDy requires to work best.

Abstract

In future power systems, the detailed structure and dynamics may not always be fully known. This is due to an increasing number of distributed energy resources, such as photovoltaic generators, battery storage systems, heat pumps and electric vehicles, as well as a shift towards active distribution grids. Obtaining physically-based models for simulation and control synthesis can therefore become challenging. Differential equations, where the right-hand side is represented by a neural network, i.e., neural ordinary differential equations (NODEs), have a great potential to serve as a data-driven black-box model to overcome this challenge. This paper explores their use in identifying the dynamics of droop-controlled grid-forming units based on inputs and state measurements. In numerical studies, various NODE structures used with different numerical solvers are trained and evaluated. Moreover, they are compared to the sparse identification of nonlinear dynamics (SINDy) method. The results demonstrate that even though SINDy yields more accurate models, NODEs achieve good prediction performance without prior knowledge about the system's nonlinearities which SINDy requires to work best.

Figures (6)

• Figure 1: Power, voltage and impedances between two grid-forming nodes.
• Figure 2: Solving the ivp with initial condition $x(t_s)$ and $\mathcal{N}_\Theta$ for $t_e$. Here, $\tilde{t}_1 < \tilde{t}_2 < \tilde{t}_3 < \dots$ denote the internal evaluation times chosen by the solver. Illustration motivated by legaard_constructing_2022.
• Figure 3: Repeatedly solving an ivp with different initial conditions and inputs to step the model forward in time. Illustration motivated by legaard_constructing_2022. Note that the portrayed nn serves as an example and does not show the correct dimensions of the actual model.
• Figure 4: Power system under investigation.
• Figure 5: rmse with \ref{['eq:rmse']} obtained over 1000 datasets. The large dot displays the median. The white area around the dot marks the interquartile range, containing the middle $50\%$ of the values. The lines, so called whiskers, extend to each side to the values deviating up to $1.5$ times to the interquartile range from the bounds of the white area. The small dots are outliers.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3