###### Abstract  
Accurate state-of-charge (SOC) estimation of lithium batteries is of great significance for electric vehicles. In this paper, a combined estimation method of multi-innovation unscented Kalman filter (MIUKF) and fractional order square root cubature Kalman filter (FSRCKF) for lithium batteries is proposed. Firstly, the adaptive genetic algorithm (AGA) is applied to carry out offline parameter identification for the fractional order model (FOM) of a lithium battery under the Dynamic Stress Test (DST). Then, battery SOC is estimated by FSRCKF, while the Ohm internal resistance \(R_{0}\) of the fractional order battery model is estimated and updated by MIUKF in real time. The results show that MIUKF-FSRCKF is better than FSRCKF, FCKF and SRCKF in estimating the SOC of lithium batteries under the Federal Urban Driving Schedule (FUDS), Beijing Dynamic Stress Test (BJDST) and US06 Highway Driving Schedule tests, especially when \(R_{0}\) is inaccurate. state-of-charge; multi-innovation unscented Kalman filter; fractional order square root cubature Kalman filter; adaptive genetic algorithm; the fractional order model +
Footnote †: journal: Applied  
was significantly higher than the other two algorithms. Ref. [12] proposed an estimation algorithm using fractional order unscented Kalman and H infinity filter (FOUHIF), and verified that the algorithm has better robustness and higher accuracy than unscented Kalman filter (UKF) and fractional order unscented Kalman filter (FUKF). Ref. [13] proposed the fractional order interpolatory cubature Kalman filter algorithm (FICKF) and discussed FUKF and FCKF as two special forms of the fractional order interpolatory cubature Kalman filter algorithm. In the literature [14], a robust fractional order cubature Kalman filter algorithm based on Masreliez and Martin was proposed, which was proved to be superior to FUKF and FCKF, and the influence of different degrees of measurement pollution noise on the estimation performance of the proposed algorithm was analyzed. In the Kalman filtering algorithms, the CKF [15; 16] uses the method of cubature selection to estimate the state variables and covariance, and its accuracy is equivalent to the third-order Taylor expansion, while square root CKF (SRCKF) [17] is transmitted in the form of square root on the basis of CKF. SRCKF can address the nonlinear transfer problem of mean and covariance, and its square root step ensures the calculation convergence; thus, the stability and accuracy of the filter algorithm are improved. Therefore, SRCKF has been applied to the SOC estimation. Ref. [18] used SRCKF to estimate SOC for lithium batteries, and verified that compared with ordinary EKF, UKF and CKF, the SRCKF algorithm has higher SOC estimation accuracy and stronger robustness. Ref. [19] proposed a method of online parameter identification combined with ASRCKF to estimate the SOC of lithium batteries, and the maximum root mean square error of the estimated SOC was controlled within 1%. Ref. [20] proposed an ASRCKF based on M-estimation, which has adequate anti-nonlinearity and anti-outlier interference ability, strong robustness and high result accuracy. At present, SRCKF has been widely used for SOC estimation of lithium batteries and FCKF has been gradually applied to SOC estimation of lithium batteries. However, to the best of our knowledge, there is a lack of studies addressing the fractional order square root cubature Kalman filter (FSRCKF). In addition, the online parameter identification of the fractional order battery model is complicated. However, the common offline parameter identification has the disadvantage of large SOC estimation error. According to the FOM of a battery, the Ohm internal resistance \(R_{0}\) directly affects the voltage by multiplying the current, thus affecting the SOC estimation. In addition, \(R_{0}\) is also a key parameter for estimating the health status of lithium batteries [21; 22].