Parameter sensitivity analysis and simplification of equivalent circuit model for the state of charge of lithium-ion batteries

Journal Pre-proof Parameter sensitivity analysis and simplification of equivalent circuit model for the state of charge of lithium-ion batteries

Xin Lai, Shuyu Wang, Shangde Ma, Jingying Xie, Yuejiu Zheng PII:

S0013-4686(19)32110-3

DOI:

https://doi.org/10.1016/j.electacta.2019.135239

Reference:

EA 135239

To appear in:

Electrochimica Acta

Received Date:

07 August 2019

Accepted Date:

06 November 2019

Please cite this article as: Xin Lai, Shuyu Wang, Shangde Ma, Jingying Xie, Yuejiu Zheng, Parameter sensitivity analysis and simplification of equivalent circuit model for the state of charge of lithium-ion batteries, Electrochimica Acta (2019), https://doi.org/10.1016/j.electacta.2019.135239

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Journal Pre-proof Parameter sensitivity analysis and simplification of equivalent circuit model for the state of charge of lithium-ion batteries Xin Lai a, Shuyu Wang a, Shangde Ma b, Jingying Xie b, Yuejiu Zheng a,* a

College of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China.

b

Shanghai Institute of Space Power-Sources, Shanghai, 200245, China.

*Correspondence: [email protected] (Yuejiu Zheng).

Highlights 1. SA of model parameters of ECM is conducted to propose a simplified model. 2. Crucial model parameters over the whole SOC range under different aging degrees are determined. 3. SOC estimation is performed using the extended Kalman filter based on the simplified ECM. 4. Effectiveness of the proposed model is verified by experiments on two types of batteries. Abstract: To ensure model accuracy, the model parameters in the equivalent circuit model (ECM) are updated frequently with varying state of health (SOH) and state of charge (SOC). In this work, the parameter sensitivity of the 2RC model with one-state hysteresis (2RCH) is investigated to determine the crucial parameters. Firstly, the model parameters of 2RCH is identified using particle swarm optimization under dynamic working conditions. Secondly, the sensitivity analysis of parameters in 2RCH for two types of batteries is qualitatively examined using the one-factor-at-atime method. Thirdly, a simplified model, in which the crucial parameters with high sensitivities are updated with SOC and SOH, while the other parameters retain their initial values, is proposed to ensure model accuracy while reducing computational complexity greatly. Finally, SOC estimation based on the simplified ECM for two types of batteries over the whole SOC range under different SOHs is performed using the extended Kalman filter. The experimental results show that the SOC accuracy obtained by updating the crucial parameters is almost the same as that obtained by updating all parameters. The simplified model is beneficial to avoid unnecessary repeated calculation of model parameters for different SOC and SOH ranges in the SOC estimation. Keywords: Equivalent circuit model; Parameter identification; Sensitivity analysis; State of charge; Lithium-ion batteries.

1

Journal Pre-proof Nomenclature

List of symbols & parameters

Acronyms & abbreviations

Ak

matrix

AHC

ampere-hour counting

Bk

matrix

BMS

battery management system

C1

electrochemical polarization capacitance (F)

ECM

equivalent circuit model

C2

concentration polarization capacitance (F)

EV

electric vehicle

Dk

matrix

EKF

extended Kalman filter

Fˆk

RMSE of the model at time step k (V)

GNL

general nonlinear

I

current (A)

HPPC

hybrid pulse power characterization

kp

decaying factor

LFP

LiFePO4

K Par

degree of bias of the model parameter

LIBs

Lithium-ion batteries

K RMSE

ratio of the RMSE

NCM

LiNixCoyMnzO2

R0

internal ohmic resistance (Ω)

NEDC

new European driving cycle

R1

electrochemical polarization resistance (Ω)

OCV

open circuit voltage

R2

concentration polarization resistance (Ω)

OFAT

one-factor-at-a-time

RCha

charging ohmic resistances (Ω)

PNGV

partnership for a new generation of

RDch

discharging ohmic resistances (Ω)

vehicles PSO

particle swarm optimization

UH

hysteresis voltage (V)

RC

resistance–capacitance

UL

terminal voltage (V)

2RCH

2RC with one-state hysteresis

U OCV

open circuit voltage (V)

RMSE

root-mean-square error

1

time constant

SA

Sensitivity analysis

2

time constant

SOC

state-of-charge

w

variance matrix

SOH

state-of-health

v

variance matrix

1. Introduction Lithium-ion batteries (LIBs) are the most promising power source for electric vehicles (EVs) due to their excellent cycle lifetime, good stability, wide range of operating temperature, and high 2

Journal Pre-proof energy density [1-4]. To ensure the safety and reliability of energy storage in EVs, a well-designed battery management system (BMS) is necessary [5, 6]. One of the critical functions of a BMS is to accurately estimate battery states, including state-of-charge (SOC), state-of-energy, and state-ofhealth (SOH) [7-9]. However, the battery state cannot be detected directly and must be inferred from limited measurements, such as voltage, current, and temperature. At present, most state estimations of LIBs are based on battery models [10]. It is well recognized that a suitable model structure and matching model parameters are very important to improve the estimation accuracy. Furthermore, model parameters are time-varying and related to many factors, such as SOC and SOH. Information on the sensitivities of the model parameters can improve our understanding of battery models and simplify them. However, numerous studies have focused on the model structure and model parameters, rather than the sensitivities of the latter. Thus, sensitivity analysis (SA) of LIB models is extremely essential and urgent. 1.1. Review of LIB models and parameter identification Several kinds of battery models have been presented in the literature previously, including the mechanism model, empirical or data-based model, and equivalent circuit model (ECM) [11, 12]. The mechanism model is based on the electrochemical or physics model, and uses a mathematical equation to capture the internal reaction processes of LIBs. It has the advantage of high accuracy, and each model parameter has a clear physical meaning. However, the model parameters are too numerous, and the calculations are too heavy to be suitable for current BMSs. The empirical or databased models do not consider the complex physical and chemical reactions inside the battery, and are based on the external experimental data to simulate the battery behavior. Neural network models are a typical representative of this kind of model. They use neural network training to simulate the external characteristics of batteries based on sample data as such training is characterized by good non-linearity and self-learning ability [13, 14]. However, this model relies heavily on the historical data of batteries and needs massive amounts of data to ensure high accuracy. The ECM consists of a simple combination of electronic components, namely the power supply, resistor, and capacitor. It has been widely used to simulate the dynamic behaviors of LIBs due to its simple structure, centralized parameters, and easy mathematical modeling [13]. The most popular ECMs include the Rint, Thevenin [15], PNGV and GNL models [16], which consist of one or more resistance–capacitance (RC) networks. In these models, the RC networks are used to describe the dynamic characteristics of the battery, such as polarization and diffusion effects. Theoretically, the higher the number of RC networks in the model, the better its accuracy. Ref. [17] confirmed that ECMs with two RC networks show good balance between accuracy and complexity. Our previous work showed that ECMs consisting of one ohmic resistance, two RC networks, and 3

Journal Pre-proof one-state hysteresis are among the best model selections for SOC estimation of LIBs in terms of accuracy, stability, and robustness [18]. However, model accuracy is not only affected by the structure of the model, but is also limited by the identification algorithm of the model parameters [5]. Various parameter identification methods for ECMs have been reported in literatures [5, 18, 19], such as genetic algorithm, least square method, particle swarm optimization (PSO). Ref. [5] compared nine parameter identification methods, and concluded that the PSO is the best choice for the parameter identification of ECMs because of its comprehensive advantages in accuracy, reliability and complexity. Thus, the combination of an appropriate model structure and matching model parameters alone can maintain high model accuracy. However, the physical meanings of the parameters of the ECM are not very clear. Moreover, model parameters are identified and optimized using experimental data. When the environment changes or batteries age, the model parameters need to be re-identified, which poses challenges to the application of the ECM. The ECM consists of many parameters, and a large amount of calculation is required if each parameter needs to be corrected frequently. Therefore, it is necessary to investigate the sensitivities of the ECM parameters and pay attention to the crucial parameters (i.e., those with high sensitivity) to improve the accuracy of the ECM and reduce the computational cost. 1.2. Review of sensitivity analyses of LIB models SA is the study of how uncertainty in the output of a mathematical model can be attributed to different sources of uncertainty in the model input factors [20]. Typically, the SA is performed on the model parameters. This process facilitates the evaluation of the model robustness, that is, it provides an understanding of how sensitive the model is to changes in its parameters. It also allows each parameter to be ranked with respect to its contribution to the uncertainty in the model output. Moreover, identifying the sensitive model parameters can reduce the focus on improving the accuracy of non-sensitive parameters, preventing waste of time on refining parameters that the output is insensitive to. To date, some studies have focused on the SA of LIB models. Ref. [21] analyzed the sensitivities of the physicochemical LIB models to determine the crucial parameters, and then predicted the surface temperature and capacity fade of the battery. Ref. [22] performed the SA of a mathematical model for lithium-sulfur batteries to investigate the effect of model parameters on the discharge current and electronic conductivity of the cathode. Ref. [23] presented a parametric low-order model through the SA of the electrochemical model. Ref. [24] presented a novel fractional-order LIB model suitable for use in embedded applications based on the SA study. Ref. [25] carried out a parameter-level SA of a simplified electrochemical and thermal model for LIBs under various operating conditions to determine the most critical parameters. Ref. [26] proposed a 4

Journal Pre-proof physical model with a low dimension based on the results of the SA. Thus, we can conclude that most previous studies focused on the SA and simplification of the electrochemical or physicochemical model parameters of LIBs, and few have conducted a SA of the ECM for LIBs. The main reason for this situation is that the electrochemical and physical models have many parameters (more than 10), and unravelling the influence of parameter mismatch on the model output is very complex. The number of parameters in the ECM model is much smaller than that in the electrochemical model. Therefore, less attention has been paid to the SA of the ECM. However, the ECM is one of the most popular battery models, and its parameters are time-varying [27]. Typically, they are functions of SOC, temperature, battery aging, and other factors. The common solution to these problems is to identify and optimize each model parameter under different influencing factors, and parameter tables or MAP diagrams are then obtained. The disadvantage of this method is that the computational process is complex and offline, thus, it is not suitable for the application of on-line state estimation. To improve the model accuracy of the ECM over the whole SOC range and entire life cycle, it is necessary to analyze the sensitivity of the ECM to clarify the influence of parameter variation on the model output. The key parameters with high sensitivity are determined by investigating the sensitivity of each parameter to prevent time wasting on insensitive parameters, that is, to minimize the calculation cost while ensuring the model accuracy. The commonly used SA methods include local and global methods [28, 29]. The most frequently used local method in SA is the one-factor-at-a-time (OFAT) method [30-32]. The basic principle of this method is to calculate the model response after varying one model input factor while keeping the other factors at their nominal values. The advantage of the OFAT method is that it can simply and quickly determine the influence of the variation of an input factor on the model output. However, as a local method, its main shortcoming is that it does not fully explore the input space since it only considers perturbations around a single “nominal value” point. Global methods consider the sensitivity over the whole input space, and numerous methods exist in the literature for implementing them (e.g., the Monte Carlo [33], derivative-based [34], and variance-based methods [35]). Although the superiority of the global methods has been extensively proven in various studies, it still suffers from some shortcomings, the primary one being that the computation load is very large. For example, the Monte Carlo method requires thousands of runs of the model. According to the above analysis, few studies have attempted SA of the ECM under different degrees of aging over the whole SOC range. To address this gap in the literature, this work conducts the SA for the frequently used ECM. It is well known that the OFAT is straightforward to establish parameter dependency of the solutions, and useful to study problems with a few uncertain parameters [30]. As the frequently-used ECMs have fewer model parameters, the simplest and 5

Journal Pre-proof frequently reported method, OFAT, is utilized in the SA of the ECM for the intuitive conclusion with less computational cost. 1.3. Contribution of this study The major contributions of this study can be summarized as follows: (1) The SA of the model parameters related to the SOC and the SOH in the selected ECM is performed for the first time using the OFAT method, and the crucial parameters in ECMs are then determined and ranked. (2) Based on the parameter sensitivity of the ECM, a simplified ECM is developed to ensure the model accuracy, reduce the parameter identification time, and improve the robustness of the model. In the proposed ECM, the model parameters with high sensitivities are updated with the SOC and SOH at a high frequency, while the other parameters with low sensitivities are kept constant. (3) The extended Kalman filter (EKF)-based SOC estimator over the whole SOC range under different aging degrees is designed based on the simplified ECM. Finally, the accuracy and robustness of the proposed simplified model and estimator are verified by the experiments on two types of batteries. 1.4. Organization of the paper The rest of this paper is organized as follows. Section 2 describes the structure and mathematical expressions of the second-order RC model (2RC) with one-state hysteresis (2RCH). In Section 3, the SA of the 2RCH is performed in detail. In Section 4, a SOC estimator based on the simplified ECM under the different aging degree over the whole SOC range is designed and verified. Finally, Section 5 concludes this paper with further discussions. 2. Equivalent circuit model ECMs are widely used in BMSs, with the nth-RC networks model being the most prevalent. This type of model describes the dynamic characteristics of the battery through paralleled RC networks. The model contains three parts. The first part is the voltage source. The second part is the internal ohmic resistance R0 . It represents the resistance between the battery components and that caused by internal side reactions. The last part is the RC network. It represents the mass transport effects and dynamic voltage performance. Theoretically, the higher the number of RC pairs, the higher the model accuracy and the lower the computational efficiency. Our previous studies [5, 18] indicate that the 2RCH exhibits optimum availability and performance for LiFePO4 (LFP) cells, and 2RC is preferred for LiNixCoyMnzO2 (NCM) cells. Moreover, these studies show that the accuracy 6

Journal Pre-proof of 2RCH is almost the same as that of 2RC for NCM cells, and complexity of 2RCH is slightly higher than that of 2RC. Thus, to make the research conclusions more representative and universal, 2RCH is selected to conduct the SA of the model parameters in this study. The schematic diagram of 2RCH is illustrated in Fig. 1.

R1

and

R2

denote the

electrochemical polarization resistance and concentration polarization resistance, respectively. C1 and C2 denote the electrochemical polarization capacitance and concentration polarization capacitance, respectively. U1 and U 2 denote the polarization voltages across R1C1 and R2 C2 , respectively. U L denotes the terminal voltage. U OCV denotes the open circuit voltage (OCV) of the battery, which is a function of the SOC. I denotes the current, which is positive in the discharging condition and negative in the charging condition. Based on the electric circuit analysis, the electrical equation of 2RCH is presented as follows: U1  IR1 1  exp   t  1  

(1)

U 2  IR2 1  exp   t  2  

(2)



U H  H 1- e

- k p It

 , H = 1-1 II  

(3)

U L  U OCV  U1  U 2  U H  IR0

(4)

where  1 and  2 are the time constants, which meet the following equations:  1  R1C1 ,

 2  R2 C2 , U H denotes the hysteresis voltage, k p denotes the decaying factor,  is adequately small and positive.

Fig. 1. Schematic diagram of the 2RCH. 3. Parameter identification and sensitivity analysis 3.1. Experimental details In this study, two types of commercially available lithium-ion cells are selected for testing. The first (Cell #1) has an NCM cathode, whereas the second (Cell #2) has an LFP cathode. Their 7

Journal Pre-proof parameters are listed in Table 1. As shown in Fig. 2(a), the test bench consists of a battery tester cycler (BT2000, Arbin Instruments, USA), which loads the programmed current/voltage and transmits the data to the host computer, a climatic chamber to regulate the operation temperature, a data acquisition unit to log the terminal voltage and battery current at intervals of 1 s, and a host computer to program and record experimental data. The testing dates are used for the model parameter identification and SOC estimation. To fully investigate the parameter sensitivity of the ECM, the basic performance and dynamic working condition experiment (Exp. #1), and aging experiment (Exp. #2) are carried out. Exp. #1 is performed on Cell #1 with an SOH of 100%. Exp. #2 is performed on Cell #2 to obtain a different aging state, and Exp. #1 is then carried out under each aging degree. The experimental process of Exp. #1 is as follows. The operation temperature is set to 25 C. Firstly, the charge/discharge test under the new European driving cycle (NEDC) is conducted. Then, the capacity and traditional hybrid pulse power characterization (HPPC) tests are performed to determine the capacity and relationship between the OCV and SOC. The flow chart of Exp. #2 is shown in Fig. 2(b), in which 16 cycles of charge and discharge form an aging experiment. To accelerate the aging of the batteries, the aging experiment is conducted at different temperatures. In this study, three temperatures (45 C, 25 C, and 0 C) are set in turn as the chamber temperatures, and two aging experiments are carried out at each temperature. Figs. 2(c) and (d) describe the voltage and current profiles of Cell #1 under an SOH of 100% over the whole SOC range. They are used to identify and evaluate the model parameters of 2RCH. Fig. 2(e) shows the OCV curve of Cell #1, which is used for the SOC estimation. The test dates of Cell #2 under different aging degrees are not plotted here for brevity. Table 1 Main parameters of the selected cells. Cell

Chemistry

Rated capacity

Nominal

Maximum charge

(Ah)

voltage (V)

current (A)

Weight (g)

Format

Cell #1

NCM

32.5

3.75

65

990

Pouch

Cell #2

LFP

3.2

3.6

1.5

49

Cylindrical

8

Journal Pre-proof (a)

(b) Exp. #1: basic performance and dynamic working conditions experiments

Climatic chamber Power connect

Ethernet cables

Set the temperature to 25� HPPC, NEDC, and Standard capacity experiments Adjust SOC to 0%

Arbin BT2000 CAN

Adjusting temperature (25� ，45� ，0� )

Data acquisition unit

PC

(c)

Signal line

Rested for 3 hours Fully charge the battery (0.9A)

Cell #1@SOH=100%

Rested of 1 hours Discharge to SOC of 0% (3A) Rested of 1 hours

Yes

Number of cycles is 16？

No

Exp. #2: aging experiment

(e)

4.2

(d)

4

Open circuit voltage (V)

Cell #1@SOH=100%

Cell #1@SOH=100%

3.8 3.6 3.4 3.2 3 2.8

0

10

20

30

40

50

SOC (%)

Fig. 2. Test bench, test process and test results of the Cell #1 over the whole SOC range. (a) Schematic of the battery test bench; (b) Test process; (c) Current profile under the NEDC test; (d) Voltage profile under the NEDC test; (e) OCV curve. 3.2. Model parameter identification using particle swarm optimization In this section, the model parameters are identified and optimized to determine their sensitivities based on the above experimental results. For the 2RCH model, eight model parameters need to be identified, which can be expressed as follows: 9

Journal Pre-proof    RCha

RDch

ka

H

1

R1  2

R2 

(5)

where RCha and RDch denote the charging and discharging ohmic resistances, respectively. In this study, the root-mean-square error (RMSE) between the model terminal voltage and measured voltage is used to evaluate the model accuracy. Hence, the fitness function can be expressed as:

 

Fˆ ˆk 

1 N

 u n

k 1

k

 

 uˆ ˆk

2

(6)

where Fˆk is the RMSE of the model at time step k. ˆ is the estimated parameter values. uk and k uˆk are the model terminal voltage and the measured model voltage, respectively. N is the data

length. The parameter optimization of the ECM is a nonlinear constrained optimization problem in the following equations:

  

arg min Fˆ ˆk

(7)

s.t. LB  ˆk  UB .

(8)

where LB and UB denote the lower and upper bounds of the parameter vector, and they can be obtained based on our experimental results. In this study,

LB and UB are set as follows:

LB = 1/1000, 1/1000, 0, 0, 0, 0, 0, 0 , and UB =  2/1000, 2/1000, 10, 0.01, 1000, 0.1,1000, 0.1 .

In this study, the PSO algorithm is utilized for global optimization of parameter identification for the 2RCH model. The pseudo-code of the PSO algorithm is listed in Table 2. To obtain high global model accuracy over the whole SOC range, a subrange identification method is employed. The process of this method is as follows. Based on the SOC value obtained by the ampere-hour counting (AHC) method, the whole SOC range is divided into 10 subranges, and the model parameters are identified using PSO in each subrange. Thus, 10 groups of model parameters can be obtained. Applying the above method, eight parameters of the 2RCH for the NCM battery (Cell #1) with an SOH of 100% are identified over the whole SOC range, and these parameters are listed in Table 3. The model parameters of Cell #2 under different SOHs are not listed here for brevity. It can be seen that the RMSE is very small in the high SOC range (SOC>20%), which indicates that the subrange identification method has good global accuracy. Note that the model error in the low SOC range (SOC<20%) is very large, which is caused by the inherent flaws of ECMs. The reason 10

Journal Pre-proof for this phenomenon can be simply described as follows: in the ECMs, the solid-phase-diffusional voltage drop U SD is described by the polarization voltage of the RC elements, and the nonlinear characteristics of the SOC-OCV curve cannot be accurately expressed in ECM [36]. When the SOCOCV curve shows good linearity, the U SD determined by ECMs illustrates high accuracy, and when the SOC-OCV curve is very nonlinear, the estimation error of U SD will be very large. For LIBs, SOC-OCV curve shows good linearity in high and middle SOC range, and consequently U SD shows high accuracy. However, the SOC-OCV curve changes rapidly with serious nonlinearity in the low SOC range, leading to a large voltage estimation error for the U SD . Moreover, the value of each parameter varies with SOC, ensuring the global model accuracy over the whole SOC range. Therefore, it is necessary to update the model parameters in real time to achieve high estimation accuracy when 2RCH is used to estimate the SOC. Note that the selection of model parameters for the all-range SOC estimation is based on the SOC range obtained by the AHC method. In other words, SOC obtained by AHC method is used to select model parameters, and then SOC estimation is carried out using other advanced algorithms based on these updated model parameters. Table 2. Pseudo-code of the PSO algorithm. Procedure PSO for each particle i Initialize velocity Vi and position Xi for particle i Evaluate particle i and set pBesti=Xi end for gBest =min {pBesti} while not stop for i=1 to N Update the velocity and position of particle i Evaluate particle i if fit (Xi)
Table 3. Parameter identification results of 2RCH (Cell #1@SOH=100%) over the whole SOC range. SOC /%

R0 /mΩ

R0 /mΩ

kp

H /V

11

R1 /mΩ

 1 /s

R2 /

mΩ

 2 /s

RMSE /V

Journal Pre-proof 90-100

1.3818

1.4194

0.0016

0.0035

0.2969

9.32

0.60414

50.38

0.00207

80-90

1.2843

1.5206

0.0018

0.0010

0.5110

172.1

0.57678

13.21

0.00255

70-80

1.2046

1.5179

0.0003

0.0036

0.6462

999.9

1.80159

66.13

0.00299

60-70

1.2153

1.6022

0.0012

0.0028

0.5429

562.0

0.54839

19.99

0.00407

50-60

1.1738

1.4227

0.0014

0.0001

0.4598

21.8

1.42303

176.6

0.00187

40-50

1.2813

1.5433

0.0033

0.0003

0.5499

49.7

2.24140

921.3

0.00231

30-40

1.2501

1.5831

0.0010

0.0003

0.5741

33.7

6.62323

999.9

0.00348

20-30

1.4787

1.5409

0.0059

0.0034

0.4790

999.9

0.59219

23.65

0.00555

10-20

1.1343

1.5372

0.0067

0.0024

0.7317

999.9

0.76069

14.97

0.01693

0-10

1.9996

1.0074

0.0034

0.0099

0.8988

999.9

2.34336

999.9

0.06531

3.3. Parameter sensitivity analysis over the whole SOC range In this section, the sensitivity of each parameter is analyzed based on the results of parameter identification. As the PSO is a stochastic optimization algorithm, the result of each identification is different. This problem can be overcome by identifying the model parameters 10 times, and thus, 10 groups of model parameters are obtained. Then, the group of parameters with the minimum model error is selected as the optimal model parameters. In this study, the parameter sensitivity is analyzed with the frequently used OFAT method; that is, when analyzing the sensitivity of a parameter, the value of that parameter is changed to approximately the normal value while retaining the values of the other parameters, and then, these parameter values are substituted in the model to obtain the corresponding model error. The parameter sensitivity is reflected by the fluctuation of the model error, that is, the higher the RMSE, the greater the parameter sensitivity. Based on the above method, the parameter sensitivity of the 2RCH model for Cell #1 under the NEDC test is obtained, and the results are depicted in Fig. 3. Note that K Par is the ratio of the value of the shift parameter to that of the normal parameter, which is used to denote the degree of bias of the model parameter. Fig. 3(a) shows the sensitivity comparison of each parameter for the NCM battery at the SOC of 50%. It is observed that the sensitivity of ohmic internal resistance is the greatest ( RCha is more sensitive than RDch ), followed by those of R1 and R2 , and the other parameters are insensitive. Fig. 3(b) shows the derivative of the sensitivity of each model parameter for the NCM battery over the whole SOC range, which dynamically reflects the stability of parameter sensitivity. It is obvious that the sensitivity change rate basically follows the law of the sensitivity of each parameter over the high SOC range (SOC ≥ 20%). Moreover, the change rates of the parameter sensitivities of  1 and  2 are the largest in the low SOC range (SOC < 20%). In this 12

Journal Pre-proof case, the model error is too large to be used in the state estimation of the LIBs. Fig. 3(c) shows the sensitivity comparison of the LFP battery (Cell #2) under the SOC of 50%. We can see that the sensitivity of ohmic internal resistance is the highest. The maximum hysteresis H is more sensitive than polarization resistances R1 and R2 . The sensitivities of the other parameters are very low. Fig. 3(d) depicts the change rate of the parameter sensitivity of the LFP battery model, and the results show that it follows the rule of parameter sensitivity. 6

10-3 RCha

RDch

ka

H

1

R1

R2

2

Cell #1@SOC 50%+SOH100%+NEDC

4 3 2 1 0.5

0.6

0.7

0.8

0.9

1.0

KPar

1.1

1.2

1.3

1.4

(a)

(b) 12

10-3 RCha

10

RDch

ka

H

1

R1

2

R2

Cell #2@SOC50%+SOH100%+NEDC

RMSE (V)

RMSE (V)

5

8 6 4 2 0 0.5

0.6

0.7

0.8

0.9

1

KPar

(c)

13

1.1

1.2

1.3

1.4

1.5

1.5

Journal Pre-proof 2

RCha

diff(RMSE) (V)

0

RDch

ka

H

1

R1

R2

2

-2 -4 -6 -8 -10

Cell #2@SOH100%+NEDC [100 90)[90 80) [80 70) [70 60) [60 50) [50 40) [40 30) [30 20) [20 10) [10 0)

SOC (%)

(d) Fig. 3. Parameter sensitivity of the NCM and LFP batteries. (a) Sensitivity comparison of the model parameter for the NCM battery (SOC=50%)；(b) Change rate of parameter sensitivity for the NCM battery over the whole SOC range; (c) Sensitivity comparison of the model parameter for the LFP battery (SOC=50%)；(d) Change rate of parameter sensitivity for the LFP battery over the whole SOC range. From the above analysis, we can conclude that the parameter sensitivities of RCha , RDch , R1 , and R2 are high for NCM battery model, while the sensitivities of the other parameters are low. For the LFP battery model, the sensitivities of RCha , RDch , H , R1 , and R2 are much higher than those of the other parameters. Next, we take the LFP battery as an example to analyze the parameter sensitivity distribution of the above-mentioned five sensitive parameters over the whole SOC range. Fig. 4 shows the calculation results. K RMSE is the ratio of the RMSE of the bias parameters to that of the normal parameters, and thus, this term represents the relative sensitivity of each parameter. The relative sensitivities of a parameter can intuitively reflect the relative influence of the fluctuation of the parameter on the model error. It is observed that the larger the parameter shift, the greater the K RMSE . Furthermore, the error fluctuations caused by parameters RCha and RDch are larger than those caused by parameters R1 and R2 in the high SOC range, and the relative

RCha

14

RDch

12

6

10

5

KRMSE

10 9 8 7 6 5 4 3 2 1

KRMSE

KRMSE

sensitivities of all the parameters in the low SOC range are low.

8 6 4

H

4 3

2

2

10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

1 10 20 30 40 50 60 70 80 90 100

SOC(%)

SOC(%)

SOC(%)

14

Journal Pre-proof 3.5

R1

3

3

2.5

KRMSE

KRMSE

R2

3.5

2 1.5

2.5 2 1.5

1 10 20 30 40 50 60 70 80 90100

1 10 20 30 40 50 60 70 80 90 100

SOC(%)

SOC(%)

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

Fig. 4. Relative parameter sensitivity of the LFP battery over the whole SOC range. The sensitivity of model parameters is relative, and the parameter sensitivity of different models cannot be compared, that is, the parameter sensitivity can only be measured and compared in the same model. Figs 3(a) and 3(c) show that the level of parameter sensitivity of each parameter is very obvious. Therefore, the parameter sensitivity in 2RCH of two types of LIBs can be qualitatively evaluated based on Fig. 3, and the results can be summarized in Table 4. The asterisk (*) symbol represents the degree of parameter sensitivity. The greater the number of asterisks, the higher the sensitivity. The biggest difference about parameter sensitivities between the two types of LIBs is that the NCM battery is not sensitive to H , while the LFP battery is very sensitive to it. These results thus indicate the most critical parameters, and the identification frequency of the model parameters with low sensitivities can be reduced, which will improve the accuracy of battery state estimation and reduce the computational cost. Table 4. Results of the SA for 2RCH over the whole SOC range. Parameter

Sensitivity NCM

LFP

RCha

*****

****

RDch

****

*****

ka

*

*

H

*

****

R1

**

**

1

*

*

R2

**

**

2

*

*

15

Journal Pre-proof 3.4. Parameter sensitivity analysis under different SOHs Battery aging is inevitable. Thus, it is very important to study the parameter sensitivity of 2RCH under different aging degrees. In this study, cyclic aging experiments are carried out on Cell #2 and NEDC tests are conducted under different SOHs. Based on these test data, the sensitivities of the parameters of the ECM model under different aging degrees are investigated. For the state estimation of the LIBs over the entire life cycle, the parameters with high sensitivities should be continuously identified and corrected, while those with low sensitivities can be identified at a very low frequency based on our previous conclusions. Fig. 5 shows the parameter sensitivities of the LFP battery under different SOHs. The following observations are made: (1) The model parameters with high sensitivities (in decreasing order) are RDch , RCha , and H ; (2) R1 and R2 have lower sensitivities; and (3) the other parameters have

much lower sensitivities. Moreover, it can be seen that the sensitivity of H increases with the aging of the battery (when SOH = 96.1% and K Par = 1.5 , the error originating from the bias of H is approximately 5 mV, and when SOH = 94.5% and

K Par = 1.5 , the value increases to

approximately 9 mV), while the sensitivities of the other parameters remain unchanged with the aging of the battery. 14 12

10

-3

RCha

14 RDch

ka

H

1

R1

2

R2

12

Cell #[email protected]%+SOC50%+NEDC RMSE (V)

RMSE (V)

-3

RCha

RDch

ka

H

1

R1

2

R2

Cell #[email protected]%+SOC50%+NEDC

10

10 8 6

8 6

4

4

2

2

0 0.5 0.6 0.7 0.8 0.9

10

1

1.1 1.2 1.3 1.4 1.5

0 0.5 0.6 0.7 0.8 0.9

1

KPar

KPar

16

1.1 1.2 1.3 1.4 1.5

Journal Pre-proof 10

14

-3

RCha

12

RDch

ka

H

1

R1

16

R2

2

Cell #[email protected]%+SOC50%+NEDC

-3

RCha

14

10

RDch

ka

H

1

R1

R2

2

Cell #[email protected]%+SOC50%+NEDC

12

RMSE (V)

RMSE (V)

10

8 6

10 8

4

6

2 0 0.5 0.6 0.7 0.8 0.9

1

4 0.5 0.6 0.7 0.8 0.9

1.1 1.2 1.3 1.4 1.5

KPar

1

1.1 1.2 1.3 1.4 1.5

KPar

Fig. 5. Parameter sensitivity of the LPF battery under the different SOHs. Fig. 6 shows the relative parameter sensitivities of the LFP battery under different SOHs. It can be observed that the relative sensitivities of all the parameters vary with the SOH, and the relationships among them are complex. In general, during the battery aging, RDch has the highest relative sensitivity; RCha , H , R1 , and R2 show higher relative sensitivities; and  1 ,  2 , and ka have lower relative sensitivities. 5

RCha

1.5

RDch

KRMSE

1.5

3 2

98

96

94

92

1 100

H

2.5

KRMSE

KRMSE

2.5 2

98

1 100

96

SOH (%)

94

1 100

92

2

2

98

96

94

92

94

92

SOH (%)

R1

1.5

1.5

1.2 1.1

SOH (%) 3

1.3

KRMSE

1 100

ka

1.4

4

2

KRMSE

KRMSE

2.5

R2

1.8 1.6 1.4 1.2

98

96

SOH (%)

94

92

1 100

98

96

SOH (%)

17

94

1 100

98

96

SOH (%)

Journal Pre-proof 2

2

1

2

1.8

1.6

KRMSE

KRMSE

1.8

1.4 1.2

1.6 1.4 1.2

1 100

98

96

1 100

94

98

SOH (%)

96

94

92

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

SOH (%)

Fig. 6. Relative parameter sensitivity of the LFP battery under different SOHs over the whole SOC range. 4. Model simplification for SOC estimation 4.1. EKF-based SOC estimator over the whole SOC range In this study, the EKF is utilized to estimate SOC based on the simplified ECM. For the 2RCH model, the state and output equations of the state-space model are expressed as follows:  0 0 0  U   U1, k 1  exp  t /  1, k  1, k  U   U  0 exp  t /  0 0    2, k 2, k   2, k 1      U H , k 1     U H ,k 0 0 exp  k p I k t 0       SOCk 1  SOCk   0 0 0 1         xk 1 xk





Ak

t        R1 1  e 1,k   0         t       0   R2 1  e 2,k    Ik      H        0 1  exp  k p I k t    t   0  C  3600  N   



(9)



Bk

U t , k 1  OCV  SOCk 1   U1, k  U 2, k  U H , k  R0, k I k { 14444444444444442 4444444444444443 { { yk 1

g  xk 1 , uk 1 

Dk 1

(10)

uk 1

where k denotes the time index, t is the sampling time interval, xk is the state vector at time k, yk is the output vector, uk is the input vector, g  xk , uk  is a nonlinear measurement function, and Ak , Bk and Dk are system matrices of the state-space model,  is the Coulombic efficiency, CN

is the capacity of the battery. Based on Eqs. (9) and (10), the detailed operation process of the EKF-based SOC estimator is summarized in Table 5.

w

and

v

are the variance matrices of the zero-mean Gaussian 18

Journal Pre-proof stochastic processes. Lk is the Kalman gain vector. The superscripts “-” and “+” indicate the a priori and a posteriori estimate, respectively. Table 5. Operation process of EKF-based SOC estimator. Definitions:

Ck 

g  xk , uk  xk

xk  xk

T  Initialization: k = 0, xˆ0  E  x0  ,  x ,0 E  x0  xˆ0  x0  xˆ0    

Iteration: For k=1,2,… Prior state: xˆk- =Ak 1 xˆk1  Bk 1uk 1 Prior error covariance:





 Ak 1  x , k 1 AkT1   w 

x , k

  Kalman gain matrix: Lk   x , k CkT Ck  x , k CkT   v   

1

Posterior state: xˆk+ =xˆk-  Lk  yk  g  xˆk- , uk   Posterior error covariance:



+ x , k

  I  Lk Ck   x , k -

4.2. SOC estimation results based on the simplified model According to the above descriptions, the low sensitivity parameters have little influence on the model accuracy. In this study, the EKF is designed for SOC estimation, and the influence of the model parameters with different sensitivities on the SOC estimation over the whole SOC range is investigated in detail. During the SOC estimation over the whole SOC range, the traditional method of working with model parameters is to update the different parameters in each subrange, which undoubtedly increases the calculation effort for parameter identification and SOC estimation. Based on the results of the SA, a simplified 2RCH model is proposed. In this model, the parameters with high sensitivities are updated in each SOC subrange in real time, while those with low sensitivities maintain their initial values under different SOHs over the whole SOC range. Fig. 7 shows the SOC estimation results of Cell #1 over the whole SOC range based on the simplified 2RCH model. The symbol “” denotes that the labeled parameters are updated in real time over the whole SOC range, while the symbol “” means that the labeled parameters retain their initial values all the time. ECM (n) represents an ECM with n real-time updated parameters. 19

Journal Pre-proof For the 2RCH model, ECM (8) is an ECM containing all real-time updated parameters. Figs. 7(a) and (b) show the SOC estimation results based on ECM (4) and ECM (8). In ECM (4), the four model parameters requiring real-time updates are RCha , RDch , R1 , and R2 , all of which have high sensitivities. It is observed that as long as these four parameters with high sensitivities are updated in real time, the SOC estimation accuracy is almost the same as that obtained by updating all the model parameters in the high SOC range (SOC > 20%). Moreover, the SOC estimation accuracy based on ECM (4) is lower than that based on ECM (8) in the low SOC range. The main reason for this result is that  1 and  2 have high sensitivities in the low SOC range, which was verified in Section 3.3. Fortunately, the ECM has low accuracy in the low SOC range and is generally not used for SOC estimation. Therefore, the simplified model has very important application value. 6

Cell #1@SOH=100%

ECM(4) ECM(8) Reference

80

SOC (%)

(a) 60

RCha RDch R1 R2

40 20 0

0

 ka  H  1  2

1

   

2

3

4

5

2

0

(c)

1

2

3

Cell #1@SOH=100%

-2

RCha RDch R1 R2

ECM(2) ECM(8) Reference

4

0

2

4

5

6 10 4

 ka  H  1  2

(d)

   

Cell #1@SOH=100% 0 -2

Low SOC

High SOC -4

Low SOC

Time (s)

SOC (%)

SOC error (%)

4

(b)

   

High SOC

6

   

 ka  H  1  2

Cell #1@SOH=100%

10 4

 ka  H  1  2

RCha RDch R1 R2

0

-4

6

6

RCha RDch R1 R2

2

-2

Time (s) ECM(3) ECM(8) Reference

ECM(4) ECM(8) Reference

4

SOC error (%)

100

0

1

2

3

Time (s)

4

5

High SOC -4

6 10 4

0

1

2

3

Time (s)

Low SOC 4

5

6 10 4

Fig. 7. SOC estimation results of the cell #1 over the whole SOC range based on the simplified 2RCH with different updating parameters. Fig. 7(c) shows the SOC error over the whole SOC range based on ECM (3). It is observed that the SOC estimation accuracy decreases when the number of updated parameters is reduced to 3 ( R1 is fixed). However, it is still within an acceptable range. When the number of updated 20

Journal Pre-proof parameters with high sensitivities continues to decrease ( R1 and R2 are fixed), the SOC error increases, as shown in Fig. 7(d). It indicates that updating only two crucial parameters does not guarantee high SOC estimation accuracy. From the above analysis, it can be concluded that for the 2RCH model, high SOC estimation accuracy can be guaranteed over the whole SOC range if only four crucial parameters with high sensitivity are updated in real-time while the other parameters are retained at their initial values. That is to say, it is not necessary to update all the model parameters in real time when estimating the SOC over the whole SOC range, which simplifies the model and saves computation load. Furthermore, LIBs age with use, and thus, real-time updating of all model parameters is necessary to ensure high SOC estimation accuracy. Next, we attempt to study SOC estimation under different SOHs when only the crucial model parameters with high sensitivities are updated while the other parameters remain unchanged. In this study, the crucial model parameters are updated with different SOHs in real time while the other parameters remain at the initial value (i.e., assuming a fresh battery; SOH = 100%). Fig. 8 shows the estimated results of the LFP battery (Cell #2) when the SOH is 92.7%. It is observed that the SOC estimation results obtained by real-time updating of five crucial parameters with high sensitivities are almost the same as those obtained by real-time updating of all the parameters (Fig. 8(a)). Fig. 8(b) shows the SOC error based on ECM (4). We can see that the SOC accuracy is lower than that of ECM (5), indicating that H is a critical parameter for the LFP battery. As shown in Fig. 8(c), when two crucial model parameters ( R1 and H ) are not updated, the SOC estimation accuracy obviously reduces. Fig. 8(d) shows that the SOC estimation accuracy is poor when only RCha and RDch are updated with the SOH.

21

Journal Pre-proof

2

 ka  H  1  2

   

0 -2 -4

Cell #2@SOH=92.7% Low SOC

High SOC

6 4 2

RCha  ka  ECM(3) ECM(8) RDch  H  Reference R  1  1 R2

10

 2 

-4

Cell #2@SOH=92.7% High SOC

Low SOC

Time(s)

10

-2

4 2

4

 ka  H  1  2

   

(b)

Cell #2@SOH=92.7% Low SOC

High SOC

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time(s) (d)

RCha RDch R1 R2

 ka  H  1  2

   

104 ECM(2) ECM(8) Reference

0 -2 -4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

RCha RDch R1 R2

0

6

(c)

ECM(4) ECM(8) Reference

2

4

0 -2

4

-4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time(s)

SOC-error(%)

6

(a)

SOC-error(%)

4

RCha RDch R1 R2

ECM(5) ECM(8) Reference

SOC-error(%)

SOC-error(%)

6

Cell #2@SOH=92.7% High SOC

Low SOC

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time(s)

104

Fig. 8. SOC estimation results of Cell #2 (SOH=92.7%) based on the simplified 2RCH with different updating parameters. Fig. 9 shows the SOC estimation results of Cell #2 under different SOHs over the whole SOC range. Here, the symbol “” denotes that the labeled parameters are updated with the SOH over the whole SOC range, and the symbol “” denotes that the model parameters do not change with the SOH; that is, their values remain the same as those of the model parameters of a fresh battery. It can be seen that in order to address the model parameter mismatch caused by battery aging, as long as the five high-sensitivity parameters of the LFP battery ( RDch , RCha , H , R1 , and R2 ) are updated with the SOH in real time, high SOC estimation accuracy can be achieved. Fig. 9(d) shows that the SOC estimation accuracy decreases significantly when the most crucial parameter ( H ) is not updated in real time.

22

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 ka  H  1  2

(a)

   

0 -2 -4

Cell #2@SOH=96.1% High SOC

0

0.5

1

1.5

2

Low SOC

2.5

3

3.5

SOC-error(%)

2

ECM(5) ECM(8) Reference

RCha RDch R1 R2

 ka  H  1  2

   

-4

High SOC

-2

Low SOC

10

(b)

   

Cell #2@SOH=94.5% High SOC

0

4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time(s)

 ka  H  1  2

0.5

1

1.5

2

Low SOC

2.5

3

3.5

4

2

ECM(4) ECM(8) Reference

RCha RDch R1 R2

 ka  H  1  2

4

4.5 10 4

Time(s)

(c)

Cell #2@SOH=93.7%

RCha RDch R1 R2

0

-4

4.5

0 -2

2

10 4

Time(s)

4

4

ECM(5) ECM(8) Reference

4

SOC-error(%)

SOC-error(%)

2

RCha RDch R1 R2

SOC-error(%)

ECM(5) ECM(8) Reference

4

   

(d)

0 -2 -4

Cell #2@SOH=93.7% High SOC

Low SOC

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time(s)

104

Fig. 9. SOC estimation results of Cell #2 under different SOHs over the whole SOC range. From the above results, we can draw the following conclusions: (1) As long as the crucial model parameters with high sensitivity to SOC distribution are updated in real time, accurate SOC over the whole SOC range is available. (2) For SOC estimation under different SOHs, high SOC estimation accuracy can be achieved by updating the model parameters with high sensitivity to battery aging. These valuable conclusions are helpful to improve the robustness of the ECM and simplify the state estimation of lithium-ion batteries over the whole life cycle. 5. Conclusion and future work In this work, parameter sensitivity studies are carried out to determine the crucial parameters of the ECM, following which the ECM is simplified for SOC estimation under different SOHs over the whole SOC range. Based on the parameter sensitivity of two types of batteries, several conclusions can be drawn, as follows: (1) For 2RCH, there are four crucial parameters with high sensitivities for NCM batteries, the ranking from high to low sensitivity being as follows: RCha , RDch , R1 , and R2 . Moreover, the sensitivity of ka , H ,  1 , and  2 is very low. There are five crucial parameters in 2RCH for LFP batteries, which are ranked from high to low sensitivity as follows: RDch , RCha , H , R1 , and R2 , 23

Journal Pre-proof and sensitivity of other parameters ( ka ,  1 , and  2 ) is very low. (2) The SA results are used to simplify the whole life cycle model for 2RCH. In the simplified model, the crucial parameters with high sensitivities update the values with the SOH and SOC at high frequencies to ensure high model accuracy, while the other parameters retain their initial values to reduce model complexity. Specifically, for the simplified model for estimating SOC over the whole SOC range, RCha , RDch , R1 , and R2 in 2RCH are updated at a certain frequency (10% SOC in this study) and other parameters remain unchanged for NCM batteries, and RDch , RCha , H , R1 , and R2 are updated and other parameters remain unchanged for LFP batteries. (3) An SOC estimator based on the simplified 2RCH model is designed to estimate the SOC under different SOHs over the whole SOC range. The experimental results show that the SOC estimation accuracy for updating the crucial model parameters is almost the same as that for updating all the parameters over the whole SOC range. Moreover, with the aging of batteries, satisfactory accuracy can be achieved based on the simplified model. As analyzed in Section 2, 2RCH is the best choice to balance the accuracy and complexity of the model for the NCM and LFP cells. Therefore, the above conclusions can be used to simplify the ECMs of the above two types of LIBs for SOC estimation with strong adaptability. Furthermore, the proposed method and the research process also provide valuable reference for the SA of other models. Further work includes: 1) investigating the global sensitivity of the model parameters in ECMs; 2) determining the relationship between temperature and sensitivity of model parameters. Acknowledgment This work is supported by the National Natural Science Foundation of China (Grant Nos. 51977131 and 51877138), the Natural Science Foundation of Shanghai (Grant No. 19ZR1435800) and Shanghai Science and Technology Development Fund (Grant No. 19QA1406200). References [1] A. Fotouhi, D.J. Auger, K. Propp, S. Longo, R. Purkayastha, L. O'Neill, S. Walus, Ieee T Veh Technol, 66 (2017) 7711-7721. [2] J.F. Yang, B. Xia, Y.L. Shang, W.X. Huang, C.T.C. Mi, Ieee T Veh Technol, 66 (2017) 10889-10898. [3] J.F. Yang, W.X. Huang, B. Xia, C. Mi, Appl Energ, 237 (2019) 682-694. [4] C. Lyu, Y.K. Song, J. Zheng, W.L. Luo, G. Hinds, J.F. Li, L.X. Wang, Appl Energ, 250 (2019) 685-696. [5] X. Lai, W.K. Gao, Y.J. Zheng, M.G. Ouyang, J.Q. Li, X.B. Han, L. Zhou, Electrochim Acta, 295 (2019) 24

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: