Online state-of-energy estimation method for lithium-ion batteries used in wearable devices based on adaptive unscented Kalman filter

doi:10.19799/j.cnki.2095-4239.2023.0721

Abstract

Abstract:

Wearable devices (WDs) with small sizes and long working time are widely used in industrial monitoring and other fields. Lithium-ion batteries provide energy for electronics used in WDs, and their online accurate estimation of state of energy (SOE) critically impacts the real-time power management and life extension of WDs. Traditional model-based estimation methods must obtain the offline relationship between SOE and open circuit voltage (OCV). However, this requires a large amount of test time and is challenging to adapt to actual working conditions, thus hindering its online application. This paper proposes an SOE estimation method based on the online identification of OCV for lithium-ion batteries used in WDs. Based on the first-order RC model of the battery, the forgetting factor recursive least squares is used to identify the OCV online and other parameters of the lithium-ion battery. After analyzing the characteristics of load changes in the WD operation, the working condition and parameter identification condition are constructed, and the experiments are conducted on the test bench. Combined with the workload characteristics of WDs, the relationship between OCV and terminal voltage is discussed, and the relationship curves between OCV and SOE are obtained online. The adaptive unscented Kalman filter is used to estimate the SOE online and is compared with the traditional method based on the offline OCV-SOE relationship. The results show that the proposed SOE estimation method based on OCV online identification has good accuracy and robustness against different initial values.

Key words: wearable device, SOE estimation, OCV online identification, adaptive unscented Kalman filter

CLC Number:

TM 912

Mingxian LIU, Jibiao LI, Bingnan TANG, Yi YANG, Renxin XIAO. Online state-of-energy estimation method for lithium-ion batteries used in wearable devices based on adaptive unscented Kalman filter[J]. Energy Storage Science and Technology, 2024, 13(5): 1688-1698.

Figures/Tables 13

Fig. 1

Table 1

Implementation of the AUKF for the SOE estimation"

针对电池等效电路模型中，非线性离散系统状态空间方程：

$x k = f (x k - 1, u k - 1) + w k y k = g (x k, u k) + v k$

式中，x是状态变量，y是可测量输出，u是输入电流I，f是非线性状态函数对应公式(16)，g是非线性性测量函数对应公式(17)，w是过程噪声，v是测量噪声。

采样时间k = 0时SOE 初始化如下：

步骤(1)：状态初始化

$x ¯ 0 = E (x 0) P 0 = E (x 0 - x ¯ 0) (x 0 - x ¯ 0) T$

式中， $x ¯ 0$ 是状态初始值x₀ 的均值, P₀是误差协方差矩阵初始值。

k ≥ 1时每个采样时刻k估算SOE，迭代执行步骤(2) ~(7)：

步骤(2)：计算采样点Sigma及加权系数

①产生Sigma点

$x ⌢ k - 1 = x ¯ 0 x i, k - 1 = x ⌢ k - 1 + (N + s) P k - 1 i i = 1,2, ⋯, N x i, k 1 = x ⌢ k - 1 - (N + s) P k - 1 i - N i = N + 1, N + 2, ⋯, 2 N$

式中， $x ⌢ k - 1$ 表示为k-1时刻状态变量的最优估计值，2N为样本个数，本工作N=2；s为尺度系数，可以表示为 $s = α 2 (N + κ) - N$ ，其中，κ为缩放参数，状态估计时通常设置为0，α为Sigma点在 $x ⌢$ 周围扩展的比例参数，用于控制采样点分布状态，取值范围为0~1，本工作α=0.01；

②计算加权系数

$w m (0) = r N + s w c (0) = r N + s + (1 - α 2 + ξ) w m (i) = w c (i) = 1 2 (N + s), i = 1,2, ⋯, 2 N$

式中，w_m, w_c为权重因子，ξ为合并高阶项中的误差的参数，本工作ξ=2。

步骤(3)：k|k-1时刻的系统状态变量均值与协方差更新，生成新的sigma点：

$x i, k k - 1 = f (x i, k - 1, u k) + q k - 1 x ¯ ̑ k k - 1 = ∑ i = 0 2 n w m (i) f (x i, k k - 1, u k) P ¯ k k - 1 = ∑ i = 0 2 n w c (i) (x i, k k - 1 - x ¯ ̑ k k - 1) • (x i, k k - 1 - x ¯ ̑ k k - 1) T + Q k - 1$

式中，q为系统过程噪声均值，Q为过程噪声的协方差。

生成新的sigma点：

$x ⌢ i, k = x ⌢ k | k - 1 x i, k = x ⌢ k | k - 1 + (N + r) P k i i = 1,2 ⋯ N x i, k = x ⌢ k | k - 1 - (N + r) P k i - N i = N + 1, N + 2 ⋯ 2 N$

步骤(4)：测量更新

$y i, k k - 1 = g (x ¯ ̑ k k - 1, u k) + r k y ¯ ̑ k k - 1 = ∑ i = 0 2 n w m (i) y i, k k - 1$

式中，r为测量噪声的平均值。

步骤(5)：误差协方差与增益矩阵 K 的更新

$P y y, k = ∑ i = 0 2 N w c (i) (y i, k k - 1 - y ¯ ̑ k k - 1) (y i, k k - 1 - y ¯ ̑ k k - 1) T + R k P x y, k = ∑ i = 0 2 N w c (i) (x i, k k - 1 - x ¯ ̑ k k - 1) (y i, k k - 1 - y ¯ ̑ k k - 1) T K k = P x y, k P y y, k$

式中，R为测量噪声协方差。

步骤(6)：状态变量测量校正及其协方差的更新

$x ⌢ k = x ¯ ̑ k k - 1 + K k (y k - y ¯ ̑ k k - 1) P k = P ¯ k k - 1 - K k P ¯ y y, k K k T$

步骤(7)：过程噪声以及测量噪声的更新

$r k + 1 = (1 - d k) r k + d k v k - ∑ i = 0 2 n y i, k k - 1 - r k R k + 1 = (1 - d k) R k + d k (v k - y ¯ ̑ k k - 1) (v k - y ¯ ̑ k k - 1) T q k + 1 = (1 - d k) q k + d k (x ⌢ k - x ¯ ̑ k - q k) Q k + 1 = (1 - d k) Q k + d k (K k (v k - y ¯ ̑ k k - 1) (v k - y ¯ ̑ k k - 1) T K k T + P k) d k = (1 - b) (1 - b k)$

式中，d是估计的残值，b是[0~ 1]范围内的遗忘因子，通常取0.95。

Table 1

Table 2

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Table 3

Fig. 9

Fig. 10

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估算方法	MAE/%	RMSE/%
端电压法	0.52	0.72
FFRLS-OCV	0.54	0.78
查表法OCV	1.00	1.36

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