Comparing analytic and data-driven approaches to parameter identifiability: A power systems case study

TL;DR

A study comparing and contrasting analytical and data-driven approaches to quantify parameter identifiability and, importantly, perform parameter reduction tasks is reported on.

Abstract

Parameter identifiability refers to the capability of accurately inferring the parameter values of a model from its observations (data). Traditional analysis methods exploit analytical properties of the closed form model, in particular sensitivity analysis, to quantify the response of the model predictions to variations in parameters. Techniques developed to analyze data, specifically manifold learning methods, have the potential to complement, and even extend the scope of the traditional analytical approaches. We report on a study comparing and contrasting analytical and data-driven approaches to quantify parameter identifiability and, importantly, perform parameter reduction tasks. We use the infinite bus synchronous generator model, a well-understood model from the power systems domain, as our benchmark problem. Our traditional analysis methods use the Fisher Information Matrix to quantify parameter identifiability analysis, and the Manifold Boundary Approximation Method to perform parameter reduction. We compare these results to those arrived at through data-driven manifold learning schemes: Output - Diffusion Maps and Geometric Harmonics. For our test case, we find that the two suites of tools (analytical when a model is explicitly available, as well as data-driven when the model is lacking and only measurement data are available) give (correct) comparable results; these results are also in agreement with traditional analysis based on singular perturbation theory. We then discuss the prospects of using data-driven methods for such model analysis.

Figures (14)

• Figure 1: The manifold boundary approximation method. A model manifold, $\mathcal{M}$, is the image of parameter space $\Theta$ in data space, $\mathcal{D}$, under the model map $\boldsymbol{Y}$. Sloppy model manifolds are long and thin, exhibiting a low effective dimensionality. Here, the corner of the manifold is stretched out to be nearly one dimensional. Segments of the boundary correspond to different approximations, such as the two singularly perturbed limits $T"_{d0} \rightarrow 0$ (red) and $T"_{q0} \rightarrow 0$ (blue). Since the model is sloppy, each of these segments capture most of the expressivity of the original model and either is a good approximation of the full manifold.
• Figure 2: Dynamic Response of the model at the default parameter values.
• Figure 3: Information Spectrum of the Synchronous Generator model for long time dynamics; the participation of the "bare" parameters in the spectrum eigenvectors is given in the form of a heat map (color scale on the right).
• Figure 4: Geodesic oriented in sloppiest direction of the full model manifold in log-parameters.
• Figure 5: Geodesic oriented in sloppiest direction of the 9 parameter manifold in log-parameters.