The internal resistance (Rint) model is the simplest of several common integral order equivalent circuits to model battery performance. Although it’s basic, it’s also surprisingly useful.
This article discusses where the integral order equivalent circuit models fit in the overall battery simulation and modeling industries. It also compares Rint with other integral order battery equivalent circuit models and considers how the Rint model performs in a practical battery simulation.
Simulating battery performance
Integral order equivalent circuit models are only one of several techniques used to simulate battery performance. Other ways to model performance include electrochemical models, which focus on internal chemical reactions. So-called “intelligence” models use artificial intelligence (AI) and machine learning (ML) techniques, and fractional order models use frequency-domain characteristics and often include nonlinearities.
Electric vehicle (EV) battery modeling is an imprecise art. Typically, the goal is to develop a model with sufficient detail to provide meaningful simulations while avoiding overcomplex approaches with high computational demands. There’s no one go-to, ideal approach. The optimal choice depends on the purpose of the simulation.
Equivalent circuit models combine basic circuit elements like resistors, capacitors, and voltage sources in a network to simulate a battery’s dynamic characteristics. The Rint model aims for simplicity and uses an ideal voltage source (Uoc) and the battery dc internal resistance (Ro) in series to model dynamic characteristics and the terminal voltage (Ut) (Figure 1).
Figure 1. A comparison of four common integer-order equivalent circuit battery models. (Image: MDPI processes)
The Thevenin model is based on the Rint model, adding a parallel RC network to simulate the battery’s polarization effect. Ro is used to simulate the abrupt resistance characteristics, and Rp and Cp are used to simulate the capacitance characteristics of voltage gradual changes.
Adding a capacitor Cb to the Thevenin model produces the Partnership for a New Generation of Vehicles (PNGV) model. Cb accounts for the change in battery open circuit voltage resulting from current integration during long-term charging and discharging. A challenge when implementing the PNGV model is that current laboratory equipment cannot detect the polarization process in detail, making it impossible to optimize the value of Cb.
Adding another RC network (Figure 1d) produces a second-order model that adds electrochemical polarization effects and concentration polarization reactions inside the battery. Multi-level RC equivalent circuit models usually include two or more sets of RC polarization parameters.
The more complex the model, the higher the expected accuracy. However, the computational implementation of the third-order and higher models is complex and limits their utility. Additionally, it’s been demonstrated that second-order and Thevenin models produce state of charge (SOC) estimators with enough accuracy to be useful in designing and engineering battery cells.
What good is the Rint model?
The basic Rint model is simple and easy to implement, but it does not describe the polarization phenomenon and other electrochemical nuances inside the battery and can be subject to errors when used to analyze battery cells.
The simplicity of the Rint model hides the fact that it can support the simulation of additional pack resistances from internal connectors, contactors, and safety components — in addition to the battery’s internal resistance.
If Ro is defined as a programmable value, the model describes the primary characteristics of battery and pack resistances when testing EV powertrain systems. More complex models are needed to understand the electrochemical processes inside the battery cells. However, they’re not necessarily better at modeling the total resistance of the pack in an EV application.
Rint models have been developed to include a programmable emulation model. This emulation mode uses Uoc and Ro (referring to Figure 1a) to determine the terminal voltage (Ut) based on the level and direction of current flow (Icharge):
Ut = Uoc + Ro * Icharge
Figure 2 shows one set of results from using this Rint model. This simulation begins with 50 A flowing from the battery, increases to 100 A, and then drops to zero (green line). The simulation included 5 mΩ of resistance for Ro. The output voltage (the dark line) tracks the current changes and provides the correct terminal voltage drop. This shows that the Rint model with current-based output-voltage adjustment accurately simulates battery pack (not necessarily cell) performance in EV drivetrains.
Figure 2. The Rint model of a 50-A step current change in an EV battery pack, which then drops to zero current (bottom line), showing the change in the output voltage (top line). (Image: NI)
Summary
The Rint model is the simplest of the common integral order equivalent circuits used to model battery performance. Its simplicity makes it unsuited for detailed modeling of the performance of individual battery cells. Still, that same simplicity makes it a powerful tool for modeling the battery pack’s performance in an EV drivetrain.
References
- Electrical Equivalent Circuit Models of Lithium-ion Battery, IntechOpen
- EV Powertrain Testing Challenges and Solutions, NI
- Review on the Battery Model and SOC Estimation Method, MDPI processes
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Filed Under: Batteries, FAQs, Testing and Safety