# Online Parameter Identification of the Lithium-Ion Battery with Refined Instrumental Variable Estimation

Authors: An Wen; Jinhao Meng; Jichang Peng; Lei Cai; Qian Xiao
Journal: Complexity
 (2020)

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## Abstract

Refined Instrumental Variable (RIV) estimation is applied to online identify the parameters of the Equivalent Circuit Model (ECM) for Lithium-ion (Li-ion) battery in this paper, which enables accurate parameters estimation with the measurement noise. Since the traditional Recursive Least Squares (RLS) estimation is extremely sensitive to the noise, the parameters in the ECM may fail to converge to their true values under the measurement noise. The RIV estimation is implemented in a bootstrap form, which alternates between the estimation in the system model and the noise model. The Box-Jenkins model of the Li-ion battery transformed from the two RC ECM is selected as the transfer function model for the RIV estimation in this paper. The errors of the two RC ECM are independently generated by the residual of high-order Auto Regressive (AR) model estimation. With the benefit of a series of auxiliary models, the data filtering technology can prefilter the measurement and increase the robustness of the parameters against the noise. Reasonable parameters are possible to be obtained regardless of the noise in the measurement by RIV. Simulation and experimental tests on a LiFePO4 battery validate the efficiency of RIV for parameter online identification compared with traditional RLS.


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## Body

## 1. Introduction

With the continuous decline of the price and the superior performance in the energy density, lithium-ion (Li-ion) battery has become an optimal choice for both the battery pack in the electric vehicle (EV) and the stationary energy storage systems in the grid [1–3]. In order to integrate the renewable energy into the grid, stationary energy storage system is critical for the inflexibility of balance and the intermittent power supply. Like the power sources, the performance of Li-ion battery also affects the acceptance of the EV produced in the market [4–8]. Especially for the pure battery-powered EV, the power capability and the energy capacity of the battery pack are closely related to the drivers’ experience. Battery Management System (BMS) is in charge of the cell operation in the battery pack, which should ensure the reliability and safety of the whole pack [9–12]. The battery’s states (State of Charge (SOC), State of Health (SOH), State of Function (SOF), etc) reflect the current status of each cell in the battery pack, which must be carefully calculated from the measurement in BMS [13–16]. Although model-based estimation seems to be a practical solution for estimating the battery states, the accuracy of the estimation is closely related to the battery model from prior knowledge [17–19]. The reliability and accuracy of the battery model depend on the choice of the parameters. Moreover, some parameters can directly reflect the battery states; for example, the internal resistance and the capacity are always treated as the effective indictors to the information of the battery SOH [20–23].Due to the electrochemical reactions inside the battery during the charging and discharging process, the parameters in the battery ECM are always changing with temperature and SOC [24–27]. Constant parameters of ECM in the battery operating process definitely reduce its accuracy. Look-up table is a possible way to improve the feasibility of the ECM under various conditions, but huge efforts are needed to build the look-up table considering the effect of temperature and SOC. Another factor should not be ignored here is that the batteries aging during their operation, which means the constant loop-up table of the parameters from the new cell may fail to accurately predict the characteristic of an old cell. The cells are connected in parallel and series into the battery pack for providing enough energy to the applications. The inconsistency of the cell leads to more workload on modelling if accurate model of each cell in the battery pack is needed [28, 29]. Therefore, updating the parameters online is necessary for modelling accuracy and states estimation [30, 31]. Recursive Least Squares (RLS) and Kalman filter are generally applied to identify the parameters of ECM [32–36]. RLS easily fails to converge to the actual value of the parameters when the measurement contains noise [11]. Kalman filter related methods usually have a higher computational complexity than RLS. As explained in [37], the system order of the Kalman filter-related methods is usually high, which makes it difficult to tune the algorithm. Extended Kalman filter (EKF) is applied to identify the parameters of the two RC ECM, 7 × 7 order matrix is calculated and updated in each step for parameters identification [38]. The parameters of the battery ECM with one RC are estimated by the moving window least square method with AutoRegressive (AR) model, but the effect of the window width on the accuracy of identified parameter have not been discussed [35]. Algorithms with more computing complexity, such as, Sigma-point Kalman filter [39] and H-infinity filter [31] are also used to online update the parameters of the battery model. Furthermore, the heuristic algorithms including Genetic Algorithm (GA) [40] and Particle Swarm Optimization (PSO) [41] with a moving window are applied to obtain the parameters in ECM. However, fewer efforts have focused on dealing with the effect of the measurement noise on the identified parameters in ECM. The noise from sensors has always existed during the measurement of the battery current and voltage, which may influence the convergence of the estimated parameters to the true values in real applications. Therefore, online parameter identification algorithm with good enough ability to obtain reasonable parameters under measurement noise conditions is still needed.In order to identify reasonable parameters of the battery ECM considering the effect of the measurement noise, the Refined Instrumental Variable (RIV) estimation is firstly applied to identify the parameters of the two RC ECM online in this paper. RIV focuses on improving the performance of RLS with the data-filtering technology in a bootstrap form. The Box-Jenkins model containing the battery system model and the AutoRegressive Moving Average (ARMA) noise model is established for the parameter identification in this paper. The parameters of the ARMA noise model are estimated from the auxiliary battery model with parameters in the previous step. High-order AR model independently generates the estimation of the input in the noise model. The data filtering technology helps the RIV estimation to obtain reasonable parameters with measurement noise.The reminder of this paper is as following. The Box-Jenkins battery model is described in Section2. The details of RIV for parameter identification is introduced in Section 3. The simulation results of the proposed method are shown in Section 4. The RIV estimation is experimentally validated on the measurements from a LiFePO4 battery in Section 5. The conclusions are given in Section 6.


## 2. Box-Jenkins Model of Lithium-Ion Battery

In order to identify the parameters of the battery model online with RIV, a suitable battery transfer function has to be established at first. Describing the output error of the battery model as an AMRA model, the form of the Box-Jenkins model is expressed as the following form [42]:(1)yk=Bz−1Az−1uk+Dz−1Cz−1ek,where e (k) is the white noise with zero mean and varianceσ2 and u (k) and y (k) are the input and output of the system. If the noise-free output of the system is defined as x (k), the Box-Jenkins model of equation (1) can be decomposed into equations (2)–(4). Among them, equation (3) can be seen as the system model and equation (4) is the noise model.(2)yk=xk+vk,(3)xk=Bz−1Az−1uk,(4)vk=Dz−1Cz−1ek.In the above equations, the termsA(z−1), B(z−1), C(z−1), and D(z−1) are normally defined as follows:(5)Az−1=1+a1⋅z−1+a2⋅z−2+⋯+ana⋅z−na,Bz−1=b0+b1⋅z−1+b2⋅z−2+⋯+bnb⋅z−nb,Cz−1=1+c1⋅z−1+c2⋅z−2+⋯+cnc⋅z−nc,Dz−1=1+d1⋅z−1+d2⋅z−2+⋯+dnd⋅z−nd.It can be seen from the structure of the Box-Jenkins model that the transfer function of the system model and the noise model are two independently parametrized rational functions. The optimal prefilter for all the related measurement in the battery model is derived from the noise model in the Box-Jenkin model.One RC ECM is easy for parameter calculation and is more stable to uncertainties [43, 44]. Compared with one RC model, one more RC network is used so that the fast dynamic and short dynamic of the battery terminal voltage can be described, respectively. Of course, more calculation is needed for two RC ECM. Two RC ECM is popular in Li-ion battery modelling area because it is a good trade-off between accuracy and complexity [45]. Thus, we choose two RC ECM to show the advantages of the methods in this paper. It should be noted that the method in this work is also suitable for one RC ECM. The structure of the two RC ECM is illustrated in Figure 1.Figure 1
The structure of two RC ECM.The two RC model consists of resistance, capacitance, and voltage source to represent the battery characteristics. The voltage source stands for the OCV-SOC relationship, andR0 is the internal resistance. The expression of the two RC ECM in Figure 1 is shown as follows:(6)U.1=−U1R1⋅C1+ILC1,U.2=−U2R2⋅C2+ILC2,Ut=Uoc−U1−U2−IL⋅R0.According to equation (6), the discretization form of the two RC ECM is as follows:(7)Utk=fSOC−ILk⋅R0−U1k−U2k,U1k=exp−Tsτ1⋅U1k−1+R1⋅1−exp−Tsτ1⋅ILk−1,U2k=exp−Tsτ2⋅U2k−1+R2⋅1−exp−Tsτ2⋅ILk−1,where Ts is the sampling time, τ1 andτ2 are the two time constants of the RC networks in the ECM.Afterwards, let us definez as the shift operator. Equation (7) can be transformed into the following form:(8)x0k=Bz−1Az−1⋅ILk=b0+b1⋅z−1+b2⋅z−21+a1⋅z−1+a2⋅z−2⋅ILk,where xo (k) is the deviation of the terminal voltage Ut and the open circuit voltage f (SOC). The coefficients a1, a2, b0, b1, and b2 in equation (8) are expressed as follows:(9)a1=−exp−Tsτ1+exp−Tsτ2,a2=exp−Tsτ1+−Tsτ2,b0=R0,b1=R0⋅−exp−Tsτ1−exp−Tsτ2+R1⋅1−exp−Tsτ1+R2⋅1−exp−Tsτ2,b2=R0⋅exp−Tsτ1+−Tsτ2+R1⋅exp−Tsτ2⋅exp−Tsτ1−1+R2⋅exp−Tsτ1⋅exp−Tsτ2−1.The noise model is selected in AMRA form with one-order AR model and MA model, which can be expressed as follows:(10)vk=1+d1⋅z−11+c1⋅z−1ek.According to the battery model in equation (8) and the noise model in equation (10), the Box-Jenkins model of the two RC ECM is(11)yk=uok+vk=b0+b1⋅z−1+b2⋅z−21+a1⋅z−1+a2⋅z−2⋅ILk+1+d1⋅z−11+c1⋅z−1ek,where the parameters in equation (11) are possible to be identified by RIV, and the noise model can be used as the prefilter for improving the robustness of RIV against the measurement noise.


## 3. RIV Estimation

RIV proposed by [46, 47] has particular advantage in dealing with the noise in the measurement. Moreover, RIV has provided a practical solution to the maximum likelihood function for the Box-Jenkins model [42] and is able to obtain more reliable results for the stiff dynamic system than the prediction error minimization method [47]. The structure of the RIV estimation is shown in Figure 2. It mainly consists of two estimation processes: the parameters in the battery model and the parameters in the noise model.Figure 2
The diagram of the RIV estimation.The noise model needs to be identified before prefiltering all the variables related to the battery model. The error function of the noise model is defined as(12)wk=Cz−1Dz−1⋅vk−ek.As described in Figure2, the prefilter of the noise model is(13)f2z−1=1Dz−1.Substitute the prefilterf2z−1 into the error function w (k); the following equation is obtained:(14)wk=Cz−1⋅vf2k−Dz−1⋅ef2k,where the vf2k and ef2k are the prefiltered variables.However,v(k) and e (k) are not known in the beginning of the parameter identification, and the profiler f2z−1 is also hardly known in advance. Auxiliary models are needed to calculate these variables in this condition. The instrumental variable v^k is obtained as(15)v^k=yk−x^k,where x^k is calculated from the auxiliary battery model (Baz−1/Aaz−1).Then,e˜kcan be independently calculated from the residuals of the high-order AR model of v(k). The unknown variables in equation (11) are able to be replaced by e˜k and v^k. Afterwards, the parameters in the noise model are able to be updated as the following recursive process:(16)ξ^k=ξ^k−1+Lek⋅e˜f2k−ψTk⋅ξ^k−1,(17)Lek=P^ek−1⋅ψ^k⋅σ2+ψTk⋅P^ek−1⋅ψ^k−1,(18)P^ak=P^ak−1−Lak⋅φTk⋅P^ak−1,where ξk=1c1d1T,(19)ψk=v^f2kv^f2k−1−e˜f2k−1T,ψ^k=v^f2kv^f2k−1−e^f2k−1T.The prefiltered outpute^f2k is calculated from the auxiliary ARMA noise model as follows:(20)e^f2k=Caz−1Daz−1⋅v^f2k,where the parameters in Caz−1 and Daz−1 are obtained from the parameters in the previous step.After calculating the parameters in the noise model, the prefilterf1z−1 can be used for the variables in the battery model. The expression of the prefilter f1z−1 is as follows:(21)f1z−1=Caz−1Daz−1⋅Aaz−1.Since the noise-free output is still unknown, the following auxiliary battery model is used to solve this problem.(22)x^0k=Baz−1Aaz−1⋅ILk,where the parameters in the auxiliary battery model come from the estimation results of the previous step.After the measurement prefiltering process, the parameters in the battery model are updated as the following equations:(23)θ^k=θ^k−1+Lak⋅yf1k−φTk⋅θ^k−1,(24)Lak=P^ak−1⋅φ^k⋅σ2+φTk⋅P^ak−1⋅φ^k−1,(25)P^ak=P^ak−1−Lak⋅φTk⋅P^ak−1,where θk=a1a2b0b1b2T, φk=−yf1k−1−yf1k−2−ILf1k−ILf1k−1−ILf1k−2T, φ^k=−x^of1k−1−x^of1k−2−ILf1k−ILf1k−1−ILf1k−2T.According to the previous steps, the RIV is able to identify the parameters of the battery model with measurement noise. The main steps of the RIV estimation are summarized as follows:Step 1. Parameters identification of the ARMA noise model: Prefilteringv(k) and e (k) forms vf2k and ef2k. Using auxiliary noise model (equation (19)) to calculate e^k and equation (15) for calculatingv^f2k. High-order AR model of v (k) is selected to obtain e˜f2k. Equations (16)–(18) are then applied to estimate the parameters in the ARMA noise model.Step 2. Parameters identification of the battery model: Prefilteringy (k), xo (k) and ILkto receive the variablesyf1k,xof1k, andILf1k. The auxiliary battery model equation (22) is used to calculate x^of1k. Afterwards, equations (23)–(25) are used to estimate the parameters in the battery model.Step 3. Update the covariance matrix: The covariance matricesP^a and P^e can be updated according to the new input φ^k and ψ^k, P^a=σ2⋅∑i=1Nφ^k⋅φ^Tk−1, P^e=σ2⋅∑i=1Nψ^k⋅ψ^Tk−1.


## 4. Simulation Validation

Current pulse profile is used in simulation to validate the performance of the parameter identification method in this paper. More specifically, the current pulse is applied to the ECM in simulation for verifying the advantage of RIV in parameters identification under measurement noise condition. RIV is compared with traditional RLS to show its improvement in parameter identification.For simplifying the parameter identification process, the voltage source in the ECM is omitted. The equivalent circuit in Figure1 is used for the validation of the parameter identification method. The true values of the parameters in simulation are listed in Table 1. The two-time constants of the RC networks are different from each other, which represent the fast dynamic and slow dynamic, respectively.Table 1
Parameters of the battery model in simulation.
ParametersR0R1C1R2C2τ1τ2Value0.01510.004512610.009959875.674559.2713White Gaussian noises with specific variances are added to the battery terminal voltage in simulation. The standard deviations of the noise are set to 1 mV and 10 mV, respectively, in this paper [48]. In order to evaluate the performance of the parameter identification method, Monte Carlo simulations are carried out for 10 times. The initial values of all the parameters are chosen to be zero and the initial covariance matrix is set to 1105 for both RIV and RLS. The order of noise model in the Box-Jenkins model is set to 1. As seen from the Bode plots in Figure 3, the identified results from traditional RLS have clearly bias error in Figure 3(a). The deviation becomes larger in all the frequencies when large standard deviations of noises are added to the measurement. Even for the lower noise level, the identified results are still obviously biased from the true system for RLS. As shown from the results in Figure 3(a), traditional RLS is quite sensitive to measurement noise. On the contrary, the results of RIV in Figure 3(b) stay close to the true transfer function in all frequencies regardless of the measurement noise, which proves the validation of the parameter identification method in this paper.Figure 3
Simulation results of the parameter identification methods with different measurement noise levels. (a) Traditional RLS. (b) RIV.
(a)(b)The simulation results of the traditional RLS and the RIV are shown in Table2. The estimation results from RIV clearly converge to the true values in the simulation, while the estimated parameters in RLS converge to the wrong values. It is clearly seen from the results with the 10 mV standard deviation of the measurement noise. The estimated parameters are further transformed into the RC parameters in Table 3. As shown in Table 3, the RC parameters from RIV are much closer to the reference than RLS.Table 2
The direct results of the parameter identification method.
Parametersa0a1b0b1b210 mVRLS0.00570.01350.01408.2493e − 040.0056RIV−1.82010.82290.0151−0.026550.011551 mVRLS−0.2201−0.25210.0146−0.0025−0.0013RIV−1.82170.82440.0151−0.026620.01160True value−1.8220.82440.0151−0.026610.01159Table 3
The identified parameters in the two RC ECM with 0.01 mV noise.
ParametersR0R1C1R2C2τ1τ2RLS0.01510.00111820.90.01002045.82.010020.5340RIV0.01510.00451260.90.00995986.75.674059.2691True value0.01510.004512610.009959875.674559.2713


## 5. Experimental Test

To further validate the parameter identification method in the real battery, the measurement from LiFePO4 battery is applied to the RIV estimation. As shown in Figure 4, the experimental tests are carried out on a battery test bench including a host computer, a MACCOR 4000 series test station, and a LiFePO4 battery. The specific current is used to charge and discharge the battery in the test chamber. The measurement data is sent back to the host computer for further analysis. The accuracy is±0.01% + 1 digit for the voltage measurement and±0.02% + 1 digit for the current measurement. The nominal capacity of the LiFePO4 battery used in this paper is 10 Ah, and the range of the battery voltage is between 2.0 V and 3.65 V.Figure 4
Structure of the test bench.In order to validate the online parameter identification method in this paper, New European Driving Cycles (NEDC) is used to discharge the LiFePO4 battery. The ambient temperature is set to 25°C in the test chamber and the sample time is 1 second for all the tests. A scaled multiple NEDC driving cycle is used to discharge the battery to its cut-off voltage. The detailed measurement during the NEDC profile is shown in Figure 5. The battery terminal voltage decreases much faster in lower SOC area. The OCV of the battery in this experiment is measured with 5% SOC interval under charge and discharge condition, respectively. The average OCV between charge and discharge condition is used to establish the OCV-SOC relationship, as shown in Figure 6. We fit an eighth-order polynomial function via quadratic optimization by using MATLAB “quadprog” for obtaining the coefficients [49]. After constraints the diminishing OCV with the decrease of SOC, the OCV-SOC function is represented as follows:(26)OCV=−330.2741⋅SOC8+1507.8350⋅SOC7−2869.7023⋅SOC6+2949.8632⋅SOC5−1773.9467⋅SOC4+632.0383⋅SOC3−128.9882⋅SOC2+13.8940⋅SOC+2.6371.Figure 5
Measurement in NEDC.Figure 6
OCV-SOC relationship.The estimated parameters of the two RC ECM during the calculating process of RLS and RIV are shown in Figures7 and 8. Although both RLS and RIV are able to accurately estimate the battery internal resistance, RIV is possible to obtain more reasonable estimation for the rest parameters. Especially, the estimation results of C1 and C2 in RLS are oscillated in a wide range in Figure 8, while the variation of the C1 and C2 in RIV are more reasonable as shown in Figure 8. In order to validate the performance of the two RC ECM, the identified parameters from RLS and RIV are substituted into the two RC ECM. The prediction of the battery terminal voltage is shown in Figure 9. The terminal voltage from the two algorithms stay close to the measurement, but the details in Figure 9 show that the two RC ECM using the parameters from RIV is closer to the reference. The results are confirmed by Figure 10, which the absolute errors of the predicted terminal voltage are calculated. As shown in the enlarged figure of Figure 10, the absolute error of the terminal voltage with the estimated parameters from RIV is lower than that from RLS. In order to verify the complexity of the proposed method, the computing time on a laptop (1.6 GHz CPU/64-bit system/MATLAB 2019b) is calculated. After 10 times evaluation, the average computing time for each step is only 0.001 s, which proves the good efficiency of the RIV estimation.Figure 7
The identified parameters from RLS.
(a)(b)(c)(d)(e)(f)(g)Figure 8
The identified parameters from RIV.
(a)(b)(c)(d)(e)(f)(g)Figure 9
Terminal voltage prediction of the two RC ECM.Figure 10
Absolute error of the terminal voltage prediction.The mean value of the identified parameters during the discharging profile and the performance of the two RC ECM are evaluated in Table4. The Mean Absolute Error (MAE) of the terminal voltage using parameters from RIV is 0.0219 V, and it is 0.0287 V for RLS. The RMSE of the RIV is also lower than using the parameters from RLS. This is mainly because the identified time constants in the two RC ECM are more reasonable in RIV as shown in Table 4. Therefore, combined with the results from simulation in the previous section, the advantages of RIV in identifying the parameters under measurement noise are proved compared with RLS.Table 4
The estimation results of RLS and RIV in NEDC.
MethodsR0R1R2C1C2τ1τ2MAE (V)RMSE (V)RLS0.01490.00230.0096179.07102674.47120.368220.74670.02870.0417RIV0.01480.00230.0117687.12097224.44771.512771.52910.02190.0313


## 6. Conclusion

This paper firstly uses the RIV to identify the parameters of the two RC ECM in battery modelling area, which facilities obtaining more reasonable parameters online with the disturbance in the measurement noise. Traditional RLS is sensitive to the noise in the measurement, which is possible to be solved by the data-filtering technology in the RIV. The battery Box-Jenkins model for parameter identification is established at first. With the benefit of the auxiliary battery model and the auxiliary noise ARMA model, RIV is able to identify all the parameters in Box-Jenkins model separately with the prefiltered measurement. High-order AR model estimation is applied to independently generate the input of the noise model. The simulation of the equivalent circuit with different noise level proves the validation of RIV in dealing with the measurement noise. While traditional RLS fails to coverage to the true value, RIV is still able to accurately estimate the parameters in simulation. The measurement from a LiFePO4 battery further proves the validation of RIV in parameter identification. The predicted terminal voltage of the two RC ECM shows that using the identified parameters from RIV is more accurate than from RLS, because the identified time constants from RIV are more reasonable. Moreover, the variations of the parameters (Figures 7 and 8) in the estimation process are more appropriate for RIV compared with RLS.Future work is needed for quantitative analysis of the measurement noise in experimental validation so that the advantages of RIV can be confirmed.



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