1. Introduction
Lithium-ion batteries (LiBs) have gained enormous popularity as energy storage elements for most rechargeable electric systems. LiBs have been implemented in small devices such as mobile phones and large systems such as electric vehicles (EVs). These types of batteries are also needed for energy storage in renewable energy sources (such as solar and wind) to improve their stability. LiBs’ popularity can be attributed to their high specific energy and high operational voltage [
1]. Unfortunately, LiBs are highly nonlinear elements with a limited operational area, which requires robust monitoring for their safe application [
2]. An LiB’s operational window can be defined by voltage and temperature boundaries. The size of the operational window will depend on the LiB’s chemistry [
3]. Continued use of the LiB outside of this operating window may cause the battery to develop dendrites over time, which increases its internal resistance and may lead to short-circuiting (e.g., dendrites may penetrate the diaphragm or battery wall). Additionally, it may start releasing toxic gases, causing it to burst into flames [
3]. Furthermore, the LiB’s maximum capacity may decrease when subjected to deep discharge cycles.
Another concern with LiB technology is the accurate estimation of its available capacity, which is typically linked to the state of charge (SOC). The SOC represents the available amount of charge in a battery during usage and is measured by ampere-hours (
). The SOC is typically defined as a percentage of current available capacity vs. the expected maximum capacity of the battery. It may also sometimes be based on the current available power vs. the expected maximum power of the battery. A value of 100% SOC refers to a fully charged battery, whereas a value of 0% SOC represents a depleted battery. Accurate SOC estimation is of great concern, since it dictates how long the system can operate. However, accurate SOC values are hard to derive due to the lack of direct measurement. The SOC can only be estimated using the voltage and current of the LiB, which are indirect measurements [
4].
In the literature, several techniques have been proposed to estimate the SOC. The most popular method has been the ampere-hour counting method. This method makes use of current measurements and the battery’s capacity to estimate the SOC of the battery [
4]. The SOC is derived by making use of an initial value of SOC from which the current value is either subtracted or to which it is added, depending on whether a charge is demanded from or supplied to the battery. Unfortunately, due to the high dependency on an accurate initial SOC value and sensor current measurements, the estimated SOC value may be subject to large errors [
3]. Moreover, the ampere-hour counting method makes use of the battery’s available capacity, which has been shown to degrade over time, resulting in an increasing amount of error from regular usage of the battery. Some of these problems may be corrected by implementing calibration techniques such as voltage-based corrections using reference tables [
3].
Even though the ampere-hour counting method has several drawbacks, great results have been obtained when combined with other techniques such as neural networks (NNs) and Kalman filters (KFs) [
5,
6]. Neural networks have shown great potential but require vast amounts of data that must be collected beforehand. Moreover, some solutions result in computationally taxing algorithms [
5]. Note also that KFs provide low computational expense and accurate solutions but make use of mathematical models [
6].
Among the developed and studied models are the electrochemical models and electric circuit models (ECMs). Electrochemical models are highly accurate in nature and provide abundant information about the LiB’s state as they are based on the underlying physics of the battery. However, these models are often composed of several partial differential equations—sometimes in the order of 10–14 equations—resulting in complex and computationally expensive models. Unless significantly reduced, these models are often only implemented in battery development research [
7,
8,
9,
10]. Meanwhile, ECMs make use of electric elements such as resistors, capacitors, and voltage sources to create a representation of the battery’s dynamics. These types of models result in low computational power use and low-complexity solutions but provide less information about the battery and have been found to be less accurate than their counterparts [
11]. Nonetheless, these attributes are highly desired in online applications.
In the literature, ECMs can be categorized by the number of resistor–capacitor (RC) branches found in the model [
12]. For example, the Rint model makes use of a voltage source and a resistor to represent the battery’s energy source and energy loss during operation, respectively. The Thevenin model adds an RC branch to capture the transient response of the battery. Models with a higher number of RC branches have been shown to allow for a better representation of battery dynamics [
12]. However, each additional RC branch increases the complexity and computational time of the model.
Kalman filter-based solutions have shown great accuracy in determining the SOC when combined with ECMs and the ampere-hour counting method [
6]. Most KF solutions make use of a nonlinear form of the KF to account for the battery’s nonlinear behavior. Some of these variations are the extended Kalman filter (EKF), unscented Kalman filter (UKF) and cubature Kalman filter (CKF) [
6,
13]. However, these solutions do not account for the aging process of the battery, which is required to ensure an accurate SOC estimation during the lifetime of the battery. For SOC accuracy to remain the same throughout the use of the LiB, a battery monitoring system (BMS) must track the natural aging process of the battery. As the battery is subject to use, the LiB’s capacity degrades over time while the internal resistance increases. The result is an LiB with less output power or lower run time [
14]. The degradation of the battery capacity is of greatest importance since the SOC estimation process is highly dependent on an accurate value of the maximum capacity of the battery. In the literature, this aging process is described by the state of health (SOH) of the battery. SOH is often defined as a ratio of the current maximum battery capacity compared to its maximum initial capacity or manufactured capacity [
15]. In addition, it has been noted in the literature that an LiB may experience a faster aging process if it is subject to aggressive current profiles, excessive cycling, or deep discharging or charging conditions [
16].
Finally, accurate tracking of the LiB’s aging process allows for an effective retirement of the battery. LiBs are often retired when their maximum available capacity value is below 80% of their designed capacity. Dual filters have been proposed for tracking the aging process of LiBs. Dual filters can track several parameters of interest, including the internal resistance and the battery’s maximum capacity [
17,
18]. In a dual-filter strategy, the normal estimation process is broken into two components, where one part estimates the SOC and other states, while the second part estimates and updates the parameter values of the model [
17]. Other proposed strategies include the use of a multiple model (MM) process. An MM process makes use of several models to describe different behaviors that the system may be subjected to. The MM algorithm then selects the best match to the current conditions, which makes the algorithm flexible and resilient to uncertainty. A variation of the MM algorithm known as the interacting multiple model (IMM) has been implemented for tracking SOH [
19]. Several models were created to represent different SOH levels of an LiB. The algorithm would then match the best model to the current conditions, thus determining the most likely current SOH of the LiB [
19].
This paper proposes an adaptive strategy that can accurately track an LiB’s battery capacity when subject to different cycling conditions. This is of particular importance for the safe and reliable operation of electric vehicles. The proposed algorithm makes use of the dual-filter architecture to allow access to the model’s parameters and combines it with the IMM method to allow access to different aging rates of the parameters. The resulting algorithm is then combined with a KF and an ECM to estimate the SOC and parameters of the battery. The final algorithm is referred to as a dual-KF-IMM if a KF was implemented. Furthermore, the proposed method is combined with a sliding innovation filter (SIF) in an effort to improve robustness to modeling uncertainties. The SIF is a predictor–corrector filter with similar estimation capabilities to the KF. However, the SIF offers robustness to modeling uncertainties at the cost of estimation accuracy (i.e., it is sub-optimal in its standard formulation). Therefore, the well-known KF and robust SIF are implemented and utilized by the IMM in this study. We were able to combine the best elements of the KF (optimality) and the SIF (robustness) for strong estimation results. This is of particular use for electric vehicles when dealing with battery health estimation for improved safety and reliability.
The first main contribution is the study of the SIF in a dual-IMM setting, which leads to a new strategy called the dual-SIF-IMM. The second main contribution is the study of the IMM algorithm when combined with the dual-filter architecture to estimate the SOH of an LiB subject to different cycling conditions. The third main contribution is the detailed comparison between the dual-KF-IMM and the dual-SIF-IMM, the results of which were obtained based on a well-known dataset used for benchmarking.
This paper is structured as follows:
Section 2 presents the battery and parameter models.
Section 3 details the experimental data and estimation algorithms.
Section 4 covers the artificial measurements.
Section 5 describes the model parameter identification results.
Section 6 presents the experimental setup and details the results of the proposed strategy.
Section 7 presents the concluding arguments of the work and proposes future research that may be explored in the area.
7. Concluding Remarks
This paper presents estimation strategies that utilize the interacting multiple model (IMM) algorithm with dual-filtering strategies to accurately estimate the state of charge (SOC) and the battery capacity of a lithium-ion battery (LiB) under cycling conditions. This study is of particular importance for electric vehicles. As the battery is subject to cycling, its capacity degrades over time. The proposed strategies made use of the standard Kalman filter (KF) and the robust sliding innovation filter (SIF) in a dual-filter strategy (dual-KF and dual-SIF) to estimate the SOC and degradation of the battery capacity. Furthermore, the IMM was combined with the dual filter to allow access to different degradation rates of the battery’s parameter values (battery capacity and internal resistance). Including the IMM structure led to two new strategies, namely, the dual-KF-IMM and dual-SIF-IMM algorithms.
The two proposed strategies were evaluated using two experimental datasets. The first dataset was used as a baseline to test the performance of the algorithm under normal aging conditions and the second dataset was used to test the algorithms under a faster aging condition. Moreover, the battery model parameters and OCV curve were identified using the baseline dataset and remained constant in the accelerated aging experiment. Lastly, the proposed algorithms were only given access to the internal resistance and battery capacity parameters to account for the faster aging conditions.
The proposed algorithms demonstrated good accuracy for the baseline case (B036 dataset). For the accelerated aging case, the dual-IMM algorithms showed greater accuracy than their dual-only counterparts. Moreover, the dual-KF-IMM showed the most accuracy and better adaptation to the faster aging conditions. Finally, the dual-IMM algorithms showed great accuracy based on the RMSE values when the minimum value in the battery capacity estimation process was selected to generate a comparison across all algorithms. Lastly, in terms of mode probability, neither dual-IMM algorithm made a complete switch to the faster aging conditions to which they had access. A reason behind that could be the indirect effect of the parameter in the estimation process.
Further study of the dual-IMM algorithms is encouraging. A new potential study could implement the dual-IMM algorithms with a neural network strategy that could update the OCV curve for different aging conditions. Updating the OCV curve to account for the battery’s new dynamics would significantly increase the accuracy of the algorithms and relax the use of the battery capacity parameter to account for the degradation rate of the battery. Additionally, other nonlinear estimation strategies such as the extended and unscented Kalman filters, particle filters, or the H-infinity filter could be implemented and studied.