# An Adaptive Observer Design for Charge-State and Crossover Estimation in Disproportionation Redox Flow Batteries undergoing Self-Discharge

## TL;DR

This paper presents an adaptive observer for estimating charge state and crossover flux in redox flow batteries, using a novel model and Lyapunov-based stability analysis, validated on a laboratory prototype.

## Contribution

It introduces a new adaptive observer design for simultaneous estimation of charge and crossover flux in disproportionation redox flow batteries, with stability guarantees.

## Key findings

- Crossover flux can be modeled as a linear function of state of charge.
- The observer design is validated on a laboratory prototype.
- Stability and convergence are proven using Lyapunov analysis.

## Abstract

This article considers a model formulation and an adaptive observer design for the simultaneous estimation of the state of charge and crossover flux in disproportionation redox flow batteries. This novel nonaqueous battery chemistry allows a simple isothermal lumped parameter model to be formulated. The transport of vanadium through the porous separator is a key unknown function of battery variables and it is approximated in the space of continuous functions. The state and parameter observer adaptation laws are derived using Lyapunov analysis applied to the estimation error, the stability and convergence of which are proved. Numerical values of observer gains are calculated by solving a polytopic linear matrix inequality and equality problem via convex optimization. The performance of this design is evaluated on a laboratory flow battery prototype, and it is shown that the crossover flux can be considered a linear function of state of charge for this battery configuration during self-discharge.