The evolution of the automotive industry is unequivocally steering towards electrification. Within this transition, the hybrid electric vehicle stands as a critical bridge technology, effectively balancing the extended range and refueling convenience of internal combustion engines with the efficiency and partial zero-emission capabilities of electric drive systems. The performance and longevity of the hybrid electric vehicle are intrinsically linked to the health of its high-voltage battery pack. Among the various battery chemistries employed, Nickel-Metal Hydride (Ni-MH) batteries have proven exceptionally reliable for hybrid electric vehicle applications due to their robust safety profile, excellent power capability, wide operating temperature range, and remarkable cycle life under the typical shallow charge/discharge regimes of hybrid electric vehicle operation.
A core parameter that serves as a key indicator for the State of Health (SOH) and remaining useful life of a battery is its internal resistance. This resistance is not a single value but a composite of several components. Broadly, it can be divided into the polarization resistance, which is related to the kinetics of the electrochemical reactions and mass transport, and the ohmic resistance. The ohmic resistance, often termed $R_0$, is a purely resistive component. It arises from the electronic resistance of the electrode materials, current collectors, tabs, and busbars, as well as the ionic resistance of the electrolyte, separator, and interfaces. Unlike polarization resistance, which can vary significantly with the state of charge (SOC), current magnitude, and temperature, the ohmic resistance is relatively stable for a given battery under static conditions. Its value is primarily determined by the battery’s physical design, materials, and manufacturing quality.
However, as a hybrid electric vehicle battery ages through numerous cycles and calendar time, its ohmic resistance tends to increase due to mechanisms such as electrolyte decomposition, corrosion of current collectors, and mechanical degradation of electrode contacts. This increase directly impacts the hybrid electric vehicle’s performance: it leads to higher voltage drops during acceleration (discharge) and regenerative braking (charge), reducing available power and increasing heat generation. Consequently, accurate knowledge and monitoring of the battery’s ohmic resistance are paramount. It enables battery management systems (BMS) to accurately estimate SOH, predict end-of-life, optimize power limits, and ensure the safe and efficient operation of the hybrid electric vehicle.
The challenge lies in measuring this parameter in-situ, during the dynamic operation of the hybrid electric vehicle. Traditional laboratory methods, such as Electrochemical Impedance Spectroscopy (EIS) or high-current DC pulse tests, are offline procedures requiring specialized equipment and controlled conditions, making them unsuitable for real-time application in a hybrid electric vehicle. Therefore, developing a reliable, computationally efficient, and model-based online estimation algorithm is essential for advanced BMS in hybrid electric vehicles.
Model Foundation: The Second-Order RC Equivalent Circuit
To enable model-based estimation, an accurate yet simple electrical representation of the battery is required. Equivalent Circuit Models (ECMs) are widely adopted in hybrid electric vehicle system simulations due to their balance between physical representation and computational tractability. For Ni-MH batteries, which exhibit dynamic voltage relaxation behaviors, a first-order RC model (one resistor-capacitor pair) is often insufficient to capture the voltage response accurately over a wide frequency range. A second-order RC model provides a significantly better approximation of the battery’s dynamics without excessive complexity.
The proposed second-order RC ECM is depicted in the figure below. In this model:
- $E$ represents the open-circuit voltage (OCV), which is a function of the battery’s State of Charge (SOC) and temperature.
- $R_0$ is the ohmic resistance, the primary parameter of interest for this estimation study.
- $R_1$, $C_1$ model the short-time-constant polarization effects, often associated with charge transfer kinetics.
- $R_2$, $C_2$ model the long-time-constant polarization effects, typically associated with mass transport (diffusion) processes.
- $I$ is the current flowing through the battery (positive for discharge, negative for charge).
- $U_{oc}$ is the terminal voltage measured at the battery poles.
The governing equations for this circuit are derived from Kirchhoff’s laws. The voltages across the two RC networks, $U_1$ and $U_2$, are state variables described by differential equations:
$$
\frac{dU_1}{dt} = -\frac{1}{R_1 C_1}U_1 + \frac{1}{C_1}I
$$
$$
\frac{dU_2}{dt} = -\frac{1}{R_2 C_2}U_2 + \frac{1}{C_2}I
$$
The terminal voltage $U_{oc}$ is given by:
$$
U_{oc} = E – U_1 – U_2 – I R_0
$$
This model forms the foundation for our estimation approach. The core idea is to exploit periods of relatively stable operation in a hybrid electric vehicle to decouple the estimation of the quasi-instantaneous $R_0$ from the slower dynamics of the polarization voltages.
Online Estimation Strategy Based on Recursive Least Squares
During typical hybrid electric vehicle driving, there are frequent periods where the battery operates in a pseudo-steady state. This means the current, while not zero, fluctuates around an average value without large, sustained step changes, and the SOC change over a short window (e.g., one minute) is negligible. Under such conditions, the open-circuit voltage $E$ can be considered constant because SOC is nearly constant. Furthermore, if the current variations are not too severe, the polarization voltages $U_1$ and $U_2$ will also settle near steady-state values corresponding to the average current.
We can define an intermediate voltage $U_s$ as the terminal voltage minus the drop across the ohmic resistance:
$$
U_s = U_{oc} + I R_0 = E – U_1 – U_2
$$
During a short, stable time window where $E$, $U_1$, and $U_2$ are approximately constant, $U_s$ can also be treated as a constant. Therefore, within this window, the relationship between measured terminal voltage $U_{oc}$ and current $I$ becomes linear:
$$
U_{oc}(k) = U_s – I(k) R_0
$$
where $k$ denotes the discrete-time sample index. This is a simple linear equation in the form $y(k) = \theta – x(k) \cdot \phi$, where $y(k)=U_{oc}(k)$, $x(k)=I(k)$, the constant offset $\theta = U_s$, and the slope $\phi = R_0$.
The Recursive Least Squares (RLS) algorithm is perfectly suited to identify the parameters $\theta$ and $\phi$ in real-time. However, for the purpose of clarity in explaining the core method, we can consider a batch processing approach over a fixed window of $N$ samples (e.g., $N=60$ for 60 seconds of data sampled at 1 Hz). The optimal estimates for $R_0$ and $U_s$ that minimize the sum of squared errors between the model and measurements are given by the standard linear regression formulas:
$$
R_0 = \frac{N \sum_{k=1}^{N} \big( I(k) U_{oc}(k) \big) – \big( \sum_{k=1}^{N} I(k) \big)\big( \sum_{k=1}^{N} U_{oc}(k) \big)}{ \big( \sum_{k=1}^{N} I(k) \big)^2 – N \sum_{k=1}^{N} \big( I(k)^2 \big) }
$$
$$
U_s = \frac{ \sum_{k=1}^{N} U_{oc}(k) + R_0 \sum_{k=1}^{N} I(k) }{N}
$$
This batch calculation can be performed periodically (e.g., every minute) by the BMS in the hybrid electric vehicle using recent data stored in a buffer. To make it truly recursive and memory-efficient, the RLS algorithm updates the estimates with each new pair of measurements $(I(k), U_{oc}(k))$, forgetting older data exponentially. The algorithm is initialized with prior estimates and a covariance matrix $P$. For each time step $k$:
- Compute the gain vector $K(k)$: $$K(k) = \frac{P(k-1) \cdot x(k)}{\lambda + x^T(k) \cdot P(k-1) \cdot x(k)}$$ where $x(k) = [1, -I(k)]^T$ is the regressor vector and $\lambda$ is a forgetting factor ($0 < \lambda \leq 1$).
- Update the parameter vector $\hat{\Theta}(k) = [\hat{U_s}(k), \hat{R_0}(k)]^T$: $$\hat{\Theta}(k) = \hat{\Theta}(k-1) + K(k) \big( U_{oc}(k) – x^T(k) \cdot \hat{\Theta}(k-1) \big)$$
- Update the covariance matrix $P(k)$: $$P(k) = \lambda^{-1} \big( P(k-1) – K(k) \cdot x^T(k) \cdot P(k-1) \big)$$
This recursive formulation allows for continuous, real-time estimation of $R_0$ in the hybrid electric vehicle’s BMS with minimal computational burden.
Experimental Validation: Offline Benchmark vs. Online Estimation
To validate the proposed online method, comprehensive testing was conducted. The first step was to establish a reliable ground truth for the battery’s ohmic resistance using a standard offline technique. A high-precision battery test bench was used to perform Hybrid Pulse Power Characterization (HPPC) tests on a commercial Ni-MH module designed for hybrid electric vehicle applications. The test was performed at a controlled temperature of 25°C. The module was set to specific SOC points (from 30% to 70%, reflecting common hybrid electric vehicle operating ranges). At each SOC, a high-current DC pulse (charge and discharge) was applied for a very short duration (100ms). The instantaneous voltage change at the very beginning of the pulse, before significant polarization develops, is attributed almost entirely to the ohmic resistance. The value is calculated as $R_{0\_pulse} = |\Delta V / \Delta I|$.
The results from this offline benchmark test are summarized in Table 1. The ohmic resistance shows minor variation with SOC and a small discrepancy between charge and discharge pulses, which is typical. An average value was computed as a reference.
| SOC (%) | Charge Pulse $R_0$ (mΩ) | Discharge Pulse $R_0$ (mΩ) | Average (mΩ) |
|---|---|---|---|
| 30 | 20.76 | 25.86 | 23.31 |
| 40 | 21.16 | 25.82 | 23.49 |
| 50 | 21.02 | 25.72 | 23.37 |
| 60 | 20.90 | 25.78 | 23.34 |
| 70 | 20.82 | 25.92 | 23.37 |
| Overall Average | 23.38 mΩ | ||
The second phase involved real-world validation on a chassis dynamometer and during actual on-road driving of a prototype hybrid electric vehicle. The battery pack’s current and terminal voltage were sampled at 1 Hz via the vehicle’s CAN network. Data from a prolonged, stable driving segment at ambient temperature was extracted. The batch least squares method (with $N=60$) was applied to sliding 60-second windows of this data to estimate the ohmic resistance online.
The results of the online estimation over a 2500-second driving period are plotted in Figure 1. The estimated $R_0$ fluctuates within a narrow band, primarily between 22 mΩ and 32 mΩ. The mean value calculated over the entire period was $R_{0\_online\_mean}$ = 25.67 mΩ. A direct comparison between the offline benchmark and the online estimates from a corresponding time period is shown in Table 2.
| Method | Condition | Estimated $R_0$ (mean, mΩ) | Notes |
|---|---|---|---|
| DC Pulse (Offline) | Controlled Lab, 25°C | 23.38 | Ground truth reference |
| LS-Based Online | Real-world HEV Driving, ~25°C | 25.67 | From vehicle CAN data |
The relative error between the two mean values is approximately 9.8%. This discrepancy is well within acceptable limits for SOH tracking in a hybrid electric vehicle and can be attributed to several factors: 1) Temperature differences between the tightly controlled lab test and the varying under-hood temperature during road operation, 2) The inherent noise and measurement tolerances in vehicle-grade sensors compared to lab equipment, and 3) The fundamental assumption of constant $U_s$ during the estimation window being only approximately true in real driving. Critically, the online estimate consistently hovers around the offline reference, demonstrating the method’s validity and practicality for deployment in a hybrid electric vehicle’s BMS.
Discussion and Comparative Analysis
The proposed Least Squares-based method offers distinct advantages for hybrid electric vehicle applications, particularly when contrasted with other estimation techniques found in literature. More sophisticated algorithms like the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), or Adaptive UKF (AUKF) can simultaneously estimate SOC and multiple ECM parameters, including $R_0$, $R_1$, $C_1$, etc. These methods are powerful and can provide high accuracy. However, they come with significant computational complexity, require careful tuning of noise covariance matrices, and their stability must be guaranteed under all hybrid electric vehicle operating conditions. For a BMS that must manage dozens of battery modules in real-time on a low-cost microcontroller, this complexity can be a burden.
Methods based on full electrochemical impedance spectroscopy are purely offline laboratory tools and are not feasible for online use in a hybrid electric vehicle. Other online techniques that inject specific probing signals into the battery to measure impedance require additional hardware and can interfere with the normal energy management of the hybrid electric vehicle.
The strength of the presented LS-based approach lies in its simplicity and directness. It targets the specific parameter of interest—the ohmic resistance—by leveraging the natural current excitations present during hybrid electric vehicle driving. It requires no additional hardware, only the current and voltage sensors already present in every hybrid electric vehicle BMS. The computational load of a simple linear regression (or its RLS form) is negligible. This makes it an extremely practical and robust solution for continuous, lifelong monitoring of battery health in a hybrid electric vehicle.
The accuracy, as demonstrated by the ~10% error compared to the lab benchmark, is sufficient for trend analysis. Monitoring the gradual increase of $R_0$ from its beginning-of-life value (e.g., 25 mΩ) to an end-of-life threshold (e.g., 40 mΩ) is a reliable indicator of SOH degradation. This information can be used to derate power capabilities, trigger maintenance alerts, or inform battery warranty and replacement decisions for the hybrid electric vehicle fleet.
Conclusion and Future Perspectives
This work successfully developed and validated a model-based online method for estimating the ohmic resistance of Nickel-Metal Hydride power batteries used in hybrid electric vehicles. By employing a second-order RC equivalent circuit model and applying a Least Squares identification technique to short periods of stable operation, the method effectively isolates the ohmic voltage drop from the overall battery dynamics. Experimental validation through both bench tests and real-world hybrid electric vehicle driving data confirmed the method’s feasibility and accuracy, showing strong agreement with conventional offline measurement techniques.
The primary advantage of this method is its practicality. It is simple to implement, computationally lightweight, and requires no modifications to the standard hybrid electric vehicle powertrain architecture. It transforms the BMS from a simple monitoring and protection device into a prognostic health management system capable of predicting battery lifespan, thereby enhancing the reliability and total cost of ownership of the hybrid electric vehicle.
Future research directions could focus on several enhancements. First, integrating temperature compensation is crucial, as the ohmic resistance of Ni-MH batteries is temperature-dependent. The algorithm could be extended to estimate a temperature-corrected $R_0$ or use a pre-calibrated look-up table. Second, adaptive logic could be developed to automatically detect suitable “stable windows” for estimation within the noisy and dynamic current profile of a hybrid electric vehicle. Finally, while this method was demonstrated on Ni-MH batteries, the underlying principle is chemistry-agnostic. With appropriate adjustment of the ECM (e.g., a different OCV-SOC relationship and polarization time constants), this approach holds significant promise for online SOH estimation in Lithium-ion battery packs for plug-in hybrid electric vehicles and battery electric vehicles as well, further broadening its impact on the electrified transportation landscape.