Online estimation of lithium battery SOC based on fractional order FOUKF‐FOMIUKF algorithm with multiple time scales

Aiming at the matter of poor precision in predicting the charge of lithium battery by applying conventional integer‐order models and offline parameter identification, this paper proposes a joint fractional‐order multi‐innovations unscented Kalman filter (FOUKF‐FOMIUKF) algorithm for predicting the cells' state of charge (SOC) online and uses the theory of singular‐value decomposition to tackle the issue of failure of the traceless transformation. Initially, the circuitry model of fractional order is built. The parameters of the model are recognized online by fractional‐order unscented Kalman filtering (FOUKF), and the obtained parameters are then transmitted to the method known as the fractional order multi‐innovations unscented Kalman filter (FOMIUKF) to calculate the SOC of the cell. The algorithm was validated under four working conditions such as FUDS (US Federal Urban Driving Distance), BJDST (Beijing Dynamic Stress Test), DST (Dynamic Stress Test), and US06 (Highway Driving Distance Test), respectively, and compared with the FOMIUKF, MIUKF, and FOUKF algorithms for offline identification. The conclusions demonstrate that the SOC estimated by the FOUKF‐FOMIUKF method is controlled within 0.5% of the mean absolute error under the four conditions and the root‐mean‐square error is controlled within 0.8%. It is not difficult to find that the FOUKF‐FOMIUKF algorithm estimates SOC with higher accuracy and robustness.


| INTRODUCTION
In today's world, due to the shortage of energy that is not renewable and the gradual deterioration of the earth's environment, countries around the world are gradually paying attention to the development and utilization of renewable energy.Wind, water, and solar energy are examples of green energy sources that have progressively become essential to our everyday existence.With low consumption, strong performance, energy saving, and emission reduction, new energy vehicles are gradually replacing fuel vehicles in society.The core of storing this electric energy is the power battery, of which lithium cells have a large capacity, low cost, long lifespan, and so forth, and already have mature production technology, as do most of the new energy vehicles as storage devices.The state of charge (SOC) of Li power cells is an essential indicator of a car's performance while driving, and an accurate SOC can help the owner plan the trip reasonably and understand the actual condition of the car.][3] There are four general categories into which methods for predicting SOC of power battery can be divided: open-circuit voltage methods, ampere-time integration methods, mechanical learning-type methods, and circuit-equivalent model-based filtering techniques.First, the battery power is estimated by applying the voltage principle of the open circuit (OCV). 4When the interior stoichiometric environment of the cell is in a steady state, the SOC value of the power cell will show a relatively stable function relationship with voltage in an open circuit; thus, the cell SOC value can be approximated indirectly from the open-circuit voltage value.The disadvantage of this method is that after each battery discharge, it requires a certain period of rest to accurately obtain the OCV relationship curve, but the battery cannot be stationary to deal with it during the actual driving of the car, which does not satisfy the actual working requirements.Second, regarding the ampere-time integration method 5,6 for estimating the battery SOC, the principle is to calculate the charge discharged from the lithium-ion cell by integrating the current against the duration, and subsequently use the initial charge condition of the cell to deduce the amount of the present SOC.The disadvantage of this method is that the process of integrating the discharge current generates accumulated errors, which makes the results of the identification deviate from the accurate SOC value.Third, the SOC value of power cells is estimated using data-driven machine learning theory, [7][8][9] a data-driven model also known as a black-box model.The researchers used some intelligent algorithms of machine learning to establish a learning model by importing a large number of SOC change rule data, setting up a suitable training algorithm, and finally forming a prediction model of SOC.Nevertheless, the algorithm is overly reliant on the training set, the reliability of the imported data dramatically impacts the reliability of the final trained model, and the training data cannot contain all the situations of the car in the actual work, which will also lead to a substantial decrease in the reliability of the SOC estimated by the model and ultimately lead to the failure of the identification.Fourth, the power cell SOC estimation is implemented using the circuit equivalent model [10][11][12] approach, which is based on the principle of constructing equivalent circuits to replace the actual operating conditions of the cell, using the relevant algorithms to solve the model, and then using the filtering algorithm to obtain an exact prediction of the SOC.The modelling approach is not only highly accurate and robust in estimation, but the parameters in the model have a clear physical meaning and a more solid theoretical basis.
In the field of equivalent circuit modelling, the modelling schemes can be classified into two main categories: one is of integer order, 13 which is characterized by the assumption of the capacitance of the energy storage element in the circuit model to be an ideal capacitive element, which makes the model computationally drastically scaled down in the process of identifying the SOCs, but this also leads to errors in the estimated results.The other category is fractional order, where the fractional order model takes into account the actual operation of the model capacitor. 14,15The constant phase element (CPE) is replaced by a capacitive element that is not compatible with battery operation, which is capable of detecting the practical parameters of the cell more accurately.
After the model has been built, suitable algorithms need to be selected for model discrimination and cell power prediction, of which the particle filter family, 16,17 the Kalman filter family, and their modifications are the most common.For parameter identification, the leastsquares method is commonly used for integer orders and genetic algorithms (GA) for fractional orders.Literature 18 proposes nonlinear recursive least-squares algorithm (RLS)-based offline identification, which improves recognition precision.However, the circuit system parameters recognized by the method are fixed, and the identification deviation is large for complex working conditions.On the basis of RLS, the XING ET AL.
| 509 literature [19][20][21] adopted the forgetting factor recursive least squares method to solve the model online, and the adaptive unscanted Kalman filter algorithm 22 was used to adaptively adjust the interference of noise, which effectively enhanced the estimation precision of SOC.However, the forgetting factor of the least squares is fixed, which is not capable of accurately describing the complex working conditions, and it will cause a major bias in the identification results.In addition, Fang et al. 23 proposed the fractional-order unscented Kalman filter (FOUKF) method to approximate the power battery SOC using FOM, which proved the effectiveness of the FOM model combined with the UKF algorithm in comparison with other algorithms.Yuan et al. 24 combined the multinomial holography theory with Kalman filtering and proposed the multi-innovations unscented Kalman filter (MIUKF) method, which successfully alleviates the issue of poor consistency in the UKF algorithm.However, the algorithm adopts an integer-order model and has limited accuracy.
To overcome the drawbacks of poor accuracy and robustness in parameter identification and SOC estimation mentioned above.In this article, we combine the online discrimination theory with the multi-novelty fractional order theory and use the principle of singular-value decomposition (SVD) 25 to overcome the phenomenon of non-positive determination of the UT transform and propose a joint estimation algorithm of the fractional-order multi-innovations unscented Kalman filter (FOUKF-FOMIUKF) for fractional-order UKF online parameter recognition, which can recognize the cell model parameters more accurately and increase the SOC prediction accuracy.First, the fractional circuit model is developed, and the adaptive genetic algorithms (AGA) method is applied to identify a set of parameters as the initial value for online discrimination.Then, the FOUKF is enabled to perform online discrimination on the model to predict the parameters, and in actual time, the latest parameters are provided to the FOMIUKF algorithm for predicting the SOC of the cell.Finally, the actual accuracy performance of the algorithm is tested under four working conditions.

| LITHIUM BATTERY MODELLING 2.1 | Fractional order theory
Since the integer-order model (IOM) circuit model does not take into account the operating characteristics of capacitors in practice, it cannot completely describe the nonlinear law of the lithium cell in the operation process.Therefore, a fractional order model is applied, in which the ideal capacitor is replaced with a CPE, and the equation of state is obtained by using the fractional order derivative defining formulae, the most common of which is the Grunwald-Letnikov (G-L) defining formulae, 26,27 which is expressed as follows: where D t a stands for the fractional order operator, h stands for the step size, a stands for the order, and ( ) stands for the Newtonian binomial in fractional order calculus.Γ(.) is the gamma function and is 28,29 denoted as

| Fractional order model
The circuit schematic is displayed in Figure 1 and the real operation of the cell is demonstrated using a fractional-order second resistor-capacitance (RC) circuit equivalent.
In the schematic diagram, open-circuit voltage is shown by U L , terminal voltage is indicated by U OC , R represents the ohmic resistance of the circuit reflecting the instantaneous change in circuit voltage at the instant of charging and discharging.R 1 and R 2 and CPE 1 and CPE 2 denote the fractional-order polarization resistor and polarization capacitance, respectively, and I is the dry circuit current in the circuit.
F I G U R E 1 Fractional order second-order RC model.

| Fractional order model expression
Referring to Figure 1, the loop equations 30 can be listed according to the fractional order theory as follows: The observation equation is Discretizing Equations ( 4) and ( 5), the final form is as follows: In the above equation, α and β denote the order of C 1 and C 2 , respectively, D α and D β denote the fractional order operators, T s denotes the sampling interval, Q n stands for the battery capacity, L is the length of the model error memory and the matrix of fractional order coefficients is set to be K j , which is given as follows: .  1.
Before the parameter identification model, the functional expressions of SOC and U oc need to be determined based on the operating conditions.In this paper, under the steady-current discharge experiment, the battery was operated at 0.5 C with a discharge interval of 0.1 SOC and left to stand for 2 h at the completion of each discharge until the cell was emptied.The corresponding values of U oc and SOC were recorded at the end of each settling, then the U oc -SOC curve was obtained by eighth-order multiform fitting, as shown in Figure 2.
XING ET AL.
| 511 The final fit was

| FOUKF online parameter identification
To address the problem of large errors in traditional offline identification, the FOUKF algorithm is used to discern the five parameters R 0 , R 1 , R where 2 , ρ k−1 is the process noise predicted by the model, ω k−1 and v k represent the process noise and observation noise of the fractional order system, respectively.N is the length of the fractional order error memories of the online identification.
First, P 0 and θ 0 are specified as the starting covariance matrix and the starting value of the parameter variables, respectively.The weighting parameters α_a = 0.05, k i = 15, β_a = 2, M = 5 were set.
Step 1: In conjunction with the SVD principle, sigma points are created for the parameter variables at the Step 2: Update the sigma point at moment k ( ) Step 3: Calculate the weights of the UT transformations Step 4: Calculate the a priori predicted values θ ˆk − Step 5: Update the sigma point y k i of the predicted end voltage Step 6: The prediction of the end voltage y ˆk and the end voltage error covariance P yy Step 7: Calculate the error covariance P xy of the parameter variables, Kalman gain Step 8: Parameter status update Step 9: Bring the updated parameter values into the SOC estimation algorithm

| Experimental verification
To confirm the precision of the fractional-order UKF online recognition, this paper uses the constant-current discharge condition for verification 31 and compares it with the AGA 32 and the GA.Furthermore, to decrease the error during the first estimation of SOC, a set of more accurate model parameters is needed as the initial value input for online identification, and an AGA algorithm will be applied to discern the fractional-order model under the DST operating situation.The estimation data can be seen in Table 2.
Figures 3 and 4 show the comparison of end-voltage prediction and end-voltage error, respectively, and Figure 5 shows the results of online identification of parameters for UKF.It can be seen that the FOUKF online identification estimates voltage values closer to the real voltage compared to the AGA and GA algorithms (Table 3).
T A B L E 2 AGA identification parameter results.

| FOMIUKF ALGORITHM SOC ESTIMATION 4.1 | Multinomial interest theory
The multi-innovation algorithm [33][34][35][36][37] updates the state variables in such a way that the residuals of the endvoltage observations y k and end-voltage predictions y ˆk at multiple moments of time are utilized to be extended into a multi-innovation matrix, which increases the traceless Kalman filter's prediction performance through the information matrix.The algorithm takes full advantage of the residual information generated by the multiple time scales to correct the current a posteriori estimates, improving the reliability and stability of the method.The relevant formulas are as follows: where μ is an adjustable character, μ = 0.5, and L is the length of the polynomial interest.

| SVD theory
Practical working conditions include running the FO-MIUKF algorithm in the process of power prediction for the battery.The covariance matrix P is susceptible to external noise interference as well as rounding errors due to system hardware calculations, resulting in a nonpositively determined P matrix.To solve the above problem, SVD theory 38,39 is introduced to solve the problem of P non-positive definite with the following equation: where ; r is the rank of the covariance matrix P.

| FOMIUKF algorithm
The traditional traceless Kalman filter algorithm has limited estimation accuracy and uses only the state value at the current moment for each posterior iteration update, resulting in the loss of previous data.In this article, the FOMIUKF algorithm is proposed to tackle the above problems by using the multi-innovation theory to correct the a posteriori estimation of the current moment through the historical end-voltage residuals to increase the estimation precision and stability of the SOC.
The steps of the FOMIUKF algorithm are as follows: 1. Set the starting values for the error covariance matrix P 0 and the state variable x 0 2. Create sigma points at the moment k − 1 using the principle of SVD, where n is the state value's dimension 3. Time update: Update the sigma point at moment k, where W is the error memory of the SOC estimate is the length 4. Find the weighting coefficients of the UT transformation 5. Calculate the a priori predicted value x ˆk − , the error covariance matrix 6. Update out the end voltage sigma point at moment k 7. Predict the end voltage value y ˆk − at moment k, and the error covariance matrix P yy k 8. Calculate the state variables, which are the Kalman gain K and the error covariance matrix P xy , respectively, 9. Generate multi-innovation matrices 10. Status updates  algorithm are verified by employing working condition data.Before starting the experiment, the power cell's SOC will start out at 0.7, and the actual charge of the battery is discharged from a SOC of 0.8.However, in operation, when the SOC of the battery is lower than 0.1, the life of the battery will be impaired due to the instability of its internal chemical reflections, so the battery power will be kept not lower than 0.1 SOC in the experiment.

| Validation under BJDST conditions
First, the FOUKF-FOMIUKF algorithm was applied to the BJDST operating conditions to demonstrate the prediction precision of the cell SOC.The relevant parameters are compared in Figure 8.
Table 4 lists the SOC and end-voltage-related parameters calculated by the four SOC estimation techniques under BJDST conditions, and Figure 8 compares the four algorithms in comparison.It is evident from the image and table that FOUKF-FOMIUKF has lower voltage error and SOC prediction error than the other three algorithms.It is easy to find that the algorithm is more accurate and more stable.

| Validation under FUDS conditions
Next, the performance of the joint FOUKF-FOMIUKF algorithm is tested under FUDS conditions, and the outcomes are compared with three other estimation methods to prove the prediction precision of the cell SOC.The comparison of the estimation of each parameter of the four algorithms can be seen in Figure 9. Table 5 illustrates the predicted voltage error and SOC error for every method used in the FUDS conditions.The experimental data indicate that the joint FOUKF-FOMIUKF method is more precise and the  SOC values are closest to the real values compared to the conventional methods.The estimation results are shown below.

| Validated under US06 conditions
The prediction accuracy of the algorithms was verified under the US06) operating condition, and the comparison plots of the four algorithms for each parameter under this condition are depicted in Figure 10A-D.The graphs and tables indicate that the FOUKF-FOMIUKF algorithm has greater estimation precision and better tracking capability (Table 6).Figure 10 displays the findings of the identification.accurate and the error analysis is plotted as displayed in Figure 12.

| CONCLUSIONS
Considering the IOM and the poor accuracy of traditional offline parameter identification, this paper proposes a FOUKF-FOMIUKF algorithm for estimating SOC for online parameter identification based on the fractional-order, second-order RC equivalent model and solves the problem of algorithmic UTtransform collapse by using the theory of SVD.In macro time, the fractional order UKF online identification will update the model parameters at regular intervals and pass the latest model parameter mentions to the FOMIUKF algorithm to estimate the SOC; The micro time is seen as the FOMIUKF algorithm updates the SOC while iterating over the micro time, triggering the UKF online recognition algorithm when the micro time reaches the macro time.The algorithm was validated under BJDST, FUDS, US06, and DST operating circumstances, and the findings indicated that the joint FOUKF-FOMIUKF algorithm estimated the SOC with an MAE value of around 0.5% and the RMSE was controlled within 0.8%.The algorithm solves the problem that the offline algorithm is detached from the actual working conditions and significantly improves the precision of SOC prediction, which is of great significance to the research of battery BMS.However, this paper only adopts the working condition data under a single temperature to confirm the precision of the algorithm model, and the applicability for different temperatures is still to be investigated.The next experiment will be carried out under different temperatures to test the practicality of the FOUKF-FOMIUKF algorithm.In addition, the online identification algorithm can be improved by using the multiinnovation theory, thus proposing a joint DFOMIUKF algorithm with higher accuracy.

T A B L E 1
Battery parameter table.

F I G U R E 3
Terminal voltage comparison for constant current conditions.AGA, adaptive genetic algorithm; UKF, unscented Kalman filter.F I G U R E 4 Terminal voltage error under constant current operation.AGA, adaptive genetic algorithm; UKF, unscented Kalman filter.

XING ET AL. | 515 4 . 4 |See Figure 6 . 4 . 5 | 7 . 5 |
FOUKF-FOMIUKF joint estimate SOC Diagram of experimental flow See Figure VERIFICATION OF WORKING CONDITIONS In the following, the FOUKF-FOMIUKF algorithm will be validated using four sets of operating conditions data measured by the CALCE battery team, namely, BJDST, FUDS, US06, and DST, and contrasted with the algorithms MIUKF, FOUKF, and FOMIUKF.The practical capabilities of the joint FOUKF-FOMIUKF F I G U R E 6 Fractional order multi-innovations unscented Kalman filter joint estimation of state of charge flowchart.F I G U R E 7 Experimental block diagram.FOUKF, fractional-order unscented Kalman filtering; SOC, state of charge.F I G U R E 8 Prediction results of the four algorithms under Beijing Dynamic Stress Test conditions: (A) state of charge (SOC) prediction results; (B) SOC error; (C) end-voltage prediction results; (D) end-voltage error.FOMIUKF, fractional-order multi-innovations unscented Kalman filter; FOUKF, fractional-order unscented Kalman filtering.

F
I G U R E 9 Prediction results of the four algorithms under US Federal Urban Driving Distance conditions: (A) state of charge (SOC) prediction results; (B) SOC error; (C) end-voltage prediction results; (D) end-voltage error.FOMIUKF, fractional-order multi-innovations unscented Kalman filter; FOUKF, fractional-order unscented Kalman filtering.

F
I G U R E 10 Prediction results of the four algorithms under Highway Driving Distance Test conditions: (A) state of charge (SOC) prediction results; (B) SOC error; (C) end-voltage prediction results; (D) end-voltage error.FOMIUKF, fractional-order multi-innovations unscented Kalman filter; FOUKF, fractional-order unscented Kalman filtering.
2 , C 1 , and C 2 online.The principle is to use the a posteriori estimate of the SOC at the current moment in time and the observed value of the terminal voltage to update the model parameters at the next moment in time.The algorithms for parameter identification and SOC estimation are not synchronized during program operation due to the different time scales; on the macroscale, the FOUKF online identification updates the parameters at regular intervals and provides the latest model parameters to the SOC algorithm.On the microscale, when the time scale of the SOC algorithm iterations reaches the macroscale, the online identification algorithm initiates the identification of the latest parameter values.The online recognition algorithm and the SOC algorithm interact and correct each other to finally estimate an accurate SOC value.The following is the T A B L E 5 SOC error and terminal voltage error for each algorithm under FUDS conditions.
T A B L E 6 SOC error and terminal voltage error for each algorithm under US06 conditions.