Lithium-Ion Battery Health Estimation Using an Adaptive Dual Interacting Model Algorithm for Electric Vehicles

Abstract

To ensure reliable operation of electrical systems, batteries require robust battery monitoring systems (BMSs). A BMS’s main task is to accurately estimate a battery’s available power, referred to as the state of charge (SOC). Unfortunately, the SOC cannot be measured directly due to its structure, and so must be estimated using indirect measurements. In addition, the methods used to estimate SOC are highly dependent on the battery’s available capacity, known as the state of health (SOH), which degrades as the battery is used, resulting in a complex problem. In this paper, a novel adaptive battery health estimation method is proposed. The proposed method uses a dual-filter architecture in conjunction with the interacting multiple model (IMM) algorithm. The dual filter strategy allows for the model’s parameters to be updated while the IMM allows access to different degradation rates. The well-known Kalman filter (KF) and relatively new sliding innovation filter (SIF) are implemented to estimate the battery’s SOC. The resulting methods are referred to as the dual-KF-IMM and dual-SIF-IMM, respectively. As demonstrated in this paper, both algorithms show accurate estimation of the SOC and SOH of a lithium-ion battery under different cycling conditions. The results of the proposed strategies will be of interest for the safe and reliable operation of electrical systems, with particular focus on electric vehicles.

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 ($Ah$). 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.

2. Battery Models

This section presents the battery and parameter models used to test the proposed strategy.

2.1. Dual Polarity (DP) Model

The DP model is an ECM that uses a voltage source, a resistor, and two RC branches. The voltage source represents the battery’s output voltage, the resistor represents the internal resistance of the battery, and the two RC branches represent the short-term and long-term transient behaviors of the battery [20]. In the DP model, one RC branch represents the battery’s concentration polarization, while the second RC branch represents the electrochemical polarization of the battery. These polarizations are responsible for a quick rise in voltage followed by a slow recovery after the battery is subject to a discharge period and the current is cut off [21]. This model was selected because it provides a good trade-off between accuracy and computational efficiency. Figure 1 depicts the circuit diagram of the DP model.

In the circuit diagram, the voltage source represents the open circuit voltage of the battery ($OCV$). The internal resistance is represented by $R_0$, the electrochemical polarization resistance and capacitance are represented by $R_{pa}$ and $C_{pa}$, respectively, and the concentration polarization resistance and capacitance are represented by $R_{pc}$ and $C_{pc}$, respectively [9].

Lastly, the system’s state space representation may be described by the following equations:

$$\left[\begin{array}{c}{U}_{pa}\\ U_{pc}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{R_{pa}{C}_{pa}}& 0\\ 0& \frac{1}{R_{pc}{C}_{pc}}\end{array}\right]\left[\begin{array}{c}{U}_{pa}\\ U_{pc}\end{array}\right]+\left[\begin{array}{c}\frac{1}{C_{pa}}\\ \frac{1}{C_{pc}}\end{array}\right]I_s$$

(1)

$$U_L=U_{ocv}\left(SOC\right)-U_{pa}-U_{pc}-I_L{R}_0$$

(2)

2.2. Parameter Model

Assuming the LiB’s parameters vary slowly (i.e., minutes to hours), the parameter changes during operation and the LiB’s aging can be represented by perturbing the states with low levels of white noise. The following equations depict the parameter model:

$${\theta }_{k+1}={\theta }_k+w_{\theta }{}_k$$

(3)

$$y_{\theta }{}_k=h\left(x_k, u_k, {\theta }_k\right)+v_{\theta }{}_k$$

(4)

$$w_{\theta }~N\left(0,Q_{\theta }\right)$$

(5)

$$v_{\theta }~N\left(0,R_{\theta }\right)$$

(6)

where $\theta$ represents the parameter vector and ${\omega }_{\theta }$ represents the level of white noise perturbation for each state with covariance $Q_{\theta }$.

3. Experimental Data and Estimation Algorithms

This section presents the selected experimental data and the estimation algorithms used for the experiments. These datasets are used for benchmarking results.

3.1. B036 (Normal Aging) Dataset

The B036 dataset contains a number of 2 Ah batteries that were run through three different operational profiles (charge, discharge, and impedance) at room temperature (24 $°\mathrm{C}$). These datasets were released by NASA’s Prognostic Center of Excellence (PCoE) and published by the Prognostic Data Repository to aid in the development of prognostic algorithms [22].

For this study, only the discharge cycles were considered since these cycles also provide battery capacity measurements. The B036 dataset provides discharging data at 2A or 1C and was used as the baseline case (normal aging) for this study. The battery was then cycled until the capacity had reduced to 1.6 Ah or a 20% fade was registered [22].

The B036 dataset registered an initial capacity of ~1.8 Ah and a final capacity of ~1.56 Ah. To ensure better adherence of the algorithms, the data was resampled to 0.6 sec. Figure 2 illustrates the battery capacity degradation after each discharging cycle.

3.2. B034 Dataset (Accelerated Aging)

The B034 dataset is also part of this small group of datasets. It provides discharging data at 4 A or 2C. This dataset was selected to represent a faster aging process.

The B034 dataset registered an initial battery capacity of ~1.68 Ah and a final battery capacity of ~1.3 Ah. As with the B036 dataset, the B034 data was resampled to 0.6 sec to allow for better adherence of the algorithms. Figure 3 illustrates the battery capacity degradation over all cycles.

3.3. Ampere-Hour Counting

As mentioned earlier, this method provides a simple way to estimate the SOC. Starting with a known initial value of SOC ($SO{C}_0$), the value is then increased or decreased based on the current profile. The ampere-hour counting formula is depicted below [3]:

$$SOC=SO{C}_0-\frac{1}{C_n}{{\displaystyle \int }}_{t_0}^{t}I d\tau$$

(7)

where $SO{C}_0$ is the initial SOC, $C_n$ is the nominal capacity of the battery, and $I$ is the discharge current.

3.4. Kalman Filter

The Kalman filter is one of the most popular solutions to the linear discrete-data filtering problem. It was presented by R. E. Kalman in 1960 [23]. The reasoning behind it is that it provides an optimal solution for a known linear system in the presence of white Gaussian noise by minimizing the state estimation error [23]. The linear system can be represented by the following two equations [24]:

$$x_{k+1}=A{x}_k+B{u}_k+w_k$$

(8)

$$z_{k+1}=C{x}_{k+1}+v_{k+1}$$

(9)

where $A$ is the dynamics matrix, $B$ is the input matrix, $C$ is the output matrix, $x$ is the system state, $z$ is the measurement output, $u$ is the input, $w$ is the system noise, and $v$ is the measurement noise.

Moreover, the KF algorithm can be represented as a two-stage process: prediction and update [24].

$$\left(i\right) Prediction Stage:$$

$${\widehat{x}}_{k+1|k}=A{\widehat{x}}_{k|k}+B{u}_k$$

(10)

$$P_{k+1|k}=A{P}_{k|k}{A}^T+Q$$

(11)

$$w~N\left(0,Q\right)$$

(12)

$$v~N\left(0,R\right)$$

(13)

$$\left(ii\right) Update Stage:$$

$$K_{k+1}=P_{k+1|k}{C}^T{\left[C{P}_{k+1|k}{C}^T+R\right]}^{-1}$$

(14)

$${\widehat{x}}_{k+1|k+1}={\widehat{x}}_{k+1|k}+K_{k+1}\left(z_{k+1}-C{\widehat{x}}_{k+1|k}\right)$$

(15)

$$P_{k+1|k+1}=\left[I-K_{k+1}C\right]P_{k+1|k}{\left(I-K_{k+1}C\right)}^T+K_{k+1}R{K}_{k+1}^T$$

(16)

where $Q$ and $R$ are the system and measurement noise covariance matrices, respectively.

3.5. Sliding Innovation Filter

The sliding innovation filter (SIF) is a recently proposed estimation strategy. It was designed using a predictor–corrector estimation architecture [25]. Much like the KF, the SIF can be described as a two-stage process, where predictions of the state estimates and error covariances are made in the first stage by using the previous time step’s information. The prediction estimates are then updated using the measurement information and a correction factor referred to as the SIF gain [25].

The SIF algorithm uses the same equations (8)–(11) as in the KF algorithm but significantly differs in the update stage. The SIF gain is formulated using the measurement matrix, the innovation, and a sliding boundary layer term [25]. In the update stage, the SIF uses the gain to drive the state estimates within the defined boundary layer, which forces the state estimates to within a region of the true trajectory [25]. The SIF algorithm can be summarized by the following set of equations:

$$\left(i\right) Update Stage:$$

$$K_{k+1}=C^{+}\overline{sat}\left(\left|{\tilde{z}}_{k+1|k}\right|/\delta \right){\widehat{x}}_{k+1|k+1}$$

(17)

$${\widehat{x}}_{k+1|k+1}={\widehat{x}}_{k+1|k}+K_{k+1}\left(z_{k+1}-C{\widehat{x}}_{k+1|k}\right)$$

(18)

$$P_{k+1|k+1}=\left(I-K_{k+1}{C}_{k+1}\right)P_{k+1|k}{\left(I-K_{k+1}{C}_{k+1}\right)}^T+K_{k+1} R K_{k+1}^T$$

(19)

Note that $C^{+}$ refers to the pseudoinverse of $C$, $\overline{sat}$ refers to the diagonal of the saturation term (value between $-1$ and $+1$), and $\delta$ is the sliding boundary layer width.

3.6. Interacting Multiple Model (IMM)

Most real-world systems can be defined by a set of operating modes. The IMM provides a solution for systems that have a multitude of operating modes. It makes use of several models and calculates the likelihood values for each model based on state estimates and error covariance [19], after which the likelihood values are processed to select the operating mode. The IMM algorithm is summarized below [19]:

$$\left(i\right) Calculation of the Mixing Probabilities:$$

$${\mu }_{i\left|j,k\right|k}=\frac{1}{{\overline{c}}_j} p_{ij}{\mu }_{i,k}$$

(20)

$${\overline{c}}_j={\displaystyle \sum }_{i=1}^r{p}_{ij}{\mu }_{i,k}$$

(21)

$$\left(ii\right) Mixing Stage$$

$${\widehat{x}}_{0j,k|k}={\displaystyle \sum }_{i=1}^r{\widehat{x}}_{i,k|k}{\mu }_{i\left|j,k\right|k}$$

(22)

$$P_{0j,k|k}={\displaystyle \sum }_{i=1}^r{\mu }_{i\left|j,k\right|k}\left\{P_{i,k|k}+\left({\widehat{x}}_{i,k|k}-{\widehat{x}}_{0j, k|k}\right){\left({\widehat{x}}_{i,k|k}-{\widehat{x}}_{0j, k|k}\right)}^T\right\}$$

(23)

$$\left(iii\right) Mode-Matched Filtering:$$

$${\Lambda }_{j,k+1}=\mathcal{N}\left(z_{k+1};{\widehat{z}}_{j,k+1|k}, S_{j,k+1}\right)$$

(24)

$${\Lambda }_{j,k+1}=\frac{1}{\sqrt{{\left|2\pi S_{j,k+1}\right|}_{Abs}}}\exp \left(\frac{-\frac{1}{2} e_{j,z,k+1}^T{e}_{j,z,k+1|k}}{S_{j,k+1}}\right)$$

(25)

$$\left(iv\right) Mode Probability Update:$$

$${\mu }_{j,k}=\frac{1}{c}{\Lambda }_{j,k+1}{\displaystyle \sum }_{i=1}^r{p}_{ij}{\mu }_{i,k}$$

(26)

$$c={\displaystyle \sum }_{j=1}^r{\Lambda }_{j,k+1}{\displaystyle \sum }_{i=1}^r{p}_{ij}{\mu }_{i,k}$$

(27)

$$\left(v\right) State Estimate and Covariance Combination:$$

$${\widehat{x}}_{k+1|k+1}={\displaystyle \sum }_{j=1}^r{\mu }_{j,k+1}{\widehat{x}}_{j,k+1|k+1}$$

(28)

$$\begin{gathered} P_{k+1|k+1}={\displaystyle \sum }_{j=1}^r{\mu }_{j,k+1}\left\{P_{j,k+1|k+1} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left({\widehat{x}}_{j,k+1|k+1}-{\widehat{x}}_{ k+1|k+1}\right){\left({\widehat{x}}_{j,k+1|k+1}-{\widehat{x}}_{ k+1|k+1}\right)}^T\right\} \end{gathered}$$

(29)

3.7. Dual Filters

In LiB literature, dual filters have been shown to be successful at tracking slow-changing dynamics that can be noticed in the long term, such as battery capacity degradation [26]. Dual filters track the slow-changing dynamics by implementing two filters [26]. One filter estimates the battery states such as the SOC, while the second filter focuses on updating the parameters within the battery model. The filter estimating for the parameters in the model makes use of a small perturbation to update the values [26]. If the EKF were to be selected for dual estimation, the result would be a dual EKF, where two EKFs would be implemented in the estimation process. The following set of equations summarize a dual-EKF algorithm for SOC estimation [27]:

$$\left(i\right) Prediction Stage:$$

$$State Filter:$$

$${\widehat{x}}_{k+1|k}=f\left({\widehat{x}}_{k|k}, u_k\right)$$

(30)

$$P_{x,k+1|k}={\widehat{A}}_{x,k}{P}_{x,k|k}{\widehat{A}}_{x,k}^T+Q_x$$

(31)

$$Parameter Filter:$$

$${\widehat{\theta }}_{k+1}={\widehat{\theta }}_k$$

(32)

$$P_{\theta ,k+1|k}=P_{\theta ,k|k}+Q_{\theta }$$

(33)

$$\left(ii\right) Update Stage:$$

$$State Filter:$$

$$K_{x,k+1}=P_{x,k+1|k} {\widehat{C}}_{x,k+1}^T{\left[{\widehat{C}}_{x,k}{P}_{x,k+1|k}{\widehat{C}}_{x,k+1}^T+R_x\right]}^{-1}$$

(34)

$${\widehat{x}}_{x,k+1|k+1}={\widehat{x}}_{x,k+1|k}+K_{x,k+1}\left[z_{k+1}-h\left({\widehat{x}}_{x,k+1|k}, u_{k+1},{\widehat{\theta }}_k\right)\right]$$

(35)

$$P_{x,k+1|k+1}=\left[I-K_{x,k+1}{\widehat{C}}_{x,k+1}\right]P_{x,k+1|k}$$

(36)

$$Parameter Filter:$$

$$K_{\theta ,k+1}=P_{\theta ,k+1|k} {\widehat{C}}_{\theta ,k+1}^T{\left[{\widehat{C}}_{\theta ,k}{P}_{\theta ,k+1|k}{\widehat{C}}_{\theta ,k+1}^T+R_{\theta }\right]}^{-1}$$

(37)

$${\widehat{\theta }}_{k+1|k+1}={\widehat{\theta }}_{k+1|k}+K_{\theta ,k+1}\left[z_{k+1}-h\left({\widehat{x}}_{\theta ,k+1}, u_{\theta ,k+1},{\widehat{\theta }}_k\right)\right]$$

(38)

$$P_{\theta ,k+1|k+1}=\left[I-K_{\theta ,k+1}{\widehat{C}}_{\theta ,k+1}\right]P_{\theta ,k+1|k}$$

(39)

where the nonlinearities of the state filter can be linearized. Furthermore, ${\widehat{C}}_{\theta ,k}$ refers to the total differential of (4). The following set of equations describe the result [27]:

$${\widehat{C}}_{\theta ,k}={\frac{\mathrm{d}h\left({\widehat{x}}_{k+1|k}, u_k, {\theta }_k\right)}{\mathrm{d}{\theta }_k}|}_{{\theta }_{k+1}={\widehat{\theta }}_{k+1|k}}$$

(40)

$$\begin{gathered} \frac{\mathrm{d}h\left({\widehat{x}}_{k+1|k}, u_{k+1}, {\widehat{\theta }}_{k+1|k}\right)}{\mathrm{d}{\widehat{\theta }}_{k+1|k}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\partial h\left({\widehat{x}}_{k+1|k}, u_{k+1}, {\widehat{\theta }}_{k+1|k}\right)}{\partial {\widehat{\theta }}_{k+1|k}}+\frac{\partial h\left({\widehat{x}}_{k+1|k}, u_{k+1}, {\widehat{\theta }}_{k+1|k}\right)}{\partial {\widehat{x}}_{k+1|k}}\frac{\mathrm{d}{\widehat{x}}_{k+1|k}}{\mathrm{d}{\widehat{\theta }}_{k+1|k}} \end{gathered}$$

(41)

$$\frac{\mathrm{d}{\widehat{x}}_{k+1|k}}{\mathrm{d}{\widehat{\theta }}_{k+1|k}}=\frac{\partial f\left({\widehat{x}}_{k|k+1}, u_k, {\widehat{\theta }}_{k+1|k}\right)}{\partial {\widehat{\theta }}_{k+1|k}}+\frac{\partial h\left({\widehat{x}}_{k|k+1}, u_k, {\widehat{\theta }}_{k+1|k}\right)}{\partial {\widehat{x}}_{k|k}}\frac{\mathrm{d}{\widehat{x}}_{k|k+1}}{\mathrm{d}{\widehat{\theta }}_{k+1|k}}$$

(42)

$$\frac{\mathrm{d}{\widehat{x}}_{k|k+1}}{\mathrm{d}{\widehat{\theta }}_{k|k}}=\frac{\mathrm{d}{\widehat{x}}_{k|k}}{\mathrm{d}{\widehat{\theta }}_{k|k+1}}-K_{x,k}\frac{\mathrm{d}h\left({\widehat{x}}_{k|k}, u_k, {\widehat{\theta }}_{k|k+1}\right)}{\mathrm{d}{\widehat{\theta }}_{k|k+1}}$$

(43)

where $K_{x,k}$ has little impact on the parameter estimates, and thus it can be neglected [27]. Removing the right term in (31) reduces the equation to $\frac{\mathrm{d}{\widehat{x}}_{k|k}}{\mathrm{d}{\widehat{\theta }}_{k|k+1}}$, which can now be estimated by recursion with an initial value of $\frac{\mathrm{d}{\widehat{x}}_{k|k+1}}{\mathrm{d}{\widehat{\theta }}_{k|k}}=0$ [27].

4. Artificial Measurements

Due to how the SIF gain was designed, individual measurements for each state estimate are required to ensure its effectiveness. In the absence of individual measurements, the generation of artificial measurements is necessary [25]. Unfortunately, LiBs do not provide direct measurements of state estimates. This section details how artificial measurements were generated using the system model equations.

4.1. State Measurement Equations

Artificial measurements were derived using Equation (2). These equations were rearranged to generate measurements for each state using the previous time-step values and current and voltage measurements. The state measurement equations are presented below.

$${\widehat{U}}_{pa,k+1}=OCV\left(SO{C}_k\right)-U_{L,k+1}-R_{0,k}{I}_{s,k}-U_{pc,k}$$

(44)

$${\widehat{U}}_{pc,k+1}=OCV\left(SOC\right)-U_{L,k+1}-R_{o,k}{I}_{s,k}-U_{pa,k}$$

(45)

$${\widehat{SOC}}_{k+1}=OC{V}^{-1}\left(U_{L,k+1}+U_{pa,k}+U_{pc,k}+R_{o,k}{I}_{s,k}\right)$$

(46)

where ${\widehat{U}}_{pa}$, ${\widehat{U}}_{pc}$, and $\widehat{SOC}$ are the measurements for each state of the battery model and $OC{V}^{-1}\left(\right)$ is the inverse function of $OCV\left(SOC\right)$.

4.2. Parameter Measurement Equations

To generate artificial measurements for the parameters of interest, $Bat{t}_{cap}$ and $R_0$, Equations (2) and (7) were rearranged to the following:

$${\widehat{R}}_o=\frac{1}{I_{s,k}}\left[OCV\left(SOC\right)-U_L-U_{pa}-U_{pc}\right]$$

(47)

$${\widehat{Batt}}_{Cap}=\frac{\Delta t\ast I_{s,k}}{abs\left(3.6\ast \Delta SO{C}_k\right)}$$

(48)

where ${\widehat{R}}_0$, and ${\widehat{Batt}}_{cap}$ represent the artificial measurements for $R_0$ and $Bat{t}_{cap}$.

5. Model Parameter Identification

For this experiment, the B036 (normal aging) dataset was used to derive the battery model. The B036 model was then used to test the proposed algorithm under faster aging conditions using the B034 data. Based on the battery model, the parameters to be identified were: $OCV\left(SOC\right), R_0, R_{pa}, C_{pa}, R_{pc}, C_{pc}$. This section presents the model’s parameter identification setup and results using the nonlinear least squares (NLLS) algorithm.

5.1. Least Squares Setup

The following relationships between the parameters and measurable data was used with the NNLS algorithm [28]:

$$OCV\left(SOC\right)={\alpha }_0+{\alpha }_{1}SOC+{\alpha }_{2}SO{C}^2+{\alpha }_{3}SO{C}^3+{\alpha }_{4}SO{C}^4+{\alpha }_{5}SO{C}^5$$

(49)

$$U_{pa}=I_L{R}_{pa}\left(1-e^{-\frac{t}{R_{pa}{C}_{pa}}}\right)$$

(50)

$$U_{pc}=I_L{R}_{pc}\left(1-e^{-\frac{t}{R_{pc}{C}_{pc}}}\right)$$

(51)

$$U_L=OCV\left(SOC\right)-I_L{R}_0-I_L{R}_{pa}\left(1-e^{-\frac{t}{R_{pa}{C}_{pa}}}\right)-I_L{R}_{pc}\left(1-e^{-\frac{t}{R_{pc}{C}_{pc}}}\right)$$

(52)

$$\theta =\left[{\alpha }_0, {\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5, R_0, R_{pa},\frac{1}{R_{pa}{C}_{pa}}, R_{pc},\frac{1}{R_{pc}{C}_{pc}}\right]$$

(53)

where $t$ represents the time vector and $OCV\left(SOC\right)$ is the $OCV$ curve approximated to a 5th order polynomial. $\theta$ represents the parameter vector. Moreover, the capacitance values are estimated using a fraction to account for their higher magnitude [28,29,30].

Finally, MATLAB^®’s $lsqcurvefit$ function was used with the data from cycle 5. Cycle 5′s voltage curve can be used to obtain the value for ${\alpha }_0$, which is when the battery has an SOC value of 0%. Furthermore, $lsqcurvefit$ requires values for the boundaries and initial conditions. The selected values are detailed in

Table 1. NLLS Boundaries and Initial Guess.

Parameters	Unit	Lower Bound	Upper Bound	Guess
$R_0$	$\mathsf{\Omega }$	0.003	0.500	0.020
$R_{pa}$	$\mathsf{\Omega }$	0.03	0.500	0.100
$C_{pa}$	$1/(\mathsf{\Omega }\mathrm{F}$)	0.00025	0.002	0.001
$R_{pc}$	$\mathsf{\Omega }$	0.03	0.500	0.100
$C_{pc}$	$1/(\mathsf{\Omega }\mathrm{F}$)	0.02	0.100	0.010

.

5.2. Least Squares Results

B036 Model

Table 2. NLLS Parameter Estimation Results.

RC Parameters	Value	OCV (SOC)	Value
$R_0$	0.0700	${\alpha }_1$	11.2906
$R_{pa}$	0.1730	${\alpha }_2$	−39.5170
$C_{pa}$	1428.86	${\alpha }_3$	65.7438
$R_{pc}$	0.4438	${\alpha }_4$	−50.5845
$C_{pc}$	52,903.10	${\alpha }_5$	14.4308

illustrates the results of the parameter identification process for the B036 model. The terminal voltage’s RMSE was 0.022 V. Figure 4 illustrates the terminal voltage’s error plot between the model and the measured terminal voltage, and Figure 5 illustrates the generated model’s terminal voltage plot vs. the measured terminal voltage.

6. Simulation Setup and Results

This section describes how the dataset was used to set up the detailed comparisons and may be used for future benchmarking.

6.1. Simulation Setup

The purpose of this experiment was to test the performance of the proposed strategy, a dual-IMM filter, against its standard form, a dual filter. The dual-IMM algorithm was first tested on normal aging conditions using the B036 dataset and implementing a KF and SIF to estimate the battery states. Then, the algorithm was tested under faster aging conditions using the B034 dataset. The IMM part of the algorithm was only implemented during the parameter estimation process, and it was given access to two different degradation rates, hereby referred to as slow and fast.

The slow mode values of the dual-IMM algorithm were set to the same values as the standard dual filter, whereas the fast mode was given values that allowed for a faster degradation rate of the parameters.

As mentioned above, the dual-filter algorithm was combined with the KF and SIF, resulting in a dual-KF and dual-SIF. When the IMM was combined with the dual strategies (dual-IMM), the resulting algorithms were the dual-KF-IMM and dual-SIF-IMM.

Table 3. Dual Filter and IMM Initial Conditions.

Model Variables	Value	IMM Variables	Value
$V_{pa}$	0	$Q_{\theta , Slow}$	$\mathrm{Diag}(1\times {10}^{-5},3\times {10}^{-5},5\times {10}^{-7}$)
$V_{pc}$	0	$R_{\theta , Slow}$	Diag(0.1,0.01,1)
$SOC$	100%	$Delt{a}_{Slow}$	$\mathrm{Diag}(5\times {10}^3,4$)
$R_0$	0.07	$Q_{\theta , Fast}$	$\mathrm{Diag}(5\times {10}^{-7},7\times {10}^{-4}$)
$Bat{t}_{Cap}$	2.00	$R_{\theta , Fast}$	Diag(0.1,0.01,1)
$Q_x$	$\mathrm{Diag}\left(1\times {10}^{-5},3\times {10}^{-5},5\times {10}^{-7}\right)$	$Delt{a}_{Fast}$	Diag(500,0.8)
$R_x$	Diag(0.1,0.01,1)	$p$	0.99
$Q_{\theta }$	$\mathrm{Diag}(5\times {10}^{-8},7\times {10}^{-5}$)	$\mu$	0.5
$R_{\theta }$	Diag(10,0.05)
$Delt{a}_x$	Diag(7,1.5,80)
$Delt{a}_{\theta }$	Diag(5000,4)

summarizes the initial conditions used for this experiment.

6.2. Simulation Results

This section presents the results obtained from the proposed strategies.

6.2.1. B036 Dataset Dual-IMM Results

The estimation results from both algorithms are presented at two cycles to demonstrate the behaviors of the algorithms at different stages of battery life.

Cycle 21 was selected to show the estimation process of the algorithms at the beginning of the experiment. At this cycle, the battery capacity was measured to be ~1.79 Ah based on the B036 dataset. Figure 6 illustrates the battery capacity estimates at cycle 21. Part (a) compares the results of the dual-KF and dual-SIF and part (b) shows the results of the dual-KF-IMM and dual-SIF-IMM. All algorithms depict a large swing in battery capacity values during the estimation process; however, the final values of the estimation process return to levels similar to the beginning of the cycle. The dual-IMM algorithms show a significantly larger swing compared to the other dual strategies, which can be attributed to their access to a faster degradation rate. Two reasons for these large swings in the experiment can be attributed to the experiment’s design. One reason could be that the battery model is forced to remain constant throughout the different stages of life of the battery. Another reason is that the algorithms were tuned and designed to place a high emphasis on battery capacity as one of the parameters to account for the aging process.

Next, Figure 7 depicts the SOC estimation results from all algorithms at cycle 21. These SOC curves demonstrate an accurate full discharge of the battery at this stage of life. It is evident that the battery is fully depleted by the end of the discharge cycle. All algorithms show similar profiles, with slight deviations from the model’s SOC curve. It is important to note that the model curve is based on the derived model from Section 5 and utilizes a constant battery capacity of 1.8 Ah.

Cycle 445 was selected to show the algorithms’ estimation process toward the end of battery life. At this cycle, the battery had a reported battery capacity of 1.6 Ah. Figure 8 illustrates the battery capacity estimates during this cycle. Part (a) compares the results of the dual-KF and dual-SIF and part (b) compares the dual-IMM strategy results. At this stage, all the algorithms showed strong accuracy in tracking the correct battery capacity.

Figure 9 shows the SOC curves of all algorithms at cycle 445. Part (a) compares the results of the dual-KF and dual-SIF and part (b) compares the dual-IMM strategy results. At this stage, all algorithms showed a significant deviation from the derived model. Furthermore, the algorithms showed an accurate SOC curve that depleted the battery by the end of the discharging cycle.

Finally, the overall battery capacity estimation results of all algorithms were compared based on the results at the end of each cycle. Due to the nature of the battery capacity curves, the mean value was selected to represent the final battery capacity estimate at the end of the cycle. It is important to note that selecting the end value yields the same results as the mean but selecting the minimum value at each cycle generated a different curve.

Table 4. B036 RMSE: Battery Capacity Estimation.

Algorithm	RMSE
Dual-KF	0.0325
Dual-KF-IMM	0.0292
Dual-SIF	0.0460
Dual-SIF-IMM	0.0469

depicts the RMSE values from each algorithm based on Figure 2. The RMSE values show good accuracy in estimating the battery capacity for all algorithms. The dual-KF-IMM had the greatest accuracy compared to the other algorithms.

Based on the LiB literature, the internal resistance should increase as the battery ages. Figure 10 shows the internal resistance results of all algorithms across all cycles. Part (a) depicts the results from the dual-KF and dual-KF-IMM and part (b) illustrates the results of the dual-SIF and dual-SIF-IMM. In part (a), both dual-KF and dual-KF-IMM show this trend of increased internal resistance as the battery ages. However, the dual-SIF and dual-SIF-IMM illustrate a downward trend in internal resistance, which could be the main contributor to higher errors compared to the KF algorithms. Based on part (a), the dual-KF-IMM makes use of the fast-aging process to show an accelerated aging compared to the dual-KF. The use of faster aging rates becomes more evident in Figure 11.

Figure 11 depicts the battery capacity estimation results of all algorithms across all cycles. Part (a) shows the results of the dual-KF and dual-KF-IMM and part (b) illustrates the results of the dual-SIF and dual-SIF-IMM. As can be seen, all algorithms were initialized with the manufacturer’s battery capacity (2 Ah) as opposed to the measured battery capacity of 1.8 Ah. Initializing the battery capacity at a different value tests the adaptability of the proposed strategies. Based on part (a), the dual-KF-IMM shows the fastest convergence to the measured values but then it slightly overestimates the battery capacity. The dual-KF on the other hand takes longer to converge to the measured values. Part (b) shows that the dual-SIF and dual-SIF-IMM take longer to converge to the measured values, which most likely is due to the internal resistance estimates.

Figure 12 depicts all the battery capacity estimation results from all algorithms across all cycles in one plot for better visualization.

Lastly, Figure 13 illustrates the mode probability results from the dual-IMM strategies. Part (a) depicts the dual-KF-IMM mode probability and part (b) shows the mode probability of the dual-SIF-IMM. Both algorithms set the slow mode degradation rate higher than the fast mode. The dual-KF-IMM mode probability remains about the same, with values of 0.68 for the slow mode and 0.32 for the fast mode, whereas the dual-SIF-IMM sets the slow mode at 0.75 and higher past the 300th cycle.

6.2.2. Dual-IMM Results with B034 Dataset

As in the previous section, cycle 21 was selected to show the estimation process of the proposed strategies at the early stages of degradation. Furthermore, cycle 445 was selected to demonstrate the behavior of the algorithms toward the end of battery life.

Based on the B034 dataset, the measured battery capacity at this cycle was 1.46 Ah. Figure 14 shows the battery capacity estimates at cycle 21. Part (a) illustrates the results of the dual-KF and dual-SIF and part (b) depicts the results of the dual-KF-IMM and dual-SIF-IMM. Here, the algorithms were still transitioning to the correct measured values. The dual-IMM algorithms showed faster convergence to lower capacity values compared to the dual-only strategies. Furthermore, at this stage, both SIF algorithms slightly outperformed their KF counterparts.

Figure 15 shows the SOC estimation results of all the proposed strategies. At this stage of the estimation process, the SOC curves did not depict a fully discharged battery, which is most likely due to incorrect battery capacity estimates.

At cycle 445, the measured battery capacity based on the B034 dataset was 1.28 Ah. Figure 16 depicts the battery capacity of the proposed strategies. Part (a) shows the estimation results of the dual-KF and dual-SIF and part (b) illustrates the results of the dual-KF-IMM and dual-SIF-IMM. Both dual strategies started to lag compared to the dual-IMM strategies. The dual-SIF showed a better estimate compared to the dual-KF algorithm. Even though the dual-IMM algorithms did not depict the correct battery capacity estimate, both algorithms performed better than their dual-only counterpart. The dual-KF-IMM showed the best estimate.

Next, Figure 17 shows the SOC curves generated by the algorithms. Part (a) illustrates the results of the dual-KF and dual-SIF and part (b) shows the dual-KF-IMM and dual-SIF-IMM results. In part (a), both dual-only strategies were only able to discharge the battery to about 10% SOC, most likely due to their high battery capacity estimate. On the other hand, the dual-KF-IMM demonstrated a better SOC curve profile compared to the other strategies, where the battery was depleted at the end of the cycle. The dual-SIF-IMM showed similar results to the dual-SIF.

Finally, in terms of results across all cycles,

Table 5. B034 RMSE: Battery Capacity Estimation.

Algorithm	RMSE
Dual-KF	0.3677
Dual-KF-IMM	0.1758
Dual-SIF	0.2891
Dual-SIF-IMM	0.2198

details the RMSE battery capacity estimation results. The dual-IMM algorithms showed better accuracy compared to their dual-only counterparts. The dual-KF-IMM showed the best accuracy compared to the other algorithms.

Figure 18 illustrates the internal resistance estimations across all cycles. Part (a) shows the results from the dual-KF and dual-KF-IMM and part (b) depicts the results from the dual-SIF and dual-SIF-IMM. All algorithms show a downward trend, which could be reflective of new battery dynamics due to the higher discharging current to which this dataset is subjected. The dual-KF-IMM makes use of different operating conditions (models) which causes internal resistance to change significantly at early cycles when compared with the non-IMM methods.

Figure 19 shows the battery capacity estimation results of all algorithms across all cycles. Part (a) illustrates the dual-KF and dual-KF-IMM results and part (b) depicts the dual-SIF and dual-SIF-IMM results. In part (a), the dual-KF-IMM shows a better adaptation to the new aging conditions compared to the dual-KF results. Analyzing part (b) and based on the peaks observed on the figure, it is evident that the dual-SIF-IMM makes use of the faster degradation rates to which it has access, resulting in a better estimation process. However, it does not show a better performance than the dual-KF-IMM algorithm.

To further compare all results, Figure 20 depicts all the battery estimation curves in one plot. It can be seen that the dual-IMM strategies performed better than their dual-only counterparts, as they had access to faster aging conditions. The dual-SIF performed better than the dual-KF because of the deltas to which it had access. Lastly, the dual-KF-IMM performed the best across all algorithms despite not having the advantage of the delta parameters. In this case, tuning of the deltas in the dual-IMM strategy might be a limiting factor in the estimation process.

It is important to mention that if the minimum battery capacity value estimated at each cycle were to be selected to generate a plot such as Figure 20, the dual-IMM algorithms completely capture the measured battery capacity, as demonstrated by Figure 21. Furthermore, the battery capacity RMSE values detailed in

Table 6. B034 RMSE: Battery Capacity Using the Minimum at each Estimation cycle.

Algorithm	RMSE
Dual-KF	0.3229
Dual-KF-IMM	0.2581
Dual-SIF	0.0831
Dual-SIF-IMM	0.0458

further demonstrate the accuracy of the proposed dual-IMM strategy.

Lastly, Figure 22 shows the mode probability results of the dual-IMM strategies. Part (a) demonstrates the mode probability of the dual-KF-IMM, and part (b) depicts the mode probability of the dual-SIF-IMM. When comparing Figure 22 to Figure 13, the dual-IMM strategies favor the fast-aging mode slightly more than the B036 dataset. However, neither algorithm makes full use of the faster degradation to which they have access (switch to fast mode). Some reasons could be due to the nature of the estimation process. Battery capacity only affects the estimation process indirectly. Furthermore, the faster degradation rates do not fully represent the new dynamics of the entire system, only partially. The faster degradation rates provide a way to adapt faster to the new conditions to which the battery may be subjected.

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.