Estimation Strategies for the Condition Monitoring of a Battery System in a Hybrid Electric Vehicle

Abstract

This paper discusses the application of condition monitoring to a battery system used in a hybrid electric vehicle (HEV). Battery condition management systems (BCMSs) are employed to ensure the safe, efficient, and reliable operation of a battery, ultimately to guarantee the availability of electric power. This is critical for the case of the HEV to ensure greater overall energy efficiency and the availability of reliable electrical supply. This paper considers the use of state and parameter estimation techniques for the condition monitoring of batteries. A comparative study is presented in which the Kalman and the extended Kalman filters (KF/EKF), the particle filter (PF), the quadrature Kalman filter (QKF), and the smooth variable structure filter (SVSF) are used for battery condition monitoring. These comparisons are made based on estimation error, robustness, sensitivity to noise, and computational time.

1. Introduction

Condition monitoring is an essential process for fault detection and diagnosis. It involves monitoring system states or parameters over an operational period, where abnormal values or significant changes would indicate a fault. Quite often direct measurements of the states are not available due to limitations in design or cost. In these cases, state and parameter estimation techniques can be used for information extraction. Condition monitoring of systems allows proper maintenance to be scheduled, which helps reduce unscheduled downtime of manufacturing equipment, as well as the cost to repair damaged systems [1, 2]. An important area for condition monitoring is energy management for hybrid electric (HEVs) and battery electric vehicles (BEVs).

In general, HEVs have two power sources: a gasoline engine and an electric motor. In full hybrid vehicles, the engine and the motor can operate separately or simultaneously. The motor is used mainly during acceleration, startup, reverse mode, and in regenerative braking. A traction battery pack is used to provide power to the motor. It is recharged by a generator or during regenerative braking. The performance of an HEV is largely dependent on a balance between the gasoline engine and the electric motor, optimized with respect to fuel consumption based on vehicle conditions [3]. Many different types of control methods have been applied to balance the power and energy requirements of HEVs, including fuzzy logic [4–6], genetic algorithms [7], dynamic programming [8, 9], Pareto optimization [10], and intelligent mechanism designs [11]. These control strategies rely heavily on the availability of battery power to balance the operation of the gasoline engine versus the electric motor. The available battery power may be obtained from the state-of-charge (SOC) information [3, 12]. Further to the SOC, the battery state-of-health (SOH) is required in order to help determine whether a battery would fail subject to a certain load [13]. The SOC cannot be measured directly with electric signals, and as such it often, needs to be estimated through current and voltage relationships [3]. Poor estimation or control of the SOC may lead to improper charging conditions and can degrade the efficiency and reliability of the batteries [13]. Hence, proper condition monitoring of batteries plays a pivotal role in the optimization of HEV performance, as well as extending the lifetime and increasing the reliability of the batteries [14].

State and parameter estimation techniques are an integral part of condition monitoring and are used when direct measurements of the states are not available. One of the most commonly studied methods for estimation is the Kalman filter (KF) for linear systems and its extended form (EKF) for nonlinear systems [15–19]. Operating conditions such as battery SOC, power fade, capacity fade, resistance, and instantaneous available power have been estimated well using the EKF [17]. Other methods such as the sigma-point Kalman filtering (SPKF) and support vector machine (SVM) have also been used for condition monitoring and fault diagnosis [20–22].

In critical applications, such as automotive that require added safety and reliability, the choice of the estimation method is very important and should be selected based on the linearity of the system, performance, robustness or sensitivity to noise and computational difficulty and time. In relation to battery condition monitoring, this paper presents a quantitative and qualitative comparison of the following methods: the Kalman and extended Kalman filters (KF/EKF), the particle filter (PF), the quadrature Kalman filter (QKF), and the smooth variable structure filter (SVSF).

2. State and Parameter Estimation Techniques

State and parameter estimation is essential for sensing and information processing in model-based condition monitoring. Estimation theory involves information extraction by tracking changes in physical parameters or operational states of the system. This paper studies four popular strategies: the Kalman and the extended Kalman filters (KF/EKF), the particle filter (PF), the quadrature Kalman filter (QKF), and the smooth variable structure filter (SVSF).

2.1. Kalman and Extended Kalman Filters

Even after 50 years, the Kalman filter (KF) remains the most studied and one of the most popular tools used in state estimation [23–26]. It may be applied to linear dynamic systems in the presence of Gaussian white noise, and provides an elegant and statistically optimal solution by minimizing the mean-squared error. The impact that the KF has had on estimation and control problems is considered by some scientists and engineers to be one of the greatest achievement in engineering and signal processing [26]. It is a method that utilizes measurements linearly related to the states, and error covariance matrices, to generate a gain referred to as the Kalman gain. This gain is applied to the a priori state estimate, thus creating an a posteriori estimate. The estimation process continues in a predictor-corrector fashion while maintaining a statistically minimal state error covariance matrix for linear systems.

The following two equations describe the system dynamic model and the measurement model used in general for state estimation. Refer to Appendix A for a description of the nomenclature 𝑥𝑧_𝑘+1𝑘+1=𝐴_𝑘𝑥_𝑘+𝐵_𝑘𝑢_𝑘+𝑤_𝑘,=𝐶_𝑘+1𝑥_𝑘+1+𝑣_𝑘+1.(1)

The next five equations form the KF algorithm and are used in an iterative fashion. Equation (2) extrapolates the a priori state estimate, and (3) is the corresponding a priori error covariance. The Kalman gain may be calculated by (4), and is used to update the state estimate and error covariance, described by (5) and (6), respectively. 𝑃𝑥̂_𝑘+1|𝑘=𝐴_𝑘𝑥̂_𝑘|𝑘+𝐵_𝑘𝑢_𝑘,(2)_𝑘+1|𝑘𝐾=𝐴_𝑘𝑃_𝑘|𝑘𝐴^𝑇𝑘+𝑄_𝑘,(3)_𝑘𝑥̂=_{𝑘+1|𝑘+1}𝑃_𝑘+1|𝑘𝐶^𝑇𝑘𝐶_𝑘𝑃_𝑘+1|𝑘𝐶^𝑇𝑘+𝑅_𝑘⁻¹,(4)𝑃=𝑥̂_𝑘+1|𝑘+𝐾_𝑘𝑧_𝑘−𝐶_𝑘𝑥_𝑘+1|𝑘,(5)_{𝑘+1|𝑘+1}=𝐼−𝐾_𝑘𝐶_𝑘𝑃_𝑘+1|𝑘.(6)

The effects due to model uncertainties can have a large impact on the stability and performance of the KF [26, 27]. For nonlinear systems, the EKF may be used. It is conceptually similar to the KF process. The nonlinear system and measurement matrices are linearized according to their corresponding Jacobian, which is a first-order partial derivative. This linearization introduces uncertainties in the estimation process; such that overlooked nonlinearities in the system may cause the EKF to become unstable [26].

2.2. Particle Filter

The particle filter (PF) has many forms: Monte Carlo filters, interacting particle approximations [28], bootstrap filters [29], condensation algorithm [30], and survival of the fittest [31], to name a few. Compared to the KF, it is newer, being introduced in 1993. Since then, the PF has become a very popular method for solving nonlinear estimation problems, ranging from predicting chemical processes to target tracking. The PF takes the Bayesian approach to dynamic state estimation, in which one attempts to accurately represent the probability distribution function (PDF) of the values of interest [32]. The PDF contains all of the pertinent statistical information and may be considered as holding the solution to the estimation problem [32]. Essentially, the distribution holds a probability of values for the state being observed. The stronger or tighter the prediction PDF, the more accurate the state estimate.

The PF obtains its name from the use of weighted particles or “point masses” that are distributed throughout the PDF to form an approximation. These particles are used in an iterative process to obtain new particles and associated importance weights, with the goal of creating a more accurate approximation of the PDF. In general, as the number of implemented particles becomes large, the PDF becomes more accurate [32]. An important step in the PF is that of resampling, which eliminates particles with low weights and multiplies those with high weights [32]. This helps to avoid the degeneracy problem with the PF, which refers to only one particle having a significant importance weight after a large number of recursions. Furthermore, it also increases the accuracy of the PDF approximation by replicating particles with high weights. The sequential importance resampling (SIR) algorithm is a very popular form of the PF and may be summarized by (7) to (10). The first equation draws samples or particles from the proposal distribution. 𝑥^𝑘(𝑛)∼𝜋𝑥_𝑘∣𝑥^{(𝑛)𝑘−1},𝑦_𝑘.(7) The next equation updates the importance weights up to a normalizing constant 𝜔^𝑘(𝑛)=𝜔^{(𝑛)𝑘−1}𝑝𝑦_𝑘∣𝑥^𝑘(𝑛)𝑝𝑥^𝑘(𝑛)∣𝑥^{(𝑛)𝑘−1}𝜋𝑥_𝑘∣𝑥^{(𝑛)𝑘−1},𝑦_𝑘.(8) Next, the normalized weights are calculated for each particle 𝜔^𝑘(𝑛)=𝜔^𝑘(𝑛)∑^𝑛𝑖=1𝜔^𝑘(𝑖).(9)

Finally, a constant known as the effective number of particles is calculated as shown in (10). Resampling is performed if the effective number of particles is lower than some design threshold 𝑁_eff=1∑^𝑛𝑖=1𝜔^𝑘(𝑖)².(10)

2.3. Quadrature Kalman Filter

Similar to the PF, the quadrature Kalman filter (QKF) is a type of Bayesian filter that is able to model dynamic processes which are nonlinear and subject to non-Gaussian noise. In 2007, it was proposed that a set of Gauss-Hermite quadrature points could be used to parameterize the PDF [33, 34]. When compared with the EKF, it was found that the QKF method provides a more accurate least-squares solution [33]. Figure 1 shows the PDF of a nonlinearly transformed Gaussian random variable. The mean of the EKF appears to be biased, and the covariance is obviously far from the true covariance. However, the QKF mean and covariance match the true values quite well, with the 5 point QKF working the best. The main drawback to this method is the fact that the number of terms used in the Gaussian sum grows exponentially, which means that more memory will be used over a longer period of time [33]. The process of this filter is similar to the PF, in the sense that it may be solved recursively using two stages (time and measurement update). Please refer to Appendix B for the full QKF algorithm.

Figure 1 

PDF of a Nonlinearly transformed gaussian random variable [33].

2.4. Smooth Variable Structure Filter

In 2002, the variable structure filter (VSF) was introduced as a new predictor-corrector method used for state and parameter estimation [27, 35]. It is a type of sliding mode estimator, where gain switching is used to ensure that the estimates converge to true state values. An internal model of the system, either linear or nonlinear, is used to predict an a priori state estimate. A corrective term is then applied to calculate the a posteriori state estimate, and the estimation process is repeated iteratively. The SVSF was later derived from the VSF and uses a simpler and less complex gain calculation [36]. In its present form, the SVSF is stable and robust to modeling uncertainties and noise, given an upperbound on uncertainties [36]. The basic concept of the SVSF is shown in Figure 2. Assume that the solid line in Figure 2 is a trajectory of some state (amplitude versus time). An initial value is selected for the state estimate. The estimated state is pushed towards the true value until it reaches a subspace around the actual state trajectory, referred to as the existence subspace. Once the value enters the existence subspace, the estimated state is forced to remain within it and into switching along the system state trajectory [36].

Figure 2 

SVSF estimation concept [36].

The SVSF method is model based and applies to smooth nonlinear dynamic equations. The estimation process may be summarized by (11) to (16) and is repeated iteratively. An a priori state estimate is calculated using an estimated model of the system. This value is then used to calculate an a priori estimate of the measurement defined by (12). A corrective term, referred to as the SVSF gain, is calculated as a function of the error in the predicted output, as well as a gain matrix and the smoothing boundary layer width. The corrective term calculated in (13) is then used in (14) to find the a posteriori state estimate 𝑥̂𝑧̂_𝑘+1|𝑘=_𝑘+1|𝑘𝐹𝑥̂_𝑘|𝑘,𝑢_𝑘,(11)𝐾=𝐶𝑥̂_𝑘+1|𝑘,(12)_𝑘+1=𝐶⁻¹||||||𝑒_{𝑧𝑘+1|𝑘}|||_𝐴𝐵𝑆+𝛾|||𝑒_{𝑧𝑘|𝑘}|||_𝐴𝐵𝑆|||_𝐴𝐵𝑆∘𝑥sât𝑒_{𝑧𝑘+1|𝑘𝑘+1|𝑘+1},Ψ,(13)𝑒=𝑥̂_𝑘+1|𝑘+𝐾_𝑘+1,(14)_{𝑧𝑘|𝑘}𝑒=𝑧_𝑘−𝑧̂_𝑘|𝑘,(15)_{𝑧𝑘+1|𝑘}=𝑧_𝑘+1−𝑧̂_𝑘+1|𝑘.(16)

Two critical variables in this process are the a priori and a posteriori output error estimates, defined by (15) and (16), respectively [36]. Note that (15) is the output error estimate from the previous time step and is used only in the gain calculation.

3. Condition Monitoring of a Battery System in a Hybrid Electric Vehicle

A variety of batteries have been studied in literature, most notably lead-acid and lithium-ion batteries [13, 15, 19, 20, 37]. Lead-acid batteries are the oldest type of rechargeable batteries, and are most commonly found in motor vehicles. Lithium-ion batteries are also a form of rechargeable battery, which contain lithium in its positive electrode (cathode). These batteries are usually found in portable consumer electronics (i.e., laptops or notebooks) due to particularly high energy-to-weight ratios, slow self-discharge, and a lack of memory effect (i.e., where a battery loses its maximum energy capacity over time) [16]. In recent years, lithium-ion batteries have slowly entered the hybrid electric vehicle market, due to the fact that they offer better energy density compared to standard batteries [38].

The operation of batteries may be studied by using the advanced vehicle simulator (ADVISOR), which was written in MATLAB and Simulink by the US Department of Energy and the National Renewable Energy Laboratory [39–41]. ADVISOR is used for the analysis of performance and fuel economy of three vehicle types: conventional, electric, and hybrid vehicles [39]. In 2001, the resistance-capacitance (RC) battery model was first implemented in ADVISOR [42]. The electrical model consists of three resistors (𝑅_𝑒, 𝑅_𝑐, and 𝑅_𝑡) and two capacitors (𝐶_𝑏 and 𝐶_𝑐). The first capacitor (𝐶_𝑏) represents the capability of the battery to chemically store a charge, and the second capacitor (𝐶_𝑐) represents the surface effects of a cell [41]. The resistances and capacitances vary with changing SOC and temperature (𝑇) [41]. ADVISOR offers two different datasets for the RC battery model: lithium-ion and nickel-metal hydride chemistries. For the purposes of this study, the lithium-ion chemistry was used in conjunction with the RC battery model. Figure 3 illustrates the equivalent circuit diagram of the RC model.

Figure 3 

RC battery circuit diagram.

A standard model of a parallel hybrid electric vehicle referred to within ADVISOR as the Annex VII PHEV was used for this study. This model has been developed by the International Energy Agency (IEA), which is an international research community for the development and commercialization of hybrid and electric vehicles [43]. The model is based on data obtained from published sources and national (U.S.) research test data [39]. The battery system of the HEV represents the battery pack which stores energy on board the HEV. The system accepts a power request and returns the available power from the battery, as well as the SOC, voltage and current [39].

The nonlinear equations that describe the system may be derived from the RC battery model of Figure 3, as shown in Appendix C. For the purposes of condition monitoring of the battery, two voltages 𝑉_𝐶𝑏 and 𝑉_𝐶𝑐 as well as two capacitance parameters 𝐶_𝑏 and 𝐶_𝑐 need to be estimated. Should a fault exist in either of the battery capacitors, one would be able to determine this given the corresponding change in the parametric value. Further to the equations found in Appendix C, a discrete-time state space model of the Capacitor voltages may be defined as follows: 𝑉𝑉=⎡⎢⎢⎢⎣−^{𝐶𝑏𝑘+1}_{𝐶𝑐𝑘+1}𝑇_𝑠𝐶_𝑏(𝑅_𝑒𝑇+_𝑠𝑅_𝑐)+1𝐶_𝑏(𝑅𝑇^𝑠_𝑒+𝑅_𝑐)𝐶_𝑐(𝑅_𝑒+𝑅_𝑐)−𝐶_𝑐(𝑅𝑇^𝑠_𝑒+𝑅_𝑐)+1⎤⎥⎥⎥⎦_𝑘+⎡⎢⎢⎢⎣𝑇_𝑠𝑉𝑉^𝐶𝑏𝑘_𝐶𝑐𝑘𝑅_𝑐𝐶_𝑏(𝑅𝑇_𝑒𝑠+𝑅_𝑐)𝑅_𝑒𝐶_𝑐(𝑅_𝑒+𝑅_𝑐)⎤⎥⎥⎥⎦_𝑘𝐼_𝑆𝑘.(17)

For parameter estimation, (17) is used to formulate a state vector that would include parameters such that 𝑥=𝑉_𝐶𝑏𝑉_𝐶𝑐𝐶_𝑏𝐶_𝑐^𝑇.(18)

By rearranging (17), and further to Appendix C, the general model used for state and parameter estimation is obtained as 𝑧𝑥_𝑘+1𝑘+1=𝑓𝑥_𝑘,𝑢_𝑘+𝑤_𝑘,=ℎ𝑥_𝑘+1,𝑢_𝑘+1+𝑣_𝑘+1.(19)

To implement the EKF method, the nonlinear system had to be linearized using a first-order Taylor series approximation. The linearized form of the system equation is provided in Appendix D.

4. Estimation Results

This paper presents a comparative performance for the application of the extended Kalman filter, the particle filter, the quadrature Kalman filter, and the smooth variable structure filter for condition monitoring of RC batteries. The study is conducted by simulation using the ADVISOR battery model. The model parameters are documented in [39].

Figures 4 and 5 illustrate the input current (𝐼_𝑠) and the output voltages 𝑉_𝐶𝑏 and 𝑉_𝐶𝑐 used in the simulation. The parameter values 𝐶_𝑏 and 𝐶_𝑐 are made to vary in order to simulate fault conditions as shown in Figures 6 and 7. Parameters 𝐶_𝑏 and 𝐶_𝑐 as well as the states 𝑉_𝐶𝑏 and 𝑉_𝐶𝑐 are estimated using the four methods described in Section 2.

Figure 4 

Supplied current input.

Figure 5 

Capacitance voltages.

Figure 6 

𝐶𝑏 simulation values.

Figure 7 

𝐶𝑐 simulation values.

A comparison and discussion of the results is provided using two simulated cases: one with noise and one with noise as well as modeling errors. Note that the sampling time used in the simulation was 0.01 seconds.

4.1. Extended Kalman Filter Results

For linear dynamic systems in the presence of Gaussian white noise, the KF provides an elegant and statistically optimal solution by minimizing the mean-squared estimation error. The EKF is used for nonlinear problems. The following covariance matrices (error, process, and measurement, resp.) were used for the EKF and were obtained by trial-and-error: 𝑄𝑃==⎡⎢⎢⎢⎣𝑅3331000=333⎡⎢⎢⎢⎣1000100033301001000001033310000001⎤⎥⎥⎥⎦⎤⎥⎥⎥⎦,.(20)

The EKF worked very well in the absence of modeling errors, as shown in Figure 8. Its performance degraded considerably when modeling errors were added, as illustrated in Figure 9.

(a) Estimation error over time

(b) Capacitor voltages

(c) 𝐶𝑏 simulation values

(d) 𝐶𝑐 simulation values

Figure 8 

Extended Kalman filter simulation results (only noise).

(a) Estimation error over time

(b) Capacitor voltages

(c) 𝐶𝑏 simulation values

(d) 𝐶𝑐 simulation values

Figure 9 

Extended Kalman filter simulation results (including errors).

As shown in Figure 10, the EKF was very sensitive to the selection of the initial conditions; however, it often recovered and performed very well. This is important to note because without selecting good starting points, the filter performed the worst. However, when good initial conditions were selected, the EKF yields the best results (based on RMSE and the assumption that there were no uncertainties in the model). When uncertainties (besides white measurement and process noise) were present in the filter model, in this example, the EKF became unstable (as shown in Figure 9) and failed to yield reasonable results. The simulation time for the EKF was one of the fastest, and compared to the other methods, it was computationally easy.

(a) 𝐶𝑏 simulation values

(b) 𝐶𝑐 simulation values

Figure 10 

EKF Simulation results with poor initial conditions (only noise).

4.2. Particle Filter Results

For the PF method, a large number of particles (500) were used. Its application results are shown in Figures 11 and 12. The PF was able to estimate fairly well in the presence of noise. However, when modeling uncertainties were introduced, the PF had difficulty tracking the true voltages. As shown in Figure 12(b), after about 20 seconds, the PF was able to recover and provide a good estimate.

(a) Estimation error over time

(b) Capacitor voltages

(c) 𝐶𝑏 simulation values

(d) 𝐶𝑐 simulation values

Figure 11 

Particle filter simulation results (only noise).

(a) Estimation error over time

(b) Capacitor voltages

(c) 𝐶𝑏 simulation values

(d) 𝐶𝑐 simulation values

Figure 12 

Particle filter simulation results (including errors).

The PF provided satisfactory results. However, the technique had the highest RMSE (for the first case). That being said, when compared to the EKF, no Jacobian matrix had to be calculated to linearize the system matrix, as weighted particles were used instead. This may be an attractive feature if the system is too difficult to linearize. A large number of particles (which were required to accurately estimate the PDF), and the resampling feature of the PF, contributed to a slower computational time. When uncertainties were added to the filter model (as shown in Figure 12), the PF was able to recover slowly from modeling errors and after some time, accurately represent the first two states. There were significant errors in the third and fourth states; however this, as previously mentioned, may be attributed to measurements being only available for the first two states.

4.3. Quadrature Kalman Filter Results

For the QKF, the initial values of the covariance matrices were set to the identity matrix. The following results shown in Figures 13 and 14 were generated using the QKF based on the simulation setup. The QKF performed very well for both cases. However, like the previous methods, when uncertainties were introduced, it was unable to accurately track the two parameters associated with the battery (𝐶_𝑏 and 𝐶_𝑐) as shown in Figure 14.

(a) Estimation error over time

(b) Capacitor voltages

(c) 𝐶𝑏 simulation values

(d) 𝐶𝑐 simulation values

Figure 13 

Quadrature Kalman filter simulation results (only noise).

(a) Estimation error over time

(b) Capacitor voltages

(c) 𝐶𝑏 Simulation values

(d) 𝐶𝑐 Simulation values

Figure 14 

Quadrature Kalman filter simulation results (including errors).

The QKF provided good results for the system. Like the PF, there was no need to calculate the Jacobian, since weighted quadrature points were used instead. One of the drawbacks of the QKF is the computational demand when a large number of states and quadrature points are used. This leads to a slower calculation time. For example, since four states were required, an array of 81 quadrature points was required (3⁴). If a more accurate model was required, five quadrature points per state could be used. However, this would further increase the computational time. For this system, it was found that increasing the number of quadrature points had a negligible effect on the accuracy. Note that when uncertainties were added to the filter model, the QKF worked extremely well. However, the slow computation time is its main hindrance.

4.4. Smooth Variable Structure Filter Results

The constant diagonal matrix (𝛾) was set to 0.4, and the smoothing boundary layer thicknesses (𝜓) were, respectively, set to 1×10⁻³ and 1×10⁻² for the two states and the two parameters, respectively. The results are shown in Figures 15 and 16, and were generated using the SVSF based on the simulation setup. The SVSF method is very robust to noise and modeling uncertainties, as demonstrated by the accurate estimation in both cases.

(a) Estimation error over time

(b) Capacitor voltages

(c) 𝐶𝑏 Simulation values

(d) 𝐶𝑐 Simulation values

Figure 15 

Smooth variable structure filter simulation results (only noise).

(a) Estimation error over time

(b) Capacitor voltages

(c) 𝐶𝑏 Simulation values

(d) 𝐶𝑐 Simulation values

Figure 16 

Smooth variable structure filter simulation results (including errors).

The SVSF yielded very good results for condition monitoring of the battery system. Similar to the previous two methods, it did not require calculation of the Jacobian matrix, and also required a time delay (signal extraction for the last two states). The SVSF simulation time was just as fast as the EKF, and the results were similar for the case with noise. When errors were added to the filter model, the SVSF worked the best (in terms of RMSE). Although there were still large estimation errors for the third and fourth states, they were finite and stable. It is important to note that the SVSF was robust and not sensitive to changes in the filter parameters and the initial conditions.

4.5. Summary of Results

The results of the simulations were compared based on estimation error, robustness, sensitivity to noise, and computational difficulty and time. As shown in Table 1, the EKF performed best interms of estimation error, in the case of only noise (referred to as Case  (1)). In the presence of modeling uncertainties (Case  (2)), the EKF’s performance severely degraded. The SVSF yielded the most accurate estimation for the case involving modeling uncertainties. Both the PF and QKF performed moderately well, but at a cost of computational time, as shown in the last row of Table 1.

Table 1 

Quantitative summary of results.

The error in Table 1 refers tothe root mean-squared error (RMSE), calculated as follows: RMSE=1𝑁^𝑁𝑖=1𝑥_𝑖−𝑥̂_𝑖².(21)

Table 2 gives a quantitative comparison of the four filters. The robustness was based on observations made on varying the noise levels and considering its impact on model errors. The sensitivities were determined by tuning the filters various parameters. Computation (simulation) time was determined for each method and may be compared on a relative basis.

Table 2 

Qualitative comparison of the filters.

5. Conclusions

This paper discussed the application of condition monitoring to an RC battery system, typically found within a hybrid electric vehicle. State and parameter estimation techniques are important as they are responsible for providing accurate estimates of the states when reliable measurements are unavailable, and hence ensure successful condition monitoring. A comparative study was presented in which the Kalman and the extended Kalman filters (KF/EKF), the particle filter (PF), the quadrature Kalman filter (QKF), and the smooth variable structure filter (SVSF) were applied for condition monitoring. These estimation methods were compared based on estimation error, robustness, sensitivity to noise, and computation time. Table 3 summarizes the results of the comparison. For the case without modeling errors, the EKF worked the best in terms of RMSE and computational speed. When modeling errors were present, which is common in physical applications, the SVSF was shown to work significantly better than the other methods in terms of stability and RMSE.

Table 3 

Ranking of the state estimation techniques.

Appendices

A. List of Nomenclature

𝐴:System matrix𝐵: Input matrix𝐶:Output matrix𝑒:State estimation error𝑓:Nonlinear system or process equationℎ:Nonlinear measurement or output equation𝑘:Time step index𝐾:Gain value (KF, QKF, or SVSF)𝑚:Number of measurements𝑛:Number of states𝑁_eff:Effective number of particles𝑃: Error covariance matrix𝑄:System noise covariance matrix𝑅:Measurement noise covariance matrixSat:Saturation function𝑡:Simulation time𝑇_𝑠:Sampling time𝑢: Input𝑣:Measurement noise𝑤:System noise𝑤_𝑙:Quadrature point weight𝑥:System states𝑦:Observed measurement or output𝑧:Measurement output𝛾:Constant diagonal gain matrix with elements having values between 0 and 1𝜔:Particle weightΨ:Smoothing boundary layer𝜋:Probability distribution𝜏:Sampling timê‌:Denotes an estimated value~:Denotes an error value⋅:On top of a parameter denotes a time derivative.

Furthermore, note that subscript 𝑘+1∣𝑘 refers to an a priori time step and that the subscript 𝑘+1∣𝑘+1 refers an a posteriori time step. A superscript of 𝑇 denotes a matrix transpose.

B. Quadrature Kalman Filter Algorithm

The following is a summary of the QKF algorithm, directly as presented in [33, 34]. This process was used to obtain the presented results. There are two updates at each step: the time update and the measurement update.

Time Update Step
(1)Assuming at time 𝑘 the posterior density function 𝑝(𝑥_𝑘−1∣𝑧_𝑘−1)=𝑁(𝑥̂_{𝑘−1|𝑘−1},𝑃_{𝑘−1|𝑘−1}) is known, then we may factorize as follows: 𝑃_{𝑘−1|𝑘−1}=𝑃_{𝑘−1|𝑘−1}𝑃_{𝑘−1|𝑘−1}^𝑇.(B.1)(2)Evaluate the quadrature points {𝑋_{𝑙,𝑘−1|𝑘−1}}^𝑚𝑙=1 as 𝑋_{𝑙,𝑘−1|𝑘−1}=𝑃_{𝑘−1|𝑘−1}𝜉_𝑙+𝑥̂_{𝑘−1|𝑘−1}.(B.2)(3)Evaluate the propagated quadrature points {𝑋^{∗𝑙,𝑘|𝑘−1}}^𝑚𝑙=1 as:𝑋^{∗𝑙,𝑘|𝑘−1}=𝑓𝑋_{𝑙,𝑘−1|𝑘−1},𝑢_𝑘−1,𝑘−1.(B.3)(4)Estimate the predicted state: 𝑥̂_{𝑘|𝑘−1}=^𝑚𝑙=1𝑤_𝑙𝑋^{∗𝑙,𝑘|𝑘−1}.(B.4)(5) Estimate the predicted error covariance: 𝑃_{𝑘|𝑘−1}=^𝑚𝑙=1𝑤_𝑙𝑋^{∗𝑙,𝑘|𝑘−1}𝑋^{∗𝑇𝑙,𝑘|𝑘−1}−𝑥̂_{𝑘|𝑘−1}𝑥̂^{𝑇𝑘|𝑘−1}+𝑄_𝑘.(B.5)

Measurement Update Step
(1)Factorize: 𝑃_{𝑘−1|𝑘−1}=𝑃_{𝑘−1|𝑘−1}𝑃_{𝑘−1|𝑘−1}^𝑇.(B.6)(2)Evaluate the quadrature points {𝑋_{𝑙,𝑘−1|𝑘−1}}^𝑚𝑙=1 as: 𝑋_{𝑙,𝑘−1|𝑘−1}=𝑃_{𝑘−1|𝑘−1}𝜉_𝑙+𝑥̂_{𝑘−1|𝑘−1}.(B.7)(3)Evaluate the propagated quadrature points {𝑍_{𝑙,𝑘|𝑘−1}}^𝑚𝑙=1 as: 𝑍_{𝑙,𝑘|𝑘−1}=ℎ𝑋_{𝑙,𝑘|𝑘−1},𝑢_𝑘,𝑘.(B.8)(4)Estimate the predicted measurement: 𝑧̂_{𝑘|𝑘−1}=^𝑚𝑙=1𝑤_𝑙𝑍_{𝑙,𝑘|𝑘−1}.(B.9)(5)Estimate the innovation covariance matrix 𝑃_{𝑧𝑧,𝑘|𝑘−1}=𝑅_𝑘+^𝑚𝑙=1𝑤_𝑙𝑍_{𝑙,𝑘|𝑘−1}𝑍^{𝑇𝑙,𝑘|𝑘−1}−𝑧̂_{𝑘|𝑘−1}𝑧̂^{𝑇𝑘|𝑘−1}.(B.10)(6)Estimate the cross covariance matrix 𝑃_{𝑥𝑧,𝑘|𝑘−1}=^𝑚𝑙=1𝑤_𝑙𝑋_{𝑙,𝑘|𝑘−1}𝑍^{𝑇𝑙,𝑘|𝑘−1}−𝑥̂_{𝑘|𝑘−1}𝑧̂^{𝑇𝑘|𝑘−1}.(B.11)(7)Estimate the Kalman gain: 𝑊_𝑘=𝑃_{𝑥𝑧,𝑘|𝑘−1}𝑃^{−1𝑧𝑧,𝑘|𝑘−1}.(B.12)(8)Estimate the updated step: 𝑥̂_𝑘|𝑘=𝑥̂_{𝑘|𝑘−1}+𝑊_𝑘𝑧_𝑘−𝑧̂_{𝑘|𝑘−1}.(B.13)(9)Estimate the corresponding error covariance: 𝑃_𝑘|𝑘=𝑃_{𝑘|𝑘−1}−𝑊_𝑘𝑃_{𝑧𝑧,𝑘|𝑘−1}𝑊^𝑇𝑘.(B.14)

C. Derivation of the RC Battery Model Equations

The output voltage may be calculated by summing the voltages of each element in the circuit. Summation of the outer loop and inner loop voltages yield two equations for the output voltage, respectively: 𝑉𝑉_𝑂=𝐼_𝑆𝑅_𝑡+𝐼_𝑏𝑅_𝑒+𝑉_𝐶𝑏,(C.1)_𝑂=𝐼_𝑆𝑅_𝑡+𝐼_𝑐𝑅_𝑐+𝑉_𝐶𝑐.(C.2)

Recall Kirchhoff’s current law, which states that the total current entering a node must equal the total current leaving, such that the supply current is defined by 𝐼_𝑆=𝐼_𝑏+𝐼_𝑐.(C.3) Rearranging for 𝐼_𝑐 in (C.3) and substituting into (C.2) yields 𝑉_𝑂=𝐼_𝑆𝑅_𝑡+𝐼_𝑆−𝐼_𝑏𝑅_𝑐+𝑉_𝐶𝑐.(C.4) Equating (C.4) with the first output voltage (C.1) and solving for 𝐼_𝑏 yields 𝐼_𝑏=𝐼^𝑆𝑅_𝑐𝑅_𝑒+𝑅_𝑐+𝑉^𝐶𝑐𝑅_𝑒+𝑅_𝑐−𝑉^𝐶𝑏𝑅_𝑒+𝑅_𝑐.(C.5) Note that 𝑉̇_𝐶𝑏=𝐼_𝑏/𝐶_𝑏 such that (C.5) becomes 𝑉̇_𝐶𝑏=𝐼^𝑆𝑅_𝑐𝐶_𝑏𝑅_𝑒+𝑅_𝑐+𝐶_𝑏𝑅𝑉_𝑒+^𝐶𝑐𝑅_𝑐−𝐶_𝑏𝑅𝑉_𝑒+^𝐶𝑏𝑅_𝑐.(C.6) Similarly, the above approach may be used to solve for the second capacitor voltage rate of change: 𝑉̇_𝐶𝑐=𝐼^𝑆𝑅_𝑒𝐶_𝑐𝑅_𝑒+𝑅_𝑐+𝐶_𝑐𝑅𝑉_𝑒+^𝐶𝑏𝑅_𝑐−𝐶_𝑐𝑅𝑉_𝑒+^𝐶𝑐𝑅_𝑐.(C.7) The output voltage may be solved by manipulating (C.1) through (C.3). Setting 𝐼_𝑐=𝐼_𝑆−𝐼_𝑏 in (C.2), and rearranging for 𝐼_𝑏, results in the following: 𝐼_𝑏=𝑅^𝑡+𝑅_𝑐𝑅_𝑐𝐼_𝑆+𝑉^𝐶𝑐𝑅_𝑐−𝑉^𝑂𝑅_𝑐.(C.8) Rearranging (C.1) for 𝐼_𝑏,and setting the result equal to (C.8) yields 𝑉_𝑂𝑅_𝑒−𝑉^𝐶𝑏𝑅_𝑒−𝑅^𝑡𝑅_𝑒𝐼_𝑆=𝑅^𝑡+𝑅_𝑐𝑅_𝑐𝐼_𝑆+𝑉^𝐶𝑐𝑅_𝑐−𝑉^𝑂𝑅_𝑐.(C.9) Manipulating (C.9) and simplifying for the output voltage results in 𝑉_𝑂=𝑅^𝑐𝑅_𝑒+𝑅_𝑐𝑉_𝐶𝑏+𝑅^𝑒𝑅_𝑒+𝑅_𝑐𝑉_𝐶𝑐+𝑅_𝑡+𝑅^𝑒𝑅_𝑐𝑅_𝑒+𝑅_𝑐𝐼^𝑆.(C.10)

D. Linearization of the Battery System

The battery model described in Section 3 requires linearization for the EKF to be implemented. The linearization was based on the Jacobian matrix and the four states (where for neatness, 𝑊_𝐶𝑏=1/𝐶_𝑏 and 𝑊_𝐶𝑐=1/𝐶_𝑐) 𝜙_𝑘=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜕𝑓¹𝜕𝑉_𝐶𝑏𝜕𝑓₁𝜕𝑉_𝐶𝑐𝜕𝑓₁𝜕𝑊_𝐶𝑏𝜕𝑓₁𝜕𝑊_𝐶𝑐𝜕𝑓₂𝜕𝑉_𝐶𝑏𝜕𝑓₂𝜕𝑉_𝐶𝑐𝜕𝑓₂𝜕𝑊_𝐶𝑏𝜕𝑓₂𝜕𝑊_𝐶𝑐𝜕𝑓₃𝜕𝑉_𝐶𝑏𝜕𝑓₃𝜕𝑉_𝐶𝑐𝜕𝑓₃𝜕𝑊_𝐶𝑏𝜕𝑓₃𝜕𝑊_𝐶𝑐𝜕𝑓₄𝜕𝑉_𝐶𝑏𝜕𝑓₄𝜕𝑉_𝐶𝑐𝜕𝑓₄𝜕𝑊_𝐶𝑏𝜕𝑓₄𝜕𝑊_𝐶𝑐⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑘,(D.1) where the functions in (D.1) are described by 𝑓₁=−𝑇^𝑆𝑉_𝐶𝑏𝑊_𝐶𝑏𝑅_𝑒+𝑅_𝑐+𝑉_𝐶𝑏+𝑇^𝑆𝑉_𝐶𝑐𝑊_𝐶𝑏𝑅_𝑒+𝑅_𝑐+𝑇^𝑆𝑅_𝑐𝑊_𝐶𝑏𝑅_𝑒+𝑅_𝑐𝑓₂𝐼_𝑆,=−𝑇^𝑆𝑉_𝐶𝑏𝑊_𝐶𝑐𝑅_𝑒+𝑅_𝑐−𝑇^𝑆𝑉_𝐶𝑐𝑊_𝐶𝑏𝑅_𝑒+𝑅_𝑐+𝑉_𝐶𝑐+𝑇^𝑆𝑅_𝑒𝑊_𝐶𝑐𝑅_𝑒+𝑅_𝑐𝑓₃𝐼_𝑆,𝑓₄=𝑊_𝐶𝑏,=𝑊_𝐶𝑐.(D.2)

The linearized form (𝜙) of the system matrix, using the state vector described by (18), is as follows: 𝜙_𝑘=⎡⎢⎢⎢⎣𝜙¹¹𝜙₂₁𝜙₁₂𝜙₁₃0𝜙₂₂0𝜙₂₄00001001⎤⎥⎥⎥⎦_𝑘,(D.3) where the elements are described by: 𝜙₁₁=−𝑇^𝑆𝑊_𝑘𝐶𝑏𝑅_𝑒𝑘+𝑅_𝑐𝑘𝜙+1,₁₂=𝑇^𝑆𝑊_𝑘𝐶𝑏𝑅_𝑒𝑘+𝑅_𝑐𝑘𝜙₁₃,=𝑇^𝑆𝑅_𝑒𝑘+𝑅_𝑐𝑘−𝑉_𝑘𝐶𝑏𝜙₂₁+𝑉_𝑘𝐶𝑐+𝑅_𝑐𝑘𝐼_𝑆𝑘,=𝑇^𝑆𝑊_𝑘𝐶𝑐𝑅_𝑒𝑘+𝑅_𝑐𝑘𝜙₂₂,=−𝑇^𝑆𝑊_𝑘𝐶𝑐𝑅_𝑒𝑘+𝑅_𝑐𝑘𝜙+1,₂₄=𝑇^𝑆𝑅_𝑒𝑘+𝑅_𝑐𝑘𝑉_𝑘𝐶𝑏−𝑉_𝑘𝐶𝑐+𝑅_𝑒𝑘𝐼_𝑆𝑘.(D.4)