State of Charge and State of Health Coestimation for Lithium-Ion Capacitor Based on Multi-innovation Filters

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Introduction
The whole world is in a critical period of energy transition due to the excessive consumption of fossil fuel and the increasing of environment temperature [1].In order to achieve the goal of reducing the environment temperature and zero carbon dioxide emissions, countries are trying to increase the penetration of renewable energy and develop green energy storage technology [2,3].The lithium-ion capacitor (LIC) is a new type of rechargeable energy storage device, which consists of a lithium-ion battery (LIB) type positive electrode, an electric double layer capacitor (EDLC) type negative electrode, and a Li-ion conducting organic electrolyte [4].As reported in literatures, the LIC combines the advantages of LIC and EDLC and has the ability to provide high-power density (3-15 kW kg -1 ) and high-energy density (10-30 Wh kg -1 ) [5][6][7].
The LIC has a wide range of application, including stationary energy storage systems and electric vehicles.To ensure the reliability and safety of LIC during its operation, a management system is indispensable, which can monitor the states of LIC in real time and make reasonable operation control strategy [8].Hence, the accurate modeling and state estimation of LIC have attracted the attention of researchers.
Similar to the LIBs, the internal electrochemical reaction of LIC can be described in detail by an electrochemical models (EMs), and the external electrical behavior of LIC can be expressed in a simple way by equivalent circuit models (ECMs) and fractional order models (FOMs) [9][10][11].In general, modeling with EMs can obtain a very high accuracy.However, plenty of partial differential equations (PDEs) in EMs makes it hard to solve in short time and thus hard to use in practical application.In contrast, the good balance between model complexity and accuracy of ECMs and FOMs has been proven by researchers, and researches aiming at online state estimation have been developed based on these models [12][13][14].
State of charge (SOC) and state of health (SOH) are two crucial parameters to battery management system (BMS) for LIBs, as they indicate the remaining capacity of current cycle and the ability of long-term cycling of LIBs [15,16].The operating mechanism of LIC is similar to LIB; therefore, the estimation of SOC and SOH for LIC is indispensable.
A number of state estimation methods have been put up for LIBs.Among methods for SOC estimation, the simplest one is the ampere hour integration (AHI) method; however, the measurement error and the initial error may make the AHI method failed [17].The open circuit voltage (OCV) is the most commonly used as the index for the look-up table method as it decreases monotonically with the SOC, but long time rest is needed for obtaining an accurate OCV, which limits its application in online usage scenario [18].Data-driven method can ignore the physical process and internal reaction of a system; however, it exclusively relies on the quality of dataset and learning method [19].Compared with the above methods, model-based methods have shown certain advantages and have also received widespread attention.Ref. [20] proposed an improved adaptive square-root cubature Kalman filter (IASRCKF) for LIB SOC estimation, and the impact of parameter settings on its performance was discussed.In Ref. [21], the nonlinear autoregressive exogenous is used to optimize the Levenberg-Marquardt training algorithm, and Bayesian regularization (BR) for LIB SOC estimation, and the author also studied the relationship between the accuracy of NARX technology and the training dataset.In Ref. [4], Yang et al. successfully applied improved cubature Kalman filter (ICKF) to the SOC estimation of LIC, and its effectiveness has been verified under various working conditions, providing a certain reference for the state estimation work of LIC.Compared to the SOC, the SOH is a slow varying parameter.Nevertheless, the online estimate of SOH is significant as it plays a decisive role in the estimation accuracy of SOC.To date, many types of coestimation methods for SOC and SOH of LIBs have been put forward.In [22], Plett achieved real-time estimation of LIB SOH using EKF by reconstructing the SOC equation.In Ref. [23], an adaptive dual square-root Kalman filtering (ADSRCKF) with the dormancy zone is proposed to achieve real-time coestimation for SOC and SOH of LIB.
We found that although there has been in-depth research on the state estimation of LIB and in order to improve the estimation accuracy, existing methods are gradually showing a trend of being more complex, which is unfavorable for future installation on BMS.Meanwhile, to the best of the authors' knowledge, there are few literatures which report the coestimation of LIC SOC and SOH.Hence, this work is aimed at proposing a joint SOC and SOH estimation method suitable for LIC, and this method is as simple and lightweight as possible, with prospects for future deployment in BMS.Main contribution of this work can be summarized as follows: (1) a Thevenin model for LIC is established to simulate its dynamic characteristics.(2) A multi-innovation filter framework is constructed for SOC and SOH estimation of LIC, where the multi-innovation linear Kalman filter (MI-LKF) is used for model parameter identification, the multiinnovation cubature Kalman filter (MI-CKF) is used for SOC estimation, and the multi-innovation extended Kalman filter (MI-EKF) is used for SOH estimation.(3) The robustness of the proposed method was verified under algorithm initial value uncertainty, state actual initial value uncertainty, and noise disturbance.
This work is organized as follows.The modeling of LIC is illustrated in Section 2. The working principles of the proposed MIF-based coestimation method for SOC and SOH are presented in Section 3. The experimental results and analysis are shown in Section 4. Finally, the conclusion is given in Section 5.

Modeling of LIC
2.1.Model Selection.Among various types of ECMs, the Thevenin model has been approved to have a good balance between model complexity, accuracy, and generalization for LIBs [24].The Thevenin model has been proved to be applicable to simulate the voltage response of LIC, and the results show that the Thevenin model can approximately describe the electric behavior of LIC under the conditions of charging, discharging, and shelving [25].The Thevenin model is shown in Figure 1, which consists of an ideal voltage source, a series resistance, and a RC network.The ideal voltage source U OC represents the OCV of the LIC, which has a functional relationship with the SOC.The series resistance (i.e., ohmic resistance) R 0 is used to characterize the transient response during charge and discharge stages.The polarization resistance R p and polarization capacitance C p are connected in parallel to form the RC network, which characterize the polarization effect during charge and discharge stages.International Journal of Energy Research Based on the Thevenin model and the Kirchhoff's law, the electric behavior of LIC can be expressed as follows: where U t represents the terminal voltage of LIC, I t represents the load current, and U p is the polarization voltage.Generally, in order to simulate the behavior of the LIC, Eq.
(1) should be expressed in discrete time form, as shown in where Δt represents the sampling time and the subscript k represents the kth step.To evaluate the impacts of OCV on the simulation of LIC states, results of the charging process, discharging process, and average of charging and discharging process are taken into account.It should be noted that the average one is calculated according to the following equation: where U OC,cha represents the OCV obtained from charging LO process, U OC,dis represents the OCV obtained from discharging LO process, U OC,ave represents the average value of the former two OCV, and these three OCV are shown in Figure 3. Finally, a polynomial of degree 8 is used to fit the functional relationship between OCV and SOC, which is expressed as where x represents SOC and a i (i = 1, 2, ⋯, 8) represents the coefficients need to be fitted.

MI-LKF-Based Parameter Identification.
Besides the OCV curve, the parameters of the electrical components used in the Thevenin model are essential as well.In order to identify the parameters in real time, the relationship between parameters and measurable signals should be expressed in linear form.By applying the bilinear transform to the transfer function, the recursive form of Eq. (1) in discrete time can be obtained.The calculation processes are shown as follows: where τ = R p × C p is the time constant of the RC network.In addition, a 0 , a 1 , and a 2 are related to R 0 , R p , and τ, and they are expressed as With Eq. ( 6) and Eq. ( 7), the model parameters can be identified by recursive algorithm, and the applicability of the liner KF (LKF) algorithm in this usage scenario has been proven in our previous work [26,27].Nevertheless, the classical filter only uses single innovation to update the estimated states, and more innovations from the model are not utilized.The multi-innovation (MI) algorithm proposed by Ding and Chen in 2007 is to address this issue [28].With the MI algorithm, the innovation scalar will be extended to a vector, which contains more innovation value.The innovation vector is expressed as where E L represents the innovation vector, L is the length of the innovation vector, and e k represents the system error at the k th iterative step.Meanwhile, the corresponding gain will be also expanded to form of vector, which is expressed as where K L represents the gain vector and K k represents the gain value at k th iterative step.Then, the state estimated by the MI-based filter algorithm should be updated by where x k represents the estimated state and x − k represents the priori estimation of the state.
In this work, the MI-LKF algorithm is implemented for parameter identification of LIC, and the recursive process of MI-LKF is given in Algorithm 1.

Coestimation
3.1.MI-CKF-Based SOC Estimation.The SOC is defined as the ratio of remaining capacity at present to its maximum available capacity, which is expressed as follows: where C cur represents the remaining capacity at present and C max is the maximum capacity available.The remaining Regression model: For k = 0, set θ 0 , P l,0 , Q l , and R l .
Step II: computation State prior estimation k φ k Innovation vector and gain vector calculation tor, K l is the Kalman gain vector of MILKF, P l is the covariance matrix, Q l is the process noise covariance matrix, R l is measurement noise covariance matrix, and I is the unit matrix.
Algorithm 1: Recursive process of MI-LKF. 4 International Journal of Energy Research capacity at present, C cur , can be calculated by the AHI method by expressing the SOC as where SOC t 0 and SOC t represent the initial and current value of SOC, respectively, and η is the coulomb efficiency.
Furthermore, the discrete time form of Eq. ( 9) is shown as follows: Integrating Eq. ( 11) with representation of the polarization U p in Eq. ( 2), the state space equation of LIC, that is indispensable for employing the CKF, can be obtained as follows: Nonlinear system: For k = 0, set x c,0 0 , P c,0 0 , Q c , and R c .
Step II: computation Time update Cubature points evaluation Measurement update Factorization and cubature points evaluation Innovation vector and gain vector calculation Here, Q c is the process noise covariance matrix, R c is the measurement noise covariance matrix, ξ i is the i th column of ξ, and n denotes the dimension of state vector and n = 2.

International Journal of Energy Research
Nonlinear system: For k = 0, set x e,0 0 , P e,0 0 , Q e , and R e .
Step II: computation Jacobian matrix calculation A k = ∂f e x e , u k /∂x e x e =x e,k−1 C k = ∂g e x e , u k /∂x e x e =x e,k−1 Error covariance prior estimation Innovation value calculation e e,k = U t,k − g e x e,k k−1 , u k Innovation vector and gain vector calculation E e,L = e e,k e e,k−1 e e,k−2 ⋯ e e,k−L+1 Error covariance posterior estimation where K e,L is the gain vector of MI-EKF, P e is the covariance matrix, Q e is the process noise covariance matrix, and R e is the measurement noise covariance matrix.
x e,k|k = x e,k|k−1 + K e,L E e,L P e,k|k = (I − K e,k C k )P e,k|k−1 Eq. ( 8) where w 1,k and w 2,k represent the state noise.
The observation equation of LIC can be expressed as where v k represents the observation noise.
The CKF was proposed by Arasaratnam et al. in 2009, which is derivative-free and is regarded as an optimal approximation to the Bayesian filter that could be designed in a nonlinear system [29].In this work, the MI-CKF is used to perform state estimation.Defining x c = SOC U p T as the system state, u = I t as the system input and z = U t as the system output, the MI-CKF based SOC estimation of LIC can be processed as per the steps listed in Algorithm 2.

MI-EKF-Based SOH Estimation.
The SOH is defined as the ratio of the current maximum available capacity to the nominal capacity, which is expressed as where C max represents the current maximum available capacity and C nom is the nominal capacity.
Considering the slow varying characteristic of the capacity, the state space equation of capacity is set as where w 3,k represents the state noise.And Eq. ( 11) is still used as the observation equation in SOH estimation.From Eq. ( 11) and Eq. ( 13), we can observe that capacity and voltage are related by a very complex nonlinear relationship.Meanwhile, it is difficult to obtain a functional relationship between voltage and capacity like OCV-SOC.The EKF made local linearization to a nonlinear system by using Taylor expansion, which provides a way to decouple capacity and voltage.Hence, in this work, according to Eq. ( 10) and the OCV curve, first-order Taylor expansion of the observation equation based on the capacity is performed at each time step, which is expressed as Furthermore, define A k = 1 and the EKF-based SOH estimation of LIC can be conducted.It should be noted that capacity is a slowly changing parameter.In order to make the estimation more efficient and reliable, this work uses MI algorithm to improve the EKF; thus, the MI-EKF can be processed as the steps listed in Algorithm 3. Based on the above framework, the online parameter identification and coestimation of SOC and SOH for LIC can be implemented follow the flow chart illustrated in Figure 4.

Experimental Analysis
In this work, a LIC with capacity of 5 Ah was tested under various conditions to validate the proposed method.The tests were carried out at room temperature under two custom dynamic tests.

OCV Curve Fitting Results
. The results of OCV curve fitting and corresponding errors are shown in Figure 5.As shown in the results, the fitting errors are limited within ± 4 × 10 −3 V, ±0.05 V, and ±8 × 10 −3 V, respectively, which indicate that the fitting curves have high accuracy.However, the three curves show a deviation within about 40 mV between each other, which may lead to significant performance differences on the estimation of states.

Parameter Identification.
The parameter identification results of MI-LKF are shown in Figure 6.With increasing time, the ohmic resistance and polarization resistance increase (i.e., SOC decreases), while the time constant decreases.Although under various test condition values of the parameters are different, overall trends are the same.It implies that under various test conditions, the proposed method has similar ability to identify parameters.The parameter accuracy is verified according to eq. ( 2), and the results are listed in Table 1.Along with the results depicted in Figure 6, it suggests that the model established with MI-LKF outperforms that with forgetting factor least squares (FFRLS) and LKF.Due to strong fluctuations in the parameters obtained by FFRLS, significant errors cannot be avoided.While the smaller error indicates that MI-LKF has better simulation ability for the model under dynamic conditions.2. As illustrated in Figure 7, the voltage estimation errors obtained by using these three OCV sources are limited within a very small range, with the difference between each other being small.It indicates well matching between the established Thevenin model and the electrical behavior of LIC.Besides, based on the slight difference between the three results, it can be inferred that the discharge OCV-based one is more suitable.
The results of SOC estimation along with the corresponding errors are shown in Figure 8, while the statistical results are shown in Table 3.In this part, impact of OCV curve on the SOC estimation error becomes particularly obvious.Adopting discharge OCV limits the estimation error within ±1% (with MAE ≤ 0 51% and RMSE ≤ 0 61%).However, switching to the average OCV results in an increase of the MAE to more than 2.46% (charge OCV in 4.33%), while the RMSE also increases to more than 2.62% (charge OCV in 4.61%).It should be noted that the initial SOC is set as 100%, implying 0% initial error.This confirms   9 depicts the comparison results between the proposed MIF method and four commonly used algorithms for SOC estimation, including the EKF [30], the H infinity filter (HIF) [29], the CKF [31], and the square-root cubature Kalman filter (SRCKF) [32], where for such algorithms, the parameters are obtained with FFRLS.In addition, the statistical results of estimation error are listed in Table 4.It can be observed that unlike other algorithms that exhibit strong fluctuations in SOC estimation under long-term constant current conditions, results obtained by MIF can converge to experimental value with high precision.The MAE and RMSE are limited within 0.47% and 0.55%, respectively.It is a significant improvement compared to the MAE of over 3% and RMSE of over 4% obtained by other algorithms.Furthermore, this indicates that the MIF method has greatly improved the accuracy of SOC estimation by incorporating more innovations, eliminating the need for more complex methods, which is of positive significance for the practical application of the model-based method.The initial value of the estimate algorithm plays a crucial role in SOC estimation, as it can significantly impact the accuracy of the estimation algorithm.To validate the reliability of the proposed method, experiments with four different initial values for MIF are conducted, and the corresponding results are presented in Figure 10.The figure clearly indicates that the convergence time is primarily influenced by the initial error.Specifically, as the initial value increases, the convergence time also increases.However, it is worth noting that the SOC estimation can converge to a high level of precision in less than 10 minutes.Furthermore, once the SOC estimation has converged, the estimation errors remain nearly the same regardless of the magnitude of the initial error.This observation underscores the robustness of the proposed method in handling initial SOC errors and highlights its ability to provide consistent and reliable SOC estimations.
In practical applications, LICs may not start working at full charge state, which means that the SOC corresponding to the actual initial operating point of the LIC may be unknown.In this work, the proposed method is validated at different initial operating points of LIC, where the actual initial values of SOC are 80%, 60%, and 40%, and results are illustrated in Figure 11.As it depicts, the initial value of MIF is set to 100%, which means that the maximum error from the actual initial operating point of SOC is 60%.Nevertheless, the proposed MIF can quickly track to the actual operating point of the LIC in a short period of time, with an error limit of ±3%, indicating the good robustness of the proposed method in dealing with such uncertainty.
The potential error caused by measurement noise cannot be ignored.This work validates the performance of proposed method under noise disturbance as well.Gaussian noise is added to the raw current and current data, where the intensity of current noise is 25 mA and 75 mA, and the intensity of voltage noise is 1 mV and 2.5 mV.SOC estimation results under such disturbance are depicted in Figure 12.Overall, the increase in noise intensity can lead to an increase in SOC estimation error, but MIF can still limit the error to within ±3%, indicating the robustness of the proposed method in dealing with such uncertainties.

SOH Estimation.
The SOH estimation results and corresponding errors are shown in Figure 13 (a1-b2).It should be noted that for comparison, the R-based estimation results are shown in Figure 13 (c1, c2) as it associated with SOH as well [33], where the internal resistance is estimated using method reported in [34].Results suggest that compared to the deviation from experimental values obtained by the EKF-CKF, the proposed MIF method can estimate SOH with higher accuracy (with error limited within ±0.5%).Meanwhile, the results of the R-based method showed  Figure 14 illustrates the SOH estimation results with different initial values.It can be seen that the estimated value of MIF needs to be iteratively calculated for a certain period of time before it can track the experimental value.This may be because the capacity of the battery is a slowly changing   14 International Journal of Energy Research parameter, and the information reflected only by the error between the battery model voltage and the actual voltage is idle.Nevertheless, even if the initial error reaches 10%, the MIF method can still limit the error to a very small range (3%), indicating that the proposed method is effective and robust in estimating the SOH of LIC.

Conclusion
As a new type of energy storage device, accurate modeling and reliable and efficient state estimation are the key to promote the application of LIC.In this work, a multifilter-based SOC-SOH coestimation method for LIC is proposed.The LIC is modeled by the Thevenin model, the model parameters are identified by MI-LKF, the SOC is estimated by MI-CKF, and the SOH is estimated by MI-EKF.Under various test conditions, when the initial capacity is correct, the MAE and RMSE of voltage estimation error are limited within 0.0010 V and 0.0172 V, respectively.The MAE and RMSE of SOC estimation error are limited within 0.47% and 0.55%.The estimation error of SOH can be constrained within ±0.5%.Robustness of the proposed method is verified under algorithm initial value uncertainty, state actual initial value uncertainty, and noise disturbance, and the results suggest that the SOC estimation error is limited within ±3%.Although SOH requires a certain number of iterations to converge, its error can also be constrained within 3%.In summary, MIF can achieve accurate SOC and SOH estimation of LIC and has satisfactory robustness, making it suitable for practical applications.

Figure 1 :
Figure 1: Scheme of the Thevenin model.

Figure 3 :
Figure 3: Results of OCV obtained from different LO tests.

1 Figure 4 :
Figure 4: Flow chart of the proposed method.

Figure 5 :
Figure 5: Results and corresponding OCV-SOC curve fitting: (a, b) based on discharging voltage; (c, d) based on average voltage; (e, f) based on charging voltage.

4. 3 .
SOC Estimation.The results of voltage estimation along with the corresponding errors are shown in Figure 7, and the statistical mean absolute error (MAE) and root mean square error (RMSE) values are shown in Table

Figure 7 :
Figure 7: Results of voltage estimation and corresponding errors: (a, b) under dynamic test 1; (c, d) under dynamic test 2.

Figure 8 :
Figure 8: Results of SOC estimation and corresponding errors: (a, b) under dynamic test 1; (c, d) under dynamic test 2.

Table 1 :
Voltage simulation error statistics.

Table 2 :
Statistical results of voltage estimation error.

Table 3 :
Statistical results of the SOC estimation error.

Table 4 :
Statistical results of the proposed method.especially during the low rate discharge stage at the end of the operating conditions.It indicates that the method proposed in this work has a more accurate and reliable estimation ability in such application scenarios.