Mitigation of state of charge estimation error due to noisy current input measurement

The key indicator to assess the performance of a battery management system is the state of charge (SoC). Although various SoC estimation algorithms have been developed to increase the estimation accuracy, the effect of the current input measurement error on the SoC estimation has not been adequately considered in these algorithms. The majority of SoC estimation algorithms are based on noiseless current measurement models in the literature. More realistic battery models must include the current measurement modelled with the bias noise and the white noise. We present a novel method for mitigating noise in current input measurements to reduce the SoC estimation error. The proposed algorithm is validated by computer simulations and battery experiments. The results show that the proposed method reduces the maximum SoC estimation error from around 11.3% to 0.56% in computer simulations and it is reduced from 1.74% to 1.17% in the battery experiment.


Introduction
Concerns regarding energy conservation and environmental protection have increased the popularity of electric vehicles (EVs).Despite their increasing popularity, their performance still need to be improved.Existing EVs have a significantly lower driving range compared to traditional internal combustion engine-based vehicles.Another issue regarding EVs is to miscalculate the remaining power, leaving passengers stranded.These are due to the lack of an efficient battery management system (BMS) that can accurately estimate the remaining power of a battery pack.In this sense, the state of charge (SoC) is a key parameter to ensure a longer driving range with reliable remaining power.Optimal SoC estimation maximises battery energy utilisation, resulting in increased driving range.It is also vital to inform the driver about the current driving range and avoid the harmful consequences of overcharging or overdischarging the battery.The majority of investigations to improve the SoC estimation accuracy primarily focus on algorithm development with no current measurement error assumption.
The SoC is the ratio of the available battery capacity to the maximum battery capacity. 1 It cannot be directly measured but can be calculated or estimated using current and voltage measurements.There are various types of SoC estimation algorithms divided into two main groups: model-free methods and model-based methods. 2 Two most widely used model-free methods are Coulomb counting (CC) and open circuit voltage (OCV) measurement methods.The CC method determines the remaining capacity of a battery by integrating the current flowing in and out of the battery over time.However, noisy current measurement is not the only disadvantage of this method but also the correct initial SoC guess is required for high accuracy. 3The OCV measurement method measures the terminal voltage when the battery is in a steady state for a sufficient time, for example, 1 h. 4 The measured terminal voltage is assumed to be equal to the OCV and then converted to the SoC through a look-up table, which is obtained by laborious laboratory work.This method is not practical since it requires a long relaxation period and a look-up table.
State of health (SoH) is another important indicator in battery systems and it refers to how well a battery is performing in comparison to its fresh condition.In literature, there are various approaches to calculating SoH accurately.In the work by Guo et al., 5 an SoH estimation algorithm using the SSA-Elman model is proposed.Battery features and capacity are better correlated using this method.In the work by Li et al., 6 an improved electrochemical impedance spectroscopy (EIS) method is introduced to estimate SoH.The improved method increases the equivalent circuit model (ECM) accuracy and reduces the error in the SoH estimation error.An attention mechanism and bidirectional long short-term memory neural network are combined in Guo et al. 7 for estimating SoH.Three features as input of the model are chosen from the incremental capacity curve.
Model-based SoC estimation algorithms require a battery model to reflect the battery dynamics.Electrochemical models and ECMs are the two most common techniques used in battery modelling.In electrochemical models, complex differential equations are used to describe the electrochemical process taking place inside the battery. 8The pseudo-two-dimensional (P2D) model is one of the electrochemical models existing in the literature.P2D models are based on the porous electrode theory, concentrated solution theory and kinetic equations. 9Another electrochemical model is the single-particle model (SPM) which was developed to simplify the P2D model.In the SPM, it is assumed that multiple uniform-sized spherical particles form the electrodes and the current distribution is uniform along both electrodes. 10The electrochemical models can explain battery dynamics in terms of the main electrochemical reactions occurring inside a battery.However, their onsite accuracy is low due to their complexity and numerous parameters to be identified in the models.They are generally used for the optimisation of the battery design. 11Unlike the electrochemical models, the ECMs are frequently used in estimating the SoC due to the advantages of low computational effort and high estimation accuracy.The ECM describes the battery dynamics via basic circuit elements.In the work by Feng et al. 12 and Hossain et al., 13 it is shown that the Thevenin ECM model with one parallel resistorcapacitor (RC) branch can accurately represent the battery dynamics.
Adopting a battery model requires model parameter identification (MPI).The MPI methods are classified into two main groups: an offline method and online methods.The offline method is an experimental method to calculate the model parameters in laboratories.The hybrid pulse power characterisation (HPPC) current profile is commonly used in the experiment. 14This method provides fixed parameter estimations.The actual model parameters change as the battery ages and the operational conditions change.Therefore, offline method cannot update the parameters according to different operational conditions and battery ageing, resulting in inaccurate SoC estimates.To adopt the parameters to the changes in operational conditions and battery ageing, online MPI methods are used.The recursive least squares (RLS) method is one of the most popular online methods found in literature. 15,16In the work by Xia et al., 17 a forgetting factor RLS method is introduced to minimise the influence of old data on the current estimate.However, the noisy current input measurement deteriorates the performance of RLS-based parameter identification methods. 18In this case, the adaptive law-based MPI method can be an alternative.This method guarantees the stability of the parametric uncertainties based on the Lyapunov direct method. 191][22][23] These algorithms first estimate the OCV and then convert it to the SoC using the nonlinear relationship between the SoC and the OCV. 24,25In the literature, the majority of available SoC estimation algorithms do not consider the input current measurement noise.The development of a more realistic SoC estimation algorithm requires taking into account the input noise.
Error is always present in the current measurement, which is the input of both current counting and voltage-based correction methods.Thus, the current measurement error causes SoC error in both methods. 26The current measurement is corrupted by the current bias noise and the white noise. 27,26The impact of these noises is significantly different.The white noise does not have a significant effect on the SoC estimation error. 28The extended Kalman filter (EKF) can accurately estimate the SoC based on the current sensor measurement with the large random white noise. 29owever, it is found that the bias noise substantially increases the error in the SoC estimation. 27In the work by Liu and He, 30 the effect of the bias noise on the SoC estimation is investigated.The bias noise of 6 10 A is injected into the SoC estimation algorithm during the simulation.It is observed that the SoC estimation error is out of the tolerable range by 5%, which may cause overcharging or discharging of the battery in real-time applications.In the work by Liu et al., 31 it is found that the bias noise may reach up to 1% in the battery experiments.It can reach up to 200 mA in practice due to the electromagnetic environment and the temperature.In the work by Liu et al., 31 the bias noise is treated as a constant parameter to be estimated with battery model parameters.The convergence of battery model parameters to their actual values is not guaranteed; therefore, the bias convergence to its actual value cannot be guaranteed.Incorrect bias estimation would lead to erroneous SoC estimation.In the work by Xu et al., 21 a dual KF algorithm is proposed to filter the SoC twice to reduce the current measurement error and battery modelling error.Despite the increased computational cost, this method cannot provide a certain mitigation of the bias noise.In the work by Hou et al., 32 the SoC is estimated for portable devices without sensing the current.The current is an unknown input that is chosen as one of the states.In the work by Chun et al., 33 a method is also developed to estimate the SoC without sensing the current measurement.The method only uses the filtered terminal voltage measurements of each cell in the battery pack.The current applied to the battery is estimated using the corresponding filtered terminal voltage measurement.To estimate the load current based on terminal voltage sensor measurements, it is necessary to use a high-quality but costly voltage sensor; otherwise, the current estimate is likely to be less accurate.An improved fuzzy adaptive KF is designed in Yan et al. 34 to estimate the SoC of EVs working under poor sensor measurements.The system noise and the measurement noise are assumed to be zero-mean white noise, and the proposed method only updates their statistical properties.However, the current sensor is also corrupted by the bias noise and it is neglected.
In literature, different noise modelling strategies have been considered.In the work by Wang et al., 35 a method is proposed to calculate the error probabilities, which characterises the estimation reliability and diagnosis accuracy stochastically.To investigate sensor measurements, data missing phenomenon is considered to address the estimation error in Chen et al. 36 In the work by Kitanidis, 37 the uncertainty in the system input is modelled as a stochastic process with a mean value that is not known and varies in time.In the work by Shu et al., 38 the corrupted measurements are characterised as a Bernoulli-distributed random sequence.In the work by Lu et al., 39 it is shown that the states and unknown system inputs can be estimated by an extended Double-Model adaptive estimation approach.The unknown time-varying input is modelled as a random walk.Then, the system and measurement models are updated based on the unknown input model.Finally, the estimation algorithm is updated based on the new system and measurement models.In this work, we modelled the current sensor as a summation of the true current, the bias noise treated as random walk and the zero-mean white noise.To the best of the authors' knowledge, our work is the first direct attempt to consider two stochastic noises in the current measurement in the SoC estimation.
The organisation of the article is as follows: section 'Adaptive battery model identification' introduces the battery modelling and online MPI method, section 'Current bias mitigation' explains the bias noise estimation method; section 'SoC estimation' presents the SoC estimation method along with the modification of the standard KF algorithm; section 'Simulations and results' presents the computer simulation and experimental results; finally, the conclusion and the future work are presented in section 'Conclusion and future work'.

Equivalent circuit battery modelling
An accurate SoC estimation algorithm can be developed based on a battery model, which is required for the safe and efficient operation of the battery.The ECM-based battery models have been used to replicate the battery dynamic behaviour. 40There are several ECMs found in the literature.Although Theveninbased ECMs do not demand comprehensive knowledge regarding battery electrochemistry, they accurately reflect the battery dynamics. 41,4245]46 Figure 1 shows the ECM used in this work.The ECM has an ideal OCV source (V oc ), an ohmic resistance (R 0 ) and an RC branch consisting of a polarisation resistor R p and a polarisation capacitor C p .The energy loss during charging or discharging the battery is caused by R 0 .The RC branch mimics the polarisation characteristic of the battery during or after the charging/discharging cycles.The load current and the terminal voltage signals are denoted by I and V t , respectively.I a is the current flowing over R p , whereas I b is the current flowing over C p .The battery reaches the equilibrium state in a sufficient enough time, for example, 1 h after the load is removed from the battery.In the equilibrium state, V t is equal to V oc .Herein, the load current has a positive sign at discharge and a negative sign at charge.Based on the Kirchhoff's law and Ohm's law, the first-order Thevenin ECM can be expressed as follows where t is the time, d=dt is the derivative, V p is the voltage across the RC branch.In equation ( 1), R 0 , R p and C p are the model parameters to be identified.Furthermore, the identification of the nonlinear SoC-OCV relationship is also necessary for the SoC estimation.This relationship can be acquired through a battery SoC drop test.

Online MPI
An online MPI method improves the real-time performance of SoC estimation algorithms.In this work, the adaptive law-based MPI method is adopted to estimate the model parameters in real time.The adaptive law calculates the current estimate of the parameters by adding the previous estimate of the parameters and the correction term.The correction term is calculated based on a difference between the calculated output signal and the measured output signal.Therefore, the model parameter uncertainty is overcome by parameter updating and correction.Discrete-time expressions of equations (1a) and (1b) are of the form where ( Á ) k represents the kth sample of ( Á ), a = e ÀDt=t , Dt is the sampling time and t is the time constant, that is, t = R p C p .Rewrite equation (2b) for step k + 1 Substitute equation (2a) into equation (3) Rearrange equation (2b) and substitute into equation (4) , k and substitute into equation ( 5) for sampling time k and k À 1 where V oc varies slowly in comparison with V t , when the sampling frequency is high enough, 47 that is, DV oc ' 0. Equation ( 6) can be approximated as follows Equation ( 7) can be written in the LPM as follows where the unknown model parameters vector is given by and the measured input vector is given by The terminal voltage difference is calculated from two measurement samples.The measurement equation for the parameter estimation is as follows where (Á) is the measurement of ( Á ) and Dv V t , k + 1 represents the measurement noise.Note that unadorned symbols represent the true values.u k is simultaneously estimated using the online parameter estimation method based on the adaptive law given as follows 48 e k + 1 = ỹk where (Á) is the estimate of ( Á ), and G is the positivedefinite adaptive gain matrix.After ûk is calculated, the model parameters R0 , Rp and Ĉp can be reversely calculated by where ( Á ) (i) for i = 1, 2, 3 is the ith element of ( Á ).Three DC motors discharge the battery with a constant current of 1.3 A when the speed is set to 50 r/min (revolution per minute).The speed remains constant at 50 r/ min during the battery experiment.The experimental conditions are summarised in Table 1.

SoC-OCV nonlinear relationship
All ECM-based SoC estimation algorithms require the nonlinear SoC-OCV relationship to convert the estimated OCV to an SoC estimate.This relationship can be obtained by a battery SoC drop test. 49First, the battery is fully charged under a CCCV regime until it reaches the higher cut-off voltage given by the battery manufacturer.At this point, the battery is assumed to be fully charged.A fully charged battery is discharged by 5% SoC intervals until the SoC decreases to 80%.Then, the discharge interval is increased to 10% and the battery is discharged until its SoC drops to 20%.Then, the discharge interval is decreased to 5% and the battery is fully discharged.The smaller SoC discharge interval is used at low and high SoC regions to observe the nonlinearity better.The sampling time of the battery SoC drop test is 0.01 s and the experiment is repeated five times.Finally, the average values are calculated based on the collected data.A battery SoC drop test was performed on the LiPo battery and the result is given in Table 2.
The OCV-SoC nonlinear relationship is modelled by the following nonlinear expression 50 V oc = a log (z) + be (z) 3 + c ð14Þ where a, b and c are the constant coefficients, z is equal to SoC [%] divided by 100 and it is in the range of ½d, 1, where d is an arbitrary small positive number.The SoC below d is considered zero.The experimental data are curve-fitted to Cho et al. 14 to calculate a, b and c.Conversely, a real-time estimation algorithm can be used to update the coefficients. 50

Current bias mitigation
In battery-powered systems, the current sensor output is the current measurement which is corrupted by two different types of stochastic noise 27 where Ĩk is the current measurement, and I k is the true current, b k is the bias noise and v i, k is the zero-mean white noise whose variance is r i .b k is a random walk and modelled as follows 39 where Db k is the variation in b from sampling time k À 1 to sampling time k.
In battery systems, the second measurable signal is Ṽt .Note that, measured Ṽt is a function of true states, including V oc , I a and true input current I.However, estimated Vt is a function of estimated states, including Voc , Îa and the measured current input Ĩ. Considering the battery states converge to their actual values, the difference between the measured Ṽt and the estimated   Vt must be caused by the current sensor measurement error.Therefore, one sampling time step difference of ( Ṽt À Vt ) provides the required information to calculate b in real time.The measurement model of the terminal voltage is given by Rewrite equation (17) for the previous sampling time Note that v V t , k is independent of v V t , kÀ1 .Subtract equation (18) from equation ( 17) where To calculate the one-step difference of the measurement residual, the estimate of DV t is required.First, the estimate of V t, k is calculated according to the noisy current input measurement as follows Similarly, the estimate of V t at sampling time k À 1 can be written as follows Subtract equation ( 21) from equation ( 20) In equation (22), to calculate the difference of OCV at two sampling points, we could use the estimated SoC at two sampling points and convert them to the corresponding OCV using the SoC-OCV relationship.However, two estimated SoC could have inconsistent values to charging (increasing SoC) or discharging (decreasing SoC) of the battery.Instead, first, the previous value of SoC is calculated using the following CC equation where Q max is the maximum available capacity and z k is substituted by the estimated z k from the KF.Then, the calculated ẑkÀ1 is transformed into the estimated Voc, kÀ1 through the SoC-OCV relationship.The estimated OCV difference is calculated by subtracting Voc, kÀ1 from Voc, k , that is, D Voc, k = Voc, k À Voc, kÀ1 .The one sampling step difference of the measurement residual can be calculated by subtracting equation (22)  from equation (19) as follows The difference Db k can be calculated as follows where e s, k = Dv V t , k = R0 À Dv i, k is a zero-mean white noise.Directly substituting Db k into equation ( 21) would amplify undesired noises in the estimated b values.Thus, the standard KF is designed for estimating b k and provided in Algorithm 1.Details of the algorithm are presented in the next section.

SoC estimation
The SoC propagation equation is given by where w z, k is a zero-mean Gaussian white noise.The current I a is propagated as follows where w I a , k is a white noise with zero mean.Note that w z, k is independent of w I a , k .The measurement equation Algorithm 1 SoC estimation algorithm based on NiKF 1: Initialise: xÀ 0 , P À 0 , bÀ 0 , p À b, 0 , u 0 2: while 0 \ z \ 1 do 3: Calculate the model parameter using ( 12) & (13) 4: Update bÀ k and p À b, k 5: Repeat is given in equation (17), where V oc is a nonlinear function of z.The SoC estimation algorithm has linear state propagation equations ( 26) and ( 27) and nonlinear measurement equation (17).Consider the following generic model for a battery system where h( Á ) is a nonlinear function x k is the state vector, y k is the measurement vector and u k is the input.w x, k and v y, k are the process and measurement zero-mean white noises with known covariance matrices Q and R, respectively.They are independent of each other and assumed to be stationary over time.
In practice, it is expected that the standard KF's performance degrades due to the noisy input current measurement.Therefore, standard KF algorithm given in Kasdin 51 is reconstructed by considering the input current measurement model given in equation (15).Note that u in equations (28a) and (28b) is replaced by Îk in battery systems.
We are to derive the standard KF based on the generic model of the battery system.The derivation of the noisy input Kalman filter (NiKF) starts with updating the priori prediction of the state vector as follows where ( Á ) À implies the priori prediction of ( Á ), K k is the Kalman gain and the (y k À ŷk ) is the measurement residual.The derivation of the update part starts with defining the posterior state estimation error given as follows where K k D( bk À b k ) term is neglected by assuming lim t!' b !b.The approximation of equation ( 31) is given by The posterior state estimation error covariance matrix is as follows where E½Á represents the expectation operator and r i, k is the variance of v i, k .Note that v y, k , v i, k and e À k are independent of each other.
The Kalman gain matrix is derived by minimising the trace of P k .The trace of P k is the sum of the mean squared errors.Expand equation ( 33) as follows Taking the trace of equation (34) gives Differentiate equation (35) with respect to K k Equalising equation (36) to zero and solving for K k yield The state is propagated as follows The posterior error covariance matrix is propagated as follows where ( Á ) À k represents the priori prediction of ( Á ) k .In equation (39), B( bk À b k ) term is neglected since lim t!' b !b in most practical cases.The final expression of the priori state prediction error is given as follows The prior covariance matrix is expressed as follows Note that e k , w x, k and v i, k are independent of each other.The NiKF is summarised in Algorithm 1.

Simulations and results
The first random noise corrupting the current measurement is the random walk bias b.The difference between b k and b kÀ1 is an independent random increment which follows the Gaussian distribution.Its mean and variance are given as follows where s b is a positive constant.The random increment is expressed by where h k is a random number generated from the Gaussian distribution.Equation (42a) must be satisfied, therefore The mean value of h k must be zero, that is, E½h k = 0.The variance of h k must satisfy the property of random increment given in equation (42a Hence, the variance of h k can be calculated by The second noise corrupting the current measurement is the white noise which is a typical sensor noise whose mean is zero and distribution is Gaussian or normal It is assumed that the h k is not correlated with the white noise v i, k .
The simulated battery has the capacity of 0.85 Ah.It is fully discharged under two different dynamic loadings shown at the bottom of Figures 7 and 8.During the process, the bias noise and the white noise are added to the current input in every 0.01 s.The true battery model parameters are chosen similar to ones in equation 52 and set to For the simulation purposes, the initial b is randomly generated within the sample space whose lower bound is 0 and higher bound is 250 mA, s b is equal to 10 À3 and the initial p b is set to 10 À3 .Db is calculated using equation (43).True b is then calculated in every calculation step by adding Db to the previous b.
The initial state I a is set to 0, whereas the initial estimate z is randomly drawn from the uniform distribution in [0,1].The initial priori error covariance matrix is set to P À 1 = ½10 À3 0; 0 10 À3 .The coefficients of the nonlinear SoC-OCV relation are as follows: a = 0:2032, b = 0:3783 and c = 7:401.The smallpositive scalar delta is set to d = 0:003.The available voltage sensors can measure Ṽt with the error of 1-2 mV. 27Therefore, we assume the 63s V = 61:5 mV, and the standard deviation in V t , s V , is calculated to be s V = 0:05 mV.We also tested the algorithm with larger s V values, such as s V = 1 and s V = 10 mV.In comparison with the current bias noise, the white noise in the current measurement does not have a significant effect on the SoC estimation. 27Therefore, the standard deviation in v i , s i , is set to the same values as s V .Q is set to ½10 À3 0; 0 10 À3 .
Figures 3 and 4 show that the online parameter estimation algorithm successfully calculates the model parameters under two different dynamic loadings.In both cases, R 0 converges to its true value fast.However, R p and C p show more tiny fluctuations depending on the current input profile.This is expected because R p and C p define the response of the battery to the dynamic loading.However, these fluctuations do not have a significant effect on the SoC estimation error.Furthermore, C p and R p have the opposite reaction to keep the time constant of the battery constant.
From MPI results, it can be concluded that the adaptive law-based parameter identification method can successfully estimate the model parameters under the noisy current input measurement.
Calculated parameters are then fed to the SoC estimation algorithm.The estimation of b and the current measurement are shown in Figures 5 and 6.Although the actual current bias does not change so drastically in EV applications, 31 an extreme case is generated to test the performance of the algorithm.The maximum true bias reaches 120 mA in the dynamic stress test (DST) simulation, whereas it is around 250 mA in the HPPC test.In both scenarios, the algorithm accurately estimated b. b converges fast and minimises the effect of the initial error.
The results show that the proposed algorithm can estimate b with reasonable accuracy without depending on the current input profile.Once the estimated b is available, the bias measurement is corrected by subtracting b from Ĩ.
Figures 7 and 8 show the SoC estimation under the measured current input and the corrected current input along with the current input estimations.The mean absolute error (MAE), root mean square error (RMSE) and maximum percentage error (MAXE) of the SoC estimation are used to quantify the performance of the proposed algorithm.The MAXE in SoC estimation is reduced from around 11.3% to 0.56% under the HPPC cycle.It is decreased from 7.2% to 0.78% under the DST cycle.The battery experiment is also conducted to validate the proposed algorithm.
Figure 9 demonstrates b and SoC estimation results under different noises in terms of the standard deviation in v V and v i .The results show that an increase in the standard deviation in v V and v i increases the fluctuations in the b estimation.However, this increase does not have a significant effect on the estimated SoC values.It can be concluded that the proposed algorithm    can remove the SoC error due to the current bias noise when the V t measurement has different noise variance values.
Figure 10 shows the parameter estimation based on the battery experimental data.The parameters converge to their values fast.R 0 shows a stable value around 0:18 O until the battery is fully discharged.In comparison with R 0 , R p value increases dramatically at the end of the battery test.This is due to an increase in the residuals of the electrochemical reaction that takes place inside the battery.C p shows the opposite trend compared to R p because of the battery's time constant.Figure 11 shows the SoC estimation result along with the beta estimation and corrected current measurement.The current drift is calculated at around 20 mA.However, it insignificantly changes around this magnitude due to the nonlinear relationship of the SoC-OCV curve.The current drift estimation result is similar to the one in Liu et al. 31 The MAXE in SoC estimation is reduced from 1.74% to 1.12% in the battery    experiment.Table 3 summarises MAE, RMSE and MAXE in SoC estimation for computer simulations and battery experiment.These results show that the proposed algorithm can successfully mitigate the input current measurement error.This significantly reduces the error in the SoC estimation.Results based on the experimental data validate the capability of the algorithm to adapt to real-world applications.

Conclusion and future work
The corrupted current sensor measurements, variations in the operational conditions and battery depredation are inevitable in battery-powered applications.Unlike the majority of the SoC estimation algorithms in the literature, the current measurement is corrupted by two stochastic noises in practice.This deteriorates the SoC estimation accuracy, resulting in shorter battery pack life or passenger safety risks due to overcharging/overdischarging.This study proposes an online SoC estimation algorithm that mitigates the input current measurement noise.The method significantly reduces the SoC estimation error and increases the reliability of the BMS.The battery model parameters are estimated online using an adaptive lawbased parameter estimation algorithm.The input current measurement is modelled by considering the following two noises: the zero-mean white noise v i with a known variance and the random walk bias noise b.Moreover, the standard KF is modified according to the input current measurement model.The current input measurement is used in the state propagation and update equations.Hence, more accurate current input measurement leads to more reliable SoC estimation.The proposed algorithm accurately estimates the bias noise in the input current and corrects the input current measurement.The proposed algorithm is assessed by computer simulations and battery experiment.It is found that the proposed algorithm significantly mitigates the current measurement error source in SoC estimation and increases the SoC estimation accuracy.The current work presents an engineering practice and a theoretical framework for estimating the SoC based on noisy input current measurements.
In conclusion, our work provides a more realistic understanding of the battery SoC estimation problem taking place in practice.
The future work will investigate the accuracy of b estimation based on the difference between the calculated SoC-OCV model and the true SoC-OCV model by further computational simulations and battery experiments.The further battery experiments will be conducted in terms of the longer time duration to observe a larger current bias noise.Then, the performance of the proposed algorithm will be assessed based on this experimental data.

Figure 1 .
Figure 1.SoC-OCV nonlinear relationship and equivalent circuit model of Li-ion battery.

Figure 2
Figure2shows a schematic diagram of the test rig designed to run three DC motors powered by a completely new lithium polymer (LiPo) battery whose

Figure 2 .
Figure 2. The schematic diagram of experimental setup.

Figure 8 .
Figure 8. SoC and I estimation results under HPPC cycle: (i) SoC estimation results and (ii) current measurement correction result.Figure10.Parameter estimation results by experiment.

Figure 10 .
Figure 8. SoC and I estimation results under HPPC cycle: (i) SoC estimation results and (ii) current measurement correction result.Figure10.Parameter estimation results by experiment.

Figure 7 .
Figure 7. SoC and I estimation results under DST cycle: (i) SoC estimation results and (ii) current measurement correction result.

Figure 9 .
Figure 9. b and SoC estimation results with different white noise standard deviations in Ṽt and Ĩ: (i-ii) b and SoC estimation results with s V = 0:0005 and s i = 0:0005, (iii-iv) b and SoC estimation results with s V = 0:001 and s i = 0:001, (v-vi) b and SoC estimation results with s V = 0:01 and s i = 0:01.

Table 1 .
The operational conditions in the experiment.

Table 2 .
The SoC-OCV relationship obtained by a battery SoC drop test.