The complete set of two-loop renormalization group equations in general gauge field theories is presented. This includes the β-functions of parameters with and without a mass dimension.

Sate of charge (SOC) accurate estimation is one of the most important functions in a battery management system for battery packs used in electrical vehicles. This paper focuses on battery SOC estimation and its issues and challenges by exploring different existing estimation methodologies. The key technologies of lithium-ion battery state estimation methodologies of the electrical vehicles categorized under five groups, such as the conventional method, adaptive filter algorithm, learning algorithm, nonlinear observer, and the hybrid method, are explored in an in-depth analysis. Lithium-ion battery characteristic, battery model, estimation algorithm, and cell unbalancing are the most important factors that affect the accuracy and robustness of SOC estimation. Finally, this paper concludes with the challenges of SOC estimation and suggests other directions for possible research efforts.

Open circuit voltage (OCV) is an important characteristic parameter of lithium-ion batteries, which is used to analyze the changes of electronic energy in electrode materials, and to estimate battery state of charge (SOC) and manage the battery pack. Therefore, accurate OCV modeling is a great significance for lithium-ion battery management. In this paper, the characteristics of high-capacity lithium-ion batteries at different temperatures were considered, and the OCV-SOC characteristic curves at different temperatures were studied by modeling, exponential, polynomial, sum of sin functions, and Gaussian model fitting method with pulse test data. The parameters of fitting OCV-SOC curves by exponential model (n = 2), polynomial model (n = 3~7), sum of sin functions model (n = 3), and Gaussian model (n = 4) at temperatures of 45 °C, 25 °C, 0 °C, and −20°C are obtained, and the errors are analyzed. The experimental results show that the operating temperature of the battery influences the OCV-SOC characteristic significantly. Therefore, these factors need to be considered in order to increase the accuracy of the model and improve the accuracy of battery state estimation.

State of charge (SOC) estimation is the core of any battery management system. Most closed-loop SOC estimation algorithms are based on the equivalent circuit model with fixed parameters. However, the parameters of the equivalent circuit model will change as temperature or SOC changes, resulting in reduced SOC estimation accuracy. In this paper, two SOC estimation algorithms with online parameter identification are proposed to solve this problem based on forgetting factor recursive least squares (FFRLS) and nonlinear Kalman filter. The parameters of a Thevenin model are constantly updated by FFRLS. The nonlinear Kalman filter is used to perform the recursive operation to estimate SOC. Experiments in variable temperature environments verify the effectiveness of the proposed algorithms. A combination of four driving cycles is loaded on lithium-ion batteries to test the adaptability of the approaches to different working conditions. Under certain conditions, the average error of the SOC estimation dropped from 5.6% to 1.1% after adding the online parameters identification, showing that the estimation accuracy of proposed algorithms is greatly improved. Besides, simulated measurement noise is added to the test data to prove the robustness of the algorithms.