1.2.4. Stochastic Battery Models

The fourth category describe the stochastic models that emphasize on "modeling recovery e ffect and describes battery behavior as a Markov process with probabilities in terms of parameters that are related to the physical characteristics of an electrochemical cell" [29]. They give a "good qualitative description for the behavior of a Li-Ion battery under pulsed discharge, for which the recovery e ffect is modeled as a decreasing exponential function of the SOC and discharge capacity" [29].

### 1.2.5. Linear Equivalent Electric Circuits and Simscape Battery Models

Finally, the last category of models is that of the linear equivalent electric circuit models (ECM), such as those discussed in the next section, Much details on these category of the models can be found in [21] for a Li-ion polymer (LiPb). The last category of models are Simscape models, described also in the next section of the present research work. The weakness and the strengths of ECM and Simscape models concerning the SOC accuracy and robustness are discussed in detail in next section.

In conclusion, this article focuses on the design and implementation of two accurate SAFT Li-ion battery models, suitable for HEV applications. For each Li-ion battery model are implemented in Part 2 three real-time SOC estimators on a MATLAB R2020a platform. The remaining sections of this paper are structured as follows. Section 2 describes the first RC ECM model attached to a SAFT Li-ion battery. Section 3 describes the second Li-ion battery model, a Simscape nonlinear model. Section 4 analyzes the SOC performance through six statistical criteria. Section 5 details the authors' contributions to this research paper.

### **2. Li-Ion RC Battery Equivalent Circuit Model—Case Study and ADVISOR Setup**

The purpose of this section is to present the case study of a small urban hypothetic car (SMCAR) which is set up using the ADVISOR 3.2 version software package, one of the most used in the automotive industry. Then, in next subsections is developed and validate an accurate Li-ion battery model that describes the dynamics of a SAFT Li-ion battery with a rated capacity of 6Ah and a nominal voltage of 3.6 V. This model is a third-order RC equivalent circuit model (3RC ECM), one of the most used in HEV applications due to its simplicity, high accuracy, and fast real-time implementation [14,17–23].

### *2.1. Li-Ion SAFT Battery and ADVISOR Small Hybrid Electric Car (SMCAR) Setup*

SAFT is one of the most prestigious research companies in the US, among the most famous battery players on the commercial market in the world. It operates "under the auspices of the United States Advanced Battery Consortium (USABC) and the New Generation Vehicle Partnership (PNGV)," developing high-power lithium-ion (Li-ion) batteries over the past two decades. These batteries currently equip most HEVs and EVs [14–19]. The Li-ion battery together with other key components of a Hydrogen fuel cell electric vehicle are distributed on the car chassis as shown in Figure 1.

**Figure 1.** The distribution of the components on the car chassis (see [3], NREL).

The key components of a Hydrogen fuel cell electric car shown in Figure 1 are described in [2] as follows:


Among the Li-ion batteries of an HEV, the one with a capacity and a nominal voltage of 6 Ah and 3.6 V respectively is used for experimental validation tests, using an advanced simulator (ADVISOR) created in November 1994 by the US National Renewable Energy Laboratory (NREL). ADVISOR has so far proved to be the most suitable tools used in the design of HEV and EV systems, very well documented in [4–7]. Thanks to a wide variety of HEVs and EVs and the multitude of "real-world" driving conditions, it has gradually improved the performance until it reached version 2003-00, as well as the latest version r0116 of 24 April 2013, as mentioned in [14–18]. After proper installation, the ADVISOR graphical user interface (GUI) is running by typing "advisor" at the command prompt in MATLAB [14–18]. The ADVISOR GUI file menu has "help buttons which will either access the MATLAB help window or open a web page with appropriate context information" [15,16]. By using the ADVISOR GUI software package for design the following steps are requested:

Step 1. Define a vehicle.

Step 1.1. Define the input HEV page setup shown in Figure 2, based on a large collection of HEVs types and characteristics contained by software.

**Figure 2.** Vehicle input page setup.

As a case study we consider a hypothetical SMCAR, powertrain control hybrid (hydrogen fuel cell electric vehicle) with the following characteristics [16]:





**Table 2.** Li-ion mechanical and thermal characteristics SAFT battery of 6 Ah, 3.6 V (cylindrical shape) [16].

\* J is the unit measure for mechanical energy (Joule), \*\* mol stands for number of molecules, \*\*\* K-stands for Kelvin degree.

The Simulink block diagram of the transmission system and Li-ion battery storage is shown in Figure 3.

**Figure 3.** Simulink diagram of the small hybrid electric vehicle (HEV SMCAR) transmission system.

Step 1.2. Drivetrain selection—selects the drivetrain configuration of the vehicle (Series, Parallel, etc.).

Step 1.3. Selecting components.

Step 1.4. Editing variables.

Step 1.5. Loading and saving vehicle configuration.

Step 2. Running simulation.

Step 2.1. Select the drive cycle—in the case study we chose the Federal Test Procedure (FTP) driving cycle used by US Environmental Protection Agency (EPA) for emissions certifications of passengers' vehicles in USA. The FTP-75 shown in Figure 4 and converted in current profile charging and discharging cycle in Figure 5 is the standard federal exhaust emissions driving cycle, which uses an Urban Dynamometer Driving Schedule (UDDS [14–18]). The FTP cycle has three separate phases: one cold-start phase (505 s), followed by a hot transient phase (870 s) and a hot-start phase (505 s) [14,16,18]. For a 10 min cool-down period between second phase and the third phase the engine is turned <sup>o</sup>ff. The first and third phase are identical. The total test time length for the FTP is 2457 s (40.95 min). The top speed is 91.25 km/h and the average speed is 25.82 km/h. The distance driven is approx. 17.7 km [14,16].

**Figure 4.** Simulink setup page.

**Figure 5.** The Simulink simulation results.

Step 2.2. Select a trip builder for a repeated cycle (if the case).

Step 2.3. Select a SOC correct options (linear or zero delta).

Step 2.4. Select interactive simulation a real-time interactive simulation interface to activate while the simulation is running.

Step 2.5. Select multiple cycles to speed up the process of running many different cycles with the same initial conditions using this functionality.

Step 2.6. Choose a test procedure to select what kind of test to run.

Step 2.7. Save Simulink setup. Step 2.8. Run the simulation and wait for the results figure to popup, as shown in Figures 4 and 5. Step 3. Looking for the simulation results.

The first graph at the top of the Simulink configuration page, shown in Figures 4 and 5 (the first graph at the top), shows the ADVISOR FTP-75 driving cycle speed profile as the input variable. In Figure 5, the second graph at the top shows the estimated ADVISOR SOC value required to validate both models of Li-ion batteries attached to the SAFT Li-ion battery. The last chart at the bottom of Figure 5 shows the conversion of the FTP-75 driving cycle speed profile to a driving cycle current profile, required in MATLAB simulations for model validation and SOC estimators, as an input variable.

The SAFT Li-ion battery electrical characteristics specifications are given in Table 1, and Table 2.
