Lithium-ion battery temperature on-line estimation based on fast impedance calculation

Journal of Energy Storage 26 (2019) 100952

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Lithium-ion battery temperature on-line estimation based on fast impedance calculation Wang Xueyuana,b, Wei Xuezheb, Chen Qijuna, Zhu Jiangongc, Dai Haifengb,

T

⁎

a

Department of Control Science and Engineering, Tongji University, 4800 Caoan Road, Shanghai, China Clean Energy Automotive Engineering Center, Tongji University, 4800 Caoan Road, Shanghai, China c Institute for Applied Materials, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany b

A R T I C LE I N FO

A B S T R A C T

Keywords: Lithium-ion battery Wavelet transform Impedance Temperature Online

Temperature affects lithium-ion battery performance, health and safety. Therefore, obtaining temperature is quite necessary for battery management system. Impedance is widely adopted and proved to be feasible to estimate temperature. In the paper, wavelet transform is applied to process the step current and the response voltage of the battery cell to obtain the impedance. And to reduce the computational complexity and process the signals automatically for online application, the basic method is improved. Considering that the step current used is a kind of non-steady perturbation and it may change the operating points of the electrochemical system insider the battery, a series of tests are performed to study the effect of the changing step current conditions on the impedance calculated. The results show that the impedance calculated at low frequencies with the method is always different from electrochemical impedance spectroscopy. The impedance in the mid-high/high frequency range is nearly not affected and provide guidance to use the impedance to estimate the temperature. With the battery impedance phase at 10 Hz calculated online, the temperature is finally estimated. The average error is 0.33 °C during step current test and max error 0.5 °C during NEDC test, showing a feasibility of the fast impedance calculation method in the battery temperature online estimation.

1. Introduction The e-mobility development promotes the wide application of lithium-ion batteries. As a basic monitoring object in the lithium-ion battery management system (BMS), temperature not only affects the battery performance and life, but also may be one of the causes of safety problems in some extreme cases, e.g. thermal runaway [1–3]. Temperature measurement of the battery by a thermocouple is the most straightforward method. However, considering the large number of battery cells inside the battery pack, temperature monitoring of every battery cell leads to high cost and system complexity. Therefore, the low-cost temperature estimation methods without sensors become an attractive solution. They mainly include the electrothermal-couplingmodel-based and the impedance-based methods [4–6]. The latter is extensively studied for it seldomly relies on the battery parameters. Srinivasan et al. [7,8] studied the relationship between the temperature and the impedance phase in 40 Hz–100 Hz of a 53 Ah LiCoO2 battery, a 2.3 Ah LiFePO4 battery and a 4.4 Ah battery with unknown material respectively. It is found that the absolute value of the impedance phase changes monotonously with the temperature from

⁎

−20 °C to 66 °C. And the temperature was finally estimated with this kind of relationship. In the work of Zhu et al. [9,10], a similar conclusion is drawn. Their further study indicates that the relationship between the temperature and the impedance phase in 1 Hz–100 Hz of an 8 Ah prismatic LiFePO4 battery is little affected by the changing state of charge (SOC) and state of health (SOH). They finally estimated the battery temperature with the impedance phase at 1 Hz, 5 Hz, 10 Hz, 50 Hz and 100 Hz. Schmidt et al. [11] studied the relationship between the temperature and the impedance magnitude at 10.3 kHz of a 2 Ah pouch battery with LiCoO2 and LiNi0.8Co0.15Al0.05O2 as the cathode material. And a method to estimate the temperature based on the impedance magnitude is proposed. The experimental results indicate that the estimation error is less than ± 0.7 °C with SOC known and ± 2.5 °C without SOC known. Raijmakers et al. [12] studied the impedance spectra of a 2.3 Ah LiFePO4 and a 7.5 Ah Li(NCA)O2 battery cell at different temperature. They obtained and verified the relationship between the temperature and the frequency corresponding to the intersection of the spectra on the real axis at different SOC and SOH. Then a temperature estimation method based on the intersection frequency was proposed. Spinner et al. [13] studied the relationship between the

Corresponding author E-mail address: [email protected] (H. Dai).

https://doi.org/10.1016/j.est.2019.100952 Received 22 July 2019; Received in revised form 11 September 2019; Accepted 11 September 2019 2352-152X/ © 2019 Elsevier Ltd. All rights reserved.

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the experiments. And Section 4 is the results and discussion. Section 5 is the conclusion.

temperature and the imaginary part of the impedance at 300 Hz of a 18,650 typed LiCoO2 battery. They improved the upper limit of the temperature estimated with the impedance from 65 °C to 95 °C, facilitating the application of the temperature estimation method for high temperature caused safety issues. Beelen et al. [14] concluded the aforementioned temperature methods and developed a comprehensive temperature estimation method for a 40 Ah LiFePO4 battery. It shows good accuracy and the average estimation error is less than 0.4 °C without SOC known. It is seen that the impedance becomes a powerful tool to estimate temperature. Before the implementation of this kind of method, on-line acquisition of the impedance is quite essential. At present, many methods for estimating and measuring the battery impedance on line are proposed. Impedance estimation methods are mainly based on the battery models, e.g. equivalent circuit models (ECMs). Recursive least square method, Kalman filter and other observers are applied for the identification of the model parameters to calculate the impedance [15–17]. The accuracy of this kind of methods is limited by the models and the parameter identification algorithms used. The simple and linear models have drawbacks to describe the battery voltage and the current characteristic while too complex and nonlinear models have the difficulties with the parameter identification [18]. In order to obtain the battery impedance credibly, the direct impedance measurement is widely accepted and studied. Applying an AC sinusoidal signal to disturb the battery to measure the impedance is a most common method, especially in laboratories. For the on-line temperature estimation of the battery, one of the biggest concerns is the cost of a well-designed AC signal generator which is needed for the on-board impedance measurement. And also, it takes a long time to measure the impedance in a wide frequency range for the measurement time increases when the impedance at lower frequency is measured. To obtain the battery impedance quickly, the multiple frequency sine, square wave, pseudorandom sequence, random noise and chirp signals are applied to disturb the battery cell as they contain rich harmonic components [19–24]. Fourier transform is frequently used to process the voltage and the current signals to obtain the impedance. However, Fourier transform has inherent disadvantages in time-frequency analysis for the non-stationary signals whose frequency changes with the time. As an alternative method, Hoshi et al. used the method of wavelet transform to process the signals of a step current and its response voltage to obtain the battery impedance. The method provides the battery impedance rapidly in 0.1 Hz–100 Hz with good consistence with Electrochemical Impedance Spectroscopy (EIS) [25,26]. However, special attention should be paid to the applications of the method. The step current is different from the sinusoidal signal. It may change the operating point of the electrode system and cause large error to the wideband impedance measurement results. Therefore, the effect of the step current conditions on the impedance should be carefully studied before the application. Besides, the analyzing moment, i.e. the jumping moment of the step current needs to be manually determined for the impedance calculation method proposed in Hoshi et al.’s work. It is nearly impossible to set the jumping moment manually for online applications. And the direct application of wavelet transform needs also a large amount of calculation as the convolution is used. For the impedance-based on-line temperature estimation, there should be proper solutions for the concerns, which are also the main contribution of the paper. As the relationship between temperature and impedance has been proved and discussed in our previous work [9], the following aspects will be mainly studied based on it in the paper. The first is to calculate the impedance online. The second is to study the effect of the working conditions on the impedance calculated. And the third is to estimate the temperature with the relationship obtained offline and the impedance calculated online. Therefore, the remainder of the paper is arranged as follows. Section 2 is the methodology with the impedance calculation and the temperature estimation method presented. Section 3 is about

2. Methodology 2.1. Basic method for impedance calculation For the impedance calculation under the unsteady operating condition, a complex Morlet wavelet is applied to obtain the real and the imaginary part of the impedance [25]. The Morlet mother wavelet is a product of a Gaussian function g(t) and a sine term h(t) expressed as Eq. (1).

t − b⎞ ψa, b (t ) = ψ ⎛ = ⎝ a ⎠

(t − b)2 ⎞ 1 ⎛ 2πfc (t − b) ⎞ exp ⎜⎛− ⎟ exp j 2f a a πfb ⎠ ⎝ b ⎠ ⎝ ⎜

⎟

(1)

g(t) and h(t) are defined as Eqs. (2) and (3) respectively.

1 t2 exp ⎜⎛− ⎞⎟ πfb ⎝ fb ⎠

g (t ) =

(2)

h (t ) = exp(j2πfc t )

(3)

For a time-domain signal x(t), the wavelet coefficient WTx(a,b) is calculated with the convolution between x(t) and the conjugate of the mother wavelet ψ*a,b(t) as is expressed in Eq. (4).

WTx (a, b) = x (t )*ψa*, b (t ) =

1 a

+∞

(t − b)2 ⎞ 1 ⎛ 2πfc (t − b) ⎞ ⎞ exp ⎜⎛− ⎟ exp j ⎟ a2fb ⎠ a πf ⎠⎠ ⎝ b ⎝ ⎝ (4)

∫ x (t )conj ⎛⎜

−∞

dt

⎜

⎟

a and fc determine the ability of the wavelet transform to extract the signal at the specific frequency. fb affects the time resolution of the signal during the time-frequency analysis by affecting the distribution width range of the Gaussian window. And b determines the specific time when the wavelet transform is performed in a sampled signal sequence. Different frequency components of the signal at different time can be analyzed by changing the value of a and b. Here, the parameters fb and fc are selected in accordance with the principle of minimizing the Shannon entropy of the wavelet coefficients. The minimum Shannon entropy means that the wavelet coefficients obtained by the combination of parameters at this time can provide a larger amount of information, that is, the ability to extract signals is strongest [27,28]. That is, fb and fc need to make the following Eq. (5) hold. M is the total number of wavelet coefficients obtained by combining fb and fc, and di is the probability of taking the ith wavelet coefficient, as shown in Eq. (6). M

min(En(X )) = − ∑ di log di i=1

di =

(5)

W (ai , b) M

∑i = 1 W (ai , b)

(6)

Wavelet transform is suitable for the process of the non-stationary signals like the voltage and the current signals of the battery during the dynamic operation. To obtain the battery impedance, the current i(t) injected into the battery and the response voltage u(t) at the battery terminals are sampled and the wavelet transform is performed on the signals. The integrated area within the ± 3σ already occupies 99.7% of the total area of the Gaussian window as Eq. (2). Therefore, we choose up to ± 4σ as the integration interval to keep the accuracy and simplify the computation. Here, we have σ2 = fb/2. The wavelet coefficients of the voltage and the current signal are calculated with Eqs. (7) and (8). 2

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U (a, b) =

1 a

b + 4σ

(t − b)2 ⎞ ⎛ 1 ⎛ 2πfc (t − b) ⎞ ⎞ dt exp ⎛⎜− u (t )conj ⎜ ⎟ exp j 2f ⎟ a a πf ⎝ ⎠⎠ b b ⎝ ⎠ b − 4σ ⎝

∫

⎜

⎟

(7)

I (a, b) =

1 a

b + 4σ

(t − b)2 ⎞ ⎛ 1 ⎛ 2πfc (t − b) ⎞ ⎞ dt i (t )conj ⎜ exp ⎛⎜− ⎟ exp j 2f ⎟ a a πf ⎝ ⎠⎠ b b ⎠ ⎝ b − 4σ ⎝ (8)

∫

⎜

⎟

Fig. 1. Improved impedance calculation method.

For the step current, the harmonic components are most significant at the jumping moment. Analyzing the signal at the moment obtains the high signal-to-noise ratio. The value of b is set at the moment manually. And then the harmonic components of different frequencies are extracted by changing the value of a. The obtained wavelet coefficients U (a,b) and I(a,b) are complex numbers. Then the battery impedance Z (a,b) is calculated with Eq. (9).

Z (a, b) =

U (a, b) I (a, b)

(9)

2.2. Improved method for online impedance calculation The wavelet transform method proposed above need to manually set the parameter b to perform the time-frequency analysis at a specific moment. In order to ensure the real-time impedance calculation, it is obviously impossible to set the analysis time manually. In other words, the algorithm needs to analyze and process the signals automatically. Besides, the Morlet wavelet adopted in this paper is such a kind of nonorthogonal wavelet that the mature Mallat algorithm [29,30] cannot be applied to accelerate the calculation. Therefore, it is necessary to design a fast, simple and practical algorithm for online impedance calculation. Here, ψ′(t) is introduced and it has the relationship with ψ*(t) described as Eq. (10).

−t 1 ψ* ⎛ ⎞ ψ′ (t ) = a ⎝ a ⎠

Fig. 2. Flow chart of online temperature estimation method, T is the battery temperature, θ is state of charge, φ is the impedance phase used to estimate temperature.

(10)

2.3. Method to estimate temperature

For the wavelet transform described as Eq. (4), we apply fast Fourier transform (FFT) to both sides of the equation and obtain Eq. (11).

 (WTx (a, b)) =  ⎜⎛ ⎝

1 a

+∞

After the calculation of the battery impedance, the flow of the method for estimating the temperature online is shown in Fig. 2. Firstly, the relationship between the battery impedance phase (φ) and the state (i.e. temperature T and state of charge θ) is obtained by off-line EIS test. Thereafter, the impedance suitable for estimating the battery temperature is confirmed, and a function relationship between the temperature and the impedance phase is established. Before using this functional relationship to estimate the battery temperature, the effect of dynamic conditions on the impedance calculated online is studied, and the impedance that is not sensitive to the working conditions is concluded. Then, the impedance phase at the selected specific frequency is calculated using the online impedance calculation method. And the relationship T = f(φ) is finally applied to estimate the battery temperature.

+∞

dt ⎟⎞ =  ⎜⎛ ∫ x (t ) ψ′ (b − t )dt ⎟⎞ ∫ x (t ) ψ* −∞ ⎠ ⎝−∞ ⎠ = X (ω)Ψ′(ω) = a X (ω)Ψ*(aω)

( ) t−b a

(11) Here, X(ω), Ψ′(ω) and Ψ*(ω) is the Fourier coefficients of x(t), ψ′(t) and ψ(t) respectively. After performing inverse Fourier transform (IFT) on Eq. (11), the wavelet coefficients are obtained and the impedance is calculated with Eq. (9). The process is similar to the windowed Fourier transform. However, by adjusting the value of parameter a, the above process acts as the bandpass filtering and only retains the harmonic components of interest. This kind of method to calculate the wavelet transform coefficient has the following advantages compared to the direct use of convolution. (1) The method makes full use of the advantages of fast Fourier transform to speed up the calculation. (2) The parameter b does not need to be manually determined and only the scale parameter a needs to be discretized according to the harmonic of specific frequency. (3) After the Fourier transform is performed, the harmonic component is detected and localized, then the wavelet transform is applied at the moment on the harmonic component. The improved calculation method is shown in Fig. 1.

3. Experiments A power type and an energy type LiFePO4 battery cells are used to conduct the experiments described in this section. The detailed specification about the battery cells is listed in Table 1. The test bench shown in Fig. 3 is used to measure the voltage and the current during the charge and the discharge processes to calculate the impedance. The battery charge/discharge test system Chroma 17,011 is suitable for the 8 Ah battery test with an output current range 3

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Table 3. Every time after the battery cells are discharged, they are charged again for the same time to keep SOC unchanged.

Table 1 Detailed specification of the two battery cells. Designation

Cell1

Cell2

Category Nominal capacity (Ah) Nominal voltage (V) Size (mm) Weight (kg) Package

power 8 3.2 V 18 × 80 × 124 0.33 prismatic

energy 40 3.2 V 133 × 16.1 × 184 0.92 pouch

3.2.2. Charge and discharge with step current of different amplitude To study the difference of the battery impedance obtained with the step charge or discharge current, an experiment with the detailed steps shown in Table 4 is designed. The step current with different amplitude is applied to the terminals of the battery cells after resting for 1.5 h. Notably, it is different that the battery cells are charged or discharged immediately after resting for 1.5 h. And the distinction is also considered in the following experimental steps.

3.2.3. Discharge with step current after different rest time The traditional EIS test is performed after sufficient relaxation. That is, the battery is at an electrochemically steady state. In practical applications, there is not long enough time for the battery to rest, especially when the battery is in dynamic operation. In order to study the effect of different rest time on the impedance calculated with the step current, an experiment described in Table 5 is performed. The battery rest time is adjusted by changing the time interval trest. With the proposed fast impedance calculation method, the impedance in a wide frequency range is obtained immediately after the short rest. The detailed steps are shown in Table 5. Fig. 3. Test bench for the impedance calculation with wavelet transform under different step current conditions.

3.2.4. Charge and discharge with step current at different temperature Temperature has significant effect on the electrochemical and the physical processes inside the battery cell. An experiment shown in Table 6 is performed to study the effect of different temperature on the impedance calculated with the step current. The purpose of the experiment is also to provide guidance for the temperature estimation with the impedance calculated online.

of ± 20 A. The Arbin BT2000 has a maximum output current range of ± 150 A and is used for 40 Ah battery test. And EIS of the battery cells is measured with an electrochemical workstation composed of a frequency response analyzer and a potentiostat-galvanostat typed with Solartron 1255B and 1287 respectively. The amplitude of the sweeping sinusoidal current is 1.500 A. A shunt of 75 mV/75 A, 1 mΩ ± 0.5% is connected in series with the battery cell to convert the charge/discharge current to a voltage signal. The terminal voltage signals of the battery cell and the shunt are measured with the data acquisition card NI USB-6210 with the sampling rate at 10 kHz. During the test, the battery temperature is controlled by a thermotank.

3.3. Experiment for impedance calculation and temperature estimation Step current test sequence shown in Fig. 4(a) and NEDC test sequence in Fig. 4(b) is injected into the 40 Ah battery. The step current test is performed under 30% SOC at 25 °C. And the NEDC test is performed under 90% SOC at 25 °C. Then the impedance is calculated with the proposed improved method after the acquisition of the current and the voltage signals. In order to measure the internal and the average temperature during operation to facilitate the subsequent studies on the temperature estimation, 15 T-type thermocouples with an outer diameter of about 0.5 mm are placed inside and mounted on the surface of the 40 Ah battery cell. The placement of the thermocouples is shown in Fig. 5. The thermocouples No. 1–10 are attached to the surface of the battery cell and No. 11–15 are placed inside it. In order to place the thermocouples inside the battery, the pouch battery cell is opened in an inert gasprotected glove box firstly, then the thermocouples are attached to the predetermined points on the separator at different layers, and finally the pouch cell is sealed and the electrolyte is poured into it.

3.1. Experiment to measure electrochemical impedance spectroscopy The EIS test at different temperature and under different SOC has two main purposes. One is to obtain the impedance spectra under the quasi-steady state of the battery and compare them with the spectra calculated by the wavelet transform under the unsteady state. The other one is to obtain the relationship between the temperature and the impedance. The discharge capacity of the used battery cells is measured at 25 °C previously, which is not listed in the table. And SOC is adjusted according to the measured capacity. The detailed steps are shown in the following Table 2. 3.2. Experiment to study effect of step current on impedance 3.2.1. Discharge with step current of different amplitude In order to study the effect of the step current amplitude on the impedance, the experiment is designed and the steps are shown in Table 2 Steps for EIS test at different temperature and under different SOC. Step Step Step Step Step

1 2 3 4 5

Adjust SOC of the battery cell to 90%; Adjust the thermotank to −5 °C, place the battery cell in it and rest for 1.5 h; Measure EIS of the battery cell; Adjust the thermotank to 0 °C, 5 °C, 10 °C, 15 °C, 20 °C, 25 °C, 30 °C, 35 °C, 40 °C and 45 °C, rest for 1.5 h and repeat Step 3 at different temperature respectively; Adjust SOC of the battery cell to 70%, 50%, 30% and 10%, repeat Step 2–4.

4

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Table 3 Steps to study the effect of different step current amplitude on impedance calculation results. Step 1 Step 2 Step 3

Cell 1 Cell 2

Step 4

Cell 1 Cell 2

Adjust SOC of the battery cell to 50%; Rest at 25 °C for 1.5 h; Discharge the battery cell with a step current of 1 A for 50 s and charge it with a step current of 1 A for another 50 s, simultaneously record the terminal voltage and the current; Discharge the battery cell with a step current of 5 A for 50 s and charge it with a step current of 5 A for another 50 s, simultaneously record the terminal voltage and the current; Change the step current amplitude to 2A, 3A, 4A, 5A, 10A,15A and 20A, and repeat Step 2–3; Change the step current amplitude to 10 A, 20 A, 30 A, 40 A, 50 A and 60 A, and repeat Step 2–3.

Table 4 Steps to study the effect of charging and discharging mode on impedance calculation results. Step 1 Step 2 Step 3

Cell 1 Cell 2

Step 4 Step 5

Cell 1 Cell 2

Step 6

Cell 1 Cell 2

Adjust SOC of the battery cell to 50%; Rest at 25 °C for 1.5 h; Charge the battery cell with a step current of 2 A for 50 s and discharge it with a step current of 2 A for another 50 s, simultaneously record the terminal voltage and the current; Charge the battery cell with a step current of 5 A for 50 s and discharge it with a step current of 5 A for another 50 s, simultaneously record the terminal voltage and the current; Rest at 25 °C for 1.5 h; Discharge the battery cell with a step current of 2 A for 50 s and charge it with a step current of 2 A for another 50 s, simultaneously record the terminal voltage and the current; Discharge the battery cell with a step current of 5 A for 50 s and charge it with a step current of 5A for another 50 s, simultaneously record the terminal voltage and the current; Change the step current amplitude to 5 A, 10 A, 15 A and 20 A, repeat Step 2–5; Change the step current amplitude to 10 A, 15 A and 20 A, repeat Step 2–5.

Table 5 Steps to study the effect of rest time on impedance calculation results. Current profile

Step 1 Step 2 Step 3

Cell 1 Cell 2

Adjust SOC of the battery cell to 50%; Adjust the thermotank to 25 °C, place the battery cell in it and rest for 3 h; Set i1 = 5A, i2 = 16A, Charge/discharge the battery cell with trest = 1 s, 5 s, 10 s, 30 s, 60 s, 300 s, respectively, simultaneously record the terminal voltage and the current; Set i1 = 5A, i2 = 40A, Charge/discharge the battery cell with trest = 1 s, 5 s, 10 s, 30 s, 60 s, 300 s respectively, simultaneously record the terminal voltage and the current.

4. Results and discussion

relationship between the battery temperature and the impedance phase in the frequency range, which is consistent with the research results of Zhu et al. To further study the relationship, the battery impedance phase at 10 Hz at different temperature is plotted in Fig. 7. Here, an exponential function expressed as Eq. (12) is applied to fit the 10 Hz impedance phase θ and the average temperature T. The fitting result is indicated as the red line in Fig. 7. The coefficient of determination is R2 = 0.9976, indicating a good fitting result.

4.1. Relationship between battery impedance phase and average temperature Zhu et al. found that there is a monotonous relationship between the impedance phase at mid-high frequencies and the average temperature of LiFePO4 battery after a large number of tests. And the relationship is not affected by the battery SOC and the aging state. They proposed a method to estimate the average battery temperature based on the battery impedance phase [31]. For the selected battery cell in the paper, EIS of the 40 Ah battery cell is measured under different state as is shown in Fig. 6. The impedance phase is sensitive to temperature change but not to SOC. Especially in the frequency range from 1 Hz to 100 Hz, SOC has nearly no effect on the impedance phase. And it shows a monotonous

(

− θ − 205 36.3100

T = 3.1390 × 1015e

2

)

(

− θ + 1.8840 24.6900

+ 23.4700e

2

)

(12)

In theory, it is only necessary to measure the battery impedance phase at 10 Hz, and then the temperature can be estimated by substituting into Eq. (12). Existing research does do this, and the vast majority are based on quasi-steady state EIS. However, considering the

Table 6 Steps to study the effect of temperature on impedance calculation results. Step 1 Step 2 Step 3 Step 4

Cell 1 Cell 2

Adjust SOC of the battery cell to 50%; Rest at 35 °C for 1.5 h; Charge/discharge with a step current of 1 A, 2 A, 10 A and 20 A, respectively, the detailed steps are referred to Step 3–5 in Table 4; Charge/discharge with a step current of 5 A, 10 A, 15 A and 20 A, respectively, the detailed steps are referred to Step 3–5 in Table 4; Adjust the thermo tank to 15 °C and rest for 1.5 h, repeat Step 3.

5

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Fig. 4. Current profile injected into the battery cell, (a) step current test sequence; (b) NEDC test sequence.

measurement/calculation of the battery impedance phase online, it is always affected by many factors, especially when calculating the impedance phase using dynamic conditions. The effect of the working conditions on the impedance calculation result should be considered. Therefore, we will study the effect of the variation of the operating conditions on the calculated impedance phase in the following. 4.2. Effect of step current conditions on impedance calculation results According to the experiments described in Section 3 and with the recorded current and voltage signals, the battery impedance under different step current conditions is calculated with the basic method proposed in Section 2. EIS of the two kinds of battery cells measured with electrochemical workstation is also plotted for comparison to show the effect of the working conditions on the impedance. Fig. 6. Electrochemical impedance spectroscopy of the 40 Ah LiFePO4 battery at 5 °C, 15 °C, 25 °C, 35 °C, 45 °C and under 90%, 70%, 50%, 30%, 10% SOC.

4.2.1. Effect of different step current amplitude The results are shown in Fig. 8. For the 8 Ah and the 40 Ah battery cells, the maximum current reaches 2.5 C and 1.5 C respectively. It is indicated that the impedance in the low frequency range lower than 0.6310 Hz gradually becomes smaller as the amplitude of the step discharging current increases. Meanwhile, the low-frequency tail of the impedance spectra is also bent toward the real axis, which indicates that the absolute value of the impedance phase in the low frequency range becomes smaller. The arc of the impedance spectra in the midhigh or high frequency range higher than 0.6310 Hz is less affected as the amplitude of the current changes. The inductive of the spectra in the high frequency range show a phenomenon of bifurcation. It is considered to be caused by the low signal-to-noise ratio due to the small response voltage. And the results show that a wideband impedance spectrum closer to EIS can be obtained by reducing the step current amplitude. Without considering the inductive part of the impedance spectra, the amplitude of the step current applied in the tests produces nearly no effect on the impedance in the frequency range higher than 0.6310 Hz for the two kinds of battery cells.

Fig. 7. Fitting of the relationship between the impedance phase at 10 Hz and the average temperature.

4.2.2. Effect of charge and discharge with step current It is seen in Fig. 9 that the effect of the charge and the discharge with step current on the impedance calculated is mainly reflected in low

Fig. 5. Placement of the thermocouples on the surface of and inside the 40 Ah battery cell. 6

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Fig. 8. Impedance spectra of the two types of battery cells calculated using the wavelet transform with the step current of different amplitude, (a) comparision of the impedance calculation results of 8 Ah battery with EIS; (b) comparision of the impedance calculation results of 40 Ah battery with EIS.

the impedance calculation results is weakened; (2) the difference between the impedance calculated by the charge and the discharge step current becomes less significant. The impedance in the frequency range higher than 1 Hz for the 8 Ah battery and 0.6310 Hz for the 40 Ah battery is not affected by the step current conditions when the temperature changes from 15 °C to 35 °C.

frequency range, i.e. the charge transfer and the diffusion processes. The impedance obtained under the charge step current is a little larger than that with the discharge current, especially for the 8 Ah battery cell as is shown in Fig. 9. And the difference is not so significant compared to that caused by the changing step amplitude. Fig. 9 shows that the impedance in the mid-high frequency range is almost the same though the direction of the step current changes from discharge to charge.

4.2.5. Discussion on effect of step current conditions The effect of the step current conditions on the impedance can be explained from the electrochemical and physical mechanism of the battery. The decrease of the impedance in the mid-low frequency range as the current amplitude increases is consistent with the work of Zhu et al. [32] and Huang et al. [33,34]. They all explain the decrease originally from the Butler–Volmer equation which basically describes the reaction dynamic on the interface of the solid particles in the electrodes. It is seen that the charge transfer resistance on the interface depends on the reaction current. And the larger the current is, the smaller the resistance becomes. It finally decreases the impedance in the mid-low frequency range as the current is enlarged. The difference of the calculated impedance in the mid-low frequency range under the step charge or discharge current reflects the asymmetry of the extraction and the intercalation of lithium ions at the positive and the negative electrodes. The results that the impedance during charge is a little larger than that during discharge are similarly presented in the work of Huang et al. [33,35] and Itagaki et al. [36,37].

4.2.3. Effect of different rest time Fig. 10 shows the impedance spectra of the two kinds of batteries calculated with the step current after different rest time. The rest time is from 1 s to 300 s. And the discharge current before the rest is 2 C and 1 C for the 8 Ah and the 40 Ah battery cells respectively. It is indicated that as the rest time is gradually shortened, the impedance in the frequency range lower than 0.6310 Hz becomes smaller. The effect makes the tail in the low frequency range be bent to the real axis. As the shortening continues, even the impedance in the mid-low frequency range from 0.6310 Hz–7.9433 Hz is affected. After the rest time longer than 5 s, the impedance in the mid-high frequency range higher than 7.9433 Hz for both the battery cells is little affected. 4.2.4. Effect of different temperature Fig. 11 shows the calculated impedance spectra of the two battery cells with different step current at 15 °C and 35 °C. It shows that as the temperature rises, (1) the effect of the large step current amplitude on

Fig. 9. Impedance spectra of two types of battery cells calculated with step charge (positive) and discharge (negative) current, (a) impedance calculation results of 8 Ah battery; (b) impedance calculation results of 40 Ah battery. 7

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Fig. 10. Impedance spectra of two types of batteries calculated with step current after different rest time, (a) impedance calculation results of 8 Ah battery; (b) impedance calculation results of 40 Ah battery.

resistance becomes smaller when the surface concentration is higher. As a result, the diffusion impedance decreases. However, as the rest time is prolonged, lithium ions gradually diffuse into the solid phase particles leading to lower lithium ion surface concentration. Therefore, the diffusion impedance becomes bigger again and remains unaffected finally. From the above experimental results, it is found that the impedance calculated online is affected by the working conditions of the battery, so temperature estimation using the relationship between the battery temperature and the 10 Hz impedance phase obtained in Section 4.1 is

When the battery is charged, the extraction of lithium ions from the solid particles is more difficult than the intercalation during discharge due to the concentration difference of the reactant. Thus, the impedance calculated during charge is bigger. The impedance calculation results after different rest time are also consistent with the work of Barai et al. [38]. Due to the slower diffusion rate of lithium ions inside the solid phase particles, a higher concentration is formed on the surface of the positive electrode solid phase particles at the end of the discharge process. The charge transfer

Fig. 11. Impedance spectra of two types of batteries calculated with different charge (positive) and discharge (negative) current at different temperature, (a)(b) impedance calculation results of 8 Ah battery at 15 °C and 35 °C; (c)(d) impedance calculation results of 40 Ah battery at 15 °C and 35 °C. 8

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Fig. 12. The sampled signals and the obtained wavelet coefficient, (a) a sequence of step current and its response voltage; (b) wavelet coefficient at 10 Hz of the current signal recognition results.

4.4. Validation of average temperature estimation based on online impedance calculation

not directly applicable to all working conditions. For the 40 Ah battery, the impedance phase calculated at least for a current amplitude of less than 20A, a rest time of more than 1 s, and a temperature range of 15 °C to 35 °C is believed to be reliable. The calculated impedance phase can be directly substituted into Eq. (12) to perform online temperature estimation of the battery. When the working conditions of the battery are outside these ranges, compensation or filtering is required to achieve an accurate temperature estimation based on online calculated impedance.

4.4.1. Temperature estimation during step current test The battery cell is continuously loaded with the dynamic current profile shown in Fig. 4(a). The temperature at different points is measured with the thermocouples embedded inside or mounted on the battery cell. The whole test process lasts for 1200 s. To make the temperature change more noticeable in the process, the thermotank temperature is set to 35 °C from about 15 °C immediately after the beginning of the test. As the test progresses, the temperature of the battery cell is continuously changing. The temperature measured with the thermal couples during the test is shown in Fig. 14. The orange line is the average temperature of the battery cell. During the test, the average temperature increases by about 10 °C, from 15.8 °C to 25.8 °C. Using the proposed on-line impedance calculation method based on wavelet transform, the impedance phase at 10 Hz is obtained as shown in Fig. 15. The measured and estimated temperature are shown in Fig. 16(a). The blue line shows the average temperature measured with the 15 thermocouples on the surface of and inside the battery cell. And the red line shows the average temperature estimated using the relationship between the impedance phase and temperature as Eq. (12). It is shown that the average battery temperature estimated using the calculated 10 Hz impedance phase is in good agreement with the real average cell temperature. The maximum estimation error is 1.45 °C, the average absolute error is 0.33 °C and the root-mean-square error is 0.43 °C as is shown in Fig. 16(b). The results show that the temperature estimation method based on the online impedance calculation during the continuous step current test is feasible.

4.3. Verification of online impedance calculation To verify the temperature estimation method with the online calculated impedance, the step current drawn as Fig. 4 is injected into the 40 Ah battery cell under 30% SOC at 25 °C. The improved battery impedance calculation method is used to calculate the impedance at 10 Hz. The sampled current and voltage signals are shown in Fig. 12(a). With fc = 3000, fb = 10−6, a = 300, the wavelet coefficient of the current signal is calculated as Fig. 12(b). The improved method can accurately identify the occurrence time of the 10 Hz harmonic component, i.e. the step current jumping time. And the corresponding 10 Hz signal intensity contained in the pulse current can also be detected with the method. Applying the improved method to the sampled voltage and the current signal from 0 s to 450 s, the change of the impedance magnitude and the impedance phase is obtained as Fig. 13. Correspondingly, the impedance magnitude at 10 Hz measured by the electrochemical workstation at 25 °C and 30% SOC is 2.0856 × 10−3 Ω and the impedance phase is −8.0715°, which is basically consistent with the impedance obtained by the on-line impedance calculation method. The relative error of the impedance magnitude and the impedance phase is within 3% and 5% respectively. It shows that the impedance at 10 Hz is limitedly affected by the changing current amplitude and rest time, which is consistent with the results and the conclusions in Section 4. And the method for on-line impedance calculation using the improved wavelet transform is feasible and with high accuracy. The computation time of the proposed fast calculation method varies according to the signal length. The algorithm is run on a laptop with an i5 8265 U CPU and 8 GB RAM using the MatlabⓇ 2018a. For the signals shown in Fig. 12(a), the 10 Hz impedance calculation for signals at 0–450 s and 0–25 s time range takes approximately 15.854 s and 0.198 s respectively. For the online application, the computation time can be decreased by shortening the signal length.

4.4.2. Temperature estimation during NEDC test The battery SOC does not change after the end of the continuous step current test. For the step current sequence, the current amplitude and the rest time are also fixed. However, when performing the NEDC test as shown in Fig. 4(b), the battery is discharged for about 6 Ah after the end of the three NEDC cycles, and SOC is reduced by about 15%. At the same time, it can be seen from the battery terminal voltage and the current of the three NEDC cycles shown in Fig. 17 that the current amplitude and the rest time are more varied than in the continuous step current test in the previous section. It is more difficult to accurately calculate meaningful battery impedance during NEDC test. The calculated impedance is greatly affected by the operating conditions, making the impedance calculation result fluctuate greatly. Therefore, a 120-point 50th percentile filter [39] is used here to smooth the calculated 10 Hz impedance phase with time. Fig. 18 shows the 10 Hz impedance phase obtained after processing. In the NEDC test, 9

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Fig. 13. The calculated impedance at 10 Hz of the 40 Ah LiFePO4 battery at 25 °C under 30% SOC, (a) impedance magnitude; (b) impedance phase; (c) relative error of impedance magnitude; (d) relative error of impedance phase.

Fig. 15. Change of the 10 Hz impedance phase of the 40 Ah battery cell with time during continuous step current test.

Fig. 14. Temperature change of the 40 Ah battery cell.

which lasted for about 1 h, the 10 Hz impedance phase of the battery was constantly changing. As can be seen from Fig. 19, the battery temperature increased by about 1.5 °C after the end of the test. The average temperature of the battery estimated by the relationship between the impedance phase and the average temperature is close to the actual average temperature, and has a similar trend with the battery test. In most cases, the average temperature absolute error obtained by the two methods is within 0.5 °C. The effectiveness of the proposed method for battery temperature estimation during NEDC test is preliminarily illustrated.

5. Conclusions In the paper, a temperature estimation method based on online impedance calculation is proposed. Unlike previous studies, the battery impedance is calculated with the voltage and the current signals in operation based on wavelet transform without the need to specifically design an excitation source for impedance measurement on board. The change of the working conditions, i.e. amplitude, charge/discharge mode and rest time affects the impedance in the low frequency range to varying degrees. This effect prevents the proposed online temperature estimation method from being applicable directly to the full range of 10

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Fig. 16. Estimation of the battery average temperature and the error during continuous step current test, (a) comparison of measured and estimated average temperature; (b) absolute error of estimated average temperature.

Fig. 17. Change of the measured average cell temperature during NEDC test under 25 °C, 90% SOC.

Fig. 18. Change of the10 Hz impedance phase of the 40 Ah battery cell with time during NEDC test, filtered by 120-points percentile filter.

Fig. 19. Comparison of measured average cell temperature with estimated average temperature during NEDC test under 90% SOC at 25 °C.

working conditions. For the 40 Ah battery cell, the average temperature is estimated with the maximum error of 1.45 °C and the average absolute error of 0.33 °C from about 15.8 °C to 25.8 °C under the continuous loading of the current pulse with amplitude of 20A. At the same time, in the NEDC test, combined with the smoothing method, the absolute error of the average temperature estimation result is less than 0.5 °C. Considering that the discharge rate of high specific energy battery is small and the operating temperature is not too low, the proposed

method will realize the temperature estimation without interrupting the normal operation of the battery and ultimately benefit the battery management. Further research will focus on the extraction of the harmonic component at certain frequency during the more complicated dynamic operation cycle with more efficient smoothing or filtering methods.

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Declaration of Competing Interest [18]

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.

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Acknowledgments [20]

This work is financially supported by the National Natural Science Foundation of China (NSFC, Grant number is U1764256).

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