## 1 Introduction
[10] used the second-order circuit equivalent model to detect the internal short circuits in short-circuit fault detection and quantitative analysis based on the variational modal decomposition mean value difference model, which proved the practicality and reliability of the second-order equivalent circuit model. In regard to the advantages of the optimization-seeking algorithm in model parameter identification, [PERSON], [PERSON] et al. [11] further compared the sparrow search algorithm with the genetic algorithm and the particle swarm algorithm, and verified the efficiency and accuracy of the sparrow search algorithm. Therefore, this paper adopts the SSA algorithm to accurately identify the parameters of the second-order equivalent circuit model of the lithium-ion battery. The SSA algorithm, with its powerful nonlinear solving ability, can provide accurate parameter values for the complex circuit model to ensure the accuracy of the model, so that a model that accurately describes the operating state of the lithium-ion battery can be obtained, laying a solid foundation for the subsequent fault diagnosis. In the joint estimation of the SOC and state of health (SOH) of lithium batteries for pure electric vehicles based on the EKF algorithm by [PERSON] et al. [12], the EKF algorithm is able to accurately estimate the SOC of lithium batteries with a small error, which proves the reliability of the EKF algorithm. [PERSON] et al. [13] demonstrated in micro-short circuit fault diagnostic tests that the smaller the short circuit resistance, the greater the deviation of the voltage profile between the short circuit and standard conditions. [PERSON] et al. [14], in the study of diagnosing micro-short circuits according to the change of relative charging time of monomer, take the monomer which reaches the cut-off voltage first as the benchmark to calculate the difference of other monomers in the relative charging time, and then use a box plot to visualize and statistically analyze the relative charging time of each monomer and accurately identify the micro-short-circuited batteries by the detected outliers, which proves the reliability and accuracy of the box plot in the diagnostic method. Therefore, this paper applies the box plot principle to anomaly identification. It serves as a powerful auxiliary tool to visualize the data distribution in graphical form, which helps us to identify the anomalies and thus the abnormal cells in the battery pack. Based on the above research results, the aim of this paper is to propose a novel fault diagnosis method that analyzes lithium-ion batteries based on the historical trajectory of their remaining discharge capacity at long time scales to identify potential faults. The method first utilizes SSA to identify the parameters of the second-order equivalent circuit model of the lithium-ion battery. Then, the EKF is employed to estimate the SOC of the Li-ion battery, which in turn calculates the remaining discharge capacity of the battery. Finally, the feature vectors are fault diagnosed on a long time scale using box plots to identify outliers and locate the battery that has failed. The various parts of this paper are organized as follows: Section 2 discusses the second-order model of lithium-ion batteries and describes how the SSA algorithm is used in the model to identify the model parameters. Section 3 provides an in-depth discussion on how the state of charge (SOC) of the battery is estimated by the extended Kalman filtering algorithm. Section 4 presents the fault diagnosis methodology, which skillfully incorporates the box plot theory to identify potential faults. Section 5 is the experimental validation, where the previous sections experimentally test the proposed method to ensure its validity and reliability. Section 6 concludes the paper.