Interacting Multiple Model Strategy for Electric Vehicle Batteries State of Charge/Health/ Power Estimation

States estimation of lithium-ion batteries is an essential element of Battery Management Systems (BMS) to meet the safety and performance requirements of electric and hybrid vehicles. Accurate estimations of the battery’s State of Charge (SoC), State of Health (SoH), and State of Power (SoP) are essential for safe and effective operation of the vehicle. They need to remain accurate despite the changing characteristics of the battery as it ages. This paper proposes an online adaptive strategy for high accuracy estimation of SoC, SoH and SoP to be implemented onboard of a BMS. A third-order equivalent circuit model structure is considered with its state vector augmented with two more variables for estimation including the internal resistance and SoC bias. An Interacting Multiple Model (IMM) strategy with a Smooth Variable Structure Filter (SVSF) is then employed to determine the SoC, internal resistance, and SoC bias of a battery. The IMM strategy results in the generation of a mode probability that is related to battery aging. This mode probability is then combined with an estimation of the battery’s internal resistance to determine the SoH. The estimated internal resistance and the SoC are then used to determine the battery SoP which provides a complete estimation of the battery states of operation and condition. The efficacy of the proposed condition-monitoring strategy is tested and validated using experimental data obtained from accelerated aging tests conducted on Lithium Polymer automotive battery cells.

INDEX TERMS Lithium-ion battery, battery management system, interacting multiple model, smooth variable structure filter, state of charge, state of health, state of power.

I. INTRODUCTION
Electric vehicles (EVs) and hybrid electric vehicles (HEVs) are creating a disruptive change in the automotive industry as they present a sustainable alternative to their fossil-fuel based counterparts. The EVs energy storage system consists of a battery pack which is the key to commercial success of such vehicles [1]. Although battery technology is thriving, lithium-ion batteries remain the most common in EV and HEV applications due to their high energy and power densities, and long lifetime. Along with the growth of battery technology, the performance of a battery management system (BMS) is critically important to ensure safety, reliability, and efficiency of the battery pack [2]. A comprehensive review of different energy management methods has been presented in [3] for EV and HEV applications. These methods can be used to optimize the performance of a battery. Included in its functionality, the BMS must also provide an accurate estimation of the State of Charge (SoC), State of Health (SoH), and the State of Power (SoP) of a battery pack [1], [4].
The SoC is an indicator of charge remaining in the battery pack, similar conceptually to the gas gauge in fossil-fuel vehicles. Since there is no sensor available to directly measure the battery SoC, it needs to be estimated from measurements such as terminal voltage, temperature and current. Therefore, an accurate and reliable estimation strategy is critical for maintaining and optimizing battery operations. It also impacts safety as an accurate estimation of SoC in the BMS can prevent the battery from being over-charged or limit the rate of current in terms of charge or discharge to maintain a safe operating temperature [5].
SoC estimation techniques can generally be classified into conventional coulomb counting, direct methods and indirect methods. The direct methods are based on direct measurements such as terminal voltage, impedance or open circuit voltage (OCV) to calculate the battery SoC [6]- [8].
Coulomb counting method is the simplest and most common technique used to calculate the battery SoC. This method can be implemented irrespective of battery chemistry and is usually employed as the base technology for SoC estimation onboard of a BMS. However, this approach requires regular calibration due to measurement errors and noise. It also needs a knowledge of the initial SoC [9].
Several enhancements on this method have been proposed in [9]- [12]. In [9], a piece wise linear approximation of the functional relation between the OCV and SoC is used to re-calibrate the battery capacity for the SoC calculation. In [10], a least-square based coulomb counting method is provided combined with the measurement of the open circuit voltage of a battery at rest for finding the initial value of the battery SoC. In [11], Peukert's law is expanded for the discharging process combined with the coulomb counting technique for the charging process to provide a SoC estimation.
Despite the proposed improvements, coulomb counting suffers from being inaccurate due to the uncertainties of measurements and determination of the initial SoC. Furthermore, a regular re-calibration is needed as the battery ages to ensure the accuracy of the SoC with respect to battery capacity [13]. Therefore, closed-loop estimation methods have been of great interest. The so-called indirect strategies are very practical for EV and HEV applications including, but not limited to, fuzzy logic-based estimation, artificial neural networks and filter/observer-based techniques. A robust and stable estimation method along with a reliable battery model must be employed to estimate the battery SoC using filter-based techniques [14]. Fusion-based methods with Machine Learning can also be used to estimate the states of a battery. However in most ML application, large volume of data is needed for initial training. In applications where a model is establish or readily identifiable, it is more convenient to use model-based strategies such as in the case of batteries where the model can provide additional insight into the internal dynamics of the system [15], [16].
Battery models can be categorized into electrochemical and equivalent circuit models (ECMs). An electrochemical model represents the internal reactions and physics of a battery cell. However, due to their high computational complexity, it is quite challenging to use them with estimation algorithms and in real-time. On the other hand, ECMs can be easily parameterized by experimental data using system identification techniques. Although the identified parameters of ECM models do not reflect the physical reaction within a battery cell, the accuracy of SoC estimation is sufficient for a BMS within bounded operating regions [17], [18]. However, the battery model considered onboard of a BMS, cannot represent the inevitable degradation happening inside the battery over time. The BMS should therefore be able to indicate the battery SoH and determine its capacity to store energy. An indicator for SoH is the internal resistance or the capacity of the battery. The aging affects the battery's characteristics and in turn its model. Therefore, the BMS must be able to update the parameters of the model as the battery ages.
Modifications to model parameters can be performed by different techniques such as parameter estimation and model selection [19], [20]. Parameter estimation techniques take available measurements of a battery to estimate model parameters over time as presented in [21]- [23]. However, adaptation based on the measurements suffer from the problem of observability and usually entails optimization. Adaptation on its own is therefore not sufficient to guarantee adaptation stability and avoid overparameterization [20]. A model selection and updating strategy would be also required to switch between predefined models usually contained within a library of models with parameters that are accessible through a look-up table. These models can be optimized to capture the changing dynamics of the battery while aging or operating at different regions. This method of model selection not only guarantees stability, but also provides information on the SoH concurrently with SoC. A post-processing method is presented in [19] to estimate parameter values of a reducedorder physics-based model for different states of a battery life.
The performance of a BMS depends as much on the accuracy of states as on the estimation of model parameters. The states of a system can be estimated using a filter, based on a dynamic model along with sensor measurements. One of the most common estimators is the Kalman Filter (KF); it has been applied to problems including state and parameter estimation, target tracking, signal processing, fault detection and diagnosis. However, this filter can be applied only when the system model is largely and known, the system and measurement noise are white, and the states have initial conditions with known means and variances. For nonlinear systems, the Extended KF (EKF) is one of the most common estimation strategies and has been widely used for management of lithium-ion batteries [24]. Other methods considered in BMS include the Particle Filter (PF), the Quadrature KF (QKF), and the Unscented KF (UKF) [25], [26].
Robust strategies have been also proposed to decrease the effect of modeling error and uncertainties such as the Variable Structure Filter (VSF) [27] and the Smooth VSF (SVSF) [28]. The SVSF is a method based on sliding mode theory which uses discontinuous gain and a smoothing boundary layer. This method enhances robustness for the SoC estimation. An improved version of SVSF is proposed in [29] using a variable smoothing boundary layer (SVSF-VBL) which is more accurate in the presence of varying noise and modeling uncertainties. However, system observability has to be guaranteed in order to use these algorithms [17].
Multiple model (MM) strategies exploit a finite number of models to provide robustness and adaptability against uncertainties. The MM methods can be considered as being adaptive techniques including renditions as static MM, dynamic MM, Generalized Pseudo-Bayesian (GPB), and the Interacting MM (IMM) [30]- [32]. In [33], the combination of IMM with the SVSF was proposed to address fault detection and diagnosis problems. The result reported from IMM-SVSF showed a significant improvement in estimation accuracy.
This paper includes the following contributions: 1) A modified equivalent circuit model formulation is presented that considers observability in the context of estimating not only the states but also parameters related to the SoH of the battery including the internal resistance, the SoC, and its bias. In the proposed model, the estimated SoC from the conventional coulomb counting method is considered as a measurement for the system. The bias resulting from coulomb counting is then defined as a state to be estimated. 2) An associated estimation strategy is employed which is a combination of IMM with SVSF-VBL approach (IMM-SVSF-VBL). It involves use of multiple models, each for a different state of life of the battery. The proposed approach provides an accurate and robust estimation for the battery states including its SoC and the internal resistance. The battery's SoH is estimated by fusing the results from two approaches: the estimated internal resistance and the model selection probabilities obtained from the IMM-SVSF-VBL strategy. 3) A combined strategy is proposed for estimating the battery SoP by using the estimated internal resistance along with the battery's SoC. The outline of this paper is as follows: Section II presents the proposed battery estimation model. Section III investigates the observability of the new model. The proposed estimation strategy for SoC is presented in section IV and the co-estimation technique is introduced in section V. The experimental and validation results are demonstrated in Section VI. Section VII contains the conclusions of the work. VOLUME 9, 2021

II. MODELING
ECMs are commonly utilized to model lithium-ion batteries as shown in Figure. 1. An ECM model links the OCV of the battery to its SoC. Its multiple Resistance-Capacitance (RC) branches are used to capture the transients and a series resistance defined as internal resistance (R in ) relates the terminal voltage to the input. A proper model structure is required to describe the behaviour of a battery especially when it ages. As batteries age, higher order models are better suited to capture their dynamic characteristics at the risk of overparametrization. In this article, a third-order model is chosen so as to minimize the possibility of overparametrization as well as providing a trade-off between complexity and accuracy for real-time BMS implementations [34].  Figure. 1 are as follows, The terminal voltage is the output of the model and is obtained as, In an electrochemical battery, the parameters vary with SoC and temperature. Therefore, model parameters can be considered constant only within a small operating range of SoC, temperature and current level [35]. Switching between models according to the operating region of the battery can be used as a mean for parameter estimation [36], [37]. If the observability condition is satisfied, model switching can be complemented by direct parameter estimation including the internal resistance (R in ). Here, the internal resistance is considered as a state described with the following equation, where w r k is white noise. The internal resistance of a lithium-ion battery reflects its aging and power capability [38].
Coulomb counting method is a common technique for SoC estimation for a BMS, defined with the following equation [1], where SOC 0 is the initial value of the battery SoC. A bias may exist on the calculated SoC with this method, due to the unknown SOC 0 and the current sensor. In addition, the nominal capacity of the battery decreases as the battery ages.
To overcome the issue, coulomb counting can be combined with model-based estimation by considering the calculated SoC as being a measurement as an extension of i(t). As such if SoC is a measurement according to equation 4 (as derived from the current), then the proposed strategy is to treat the SoC bias as a state with slow dynamics to be estimated as follows, where w b k is white noise. Then SoC measure includes this bias such that, The state-space form of the proposed model can be described as, where x ∈ X , u ∈ R, y ∈ R m , f : X → R n , g : X → R n and h : X → R m are all differentiable functions. The state and measurement vectors are is linear and f (x k ) = Ax k and g(x k ) = B are given as, where G 3 ∈ R 3×3 is defined a s, The output equations are nonlinear functions as defined by equations 2 and 6.

III. NONLINEAR OBSERVABILITY
An estimation method cannot be employed unless the system observability is guaranteed. This section investigates the observability of the proposed battery model.

A. DISTINGUISHABILITY AND OBSERVABILITY
A full rank observability matrix can guarantee the global observability of a linear time-invariant system defined by a state-space representation. Determining observability of a nonlinear system is more challenging than a linear one. Thus, only local observability can be analyzed. To illustrate the observability of a nonlinear system a few concepts are required to be introduced.
Definition 1 (See [39]): For a system represented by equations 7 and 8, x 0 and x 1 are declared to be distinguishable states if there exists an input function u(.) such that for a finite time. The system is locally observable at x 0 ∈ X if there exists a neighborhood N of x 0 such that every x ∈ N excluding x 0 is distinguishable from x 0 . Therefore, the system is called locally observable if it is locally observable at each x ∈ X .
A system is globally observable if every pair of states (x 0 , x 1 ) with x 0 = x 1 is distinguishable. It can be demonstrated that two states are distinguishable for a linear system if y(k, x 0 , u) = y(k, x 1 , u) condition holds for any u. Furthermore, it can be proven that for a linear system, local observability leads to global observability. However, this is not guaranteed for a nonlinear system.
The observability of a nonlinear system can be illustrated using extended Lie-derivative.
Definition 2 (See [40]): T is a m dimensional vector function on x and u. The gradient of h j , j = 1, . . . , m denoted by dh j is a form of, Then the extended Lie-derivative of h with respect to f is, The Lie-derivatives for higher order than one are obtained as, The following theorem gives the sufficient condition for local observability based on the given definitions. The condition should be guaranteed for a nonlinear system to show observability.
Theorem 1 (See [39]): For a system described by equation 7 and 8 with the assumption of given x 0 ∈ X . Consider the form, evaluated at x 0 where i = 1, . . . , s, j = 1, . . . , m and for s = 0 the expression is equal to dh j (x 0 ). Suppose there are n linearly independent row vectors in this set. Then the system is locally observable around x 0 .
The observability condition for a linear system can also be derived from theorem 1. Based on the theorem, O I (x, u) for a general nonlinear system is defined as, Therefore, the model described by equations 7 and 8 is locally observable if O I (x, u) has n linearly independent row vectors.

B. OBSERVABILITY ANALYSIS OF THE BATTERY MODEL
Based on the notation on section III-A, dh(x, u) for the proposed battery model in section II is, The Lie-derivative of dL f h with respect to f and g can be found as, Based on the observability matrix defined in equation 16, the battery model represented in section II is locally observable if there is n = 5 independent row vectors. This condition is satisfied if In a physical sense, if two RC pairs are equal, they can be combined into one and therefore the voltage across them are not distinctive. Since each RC pairs can be determined uniquely, then if there exists a k ∈ Z such that ∂V k ocv ∂SOC k = 0 the local observability of the battery model can be guaranteed.

IV. SVSF-BASED INTERACTING MULTIPLE MODEL
The IMM-SVSF-VBL strategy is an adaptive method that relies on a finite number of models instead of a single one. The approach can be utilized for SoC/SoH and SoP estimation since use of multiple models (each for a different age or operating point) achieves better accuracy and robustness in estimation. With battery degradation, the procedure can adjust to any changes in battery dynamics such as capacity fade or changes in internal resistance. The use of ECM VOLUME 9, 2021 with the proposed technique makes the strategy feasible for real-time application in a BMS.

A. SVSF-VBL
The SVSF approach is a predictor-corrector method based on sliding mode control (SMC) and it was presented in [28]. The stability and robustness of the SVSF method has been demonstrated against uncertainties and noise in relation to the filter model. The SVSF employs a smoothing boundary layer ψ and a discontinuous gain which is similar to the SMC. The SVSF gain forces the states to converge to a neighborhood of the true value. The SVSF is applicable to any observable and differentiable system with the following class of nonlinear equations, The SVSF was later improved with several advancements, including the covariance formulation, time-varying smoothing boundary layer (SVSF-VBL) and combinations with different filters such as KF, EKF, UKF, Particle filter (PF) and more [29], [41]- [43]. This paper employs the SVSF-VBL with a time-varying smoothing boundary layer to enhance estimation accuracy. The width of the boundary layer depends on the uncertainty of the filter model, as well as the system and measurement noise. The SVSF-VBL algorithm uses a time-varying boundary layer with saturated limits to guarantee stability and estimation convergence of the method [29]. Figure. 2 provides an overview of the SVSF-VBL strategy. The SVSF-VBL estimation process and equations are as follows, [29]: • Prediction: The a-priori state estimate is obtained by using an estimated filter model.
• Correction: The updated state is acquired using a gain to refine the a-priori estimate into its a-posteriori form.
The SVSF gain applied to update the states for the case when ψ k+1 ψ lim is then evaluated as follows, And for the case with ψ k+1 < ψ lim the SVSF gain is, Finally, the a-posteriori parameters are calculated as, Equations 21 to 33 summarize the SVSF-VBL strategy.

B. IMM PROCEDURE
The IMM approach employs multiple models to capture the changing dynamics of a system. Since a battery cell ages over time, a new model is needed to describe changes in its dynamics over time. The IMM uses a library of predefined models which in this case are used in SVSF-VBL estimation filters running in parallel. The error covariance matrix quantifies the estimation error for each of these parallel filters and quantifies the applicability of the model. The error covariance matrix is used to evaluate the mode probabilities, which represents how close the filter model is to the true characteristics of the battery [29], [44]. In this paper, multiple models are used to characterize the aging of the battery. The mode probability indicates the suitable model that captures the aging of the battery and by extension SoH of the battery [29], [44], [45]. The IMM-SVSF-VBL consists of five main steps: 1) calculation of mixing probabilities; 2) mixing stage; 3) mode-matched filtering with the SVSF-VBL; 4) mode probability determination; 5) calculation of the a-posteriori estimate and error covariance matrix. Figure. 3 shows the strategy employed to estimate the states of a battery using 3 different models at various states of life, namely: new battery (100% capacity), mid-aged battery (90% capacity) and aged battery (80% capacity). The algorithm is demonstrated by the following equation.
• Calculation of mixing probabilities: The mixing probability can be written as, where the probability mass function prediction is, The transition probability matrix is defined as, • The Mixing step computes the mixed initial condition for the filters as, • This step involves mode-matched filtering where one iteration of the SVSF-VBL evaluates in mode j to produce a new state estimate by the provided values of the state and covariance from the mixing step. The likelihood functions of filters in mode j are then calculated as, • Mode probability is updated as follows, where the normalizing constant is defined as, • The a-posteriori estimated states and covariances from each filter are then combined by their mode probability to compute the outputs of the algorithm,

V. BATTERY STATES CO-ESTIMATION ALGORITHM
This section presents the co-estimation procedure proposed in this paper to estimate the SoC, SoH and SoP of a battery. The battery SoC is estimated by the IMM-SVSF-VBL presented in section IV along with its bias and the internal resistance of the battery. The SoH and SoP of the battery are then evaluated by the estimated states of the system and measurements. Figure. 4 illustrates the proposed strategy.

A. SOH ESTIMATION
The SoH is a measure of battery aging. It is related to battery capacity that is 100% for a new battery and for automotive application can drop to 80%. The SoH is the ratio of a full charge capacity of a battery to its nominal capacity.
SoH has been linked to different battery parameters including its impedance, coulombic efficiency, internal resistance and self-discharge rate [46]. The SoH is commonly measured through battery capacity and internal resistance which reflect the energy and power potential of a battery, respectively. The proposed method in this paper estimates the internal resistance of a battery in real-time as a measure of SoH. This can be stated as, where R EOL is the internal resistance at the end of battery life, R new is the internal resistance for a new battery provided by VOLUME 9, 2021 the manufacture and R in is the estimated internal resistance using the IMM-SVSF-VBL technique. An accurate estimation of the SoH can help to prevent a sudden degradation of the battery and any potential failures. Additional information can lead to a more precise prediction of the SoH. As it was mentioned in section. IV, the mode probability indicates the closeness of the battery dynamics to a specific filter model. Since each model reflects a battery state of life, the mode probability can be employed as another indicator of the SoH. Based on the SoH of the predefined models, a weighting vector is specified to determine the SoH of the battery.
where 0 < SOH i < 1, i = 1, . . . , r presents the SoH for mode i and can be defined based on the models described for the IMM-SVSF-VBL method. In the proposed strategy and further to equation 45 and 46, the SOH R and SOH µ are combined to provide a measure of the battery SoH onboard of a BMS. An adaptive weighted average is introduced to present a better and smoother outcome for the SoH estimation in the presence of noise or uncertainties.
The parameter α is found adaptively through the following optimization problem.
where ∈ R is a small positive number. The second constraint of the optimization problem can be followed from equation. 47.

B. SOP ESTIMATION
The SoP is determined as the ratio of peak power to nominal power. The peak power is determined as the maximum power that a battery can persistently provide over a period of time.
The SoP describes the power demands of a battery. The maximum available power for a battery is limited because the battery terminal voltage and current are always restricted within a range for battery safety. These limitations are to avoid any over-discharge or over-charge [47]. Different methods have been presented for SoP estimation considering the terminal voltage, SoC and current limitation of a battery [48]- [50]. This paper considers the available current under a specific voltage and in conjunction with limiting factors for current supply usually imposed by the BMS to estimate the SoP. The maximum available charge and discharge current under the terminal voltage limits are computed as follows, (50) where V max and V min denote the voltage limits and V ocv is obtained by the SoC estimation results using a look-up where I cha max,cur and I dch max,cur are the charge and discharge current thresholds, respectively. Therefore, the maximum available charge/discharge power can be determined as,

VI. EXPERIMENTS
An aging study was conducted over 12 months and used to evaluate the proposed methodology for battery SoC, SoH and SoP estimation. The experimental setup consisted of three-channel Arbin BT2000 cycler, three NMC Lithium Polymer battery cells, three environmental chambers namely Espec and Thermotron, an AVL Lynx data acquisition system, and AVL Lynx user surface software as shown in Figure. 6. AVL Lynx software was used for data acquisition and setting up of the test procedures. Three battery cells were tested separately in this study in an environmental chamber [35]. Three categories of tests were conducted, namely: characterization, aging, and reference tests. The characterization tests include static capacity, hybrid pulse power characterization (HPPC), and efficiency tests. Aging tests, including cycle life and calendar life, are utilized to predict battery performance. Cycle life aging tests perform accelerated aging in a short period of time.
A mid-sized EV model as derived in [35] and as shown in Figure. 5 is used to generate the current profile from the velocity profile of a combined driving cycle. The profile is a mixture of three common driving cycles including an Urban Dynamometer Driving Schedule (UDDS); a light duty drive cycle for high speed and high load (US06); and, a High Fuel Economy Test (HWFET) drive cycle. These simulate different driving habits as presented in [35]. The UDDS cycle can present the driving habits of an average driver in the city. The US06 provides a high acceleration associated with an FIGURE 5. All-electric mid-size sedan simulation model [35]. aggressive driving habit, and the HWFET presents highway driving conditions. The pack current profile is then scaled down to obtain the cell-level current profile. The aging data was collected for battery cells over time at elevated temperatures ranging from 35 • to 40 • scanning the entire range of SoC from 90% to 20% [1], [35].
Reference performance tests (RPTs), track changes in battery characteristics, is considered in this study for various states of life. Associated characterization tests conducted in series to cycling included static capacity test, OCV-SoC test, pulse charge/discharge test, and driving cycle tests [1], [4], [35]. Figure. 7 depicts the OCV-SoC curve of the battery obtained for different SoH. The figure illustrates that the OCV curve is correlated with SoH. The driving profile considered in this paper is shown in Figure.8. The driving profile is applied at different states of life including 100%, 90%, and 80% to explore the degradation behavior of the battery. Figure. 9 demonstrates the measured terminal voltage and SoC at different SoH. It is evident from Figure. 9a that the terminal voltage is dropping as the battery ages and therefore the battery depletes faster (as shown in Figure. 9b). The measured SoC is evaluated from the cycler using the coulomb counting method.
An important element of the estimation is the determination of the filter model. For better performance, battery model parameters are required to change with SoC as well as SoH. A least-square optimization methodology is used with the collected data to identify the parameters of the battery in a  range of SoC at different SoH. Table. 1 shows the bounds of time constants (τ i = R i C i , i = 1, 2, 3) and resistances of the equivalent circuit model described in section. II.
In this paper, three models corresponding to 100%, 90% and 80% capacity of SoH are used and the associated bound of parameters in Table. 1 define the three models of the IMM-SVSF-VBL approach as explained in section. IV. The IMM-SVSF-VBL strategy estimates the states of a battery including the internal resistance, the SoC and the SoC bias. The tuned parameters for the SVSF-VBL filters are as follows: the initial noise covariance matrix and the initial system error covariance matrix are selected as R = diag [5, 0.095] and Q = diag[10 −8 , 10 −8 , 10 −8 , 10 −10 , 10 −10 , 10 −10 ], respectively. The initial value of the state error covariance matrix is set to P = I 6 . The SVSF-VBL convergence and the initial boundary layer are chosen as γ = 0.38 and ψ = 3.2I 2 , respectively. A genetic algorithm is used to tune the noise covariance matrices of the Kalman. The same noise covariance matrices are used for the SVSF-VBL filters for providing a direct comparison between the performance of the two filters. For the IMM settings, the mode transition matrix and the initial mode probability are required to be initialized. The initial pertinent filter model is unknown at the start of the estimation process and, therefore the mode probabilities are initially selected to be equal for each mode. The time-invariant mode transition probabilities provide a mechanism for adjusting the speed at which the IMM can switch between different modes. The selected values present a moderate speed to allow a smooth mode transition and switching behavior for the IMM.  Figure. 10 shows the terminal voltage error and the percentage of SoC estimation error for the proposed strategy using the experimental data at different SoH. Two different initial conditions are considered for the battery SoC as SOC 0 = 75%, 85% to verify the proposed methodology. It can be seen from Figure. 10b that the proposed strategy can keep the percentage of SoC error to less than 1% with the terminal voltage error of less than 0.03 with various initial conditions over an entire range of SoC.
A comparison of the proposed procedure and the IMM-EKF method is then conducted. A simple third-order ECM structure has been considered for the IMM-EKF method. The proposed IMM-SVSF-VBL strategy uses the same initial parameters as the IMM-EKF except for the additional parameters specific to the SVSF-VBL method, namely γ and ψ. The root mean square error of the results are presented in Figure. 11. The results indicate that the proposed algorithm using IMM-SVSF-VBL provides a more reliable performance. Figure. 12 also shows the mode probability of the proposed strategy in contrast with the IMM-EKF. It is demonstrated that the proposed strategy is superior in identifying the correct model compared to the IMM-EKF method.  The estimated internal resistance is displayed in Figure. 13 which is used to estimate the SoH of the battery along with the mode probability of the IMM-SVSF-VBL demonstrated in Figure. 12. Figure. 13 illustrates that the internal resistance increases as the battery ages. Reference points are provided in Figure. 13, based on the characterisation tests for various SoH, to verify the estimated values of the internal resistance. The internal resistance is not always a constant value which affects the estimation of the SoH. The mode probability is also considered to enhance the performance of the SoH estimation as it can identify the operating mode of the battery using equation 46 where SOH 1 SOH 2 SOH 3 = 1 0.9 0.8 . The estimated SoH obtained by the proposed method in section. V-A is shown in Figure. 14. Figure. 15 shows the estimated SoC bias which is in turn used to refine the measure of SoC obtained by coulomb counting according to equation 4. A larger SoC bias will increase the SoC estimation error which is compensated by the estimated SoC bias. The existing bias could affect the real-time estimation of the states. However, the proposed strategy is able to provide an accurate estimation of the states of the battery by refining the measured value of SoC.
Since the internal resistance and the SoC of the battery are estimated accurately, the SoP can be calculated as described in section. V-B. Figure. 16 shows the maximum power for charge and discharge at different SoH of the battery.
A subset of driving profile is also considered to assess the efficacy of the proposed strategy to recognize the battery SoC,    and SoH for unknown cases as shown in Figure. 17. Figure. 18 indicates the mode probability when the battery SoH is in between the level associated with the filter models.   Figure. 18b displays the mode probability where the actual SoH is about 85%. Figure. 19 confirms the changes in the estimated internal resistance as the battery ages. The estimated SoH using the mode probability and the internal resistance is demonstrated in Figure. 20 for the test profile presented in Figure. 17.
The efficacy of the proposed strategy has been presented for a single battery cell. However, this algorithm can be expanded to a pack level as voltage and current measurements are readily available in Battery Management Systems and given that with the proposed method, no additional measurements are needed. The estimated SoC bias is used here to refine a measure of the SoC obtained by coulomb counting. Moreover, the battery's SoH has a slow dynamic and is not required to be estimated continuously if there are constraints associated with the availability of computational resources.  . Estimated internal resistance for new (100% capacity) and aged (95%, 90%, 85% and 80% capacity) battery using the validation data.

VII. CONCLUSION
This paper proposed an interacting multiple model framework to estimate the SoC, SoH, and SoP of a battery. A modified third-order ECM model is proposed to enable the combined estimation of the internal resistance, the SoC, and the SoC bias. The observability of the proposed model is investigated and guaranteed. The observability ensures that states of the system can be uniquely extracted from measurements. Since the characteristics of a battery changes over time, an Interacting Multiple Model (IMM) strategy with the Smooth Variable Structure Filter with Variable Boundary Layer (SVSF-VBL) is used to track and adjust to the aging of the battery. The estimated SoC bias is used to update the measure of SoC obtained by coulomb counting which leads to a more accurate SoC estimation. Moreover, the estimated internal resistance and the mode probability of the IMM-SVSF-VBL are utilized to present an accurate estimation of the battery SoH. The estimated states of the battery are then used for obtaining the battery SoP. The proposed combined method for SoC/SoH/SoP estimation is validated with experimental data.
SARA RAHIMIFARD received the B.Sc. degree in electrical engineering from Shiraz University, Shiraz, Iran, in 2011, and the M.Sc. degree in electrical engineering from Amirkabir University of Technology, Tehran, Iran, in 2015. She is currently pursuing the Ph.D. degree in mechanical engineering with McMaster University, Hamilton, ON, Canada.
After completing her master's degree, she spent more than two years working as a research engineer in Iranian automotive industry. Her research interests include electric and hybrid technologies, battery management systems, battery modeling, estimation theory, nonlinear, and adaptive control. ter University, where he is also the Director of the Centre for Mechatronic and Hybrid Technology (CMHT). He has developed his own theory related to estimation referred to as the smooth variable structure filter (SVSF). He has a number of patents and has coauthored close to 200 refereed articles. His academic research interests include intelligent control, state, and parameter estimation, fault diagnosis and prediction, variable structure systems, actuation systems, mechatronics, and fluid power. The application areas for his research have included automotive, aerospace, and robotics. He is a professional engineer and a fellow of ASME and CSME.