Differential Diagnostics for Lithium Ion Battery Cells Connected in Series

This paper presents a new method for estimating the capacity of a lithium ion battery cell in the presence of a reference cell the parameters of which are well characterized in series with it. The method assumes that both cells are cycled using the same current trajectory starting from the same state of charge (e.g. fully charged). Voltage measurements for both cells as well as current measurements for the series string constitute the input to a nonlinear least squares minimization problem. The goal of this problem is to estimate the capacity of the cell given the difference between its voltage and that of the reference cell. We refer to this as the differential estimation problem, and use Monte Carlo simulation to compare it to the more traditional approach of estimating the capacity of each cell in a battery string independently using its current/voltage measurements. Two key conclusions emerge from this simulation. Compared to traditional estimation, differential estimation results in capacity estimates whose variance is (i) twice as sensitive to voltage measurement noise but (ii) significantly less sensitive to current measurement noise. This makes differential estimation more appealing for battery packs with high current measurement noise and low voltage measurement noise. ∗Address all correspondence to this author. NOMENCLATURE st The state of charge of a cell at time t s0 Initial state of charge Q(i) Charge capacity of the ith cell It True value of current at time t It = ∑t−1 k=1 Ik Ĩt Measured value of current at time t. Ĩt = It +wt wt ∼N (0,σ2 w) σw Standard deviation of current measurement noise wt Ĩt = ∑t−1 k=1 Ĩk R(i) Resistance of the ith cell. y t True voltage response of the ith cell at time t ỹ t Measured voltage of the ith cell at time t. ỹ (i) t = y (i) t + v (i) t v t ∼N (0,σ2 v ) g(s) The open circuit voltage map of the battery Q̂(2) Estimate for cell 2’s capacity ŷ t Predicted voltage for the ith cell at time t σQ2 Standard deviation of Q̂ (2)