Battery management systems typically employ equivalent circuit models (ECMs) due to their low computational cost and simplicity. However, the development of new estimation methods is desirable to unlock the power of physics-based models and improve battery performance and longevity. We test the capabilities of a new mathematical approach to optimisation, based on recent developments in the field of measure-moment theory [1], and apply the approach to foundational problems in the field of battery management. In contrast to a Kalman filter-type approach [2], this method does not require discretisation in time to estimate the evolution of the states and parameters. Instead, a linear matrix inequality (LMI) hierarchy is constructed in terms of their moments.

We will begin by explaining the unique advantages of our approach and the types of (nonlinear) optimisation problem that can be addressed. We then demonstrate the utility of the method by numerically solving [3] a selection of battery management optimisation problems. For this, we will use the Thévenin model with a variable series resistance and synthetic data. Our first example is state-of-charge (SOC) estimation from current-voltage data perturbed by unknown measurement noise [2]. Secondly, we tackle a joint state-parameter estimation, in which both of the model states (SOC and voltage across the RC pair) are estimated along with an unknown parameter (the variable series resistance) from measurements of the output voltage and input current. The series resistance is considered a function of time as well as SOC and is identifiable when the input current is non-zero. Our third application is a constrained, optimal fast-charging problem where the aim is to identify the optimal input current profile to achieve the shortest possible charging time subject to a given set of constraints [4]. In each case, we benchmark the approach in terms of computational speed and accuracy.

References:
[1] S. Marx, E. Pauwels, T. Weisser, D. Henrion, & J.-B. Lasserre (2021). Semi-algebraic Approximation Using Christoffel–Darboux Kernel. Constructive Approximation. https://doi.org/10.1007/s00365-021-09535-4
[2] G. L. Plett (2019). Review and Some Perspectives on Different Methods to Estimate State of Charge of Lithium-Ion Batteries. J Automotive Safety and Energy, 10:3, 249-272. https://doi.org/10.3969/j.issn.1674-8484.2019.03.001
[3] D. Henrion , J.-Be. Lasserre, & J. Löfberg (2009). GloptiPoly 3: moments, optimization and semidefinite programming. Optimization Methods & Software, 24:4-5, 761-779. https://doi.org/10.1080/10556780802699201
[4] A. Tomaszewska, Z. Chu, X. Feng, S. O’Kane, X. Liu, et al. (2019). Lithium-ion battery fast charging: A review. eTransportation, 1, 100011. https://doi.org/10.1016/j.etran.2019.100011