A comparative study of different equivalent circuit models for

Electrochimica Acta 259 (2018) 566e577

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Electrochimica Acta journal homepage: www.elsevier.com/locate/electacta

A comparative study of different equivalent circuit models for estimating state-of-charge of lithium-ion batteries Xin Lai, Yuejiu Zheng*, Tao Sun College of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 July 2017 Received in revised form 24 October 2017 Accepted 24 October 2017 Available online 26 October 2017

An appropriate model is a prerequisite for accurate state-of-charge (SOC) estimation. The widely used equivalent circuit models (ECMs) employ a variety of forms; thus, to find the optimum ECM is a primary task for SOC estimation. In this work, we examined eleven ECMs to fulfill the following goals: (1) to compare the typical ECMs for accuracy, stability, and robustness of model and SOC estimation; (2) to compare and evaluate the robustness of the ECMs considering model and sensor errors. The results indicate that the model accuracy does not always improve by increasing the order of the RC network. Conversely, over-fitting problems appear with a certain probability. The first- and second-order RC models are the best choice owing to their balance of accuracy and reliability for LiNMC batteries. The higher-order RC model has better robustness considering the variation in model parameters and sensor errors. Independently of the ECM adopted, an accurate OCV-SOC curve and high precision sensors are essential. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Lithium-ion battery SOC estimation Equivalent circuit model Extended Kalman filter Comparative study

1. Introduction Over the past decade, energy shortage and global climate warming have provided a good opportunity for the rapid development of electric vehicles (EVs) [1e3]. Battery, motor and electric control are the three key technologies applied in EVs, and the battery represents the main factor restricting the expansion of EVs in the marketplace [4,5]. Among all types of power batteries for EVs, the lithium-ion battery (LIB) is widely accepted because of its high energy density, long lifespan, high efficiency, and low selfdischarging [6e8]. The battery management system (BMS) is used to maximize battery power performance and prolong its service life. The state-of-charge (SOC) is the key parameter in a BMS. Accurate SOC estimation is crucial to guarantee the safety and reliability of LIB in Evs.

1.1. Review of the ECM and SOC estimation method A reliable and accurate battery model is crucial for the modelbased SOC estimation methods [9]. The battery models can be categorized into three types in terms of control theory, namely,

* Corresponding author. E-mail address: [email protected] (Y. Zheng). https://doi.org/10.1016/j.electacta.2017.10.153 0013-4686/© 2017 Elsevier Ltd. All rights reserved.

white box model, grey box model, and black box model. The white box model mathematically describes the chemical reactions inside batteries on the basis of electrochemical theory [10,11]. The advantage of this model is that it is able to clearly present the basic pattern of battery decline. While its disadvantage is that the mathematical equations are fairly complex with multiple parameters, limiting its practical application [12,13]. The grey box model is actually an empirical model based on the experimental results. Most empirical models have relatively low calculation load and therefore is suitable to be applied for online SOC estimation. While the drawbacks are also obvious: some empirical models such as the OCV based model cannot be used for real-time estimation due to the working condition restriction; others may suffer from the estimation accuracy which highly depends on the state-of-health (SOH) and temperature. The well-know ECMs blong to this model [14,15]. The black box model, mainly represented by neural network models [16e18], realizes battery management and state estimation through input/output data training and self-learning. It can simulate the high nonlinearity of LIBs, but requires a large number of training samples. Moreover, it has a high computational demand and poor practical applicability. ECM is the most widely used battery models for online vehicle applications [19e21]. The ECM do not consider the chemical components inside batteries or the corresponding chemical reactions, and mostly utilize the electrical elements such as

X. Lai et al. / Electrochimica Acta 259 (2018) 566e577

resistance and capacitance to describe the electrical behavior of a battery, making it a preferred choice for SOC estimation with more precision and lower computational cost. There are several battery ECMs presented in many literature, such as the Rint model [14,22,23], Thevenin model [24], PNGV model [1], and GNL model [20]. According to the practical applications, the resistancecapacitance (RC) network models were also widely studied [21,25e27]. Theoretically, the higher ordel of the RC models, the higher the accuracy of the model that can be obtained. However, a complex ECM usually have a significant increase in complexity and therefore is difficult to be used for online applications. Besides, in real applications, model accuracy highly depends on model parameters. A complex ECM with more model parameters may cause paramete mismatch, which lead to poor accuracy. Therefore, the selection of the ECM is the balance between accuracy and complexity. The SOC cannot be measured directly using physical sensors; rather, it must be estimated using other measurable quantities. Moreover, the internal chemical reactions in a LIB are nonlinear and time-variable, and are affected by many factors, including ambient temperature and battery ageing. Overall, the accurate estimation of SOC is a challenging task. A number of SOC estimation methods were proposed, each with its own strengths and limitations, and these methods can be summarized as follows [28]: (1) Conventional method, which uses the physical properties of the battery (voltage, discharge current, resistance, and impedance). This method mainly includes the coulomb counting method, open circuit voltage (OCV) method, electromotive force (EMF) method and internal resistance method. (2) Adaptive filter algorithm (AFA) [1,29e31], which mainly includes the Kalman filter (KF), extended Kalman filter (EKF), unscented Kalman filter (UKF), sigma point Kalman filter (SPKF), and particle filter (PF). (3) Learning algorithm. The most frequently used algorithms of this method are neural network, fuzzy logic [32], and support vector machine [33]. (4) Non-linear observer. This method mainly includes the Luenberger observer [34], sliding mode observer [35], proportional-integral observer [36], and non-linear observer. (5) Others [37]. These include multivibrate adaptive regression splines, bi-linear interpolation, and hybrid method that consists of two or three of the above algorithms. Most of these methods have been widely used and they achieve acceptable results in different applications. Among these methods, EKF considers the noise characteristics of the current and voltage sensor, can effectively overcome the problems of random error and lack of sensor resolution. In addition, its complexity is relatively low. For these reasons, EKF is one of the most popular methods to estimate battery state in EVs. Therefore, we select EKF for SOC estimation to evaluate the SOC accuracy, reliability, robustness of each ECMs in this paper. The structure and complexity of the ECM are critical for SOC estimation. Thus, it is important to study representative ECMs comparatively. The available comparative studies about ECM and SOC are divided into two categories: one is the comparison of SOC algorithms based on a fixed ECM [29e31,38,39], such as the robust EKF, standard extended EKF, UKF, and PF. In Ref. [40], the authors compared 18 types of KF algorithm. The other category is the comparison and evaluation of model error based on various ECMs. The basic method is to use some algorithms to identify and optimize the model parameters, thus determining the model error. Ref. [27] used a modified particle swarm optimization (MPSO) algorithm to compare the accuracies of twelve ECMs. Ref. [41] used three algorithms (gradient descent (GD), genetic algorithm (GA), and prediction error minimization (PEM)) to identify the parameters of three ECMs. Ref. [15] carried out a systematic evaluation on

567

the seven typical battery models based on recursive least square with an optimal forgetting factor (RLSF) method. From these literature, we can see that the previous studies on the comparison and evaluation of LIB models have paid little attention to the SOC estimation error and its distribution. In addition, stability is as important as accuracy for SOC in a BMS, and thus, it is necessary to investigate the stability of SOC estimation based on various ECMs. As the parameters of the ECM are affected by many factors, such as working current, temperature and battery ageing, it is essential to study the robustness of various ECMs in which the model parameters change. 1.2. Contribution of the study In this study, we consider a LIBs with cathode of LiNixCoyMn1-x-y (LiNMC) as research object, and conduct a comparison of eleven typical ECMs in terms of accuracy, stability, and robustness. The key contributions of this work can be summarized in the following three items: (1) To evaluate the accuracy of eleven ECMs, a systematic comparative study in terms of model and SOC estimation errors is conducted. (2) The distribution of the model and SOC errors is investigated, then the stability of the ECMs is analyzed. (3) A comparative study concerning the influences of model parameters on SOC with various ECMs is carried out to check the robustness of the ECMs. The aim of our study is to provide guidance in model selection for SOC estimation. In Ref. [18], the authors concluded that the estimation accuracies of all the KF algorithms were similar. To facilitate the comparison of the SOC estimation errors of various ECMs under a unified benchmark, we therefore employed the EKF algorithm, which is frequently used. 1.3. Organization of the paper The remainder of this article is organized as follows: In Section 2, the experimental device and protocols are described. In Section 3, eleven typical ECMs are introduced, a GA is employed to identify their respective parameters, and the EKF algorithm is introduced to estimate SOC. In Section 4, the accuracy and reliability of model and SOC are compared for various models. In Section 5, the impact of changing model parameters on SOC estimation error under various models is studied. The final section includes some conclusions and final remarks. 2. Experiments The LiNMC battery used in this experiment and its basic parameters are listed in Table 1. The battery test system shown in Fig. 1, comprises a battery test system, a thermal chamber for environmental temperature control, a data acquisition unit for signal collection, a computer for online experiment control and data recording, and the battery. Charging and discharging of the battery are controlled by the battery test system according to the driving cycle. The battery terminal voltage and current, and the real value of the SOC (obtained using the Ampere-hour (Ah) counting method) are recorded by the data acquisition unit using a sampling frequency of 1 Hz. It is also worth noting that the reasons for taking the SOC value obtained by the Ah integral method as the real SOC are as follows: The results of Ah integral method are mainly affected by the SOC initial value, battery capacity, and current measurement error. First, the battery is charged to the cut-off voltage by the standard charging method in our experiment. According to the definition of SOC, it can be identified as 100%; then this value is set to initial SOC,

568


Table 1 Main parameters of the LiNMC battery cell. Type

Nominal Capacity (Ah)

Nominal Voltage(V)

Lower cut-off Voltage(V)

Upper cut-off Voltage(V)

Maximum charge current(A)

LiNixCoyMn1-x-y

32.5

3.75

2.5

4.15

65

Fig. 1. Schematic of the battery test system.

which is highly accurate. The battery capacity is obtained through standard capacity experiments, and is therefore very accurate. In addition, the SOC error caused by the current error is very small in a short time. Assuming that there is a current error of 0.1%, i.e., the maximum current error does not exceed 65 mA, the error of SOC caused by the current error will not exceed 0.2% in 1 h. Therefore, we can conclude that it is reasonable to use the SOC value obtained by the Ah integral method as the real SOC. Moreover, the SOC value obtained by the Ah integral method is regarded as the reference SOC in numerous studies [42,43]. The new European driving cycle (NEDC) was used in the experiment, and the experimental results are shown in Fig. 2. These data were used to identify the model parameters and estimate SOC. Fig. 2(d) shows the relationship between the SOC and the open circuit voltage (OCV), which was measured experimentally. As it represented the reference of the SOC, the SOC-OCV curve is particularly important for SOC estimation.

3. ECMs and the SOC estimation algorithm 3.1. ECMs The ECM has advantages such as simplified structure, centralized parameters, easy identifiability, clear physical meanings, and is therefore widely used in SOC estimation [20]. However, the structure of the ECM can take many forms, making it very important to select the appropriate model structure and accurate model parameters. A typical ECM generally uses the RC network comprising the resistor and capacitor parallel circuits to simulate the dynamic characteristics of the battery. Fig. 3 presents an ECM with nth-RC networks, named the nRC hereafter. The output voltage of the battery can be expressed using the

Kirchhoff voltage law as follows:

" Vi ¼ OCV I R0 þ

n X

Ri 1 et=Ri Ci

# (1)

i¼1

where Vi denotes the battery terminal voltage, OCV denotes the open circuit voltage, R0 denotes ohmic resistance, Ri and Ci are the i-th polarization resistance and i-th polarization capacitance, respectively. Some of the frequently used ECMs are included in the nRC. Actually, the 0RC model is the Rint model, the 1RC model is the Thevenin model, and the 2RC model is the DP model [20]. As the number of RC networks increases, the mathematical representation of the model becomes more complex, which makes the model parameter identification and SOC estimation more difficult and not conducive to the calculation of the BMS [44]. Therefore, the number of RC network in the ECM is limited to four in this paper. Hysteresis, the phenomenon by which the OCV of the battery is inconsistent during the process of charging and discharging, is a basic characteristic of a LIB. The hysteretic model used in this paper can be expressed as:

uh;k ¼ H 1 ejkp It j H¼

M I 0 M I>0

(2)

(3)

where, uh;k is the hysteresis voltage and M is the maximum hysteresis voltage. Based on the combination of the above nRC (0 n 4) model and hysteresis model, ten ECMs were constructed. Moreover, the PNGV model [1] is widely adopted in the literature. Therefore, the


569

Fig. 2. Battery charging and discharging experiments in NEDC: (a) Measured current; (b) Measured voltage; (c) True SOC; (d) OCV-SOC curve.

eleven ECMs are chosen as the research objects in this study, as shown in Table 2. In Table 2, the nRC model with one-state hysteresis model is marked as nRCH.

3.2. Identification of model parameters Prior to SOC estimation, it is necessary to identify the parameters of the ECM and determine the value of electronic components for the models shown in Fig. 3. In this paper, the genetic algorithm (GA) was used to find the global optimal parameters of the ECM. The GA is described in detail in Refs. [45,46]. For the ECM of battery, the model parameters to be optimized can be described as follows

"

q ¼ Rþ R kp H 0 0

t1 R1 |fflfflfflffl{zfflfflfflffl}


1st RC

2nd RC

/


# (4)

nth RC

where Rþ 0 and R0 are the ohmic resistances during charging and discharging, respectively. In this work, the root-mean-square error (RMSE) between the model terminal voltage and the measured voltage is used to evaluate the fitness of model parameters. Correspondingly, the fitness function of the GA can be expressed as

MRMSE

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X b i;k 2 ¼t ui;k u n

(5)

k¼1

where MRMSE is the RMS error of the model, ui;k is the model terb i;k is the measured voltage. minal voltage, and u The optimization objective function of the GA can be expressed as

Fig. 3. Schematic diagram of the nRC ECM.

o n ðkÞ min MRMSE qj ðkÞ

(6)

where qj denotes the estimation value of the current population j at generation k. By setting the fitness function, variable name and number, and variable upper and lower limits, the parameters of the ECM can be identified by the GA over a global scope. Taking the 4RCH model as an example, the parameters of the model identified by GA in the NEDC are listed in Table 3. The results of the model identification are shown in Fig. 4, from which it is evident that the model voltage is very close to the experimental voltage within an error of ±0.01 V. The RMSE of the model is 4.551 mV, indicating that the application of GA in identifying the ECM parameters has high recognition accuracy under a complex driving cycle. Based on these results, we believe that it is reasonable to use the GA to identify the parameters of each model and compare the characteristics of the respective models.

570


Table 2 The discretization equations of the eleven equivalent circuit models. Model number 1

0

Model name

Model equations and description

0RC (Rint model)

Vk ¼ OCV ðZk Þ Ik R0 where ðZK Þ represents the dependence between OCV and SOC in the form of a table. 0 Vk ¼ OCV ðZk Þ Ik R0 þ uh;k uh;k ¼ expð kp Ik Dt Þuh;k1 þ ½1 expð kp Ik Dt H, H ¼ M where kp is the decaying factor; Dt is the sampling time. Vk ¼ OCV ðZk Þ Ik R0 þ u1;k , u1;k ¼ expðDt=t1 Þu1;k1 þ R1 ½1 expðDt=t1 ÞIk where u1;k and t1 ¼ R1 C1 are the voltage and time constant of the RC network. Vk ¼ OCV ðZk Þ Ik R0 þ u1;k þ uh;k

2

0RCH

3

1RC (Thevenin model)

4 5

1RCH PNGV

6

2RC (DP model)

7 8

2RCH 3RC

9 10

3RCH 4RC

11

4RCH

Ik 0 Ik > 0

Vk ¼ OCV ðZk Þ Ik R0 þ u1;k þ ucb;k ucb;k ¼ ucb;k1 þ C1b R1 ½1 expðDt=t1 ÞIk where Cb is the equivalent capacitance, ucb;k is the voltage of the equivalent capacitance. Vk ¼ OCV ðZk Þ Ik R0 þ u1;k þ u2;k , u2;k ¼ expðDt=t2 Þu2;k1 þ R2 ½1 expðDt=t2 ÞIk where u2;k and t2 ¼ R2 C2 are the voltage and time constant of the second RC network. Vk ¼ OCV ðZk Þ Ik R0 þ u2;k þ uh;k Vk ¼ OCV ðZk Þ Ik R0 þ u1;k þ u2;k þ u3;k u3;k ¼ expðDt=t3 Þu3;k1 þ R3 ½1 expðDt=t3 ÞIk where u3;k and t3 ¼ R3 C3 are the voltage and time constant of the third RC network. Vk ¼ OCV ðZk Þ Ik R0 þ u1;k þ u2;k þ u3;k þ uh;k Vk ¼ OCV ðZk Þ Ik R0 þ u1;k þ u2;k þ u3;k þ u4;k u4;k ¼ expðDt=t4 Þu4;k1 þ R4 ½1 expðDt=t4 ÞIk where u4;k and t4 ¼ R4 C4 are the voltage and time constant of the fourth RC network. Vk ¼ OCV ðZk Þ Ik R0 þ u1;k þ u2;k þ u3;k þ uh;k

Table 3 Parameter identification results of the 4RCH model. Model

Parameters and units

4RCH

Rþ 0 ¼ 0:0013U R1 ¼ 6:731e 04U R3 ¼ 0:0015U

R 0 ¼ 0:0013U C1 ¼ 3:436e þ 04 F C3 ¼ 2:139e þ 05 F

kp ¼ 0:1763

H ¼ 4:5799e 05

R2 ¼ 0:0021U R4 ¼ 0:0011U

C2 ¼ 1:488e þ 05 F C4 ¼ 4:130e þ 05 F

3.3. SOC estimation algorithm The EKF algorithm is used to estimate the SOC in this paper. The EKF is based on dynamic equations. Assuming that k is the discretetime index, xk is the state vector to be estimated, yk is the output vector, and the system input vector is uk . The battery model can be expressed by the following state equations:

xkþ1 ¼ f ðxk ; uk Þ þ wk

(7)

yk ¼ gðxk ; uk Þ þ vk

(8)

where wk denotes random process noise, vk denotes measurement error, f ðxk ; uk Þ is a state transition function, and gðxk ; uk Þis a measurement function. The state equations of the EKF algorithm can be expressed as:

xk ¼ SOCk ; u1;k ; u2;k ; /; un;k

yk ¼ OCVðZk Þ

n X

ui;k R0 Ik þ vk

(10)

(11)

i¼1

uk ¼ Ik

(12)

The flowchart of the EKF algorithm based on these state equab ,B b , and D b ,C b tions is shown in Fig. 5. The coefficient matrices A k k k k of the state equations can be expressed as:

Fig. 4. Identification results of 4RCH model in NEDC: (a) Model voltage and experiment voltage; (b) Estimation error.


b ¼ vf ðxk ; uk Þ A k vxk xk ¼bx þ k 0 1 0

B 0 exp Dt t1;k ¼B @« « 0 0

/ / 1 /

571

state estimation and can be used to estimate the error bounds. In this discrete EKF algorithm, the state and covariance matrices are updated twice at each time step. The first update is the a priori P b estimate before measurement represented by x k and ~ ;k , and the x second is the a posteriori estimate after the measurement can be Pþ bþ represented by x . k and ~ ;k x The RMS error of SOC estimation can be expressed as

1 0 C 0 C A «

exp Dt tn;k SOC¼S b OC þ (13)

SOCRMSE

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X ¼t ðSOCreal SOCest Þ2 n

(17)

k¼1

2 6 b ¼6 B k 4

0

hDt Qn ,3600

1

B tDt C 1;k C R1 B @1 e A

dOCV b ¼ vgðxk ; uk Þ ¼ C ; k þ vxk dSOC xk ¼b x

0

1

B tDt C 2;k C R2 B @1 e A

0 /

B tDt C 7 n;k C 7 Rn B @1 e A5

1;

/;

1

(15)

k

( b ¼ D k

Rþ 0 ðIk 0Þ R 0 ðIk < 0Þ

1 3T

(16)

In Fig. 5, the superscript “-” and “þ” indicates a priori estimate and a posteriori estimate at time step k, respectively. Lk is the P P Kalman gain, w and v are the covariance matrices of the input P and output measurement noise, respectively. x~;k is the covariance matrix of the state estimation error, which shows the uncertainty of

(14)

where SOCreal and SOCest denote the real and estimated values of the SOC, respectively. The NEDC was then consulted to verify the accuracy and robustness of the EKF algorithm. Applying the best parameters identified by the GA to the 2RCH for the SOC estimation, the results and errors are shown in Fig. 6. These results show that the EKF algorithm has good convergence, even if the initial SOC error is introduced (the initial error was set at 5%). When there is no initial error, the SOC estimation error is uniformly less than 2%, and the RMS error of the SOC is 0.00752 V. The EKF algorithm is thus seen to work very robustly and accurately, and this validation lays the foundation for the following research.

Fig. 5. Schematic diagram of the EKF algorithm.

572


Fig. 6. SOC estimation results.

4. Comparative analysis of the model and SOC estimation errors 4.1. Comparison of the accuracy and reliability of ECMs In this section, the parameters of the eleven commonly used ECMs shown in Table 2 are identified, and the corresponding model errors and distributions for each model are calculated to produce references for the selection of ECMs in the SOC estimation. The GA was applied to conduct parameter optimizations. In this paper, the accuracy of each ECM is measured by the RMSE between the test datasets and the output from the optimized models. The reliability of each ECM is measured by the minimum times of reduplicate identification (Na ) under certain accuracy and probability requirements. The larger the Na , the lower the reliability of the model. To evaluate comprehensively the model accuracy and reliability, all parameters of each model in Table 2 were calculated 400 times by the GA in the NEDC to derive 400 groups of optimal parameters for each model. The respective model errors were then calculated. We used the RMSE between the test datasets and the outputs from the optimized models to measure the accuracy of each model. The maximum, minimum and average RMSEs of the eleven models are shown in Fig. 7(a), which shows that the model error is larger when the ECM does not contain RC networks, because such models do not reflect the dynamic characteristics of the battery. In addition, by increasing of the order of RC in the ECMs, the maximum RMSE will increase, and the minimum and average RMSE remain nearly constant. Therefore, we can conclude that the model accuracy will not always improve by increasing the order of the RC network. The 1RC, PNGV, and 1RCH have excellent stability, but their accuracy is not the best, and the second- and higher- order models have excellent model accuracy, but their stability is not ideal. The distribution of RMSE for each ECM is shown in Fig. 7(b), which indicates that adding RC networks can improve model accuracy. However, adding a higher than second-order RC is not helpful, as the probability of outliers rises with the increases of the order of model. The main reason for this phenomenon lies in numerical computation and optimization problems. There are additional optimization parameters and a heavier calculation load is needed for high-order RC models, and although the GA is a global optimal algorithm, the solution is more likely to be trapped into a local optimum, which can lead to the problem of parameter overfitting.

Fig. 7. RMSE and statistical distributions of ECMs: (a) RMSE; (b) statistical distributions of RMSE.

Furthermore, appropriate model parameters can be obtained with reduced identification times for the low-order RC models. By contrast, it is necessary to increase the times of reduplicate identification for higher-order RC models to free the GA from the local optima that cause a large model error. Assuming that the maximum allowable model RMSE is 5 mV, and to ensure the probability of 99% for the RMS error less than 5 mV, the minimum number of model parameters x can be calculated by:

Na x 1 ¼ 0:01 Nb

(18)

where Nb is the total number of identifications (400 in this case). Eq. (18) can be used to measure the identification reliability of the ECM. It was applied to the 400 groups of model parameters above to calculate the reliability of each model, as shown in Table 4. It is obvious from the results that the increasing model complexity lowers the reliability, and adding a one-state hysteresis to the model has an unsatisfactory effect on its reliability. Moreover, the parameters of the first-order RC models (1RC, 1RCH, PNGV) are identified only 2 times, and the second- and third-order models need to be identified more than 3 times, while the fourth-order models require at least 6 times for guaranteeing that the probability of the optimal parameters is over 99%. Based on the above analysis, we can conclude that the 1RC and 2RC are preferable for the LiNMC battery, owing to their balance between accuracy, reliability and complexity. It is worth noting that


573

Table 4 Reliability of various models. Model name

1RC

1RCH

PNGV

2RC

2RCH

3RC

3RCH

4RC

4RCH

Na

1.313

1.356

1.4025

2.031

2.236

2.559

2.751

5.688

6.057

certain probability, with SOC estimation error also appearing when more complex models were used. The distribution of the RMSE of the SOC for various ECMs is shown in Fig. 8(b), which shows that the EKF algorithm has no obvious effect in terms of eliminating or reducing the influence of model error in SOC estimation. In an actual BMS, the computation time of the SOC estimation is as important as its accurate. To compare the computation time, the SOC estimation based on eleven models was implemented using MATLAB version 2014a and tested on a computer with a 3.1 GHz processor, 4 GB of memory, and a 64-bit operating system. The computation time was started right before the iterative operation of the EKF algorithm and ended right after the conclusion of iterative computing for each model. Table 5 lists the computation times for SOC estimation using the various ECMs. We note that the computation times increased with model complexity. We can conclude that complex ECMs do not improve the accuracy of SOC estimation; instead, the first-order RC model is preferable for the LiNMC battery owing to its ideal accuracy and computation time. In addition, from the above analysis, we can see that the PNGV model is very similar to the 1RC model. Therefore, we will not further discuss the PNGV model in the following analysis. 5. Comparative study of the robustness of the ECMs

Fig. 8. RMSE and statistical distributions of SOC estimation: (a) RMSE; (b) statistical distributions of RMSE.

the 1RC has excellent reliability, while the 2RC has better accuracy, and 1RCH and 2RCH are not satisfactory choices. The accuracy of the PNGV model is similar to that of 1RC, but the stability of the PNGV is lower than that of 1RC, so the PNGV model is not recommended. 4.2. Comparison of the accuracy and reliability of SOC estimation In this section, the SOC estimation errors of various models are compared. The EKF algorithm was employed to estimate the SOC, and the RMSE was used to evaluate SOC estimation accuracy. Based on the 400 groups of parameters for each model obtained in Section 4.1, the SOC was estimated 400 times for each model under the NEDC. The maximum, minimum and average RMSEs of the SOC for ECMs were then obtained, as shown in Fig. 8(a). The results clearly show that the 1RC model is the best choice for the LiNMC battery. In line with Section 4.1, abnormal model parameters appeared with a

The external relationship between battery voltage and current is described by the configuration of the ECM. However, the identified parameters may not be exactly equal to their true values in the actual system for the following reasons: (1) battery manufacturing process variances may cause inconsistencies among the battery equivalent internal resistances. If the same model parameters are used in the SOC estimation for all batteries, error is inevitable; (2) battery parameters vary with battery age and environment temperature. Hence, for the model selection, we should not only consider the accuracy and reliability of the ECMs under the best model parameters, but also consider the robustness regarding the model and sensor errors. In this paper, the impact of model errors caused by the ohmic resistance and equivalent impedance of LIB on SOC are studied, respectively. The same study is carried out about the sensor error caused by the voltage and current sensors. 5.1. Robustness of the ECMs considering model error 5.1.1. Impact of the ohmic resistance on SOC estimation In Refs. [47e50], the authors confirmed that the evolution of ohmic resistance is one of the main causes of the global impedance increases as a battery ages. For instance, ohmic resistance increased fourfold over the course of 200 days [47]. In this section, the impact of ohmic resistance on the SOC estimation error is studied for the various ECMs. As the SOC estimation of 0RC and 0RCH does not meet the requirements here, we do not consider these ECMs in this section. In general, the model parameters used in SOC estimation for each model are optimal, and the conclusion of Section 4.2 indicated that, in the case, the SOC estimation error is nearly equal for all ECMs. Therefore, each model is assumed to use its optimal model parameters for the purposes of this comparison. Fig. 9 shows the relationship between SOC estimation error and ohmic resistance

574


Table 5 Computation times of various models. Model name

1RC

1RCH

PNGV

2RC

2RCH

3RC

3RCH

4RC

4RCH

Computation time(s)

1.657

1.659

1.667

1.681

1.685

1.687

1.689

1.692

1.695

Fig. 9. Relationship between the internal resistance and the SOC error.

using the EKF algorithm for various ECMs under the test condition in which the ohmic resistance gradually increases to three times its initial value. It is shown that the effect of ohmic resistance on SOC estimation error is not significant. For instance, when tripling the ohmic resistance, the RMSE of the SOC estimation increases by less than 0.4%; similarly, when the ohmic resistance was increased by a factor of 1.5, the RMSE of all ECMs remained almost unchanged. Furthermore, it should be noted that the SOC estimation errors of ECMs with a large numbers of RC networks are relatively small. Thus, we can conclude that the higher-order RC model has better robustness. 5.1.2. Impact of the inaccurate impedance on SOC estimation In a real BMS, the change of impedance of LIB is inevitable. To

Fig. 10. Relationship between equivalent impedance and SOC estimation error.

better understand the effects of equivalent impedance change on the SOC estimation of various models, we examined several major ECMs to develop a reference for model selection for estimating the SOC. In this section, we primarily address the effects of polarization resistance and capacitance on SOC. A changing number and a variety of polarization resistors and capacitors can be found in different ECMs. As it is inconvenient to analyze the influence of each resistor and capacitor on an individual basis, we performed a global equivalent analysis of these components by assuming that, for an equivalent impedance error of þa%, each polarization resistance and capacitance in the model would be increased by a%. As described in Ref. [50], the impedance growth increases significantly when the temperature and aging is increased. Fig. 10 shows the influence of equivalent impedance on the SOC estimation error of the ECMs. In Fig. 10, RMSEeqi0 is the initial RMSE of SOC under the condition of the equivalent resistance error is 0%, RMSEeqi is the RMSE of SOC under a variety of equivalent impedance errors. Therefore, the coefficient KSOC eqi (KSOC eqi ¼ RMSEeqi =RMSEeqi0 ) can be used to measure the relation between the SOC estimation error and the equivalent impedance error. The results show that the equivalent resistance error increases from 0 to 20%, the RMSE of SOC significantly increases, indicating that equivalent resistance has a great influence on SOC. Furthermore, it is very clear that the greater the number of RC networks in the models, the smaller the SOC estimation error caused by inaccurate impedance. Therefore, choosing a larger number of RC models can reduce the SOC estimation error owing to increase in impedance. However, when the parameter identification is repeated, this will cause parameters mismatch with a certain probability.

5.1.3. Impact of inaccurate SOC-OCV curve on SOC estimation The precise relationship between SOC and OCV is necessary for accurate SOC estimation in BMS and these nonlinear monotonic relationships are obtained by experiment. The main factors influencing the SOC-OCV correlation are battery ageing, ambient temperature, and history of the battery. Refs. [48e50] indicated that the temperature and ageing of the battery influence the SOC-OCV

Fig. 11. SOC estimation error caused by inaccurate SOC-OCV curve.


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Fig. 12. Comparison of the SOC estimation profiles with þ5 mV drift voltage for various models.

relaxation significantly, with experimental results showing that the OCV of LiNMC drops 20 mV over three months, whereas the value of OCV increases by about 20 mV for a temperature increase of 30 C. According to Ref. [49], the OCV of LiFePO4 battery decreases by 60 mV when the temperature decreases from 45 C to 25 C. In this section, a comparative study of the SOC estimation error is performed for the ECMs under the assumption that the SOC-OCV curve is not accurate. It is assumed that the drift voltage in SOCOCV curve varies from 0 mV to 50 mV, then the SOC estimation error of all ECMs are calculated under these drift voltage conditions, respectively. The results are shown in Fig. 11. It can be seen that increasing the number of RC networks slightly increases the SOC estimation error. However, the impact of OCV offset on SOC estimation is extremely large for all ECMs. For example, at a drift voltage of 20 mV, the RMSE of SOC estimation is about 5%. Furthermore, the RMS errors in all ECMs follow nearly the same pattern. From these results, we can conclude that the structure and complexity of the ECM have little or no effect on reducing or eliminating the SOC estimation error caused by the inaccurate SOC-OCV curve. Therefore, to ensure SOC estimation accuracy, it is necessary to ensure the accuracy of the SOC-OCV curve. 5.2. Robustness of the ECMs considering sensor error Sensor measurement error is widely prevalent in BMSs and is one of the most significant causes of SOC estimation error. Because voltage and current are the primary signals required for SOC estimation, in this section, we study the SOC estimation errors in different ECMs arising from voltage and current errors. 5.2.1. SOC estimation error caused by voltage sensor error In an actual BMS, the measurement errors in a voltage sensor can be divided into two categories: system and random errors. A system error can be regarded as a fixed offset of the measured value of the voltage sensor; that is, a fixed drift voltage is add to the measured voltage at any given time. A random error is primarily caused by the resolution of the voltage sensor and can be regarded as a Gaussian noise. In current BMSs, because the KF has a strong inhibitory effect on random noise, the impact of random error on SOC estimation is small. Therefore, we focusssed on the impact of system error on SOC estimation in the ten types of ECMs. The system error of the voltage acquisition chip commonly used in BMS is generally limited to within ±10 mV; for example, the approach system error of LTC680X series chips is 8 mV, with a minimum error of 1.2 mV. To estimate system error, we used the

following approach: a fixed system error was artificially added to the voltage measurement in the NEDC to obtain estimation errors for each model. To facilitate comparison, the optimal parameters of the ECMs were selected from the 400 groups of parameters given in Section 4.1. Fig. 12 shows the RMSE of the SOC estimation for various models at a voltage system error of þ5 mV, in which the blue and yellow section represent the SOC estimate values without drift voltage and with þ5 mV drift voltage, respectively. The percentage numbers in green font represent the increasing in RMSE of the SOC estimation caused by drift voltage. It is shown that the SOC estimation errors of all ECMs remained within an acceptable range except 0RC and 0RCH. The increase in the RMSE of the SOC estimation, which ranged from 30.4% to 34.8%, indicates that increasing the number of RC networks in ECM cannot eliminate or decrease the SOC estimation errors arising from the system error of voltage measurement.

5.2.2. SOC estimation error caused by current sensor error Similar to the case of voltage measurement error, the current measurement errors can be divided into system and random errors, and the random error can be ignored. Thus, we analyze here the relationship between SOC estimation error and current measurement error for various ECMs. Fig. 13 shows the SOC estimation profiles with þ0.1 A drift current for various ECMs. It is evident, in line with the findings given in Section 5.2.1, that increasing the number of RC networks in the ECM cannot eliminate or decrease the SOC estimation errors arising from the system error of current measurement. From the preceding analysis, it can be observed that the measurement error of sensors in a BMS has a significant influence on the SOC estimation, with no direct relationship to the complexity and structure of the ECM. Therefore, improving the accuracy of SOC estimation requires improvements in sensor measurement accuracy.

6. Conclusions This paper presents a systematic comparative study of eleven ECMs in which the GA algorithm was applied to carry out parameter identification and optimization. The EKF algorithm was used to estimate the SOC for a LiNMC battery in the NEDC. Based on the comparison results of the models and SOC estimation errors in terms of accuracy, stability, and robustness, several conclusions can be drawn as follows:

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Fig. 13. Comparison of the SOC estimation profiles with þ0.1A drift current for various models.

(1) Increasing the order of RC in ECM cannot always improve the accuracy of model. Instead, owing to the large number of unknown variables, over-fitting problems appear with a certain probability when identification and optimization are conducted repeatedly. Our results show that ECMs with firstand second-order RC models are consistently more accurate and reliable than other models. Specifically, the 1RC has excellent reliability, while the 2RC has better accuracy. On the contrary, 1RCH and 2RCH are not satisfactory choices because adding a one-state hysteresis to the models does not improve their accuracy; instead, it reduces the reliability of the model parameter identification. The reason for this phenomenon is that with the increase of the order of model, more parameters need to be identified, and the over-fitting problem becomes increasingly serious; thus, reliability is reduced. (2) Using the EKF algorithm cannot eliminate or weaken the influence of model error on SOC estimation. With a certain probability, the SOC error will become increasingly large when using a high-order RC model. Thus, in consideration of the accuracy, stability, complexity and computation burden of SOC estimation, it is best to employ the first-order RC model for a LiNMC battery. (3) Increasing the order of RC networks in the ECM can weaken the adverse effects of the SOC estimation error arising from an increase in ohmic resistance. The influence of the impedance of various models on SOC estimation was studied by using equivalent impedance and it was concluded that the SOC estimation error of ECMs with higher-order RC networks is smaller when the equivalent impedance increases. (4) Independent the ECM adopted, an accurate OCV-SOC curve is essential for SOC estimation. Therefore, the accuracy of the SOC-OCV curve must be verified. (5) Random error of sensor has a negligible effect on SOC estimation, because the EKF algorithm is used. However, our calculations show that system error of sensor does have a significant effect on SOC estimation, and these effects cannot be eliminated or weakened by increasing the complexity of the ECM. It is noted that our conclusions are obtained based on the experimental data of the tested LiMNC cell. There may be some differences for LiMNC batteries produced by different manufacturers. However, our research provides a valuable reference for selecting a suitable ECM for SOC estimation, because the methods and processes to determine the suitable ECM are the same as

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