Electrothermal Coupling Modeling and Electrothermal Dynamic Characteristics of Nickel-Cadmium Battery

Although Nickel-Cadmium batteries are widely used, there is a general problem of thermal failure and even thermal safety, and a lack of thermal characteristics and temperature rise law research. Accordingly, a typical Nickel-Cadmium battery’s equivalent circuit model, thermal model, and electrothermal coupling model were established in this article. The model’s open-circuit voltage, resistance, and capacitance, as well as its entropy heat coefficient, thermal conductivity and heat transfer coefficient were measured, and finally, according to these parameters, the verification analysis of the electrothermal coupling model was carried out in Simulink under the HPPC working condition and constant current working condition. The results show that the simulation results are basically consistent with the experimental data, the absolute value of voltage error is less than 0.02V under HPPC condition, the maximum voltage error is only 0.7% under constant current condition, and the temperature maximum error is only 3%. Consequently, the constructed electrothermal coupling model can well reflect the electrothermal characteristics of the battery, which provides data support and reference for solving the thermal failure problem.


Introduction
Compared with lead-acid batteries and lithium batteries, Nickel-Cadmium (Ni-Cd) batteries have stable performance, simple maintenance, strong resistance to over-charging and over-discharging, and good low-temperature performance, and are widely used in DC power supply of substations in various fields, as well as on-board starter or emergency power supply of rail transit trains, mining vehicles, and heavy-duty vehicles [1].Especially, in the high-speed trains, Ni-Cd batteries are used to provide power for control equipment and pantograph lifting, and emergency power for important loads, which is important for the safe and stable operation of the train and the passenger travel experience [2], [3].
However, thermal failure of Ni-Cd batteries due to high temperature may occur with the extension of service time, and even occasionally causes combustion thermal safety accidents [4].For this reason, manufacturers and users usually remove and scrap the battery as long as the temperature reaches 60 ℃, resulting in economic losses.At the same time, the temperature has a critical impact on the Ni-Cd battery state of charge [5], service life [6].Therefore, it is crucial to study the electrothermal characteristics of Ni-Cd batteries, providing a basis for the design and effective control [7], [8].
The change rule of battery temperature is often studied and analysed by thermal model [9], [10].LI et al. [11] considered the influences of SOC, current and temperature on the resistance and heat production rate of lithium battery, built a dynamic electrothermal coupling model to investigate the influences of heat production rate and temperature distribution; XIE et al. [12] developed an improved electrothermal coupling model that takes into account the current distribution in the battery module and predicted the inconsistency of the battery temperature; GAO et al. [13] proposed an electrothermal coupling model for lithium battery considering high rate operating situation, and investigated the changing law of battery parameters and temperature rise under high rate current operating situation; LIU et al. [14] combined the traceless Kalman filter with the battery electrothermal coupling model, to estimate the SOC of the battery and the real-time surface temperature; SHI et al. [15] considered the hysteresis effects and the influences of temperature on battery model's accuracy, combined with twostate thermal model to build a battery electrothermal coupling model; CHARLES et al. [16] developed a two-dimensional electrothermal coupling model to evaluate the thermal behaviour of battery overcharging in a high temperature environment.
The above studies mainly focused on lithium batteries, while there are significant differences in the operating characteristics of Ni-Cd batteries [17], [18], but few studies have been reported on the thermal model and thermal characteristics of Ni-Cd batteries.For this reason, this article identifies the Ni-Cd battery model parameters at different temperatures and SOC through HPPC experiments, analyses the heat production and heat dissipation characteristics of the discharging process, measures the parameters required for the thermal model, constructs a Ni-Cd battery electrothermal coupling model, which is used to predict the temperature change, and tests the model's correctness through experiments to provide the theoretical and data foundation for Ni-Cd batteries.

Experimental Platform
The experimental platform consists of Ni-Cd batteries, battery charging and discharging equipment, accessory channel, and an incubator, shown in figure 1. Battery charging and discharging equipment for the new weir power battery test equipment (CT-4002-5V300A-NTFA), voltage and current range of 0~5V and 0~300A, thermocouples connected to auxiliary channels (CA-4008-1U-VT-JX), collecting the Ni-Cd batteries surface temperature change rule of law, and the measurement error of ±0.1℃; incubator maintains temperature equilibrium between the inside and outside of the battery, the temperature range is from 0℃ to +105℃.The test condition mainly adopts HPPC (Hybrid Pulse Power Characteristic) test and constant current test.

Equivalent Circuit Model
The Thevenin model, which uses an RC parallel network to describe the battery polarization effect and characterize the bounce characteristics of the battery voltage, is the most used equivalent circuit model and is based on Davinan Circuit Principle [18].This article chooses the first-order RC equivalent circuit model, as shown in figure 2. In figure 2, UOCV stands for the Ni-Cd battery's open-circuit voltage (OCV), R1 for the battery's ohmic internal resistance, and R2, C2 for the battery's resistance and capacitance for the electrochemical polarization and the concentration polarization effects, which are used to describe the battery's transient response and the polarization effects [19], [20].The UOCV, R2, and C2 are all affected by temperature and SOC variations and are variables concerning temperature and SOC variations.

Thermal Model
Ohmic heat, polarization heat, reaction heat, and side reaction heat are the main types of heat generated during battery operation [21], [22].Normal operation often just takes into account ohmic heat, polarization heat, and reaction heat, and side reaction heat is negligible, i.e.: Polarization heat Qp is the heat produced by the internal polarization effect by the polarized internal resistance.Ohmic heat Qj is the heat produced when the current travels through the ohmic internal resistance of each component of the battery.Therefore, the ohmic heat and polarization heat are irreversible heat, and the heat generated when the charging and discharging current I flows through the internal resistance R is: Chemical reaction's heat Qr is produced by the electrode reaction and is a reversible reaction heat that is positive when the battery is being discharged and negative when it is being recharged: where E is electromotive force of the battery, T is battery temperature, and dUOCV/dT is entropy heat coefficient.According to ohm's theorem, the difference between the battery's UOCV and its terminal voltage U can be used to replace the Rj+Rp multiplying current in the battery's Joule heat and polarization heat in order to calculate the battery's overall heat output: ( ) From equation ( 4), the rate of heat production of the battery during charging and discharging is related to the charging and discharging currents, the ohmic and polarization internal resistance, the temperature, and the entropy heat coefficient.
According to the theory of heat conduction, Ni-Cd batteries lost mainly through heat conduction, heat convection and heat radiation.When considering the heat transfer between the surface of the battery and the external environment, the thermal radiation is often ignored because the battery thermal radiation is much smaller than the convective heat transfer, and the heat dissipation on the surface of the battery is mainly determined by the convective heat transfer.Assuming that the heat convection between the battery and the environment is stable, the convective heat transfer heat can be obtained by Newton's cooling formula as follows: where A is the convective heat transfer area, Tc the surface temperature of the battery, Tf the ambient temperature where the battery is located, and h is the heat transfer coefficient.

Electrothermal Coupling Model
In battery modeling, the electrical model can identify parameters such as internal resistance, capacitance, and terminal voltage, and uses these parameters along with the current through the thermal model to calculate the actual temperature of the battery.Operating temperature has an impact on battery parameters, and the battery thermal model can represent the battery's heat production characteristics, reflecting the battery's temperature change characteristics [23].As the heat generated by the battery accumulates and corrects model errors, thereby improving model accuracy.Therefore, the output parameters of the electric model are used as inputs for the thermal model in the battery electrothermal coupling model, and the output temperature parameters of the thermal model are used as inputs for the electric model.The two models are coupled together, and the fundamental structure of the electrothermal coupling model is shown in figure 3.

Open-circuit Voltage
There is a monotonic mapping between open-circuit voltage UOCV and SOC, and the SOC-OCV function is a nonlinear monotonic function with respect to SOC and OCV, which describes the battery's electrochemical process at various SOCs, and also widely used for SOC and capacity estimation.The ambient temperature affects the measurement results of the SOC-OCV curve, and therefore, the SOC-OCV curves at different temperatures need to be measured to improve the battery modeling accuracy.The detailed experimental steps are as follows: 1) Charge the battery with 0.2C constant current and constant voltage, and place it at 15℃, 25℃, 35℃ and 45℃ ambient temperatures respectively, and wait for the temperature of each part of the battery to be consistent and stable.2) Take 10% of the rated capacity of the battery as the interval, discharge at 0.2C to a specific SOC value and set aside for 1h and record the battery's OCV. 3) Repeat step 2) until the battery discharge voltage reaches the cut-off voltage of 1.0V.Considering the accuracy requirements of the SOC-OCV curve, the sixth-order fitting curve was used, and its fitting curve Coefficient of Determination (COD) R 2 reached 0.999, which indicates that the fitting curve can accurately reflect the SOC-OCV relationship, and the polynomial fitting obtained the following SOC-OCV relationship equations at different temperatures: where a0~a6 are polynomial coefficients.
As shown in figure 4, the OCV of Ni-Cd batteries varies linearly with SOC in the intermediate SOC region, and the growth rate is accelerated in the low SOC region and high SOC region.In the high SOC region, the increase in temperature causes the open-circuit voltage of the battery to increase, and it is basically unchanged in the low or room temperature environments; in the intermediate SOC region, the OCV of the temperature rise condition is similar with room temperature, and it decreases significantly in the low-temperature environment, and in the low SOC region, temperature increasing and decreasing lead to a smaller variation in the OCV.All in all, the OCV is similar between high temperature and room temperature, and the OCV has the largest difference with room temperature at low temperature, and the low temperature has a greater effect, so it is crucial to study the effect of temperature factors on the SOC-OCV curves.

Battery Model Parameters
HPPC pulse cycle testing is currently the most effective experimental method for identifying equivalent circuit model parameters.According to the test results from the characterization experiment, parameter identification can be used to determine the equivalent circuit model's resistance and capacitance data.The recursive least squares method is used in this article to identify the battery parameters.The detailed steps are as follows: 1) In a constant temperature environment, fully charge the battery with 0.2C and constant voltage, at which time the SOC is 100%, and leave it for more than 1h to ensure that the battery is stable.2) Discharge at 1C constant current for 10s and leave for 40s.
3) Charge at 0.75C constant current for 10s and leave for 40s.4) Discharge at 0.25C constant current until the capacity is 10% of the maximum usable capacity and leave it for 30min.5) Repeat steps 2) to 4) to perform characterization experiments at different SOCs until the battery discharge voltage reaches the cut-off voltage of 1.0V.Based on the recursive least squares method of solving to get the ohmic internal resistance R1, polarization internal resistance R2, polarization capacitance C2 curves are plotted for each battery parameter as shown in figure 5.

Determination of Entropy Heat Coefficient Parameters
The entropy heat coefficient, also known as the OCV coefficient, is the derivative of the OCV to temperature, which is mainly influenced by the battery temperature and SOC.The entropy heat coefficient is also influenced by the battery's charging and discharging states; when the battery is charging, the entropy heat coefficient is greater than zero and is exothermic reaction, while entropy heat coefficient is less than 0 and is absorbing reaction; and vice versa when the battery is discharging.This article uses a widely accepted technique for measuring entropy heat coefficient, which is obtained by measuring battery OCV values under various temperatures and analysing the correlation between voltage and temperature.The battery will be put in 25℃ in the constant temperature incubator, the detailed steps are as follows: 1) Firstly, the battery is fully charged under constant current and voltage, then stewing for 1h.
3) Then keep the battery at 25°C for 3h.4) Discharge 10% of the rated capacity with 0.2C constant current and repeat steps 2) to 3) until the battery discharge voltage gets 1.0V.Based on the experimentally measured entropy heat coefficient under different SOCs, due to the certain smoothness of the SOC-dUOCV/dT curve, this article applies the polynomial fitting method to obtain the correlation between the dUOCV/dT and SOC, and the relationship curve between SOC and entropy heat coefficient is fitted by polynomial data, as shown in figure 6.

Measurement of Heat Exchange Coefficient
The heat transfer coefficient is obtained by using the cooling temperature curve of the battery after a sharp increase in high rate discharge temperature [13].The battery is first placed in the incubator at 25°C, fully charged the battery with constant current and constant voltage charging, left static for a sufficient amount of time to allow the battery temperature to return to the ambient level, and then discharged at a high rate of constant current while recording the battery's temperature changes, at this time, the battery temperature rises sharply, after ending the discharge, the battery is static cooling until restored to the battery temperature to maintain stability, as shown in figure 7, and then the equation ( 7) is obtained.
where K is the exponential function coefficient, S the surface area, and c the specific heat capacity.
From the cooling curve at a constant temperature of 25°C, the heat transfer coefficient h=8.473W•(m 2 •°C) -1 was calculated.

Measurement of Thermal Conductivity
Thermal conductivity affects the heat dissipation characteristics of the battery, describes the amount of heat conducted per unit cross-section or length of a cell per unit time., and is related to the structure and composition of each component of the battery, as well as temperature and other factors.Because the inside of the battery is made of different materials stacked on top of each other, the battery's thermal conductivity varies in each direction, and because the Ni-Cd battery is cuboid in shape, the temperature inside the battery is conducted along the three directions of the x, y and z axes.
The thermal conductivity of Ni-Cd batteries can be calculated by superimposing the relevant parameters of each component, as shown in equation (8).
where i is the ith layer material of the battery, Li is the thickness, and λi is the thermal conductivity coefficient, but this method is not applicable to the laboratory test, and in this article, we use the simulation method to solve the battery's thermal conductivity coefficient.
The simulation method heats the battery through experiments, reduces convective heat transfer, measures the temperature of the battery's characteristic points, and solves the thermal conductivity of the battery in the x, y, and z directions using Fluent software.The battery is set aside in the 25℃ incubator, the heating plate is fixed on the surface side of the battery, then heats a localized area of the battery at a constant temperature, the temperature rises in all places, heat evenly at the end of heating, the battery cooling naturally, achieving internal thermal equilibrium, and then measures the steadystate temperature of the battery at each characteristic point.
Five temperature measurement points are set on the battery's surface, as displayed in figure 8, labelled T1~T5, and one measurement point is marked on the opposite side of the battery, labelled T6.The battery thermal conductivity is solved by using the steady state value, and there is no heat source in the battery, so that the thermal conductivity equation of the battery is shown in equation (9).(10) where Tx, Ty and Tz are the planes of the battery perpendicular to the x, y and z axes respectively, the internal temperature field distribution is calculated by using Fluent software simulation, matching the temperature data measured in the experiment, the λx, λy and λz thermal coefficients can be solved:

Model Verification and Analysis
According to the model parameters identified earlier, the simulation model of Ni-Cd battery was built in Simulink.To validate the electrothermal coupling modeling accuracy, the HPPC test and constant current discharging test are verified, as shown in figure 9 and figure 10. Figure 11.Temperature for constant current test.Electrothermal coupling model simulation results and experimental data basically match, the difference between experimental and simulation results is basically maintained below ±0.02V.This is because the capacitance and resistance characteristics are not in a rational state in actual operation, and the voltage does not undergo a sudden change when the working conditions change, so there is a certain error in the simulation and experimental voltage data, and the voltage change is steep when discharged to below 10% SOC, and the maximum simulation error is 7.5%.
In a constant temperature environment at 25℃, 1C constant current discharge is used until the battery terminal voltage reaches 1.0V cut-off discharge voltage and records the voltage and temperature changes at different locations during the battery discharge.Battery constant current discharge voltage experimental data and simulation data comparison results are shown in figure 10, the temperature change curve is shown in figure 11, constant current discharging voltage and temperature simulation results of the change trend are basically consistent with the experimental data, at the beginning of the discharging, the battery voltage changes faster, the simulation results of the error is too large, the voltage of the maximum error of 0.7%, the temperature of the maximum error of 3%, but in the permissible range, the electrothermal coupling model basically reflects the battery's discharge characteristics.

Conclusions
The ohmic internal resistance decreases slightly with the increase of the SOC, which is significantly different from that of the Li-ion batteries and in the SOC is less than 0.2 when the decrease is relatively more obvious, and overall ohmic internal resistance is insensitive to the changes of SOC, but as the temperature decreases the ohmic internal resistance increases sharply, indicating that low temperature has a great impact on Ni-Cd battery.
The SOC-OCV model is constructed, the research results show that: as the increase of SOC, OCV increases, and OCV increases rapidly with the increase of SOC when SOC is greater than 0.8, and decreases rapidly with the decrease of SOC when SOC is less than 0.2; at the same time, the temperature is in the range of 25℃~45℃, the effect of different temperatures on OCV is not obvious, while at low temperatures, the OCV decreases as the temperature decreases; it should be emphasized that the result of entropy heat coefficient measurement shows when the SOC is greater than 0.4, OCV is basically stable for the derivative of temperature, while when the SOC is less than 0.4, the derivative of OCU with respect to temperature increases rapidly as the SOC decreases, which indicates that the OCV is most affected by the temperature at low SOC.
The establishment of a heat production model and a heat dissipation model for the charging and discharging of Ni-Cd batteries, where the heat production model takes into account ohmic heat, polarization heat, and reaction heat while the heat dissipation model primarily considers convective heat transfer; based on the design of the test platform, the first measurement of Ni-Cd batteries for locomotives: the thermal conductivity coefficients λx=0.85W•m - •K -1 , λy=0.85W•m -1 •K -1 , λz=2.02W•m - 1 •K -1 , and the heat transfer coefficient h=8.473W•(m 2 •°C) -1 .
The battery voltage difference under the HPPC test is basically maintained below 0.02V, and during the constant current discharge process, the maximum voltage error is 0.7%, and the maximum temperature error is only 3%, and the simulation results are in general agreement with the experimental data, which indicates that the electrothermal coupling model basically reflects the discharge characteristics and temperature rise law of the battery, and it has a good modeling accuracy.

Figure 4 .
Figure 4. SOC-OCV curves of Ni-Cd battery at various temperatures.

Figure 5 .
Figure 5. RC change curves of Ni-Cd battery.

Figure 8 .
Figure 8. Thermal measurement points.Figure 9. Comparison of HPPC test.The boundary conditions are obtained by assuming that the heat transfer coefficients of the six surfaces of the battery remain the same during the solution process, as shown in equation (10).

Figure 9 .
Figure 8. Thermal measurement points.Figure 9. Comparison of HPPC test.The boundary conditions are obtained by assuming that the heat transfer coefficients of the six surfaces of the battery remain the same during the solution process, as shown in equation (10).

Figure 10 .
Figure 10.Constant current test verification.Figure11.Temperature for constant current test.Electrothermal coupling model simulation results and experimental data basically match, the difference between experimental and simulation results is basically maintained below ±0.02V.This is because the capacitance and resistance characteristics are not in a rational state in actual operation, and the voltage does not undergo a sudden change when the working conditions change, so there is a