## 1 Introduction
These uncertainties can also cause the users to be over-conservative and increase the number of charge-discharge cycles of LiBs and accelerate the degradation of LiB as well. In this work, we focus on the estimation of SoH as it forms the basis for accurate SoS, SoC, and SoE estimations. There are many methods reported for measuring battery capacity. Among the electrical methods, the common methods are Coulomb counting [8], open-circuit voltage (OCV) [9], extended Kalman filter (EKF), and the neural network [10]. The Coulomb counting is simple and easy to implement, but it is also easy to have accumulated error due to measurement error/uncertainty of the factors, such as current, temperature, etc. OCV is another common method, but it can only be applied in the open circuit condition. In addition, the flat portion of the OCV curve with SoC can lead to significantly inaccurate estimation. EKF is a modification of the Kalman filter to ease its implementation, and thus it offers a computationally efficient option for runtime SOC evaluation for on-board vehicles. However, the error-prone calculations of the Jacobian matrix, as well as the lack of robustness due to the linearization process and instability of the filters, are the shortcomings of the EKF method. The neural network has adaptability and a self-learning ability, but it needs large numbers of experimental data to train the network to acquire good prediction results. As battery cells are degraded, training is again required, making the training resources expensive. Many physical models based on the dynamics of the battery have been purposed recently. [PERSON] et al. [11] and [PERSON] et al. [12] developed an equivalent circuit based model to estimate battery dynamics. [PERSON] et al. [13] defined a new variable for the variation of the potential to perform data regression on SoC and SoH estimation. The electrochemistry-based electrical (ECBE) model that was developed in our previous work is an electrical model developed based on the first principle of electrochemistry [9; 14]. This model can compute the internal parameters of a cell, such as \(\text{Q}_{\text{m}}\), \(\text{m}_{1}\), \(\text{m}_{2}\), and, k, from its discharge curve and the change in these parameters, can be used to describe the degradation (including SoH degradation) of an LiB. Here \(\text{Q}_{\text{m}}\) is the maximum charge storage of a battery, and it is an important factor for calculating the SoH. The total amount of charge extracted by the external circuit is not equal to the maximum charge stored due to the non-zero cut-off voltage for safety consideration and the electrochemical impedance inside a battery caused by cycle aging. m\({}_{1}\) expresses the effectiveness of a cathode in storing Li-ions and its decrease indicates a degradation in the ability of the cathode to effectively accept Li-ions. m\({}_{2}\) expresses the effectiveness of the graphite anode to providing its stored Li-ions, and its decrease indicates degradation in the ability of the anode to effectively deliver Li-ions. k is the rate of charge transfer from electrodes to the electrolyte, and it can be used to measure microscopic polarization properties, and its reduction represents degradation at the electrodes' interface that affects the kinetics of electrochemical reactions at the electrodes [15]. While the ECBE model is accurate and comprehensive, the required computation time to obtain the values of the model parameters is excessive and renders its unsuitability for in-situ SoH estimation. In this work, a semi-empirical capacity fading model [16] is investigated to determine the SoH of the cells under cycle aging test. This model can describe the cell's degradation behavior due to discharge rate, temperature, and cycle numbers. We use the ECBE model as a reference to assess the accuracy of the semi-empirical model in this work. In other words, the ECBE model has a more complex structure and requires longer computation time, rendering it difficult to implement practically.