A New State of Charge Estimation Algorithm for Lithium-Ion Batteries Based on the Fractional Unscented Kalman Filter

An accurate state of charge (SOC) estimation is the basis of the Battery Management System (BMS). In this paper, a new estimation method which considers fractional calculus is proposed to estimate the lithium battery state of charge. Firstly, a modified second-order RC model based on fractional calculus theory is developed to model the lithium battery characteristics. After that, a pulse characterization test is implemented to obtain the battery terminal voltage and current, in which the parameter identification is completed based on least square method. Furthermore, the proposed method based on Fractional Unscented Kalman Filter (FUKF) algorithm is applied to estimate the battery state of charge value in both static and dynamic battery discharging experiment. The experimental results have demonstrated that the proposed method shows high accuracy and efficiency for state of charge estimation and the fractional calculus contributes to the battery state of charge estimation.


Introduction
In response to the concerns of the energy depleting and environment protection, electric vehicles and hybrid electric vehicles are proposed as popular substitutions for conventional fossil fuel vehicles for a wide range [1].Battery Management System (BMS) is one of the core technologies of electric vehicles.An advanced BMS can avoid the potential safety hazard and ensure efficient operation as well [2].State of charge (SOC) estimation performs as a significant factor of BMS performance, which is defined as the percentage of the left battery coulomb to the rated capacity and SOC cannot be measured directly [3], thus, SOC estimation is usually achieved by the measurable battery terminal voltage and the battery current.
The accurate state estimation with the terminal voltage and the current is a complicated issue [4], because the battery is a strong non-linear system within the charge and discharge process [5].In recent years, many scholars have proposed substantial methods to improve the accuracy of SOC estimation, such as Ampere hour counting [6], Kalman Filter (KF) [7], H∞ observer [8], sliding model observer [9], etc. Ampere hour counting is an extensively utilized method, which applies integrals with respect to the battery current, but it heavily depends on the accuracy of the current measurement and the initial SOC value [10].The errors of current measurement and initial SOC value can cause the accumulation of the SOC estimation error, which cannot be corrected.KF is another widely used method to estimate SOC, which sets recursive equations to minimize the state error [11].KF is not sensitive to the noise because it considers the mean and covariance of the process noise and measurement noise [12].Plett adopted Extended Kalman Filter (EKF) on SOC estimation of Li-ion battery [7].The SOC estimation of the battery is a non-linear system problem [13].EKF linearizes the non-linear battery model with Taylor expansion.Usually only the first order term of Taylor expansion is used to avert the mass calculation, thus this causes the accumulation of the state error in the process of iteration and the derivation of the Jacobian matrices [14].Unscented Kalman Filter (UKF) has also been investigated for non-linear system [15,16].Compared with EKF, UKF with Unscented Transformation (UT) doesn't contain the linearization of the system and it provides a series of carefully chosen sigma points [13], which offers better covariance approximation and higher precision than EKF due to its at least second order approximation items of nonlinear system [17].D'Alfonso et al. have utilized UKF in mobile robot localization problem [18].Miyabayashi et al. have adopted UKF on state estimation of tubular microreactors [19].He et al. have used UKF in battery SOC estimation [20].All of their papers demonstrate that UKF shows higher precision than EKF in different real systems.
In recent years, many scholars, such as Monje, found that non-linear systems have fractional properties [21].Fractional calculus, which is an expansion of the integer calculus, depicts the non-integer order integration and derivative, and fractional calculus describes the memorial and hereditary properties of systems [22].Ortigueira and Machado point out that fractional models can describe dynamical behaviour of the systems and they exhibit better capability of fitting experimental data [23].Thus, fractional calculus is widely used in non-linear systems, including signal processing [24], as well as SOC estimation.Zhang et al. established a fractional-order model using a Walburg-like ultracapacitors, and adopted fractional Kalman filter to estimate the battery SOC [25].Liu et al. established a fractional-order PNGV model and Extended Fractional Kalman Filter (EFKF) to implement the state estimation [26].These scholars have proved that fractional calculus is capable of accurately depicting system performance in real system.Thus, considering the advantages of UKF in contrast to KF and EKF in integer systems mentioned above, it is reasonable and practicable to combine fractional calculus with UKF to propose a new method, which may be useful for battery SOC estimation accuracy.Therefore, the main contributions of this paper are: (1) developing an equivalent second-order resistance-capacitance (RC) circuit model with fractional operator; (2) developing a UKF algorithm combined with fractional calculus to estimate the battery SOC.The combination of fractional calculus and UKF takes full advantages of the merits of them.The new proposed algorithm contains more battery model parameters identified, and this helps the accuracy of SOC estimation.In addition, the proposed FUKF in this paper is also helpful for other filtering problem, which is meaningful for practical engineering application.
In this paper, a fractional second-order RC model is established in Section 2 and Fractional Unscented Kalman Filter (FUKF) is developed in Section 3. The procedure of parameter identification for the fractional model is given in Section 4. The results in Section 5 confirm that FUKF is advantageous to SOC estimation.

Battery Modeling
SOC estimation is one of the most important issues of an electrical vehicle BMS, and the foundation of precise SOC estimation is an accurate battery model.The first-order RC model, the simple model etc. are extensively used in battery SOC estimation [27].The second-order RC model is utilized in this paper because it provides an accurate approximation for Li-ion battery dynamics [28].

Integer Second-Order RC Model
The model contains two capacitors, which are also suitable for the use of fractional calculus.Meanwhile, because of the model's simplicity and universality, this model is adopted as equivalent circuit model in this paper.The structure of second-order RC model is shown as Figure 1.In Figure 1, E represents the ideal battery electrodynamic force.The battery inner resistor R 0 accounts for the resistance between the battery plate, electrolyte and electrodes.R 1 and C 1 describe the resistance and capacitance of the activation polarization effect, respectively.R 2 and C 2 depict the resistance and capacitance of the battery concentration polarization effect, respectively.All parameters are nonlinear functions of SOC value.The state-space equation of the second-order RC model is: where I and U are the battery current and the terminal voltage, respectively.U 0 is the voltage of R 0 .U 1 and U 2 are the voltages of the two RC loops, respectively.

The Definition of Fractional Capacitor
Westerlund and Ekstam pointed out that the majority of capacitors shows fractional characteristics in practical dynamic conditions [29].A fractional capacitor model, which is formed in frequency domain, is established to represent the fractional properties of the capacitors.The definition of the fractional capacitor is given as: where C f is the capacitance of the capacitor, and n is the fractional order of the fractional capacitor.

The Definition and Properties of Fractional-Order Calculus
The fractional order Grünwald-Letnikov calculus formulation is given by the following equation [26]: where D α t represents the fractional calculus operator with respect to the variable t; α is the order of the system; h is the sampling interval time; L is memory or window length.The factor w α j = (−1) j ( α j ), and ( α j ) is Newton binomial coefficient, thus this derives w α 0 = 1; w α j = (1 − α+1 j )w α j−1 , j = 1, 2, ..., k.
In Equation (3), the value of α determines the type of the function.When α > 0, the function is differential.When α = 0, the function is the original function.When α < 0, the function is integral.The formula is shown as the followings:

Fractional Second-Order RC Model
According to the equivalent circuit shown as Figure 1, the fractional second-order RC model is established when the capacitors of the second-order RC model are replaced by the fractional order capacitors.Then the state-space equation is derived as the followings: The widely used SOC estimation method is the Ah (amper/hour) counting method shown as the followings: where SOC 0 is the initial SOC value.Q N is the rated battery capacity.η is the charge-discharge efficiency, which is related to the battery working temperature, charge and discharge rate etc.The value of η can be set into the range of 0.92 to 0.998 [30].In this paper, it is set to 0.98 for the convenience of the calculation.Equation ( 5) with Equation ( 6) are combined, and the discrete fractional second-order RC model is obtained as the followings: where T , and the fractional order factor K j = diag{w α j , w β j , 0}.Theoretically, the entire past estimated states should be considered.However, this may cause the computation burden in real system, thus the past states are shortened to avoid the computation burden according to short memory principle proposed by Caponetto et al. [31].E is the electromotive force of the battery.w k and v k are the system noise and measurement noise, respectively.They obey normal distribution and both of them are independent zero-mean Gauss white noise with covariance matrices Q k and R k , respectively.In Equation ( 7), all the parameters including R 0 , R 1 , R 2 , C 1 , C 2 , E vary with the SOC value, thus, the model is nonlinear.FUKF is then proposed by applying UKF to this nonlinear fractional second-order RC model.

Fractional Unscented Kalman Filter
For fractional second-order RC model, normal integer UKF is not applicable, thus UKF needs to be modified to fit the fractional second-order RC model.FUKF is a method based on fractional calculus and UKF and it is specially proposed in this section for the fractional battery model, including fractional second-order RC model.The observability is discussed and the detail process of FUKF is presented in this section.

The Observability of the Battery Model
As Equation (7) shows, the battery SOC value is included in the state vector.This explains that the estimation of the battery SOC value equals the observation of the state vector, thus it is important to ensure the observability of the model before the SOC estimation.For the discrete-time model, the model is observable if for any initial state x 0 and some final time k f the initial state x 0 can be uniquely determined by knowledge of the input From Equation ( 7), the derivation is performed to separate x 0 out as the followings: Thus, the nonlinear equation can be obtained as: where Noticing that M is the linear part of the model, and G(x) is the nonlinear part of the model.M T M is an invertible matrix, thus, As Brouwer Fixed Point Theorem illustrates, every continuous function mapping the disk to itself has a fixed point [33], thus the nonlinear equation Equation ( 8) has a unique solution, which means the initial state x 0 can be uniquely determined.This demonstrates that the discrete-time model in Equation ( 7) is observable.

Details of Fractional Unscented Kalman Filter
FUKF is based on unscented transformation which approximates the probability distribution of the variable by using sigma points [34].For convenience, the n-dimensional discrete fractional model in Equation ( 7) is rewritten as Equation ( 9).
where u k is the input of the model, which is equal to i k in Equation (7).y k is the output of the model, which is equal to U k in Equation (7).and the process steps of FUKF are given as the followings: Step 1: Initialize the state vector and the covariance matrix: Step 2: In FUKF, a series of sigma points are adopted to estimate the system mean and covariance matrix at time k according to the optimal mean and covariance estimation at time k − 1.The number of the sigma points is 2n + 1, where n is the scaling parameter of the state vector. x(0) where λ = a 2 (n + κ), and a is the parameter that determines the distribution of the sigma points.
The constant κ is generally set to 3 − n. ( (n + λ)P + k−1 ) i is the ith column of the Cholesky decomposition of the matrix (n + λ)P + k−1 [35].
Step 3: Transform sigma points from time k − 1 to time k: Step 4: Calculate the mean and the covariance matrix of the sigma points for the prior state estimation at time k and it needs optimal state variable x+ 1 , ..., x+ k−1 : x where the weights of the sigma points are given as the followings: where b is set to 2, as the optimal value to incorporate the prior knowledge of Gauss distribution of the state vector [35].In addition, the covariance matrix is: According to Equation ( 12), we substitute f ( x(i) k−1 , u k ) with x(i) k in the equation and expand the expectation in the equation, substitute E() with i , thus, the covariance matrix can finally be displayed as: Step 5: Transform the sigma points to the measurement estimation points: Step 6: Calculate the mean and the covariance matrix of the measurement estimation: and the covariance matrix of the measurement estimation is: Step 7: Calculate the covariance between x− k and ŷk : Step 8: The process of the state measurement update is:

Model Parameter Identification
The accuracy of SOC estimation depends heavily on the estimation accuracy of the battery capacity and precise model parameters.The battery parameters R 0 , R 1 , C 1 , R 2 , C 2 , α and β shown in Equation ( 7) are related to SOC value.The battery parameters cannot be measured directly.Thus, the unknown parameters are to be identified and calculated in this section to acquire the equivalent circuit information.

Test Bench
A test bench is established to verify the effectiveness of the proposed method.The test bench is taken under the temperature at 25 ± 2 • C and by the laboratory equipments of the BTS-4000 battery testing system(BTS-4000-6V4A-CCDC-USB, NEWARE Electronic Co., Ltd, Shenzhen, China), which have a measurement current range of 0-3000 A and voltage range of 0-110 V and temperature range of −25-100 • C with the sampling period 100 ms.Moreover, the SOC value can be obtained through the Ah counting by the software after configuring some test profiles and parameters.The test object is a ternary manganese-nickel-cobalt (MNC) lithium battery (Sony Corporation, Tokyo, Japan) pack shown in the red oval coil of Figure 2, which consists of ten Sony commercial cells (US18650GR G7, Sony Corporation, Tokyo, Japan) in parallels and has a nominal capacity of 24 Ah, a nominal voltage of 3.7 V, a charging cut-off voltage of 4.2 V and a discharging cut-off voltage of 3.0 V. Some prior discussions are provided firstly as followings: 1.When the battery capacity is measured with 1 C, 2 C and 3 C discharge rates.The corresponding results are 23.587Ah, 23.294 Ah and 22.460 Ah, respectively.It is shown that the discharge capacity at higher rates is lower than that at low rates and it is different from the nominal value 24 Ah.For the non-fresh lithium battery pack, it is acceptable that the discharge capacity is slightly less than 24 Ah of the nominal capacity.Hence, the rated capacity of the battery is considered as 23.587 Ah instead of 24 Ah. 2. Since the measurement error of the current and voltage of the test platform is less than 0.1% of full scale, it is feasible to assume that the battery SOC value can be obtained by this high-precision battery testing system BTS-4000.Hence, the obtained SOC is used as a true SOC value although some sensor errors exist.3.For convenience, the integer order model can be obtained by setting the orders of the fractional capacitors α = β = 1.The physical meanings of all the seven parameters of the battery second-order RC model are identical no matter whether the model is a fractional order model or an integer order model.
A universal method to identify the battery model parameter is the battery pulse characterization experiment.A pulse characterization experiment which is derived from discharging and standing process is intended, and it is shown as Figure 3.With the voltage and current curves of the pulse characterization experiment, the seven unknown parameters can be determined and this is the previous work of the battery SOC estimation with FUKF.

The Identification for Resistor R 0
The instantaneous voltage drop at the beginning of the discharging process is caused by the resistor R 0 .The terminal voltage before the current discharge pulse appears is considered as the battery electrodynamic force, E, because the voltages of the capacitors C 1 and C 2 can be considered to 0 V. Thus, with the terminal voltage U at the beginning of the discharging process, the resistor can be determined as:

RC Loop Identification
The responses of the two RC loops in Figure 1 are as: The response consists of the zero-input response and the zero-state response, and the parameters 2 are the time constants of the two RC loops.These two time constants represent the rate of convergency of the terminal voltage curve at the discharge process and standing process.At the discharging process, U 1 (t 0 ) and U 2 (t 0 ) equal zero, and according to the model voltage in Equation ( 1), the battery voltage are: thus the least square method [36] is used to fit the terminal voltage U(t) of discharging process curve.With the exponential fitting, the time constants τ 1 and τ 2 can be determined.At t f , which is the time of the end of the discharging process, U 1 (t f ) and U 2 (t f ) can be determined, thus, R 1 , R 2 , C 1 , C 2 can all be identified.
Then considering the voltage U 1 (t) and U 2 (t) with the fractional order α and β, and integrating Equation ( 5), this derives: According to Equation ( 5), the battery terminal voltage in fractional form at the standing process can be derived as: and also the least square method is used to find a group of values of α and β.The values aim to minimize the error between U i (t) and U f (t) with the performance indicator [26]:

Experimental Verification
To illustrate the effectiveness of fractional second-order RC model and FUKF, three experiments are developed on the battery test bench for the MNC lithium battery pack.The three experiments are pulse characterization experiment, static discharge experiment and dynamic discharge experiment, respectively.Pulse characterization experiment is performed for the parameter identification of the battery model, and static discharge experiment and dynamic discharge experiment are performed to validate the proposed SOC estimation method based on FUKF.These two experiments are operated with a discharge current rate of 1C and the dynamic pulse condition, respectively.

Pulse Characterization Experiment
A pulse characterization experiment such as in Figure 3 is developed to identify the battery model parameters.The battery initial SOC is set to 100%.The current of the discharging process is set to 1 C.The discharging time is set to 180 s to bring the battery SOC with a 0.05 reduction.The standing process time after the discharging process is set to be 1500 s.According to Equations ( 21)-( 23), R 0 , R 1 , C 1 , R 2 , C 2 can be identified.The results of the parameter identification are shown as Figure 4.The capacitances and resistances in the model are treated as identical for both the integer and fractional models, because the difference of the integer and fractional model is the orders of the capacitors according to the definition of the fractional capacitor model.According to Equation (24), the orders of C 1 and C 2 are determined by minimizing the performance indicator in Equation (26).From Figure 4, the battery model parameters vary with the value of SOC.When the SOC value decreases from 1 to 0, the resistances of the battery model including R 0 , R 1 and R 2 fluctuate slightly.However, the capacitances of the battery model including C 1 and C 2 ascend first and then descend and the peak values appear when SOC value is 0.6.This phenomenon is caused by the battery polarization effect and the capacitors of the second-order RC model successfully explains the effect.When the SOC value is less than 0.2, the battery dynamic becomes different, thus the parameters, especially R 1 and R 2 fluctuate steeply.The orders of the two capacitors, α and β, hint a bigger time constant, thus the second-order model can depict a more obvious polarization effect with the fractional order calculus.In addition, the results of the orders α and β of capacitors C 1 and C 2 are given as Figure 5.The parameter identification is developed by the discharging process curve, thus the orders remain constant within a circle of the discharging process.From Figure 5, the order of C 1 is larger than 1 mostly.Radwan et al. have pointed out that fractional capacitors whose order is larger than 1 are still fractional capacitors and C 1 can be regarded as a fractional capacitor whose order is less than 1 in series with an integer capacitor [37].

Static Discharge Experiment
To prove the static characteristic of the battery model and FUKF, the static experiment with a discharge current rate of 1 C is adopted.The static experimental validation current profile is shown as Figure 6.
To validate the convergent properties of FUKF and UKF, it is assumed that the initial SOC value is unknown, thus the estimation values of initial SOC are both set to 0.9 intentionally while the actual SOC value is 1.0, which means an initial SOC value error exists.The results under static discharge experiment are given as Figure 7. From Figure 7, the static discharge experiment result comparison of FUKF and UKF is shown as Table 1.In static discharging experiment, the fractional second-order RC model and FUKF algorithm show a more accurate performance on the estimation of battery SOC and battery terminal voltage, where the SOC error and the battery terminal voltage error are 5.36% bound and 0.0296 V bound, respectively.However, the SOC error and the battery terminal voltage error of the integer model and UKF are 9.47% bound and 0.0328 V bound, respectively.In real system, to illustrate the robustness and stability of SOC estimation for most of the working time, mean error is utilized as an indicator instead of maximum estimation error.Thus, the mean errors of SOC estimation of FUKF algorithm and UKF algorithm are given as 2.56% and 3.65%, respectively.

Dynamic Discharge Experiment
To verify the dynamic characteristic of the battery model and FUKF, the dynamic pulse condition is adopted.The magnitude of the current profile is scaled to obtain the voltage response with respect to the whole SOC range.The current profile of dynamic experimental validation is shown as Figure 8.The results under dynamic discharge experiment are given as Figure 9.The dynamic discharge experiment result comparison of FUKF and UKF is shown as Table 2.   9, there are some results as: 1.The fractional second-order RC model and FUKF algorithm track the measured SOC and terminal voltage well.The SOC error and the battery terminal voltage error are 11.55% bound and 0.2712 V bound, respectively.On the contrast, the SOC error and the battery terminal voltage error of the integer model and UKF are 14.20% bound and 0.3568 V bound, respectively.The mean errors of SOC estimation based on FUKF algorithm and UKF algorithm are also given as 2.88% and 5.92%, respectively.Although the maximal error of the SOC estimation based on FUKF algorithm is quite large, the mean error is mild enough.Thus, for most time, FUKF algorithm presents extraordinary precision on SOC estimation.2. In both static and dynamic experiments, the result curves of fractional order model and integer order model can both converge to the measurement curve even though the initial SOC values are incorrect and the curve of fractional order model recovers the initial SOC error on a larger scale and with less time according to the enlarged drawings.This demonstrates better stability and robustness of FUKF. 3. In both static and dynamic discharge experiment, both UKF and FUKF show a convergent performance.However, when the remaining capacity of the battery is little, due to the enlarged battery polarization effect, the battery model parameter estimation becomes inaccurate.This leads to the error of the space state estimation.UKF loses its convergence and performs divergently.Nevertheless, the fractional second-order RC model and FUKF algorithm could still make precise estimation of the battery parameters and track the actual value of SOC. 4. Compared with the integer model, the fractional orders of the capacitors C 1 and C 2 are more capable of reducing the performance indicator in Equation ( 26) with least square method, because the fractional order model and fractional parameter identification reflect the system performance more precisely.5.It is obvious that some sharp peaks of the terminal voltage error appear at the beginning and the end of the discharge process in Figure 9d.That is because the battery model parameters suddenly change at the instant the battery current pulse appears or disappears.This phenomenon is caused by the battery polarization effect.The fractional order model and FUKF exhibit a more accurate performance on the suppression of sharp peak error.In dynamic operation condition, the error of the terminal voltage estimation is more severe than in the static operation condition, because the battery polarization effect is more obvious in the rapidly changing operating conditions.This effect causes the dynamics which cannot be modeled.However, FUKF is still effective with this phenomenon.
The characteristics of the battery can be described by the fractional order model more accurately.FUKF algorithm can effectively guarantee the state estimation accuracy.The precision of the c estimation can be improved significantly by combing the fractional model and FUKF algorithm.The proposed method performs accurately and reliably on terminal voltage and SOC estimation in complicated working condition.The proposed method can be applied in real BMS.

Conclusions
In this paper, a fractional second-order RC circuit model is presented, and an algorithm combining UKF and fractional calculus is proposed to estimate the battery SOC.The algorithm can be easily implemented and applied by iteration, and it is suitable for the SOC estimation of electric vehicle BMS.The algorithm, FUKF, has been proved to guarantee the performance of the SOC estimation.The fractional battery model and FUKF fit the dynamic characteristics precisely.Their behavior of battery parameter identification is better than the integer one.These properties explain the precision of SOC estimation and terminal voltage estimation of FUKF.The experimental results show robustness on suppressing the noise and disturbance caused by the inexact or unknown statistical properties of the system modeling error and measurement error.

Figure 1 .
Figure 1.The circuit of the second-order RC model.

Figure 3 .
Figure 3.The part of the pulse characterization profile.

Figure 4 .
Figure 4.The results of parameter identification.(a) The curve of R 0 with respect to SOC; (b) The curve of R 1 with respect to SOC; (c) The curve of C 1 with respect to SOC; (d) The curve of R 2 with respect to SOC; (e) The curve of C 2 with respect to SOC.

Figure 5 .
Figure 5.The orders of the capacitors.(a) The order α of C 1 with respect to SOC; (b) The order β of C 2 with respect to SOC.

Figure 8 .Figure 9 .
Figure 8.The current profile of dynamic experimental validation.

Table 1 .
Static discharge experiment result comparison.

Table 2 .
Dynamic discharge experiment result comparison.