**1.2 Statistical criteria for evaluating the performance of Li-ion battery – SOC and terminal voltage**

Based on MATLAB simulation results useful information on SOC and battery terminal voltage accuracy can be extracted based on SOC and terminal voltage residuals and based on four statistic criteria values shown in eight tables, defined in [29, 30], and grouped as:


For each Li-ion battery model developed in this research, the SOC and terminal voltage performance are evaluated by simulations conducted on MATLAB Simulink platform. The information extracted from these simulations is beneficial for a rigorous comparison of performance, so that the reader has a better perspective on the modeling, design, and implementation of the battery. From a variety of battery models, the reader has the ability to decide which model and estimator are best for a particular application.

#### **1.3 Manuscript structure, objectives, and performed results**

The paper is organized as follows: Section 2 gives a brief description of Rint Li-ion SAFT 6 Ah 11 V nominal voltage and battery selection for the case study, model option, and its validation using the National Renewable Energy Laboratory (NREL) ADVANCED SIMULATOR (ADVISOR) 2003 for HEVs and EVs design. An equivalent electric circuit model (ECM) for the Li-ion battery SAFT Rint model is preferred due to its simplicity and ability to capture all the battery dynamics such that its SOC and the predicted terminal cell voltage are of high accuracy. The dynamics battery part ARX and ANFIS models, the ANFIS battery OCV(SOC) model, their order

selection, parameters identification and model implementation, ANFIS models' generation and performance, and the MATLAB simulation results are shown and discussed in Sections 2 and 3, respectively. For the ARX model and hybrid combinations of the ARX model and ANFIS model of OCV(SOC) nonlinear block, an overall ANFIS battery model consisting of ANFIS dynamic battery part model and ANFIS OCV(SOC) nonlinear battery block models are considered. A rigorous analysis of AEKF SOC estimator adaptability for all these models is evaluated based on simulation results conducted on MATLAB R2021b and Simulink environments that provide valuable information on SOC and battery terminal voltage accuracy and robustness performance. Also, a comparison of the evaluation of the results is made based on the SOC and battery voltage residuals and statistics criteria values summarized in almost eight tables in the last subsection 2.1.3 of Section 2 and all Section 3. Section 4 is dedicated to Conclusions supporting all the previous MATLAB results. In summary, in this research paper an impressive number of investigations are done on the accuracy and adaptability of six alternative Li-ion batteries models to the original NREL Li-ion Rint SAFT-type 6 Ah 11 V rated voltage battery model. The main reason of this selection is that the Li-ion batteries are very common in a wide variety of HEVs/EVs applications in the automotive industry, Also, it is a beneficial option to be used as a baseline model of Li-ion battery for performance comparison and validation of alternative models, among them the linear, simple, and accurate 3RC ECM Li-ion battery model developed in this research work. This alternative model is designed by using one of the most used designing tools very spread in the automotive industry, created by NREL, known as ADVANCED SIMULATOR (ADVISOR) with the final launch in 2003 for HEVs/EVs design. The Li-ion battery model 3RC ECM model is selected for simulation purpose and "proof" concept for developing new alternative battery models. Five alternative models that derivate from 3RC ECM Li-ion battery model are developed in this research work: (a) ARX-ECM that models the dynamic part of the battery represented by the Rint-3RC circuit using an equivalent ARX model; (b) ANFIS-ECM replacing the Rint-3RC circuit with an ANFIS model; (c) ARX-ANFIS hybrid structure that is a combination of ARX model for Rint-3RC dynamic part and an ANFIS model of the nonlinear static block OCV (SOC); (d) Rint-3RC -ANFIS model that keeps the dynamic part of the battery combined with the ANFIS model of OCV(SOC) static block; (e) full ANFIS model structure derived from ARX-ANFIS, which replaces the ARX model with an ANFIS model. The MATLAB simulation results for each model provide a large database stored in 10 useful tables for a rigorous analysis of the performance of the Li-ion battery in terms of SOC accuracy and terminal voltage, as well as the robustness of the AEKF algorithm for estimating the SOC of the selected battery.
