Keywords: battery inconsistency, equalization converter, staggered parallel topology, jumper switches, small-signal model Goodarzi, S.; Beiranvand, R.; Mousavi, S.M.; Mohamadian, M. A new algorithm for increasing balancing speed of switched-capacitor lithium-ion battery cell equalizers. In Proceedings of the 6th Power Electronics, Drive Systems & Technologies Conference (PEDSTC2015), Tehran, Iran, 3–4 February 2015; pp. 292–297. [ Google Scholar] [ CrossRef] Support wireless cascading, the number of cascading series=A equalization board series+B equalization board series-1

The MOSFET employs switching frequency to control the current during equalization. The maximum withstanding voltage of the MOSFET must be greater than the voltage applied at both ends, preventing it from breaking down. As seen in Figure 5, the voltage across the MOSFET at the disconnection moment is the battery voltage. However, the MOSFET in each layer withstands the different voltage, increasing with the number of layers. The maximum withstand voltage of the MOSFET is where V max is the maximum voltage of an individual cell, and m is the number of equalization layers. 3. SOC Estimation Based on AUKF Signal waveform of inductors (L)1 and (L)2. 2.2. Equalization Circuit Parameters 2.2.1. PWM Duty Cycle Lu, C.; Kang, L.; Luo, X.; Linghu, J.; Lin, H. A novel lithium battery equalization circuit with any number of inductors. Energies 2019, 12, 4764. [ Google Scholar] [ CrossRef][ Green Version]Furthermore, the equalization loss [ 44] of the battery is where E L is the total energy consumption of the battery, is the energy of the i-th battery cell before equalization, denotes the energy of the i-th cell after equalization, and is the average energy of all battery cells. Lu, L.; Han, X.; Li, J.; Hua, J.; Ouyang, M. A review on the key issues for lithium-ion battery management in electric vehicles. J. Power Sources 2013, 226, 272–288. [ Google Scholar] [ CrossRef]

By the current integration method, we have where SOC 0 is the initial SOC, Q N is the normal capacity of the battery, and η is the charging and discharging efficiency of the battery. Manenti, A.; Abba, A.; Merati, A.; Savaresi, S.M.; Geraci, A. A new BMS architecture based on cell redundancy. IEEE Trans. Ind. Electron. 2010, 58, 4314–4322. [ Google Scholar] [ CrossRef]How, D.N.; Hannan, M.; Lipu, M.H.; Ker, P.J. State of charge estimation for lithium-ion batteries using model-based and data-driven methods: A review. IEEE Access 2019, 7, 136116–136136. [ Google Scholar] [ CrossRef] For a nonlinear discrete-time system, the state equation and the measurement equation are as follows: where x k is the system state vector, y k is the measurement vector, u k is the known input vector, is the process Gaussian noise, is the measurement noise Gaussian, and f( x k, u k) is a nonlinear measurement function. The AUKF for the battery SOC estimation is illustrated as follows: Step 1: Initialization and state extension where and are the initial state estimate and error covariance matrix. The state vector X k is defined as with the state variables SOC(k), U e(k) and U d(k). Step 2: 2L + 1 sampling (Sigma) points establishment Step 3: The weight calculation The calculation of the mean weight and covariance weight can be presented as where is a scale parameter to eliminate the total prediction error, α determines the distribution of sigma points around and is usually a small positive value ranging from 1 to e −4, takes values of 0 to 3- n generally, β is used to integrate the X prior estimation and generally selected to be 2, and and are weighting factors to calculate the mean and covariance of the i-th sigma point, respectively. Step 4: State estimation time updating Step 5: Covariance matrix time updating where Q k is the covariance of the process Gaussian noise . R k is the covariance of measurement noise Gaussian , without relationship between them. Step 6: New sigma points attainment at time k| k-1 Step 7: Measurement estimation Step 8: Kalman gain matrix where Step 9: Residual calculating Step 10: State estimation measurement updating Step 11: Covariance matrix measurement updating The above steps constitute the UKF. The AUKF contains an additional step to adjust the process noise covariance and measurement noise as follows. Step 12: Adaptive adjustment of Q and R The equalization strategy is developed on the basis of the equalization circuit to optimize the performance of the equalization circuit, so that the overall equalization performance can be maximized [ 39]. However, controlling the magnitude of the equalization current is a very complex nonlinear control problem. Model neural networks (NNs) can achieve nonlinear mapping with arbitrary accuracy. However, the NN requires a large amount of data to train. In this paper, an adaptive fuzzy neural network is used to control the value of the equalization current in the equalization process.