This paper describes a geometric approach to parameter identifiability analysis in models of power systems dynamics. When a model of a power system is to be compared with measurements taken at discrete times, it can be interpreted as a mapping from parameter space into a data or prediction space. Generically, model mappings can be interpreted as manifolds with dimensionality equal to the number of structurally identifiable parameters. Empirically it is observed that model mappings often correspond to bounded manifolds. We propose a new definition of practical identifiability based the topological definition of a manifold with boundary. In many ways, our proposed definition extends the properties of structural identifiability. We construct numerical approximations to geodesics on the model manifold and use the results, combined with insights derived from the mathematical form of the equations, to identify combinations of practically identifiable and unidentifiable parameters. We give several examples of application to dynamic power systems models.
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Parameter Estimation in the Age of Degeneracy and Unidentifiability
Parameter estimation from observable or experimental data is a crucial stage in any modeling study. Identifiability refers to one’s ability to uniquely estimate the model parameters from the available data. Structural unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern. Therefore, the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments. Classical studies on structural unidentifiability focused on the notion that one can at most identify combinations of unidentifiable model parameters. We have identified a different type of structural degeneracy/unidentifiability present in a family of models, which we refer to as the Lambda-Omega (Λ-Ω) models. These are an extension of the classical lambda-omega (λ-ω) models that have been used to model biological systems, and display a richer dynamic behavior and waveforms that range from sinusoidal to square wave to spike like. We show that the Λ-Ω models feature infinitely many parameter sets that produce identical stable oscillations, except possible for a phase shift (reflecting the initial phase). These degenerate parameters are not identifiable combinations of unidentifiable parameters as is the case in structural degeneracy. In fact, reducing the number of model parameters in the Λ-Ω models is minimal in the sense that each one controls a different aspect of the model dynamics and the dynamic complexity of the system would be reduced by reducing the number of parameters. We argue that the family of Λ-Ω models serves as a framework for the systematic investigation of degeneracy and identifiability in dynamic models and for the investigation of the interplay between structural and other forms of unidentifiability resulting on the lack of information from the experimental/observational data.
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- NSF-PAR ID:
- 10387689
- Date Published:
- Journal Name:
- Mathematics
- Volume:
- 10
- Issue:
- 2
- ISSN:
- 2227-7390
- Page Range / eLocation ID:
- 170
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/math10020170
