With the intermittent nature of renewable energy generation such as solar and wind power, energy storage systems are required to ensure the frequency stability and power balance of the power grid. Retired power batteries can play a role in smoothing power fluctuations in the new energy power generation system, and a reasonable configuration can achieve better benefits for the power generation system. The retired power batteries used in the energy storage devices of the power grid and new energy power generation usually need to undergo certain screening and reorganization.
However, the internal electrochemical reactions of lithium-ion batteries are relatively complex and have strong nonlinear characteristics. Only the battery terminal voltage and terminal current can be detected from the outside, making it a typical black-box system. Establishing an accurate equivalent model is the prerequisite for making a reliable judgment on the working state of the battery to ensure the safe operation of the energy storage system.
With the development of battery modeling theory, according to the degree of description of the internal mechanism of the battery, the commonly used battery models can be divided into two categories: electrochemical models and equivalent circuit models. The electrochemical model can describe the comprehensive electrochemical reactions of the battery, with high precision, but the calculation process is complex. The equivalent circuit model is composed of ideal circuit elements such as resistors and capacitors. Each element has a clear physical meaning, can clearly reflect the electrical characteristics of the battery, and the model is easy to establish and implement. It can be applied in the circuit simulation environment. Common equivalent circuit models include the Rint model, the Thevenin model, the PNGV model, and the second-order RC model. The second-order RC model contains two RC resistance-capacitance networks, which can more accurately describe the internal response characteristics of the battery and is also easy to implement in practical engineering. Considering the simulation accuracy and computational complexity of the model, this paper chooses the second-order RC equivalent circuit model as the basic research model for the battery.
The accuracy of the basic equivalent circuit model is easily affected by various factors, such as capacity attenuation, temperature, current rate, cycle number, and self-heating. The traditional offline parameter identification method needs to repeat the parameter identification experiment on the battery for a certain influencing factor to obtain the adapted values of the model parameters under different operating conditions. This method is still feasible for batteries with a shallow degree of aging, but for retired batteries, repeated charging and discharging experiments will undoubtedly damage the service life of the battery, reduce the number of cycles of the battery, and even reduce the cascade utilization efficiency of the retired batteries, thereby affecting the reliability of the energy storage system.
In this paper, to address the problem that the accuracy of the basic model of the battery is easily affected by the state of charge (SOC), temperature, and current rate, and to reduce the further aging of the retired batteries caused by the traditional offline parameter identification experiment, a model improvement method for sample data expansion is proposed. Only using the model terminal voltage error data under small-range temperature and small discharge rate sample conditions, a back propagation (BP) neural network is created to predict the model error under a wide range of temperatures and various charge and discharge rates, and to achieve dynamic compensation of the model. This method can avoid repeating the model error acquisition experiment under different temperatures and current rates for the battery, and to some extent shortens the time for manually improving the accuracy of the model, which has certain practical engineering significance in the application of the new energy power generation battery energy storage system.
1. Battery Equivalent Circuit Model and Parameter Identification
1.1 Second-order RC Equivalent Circuit Model
The circuit diagram of the second-order RC equivalent circuit model is shown in Figure 1. The model includes two RC resistance-capacitance networks: the first RC resistance-capacitance network describes the impedance during the transmission between the electrodes, and R1 and C1 represent the activation polarization internal resistance and activation polarization capacitance, respectively; the second RC resistance-capacitance network describes the impedance when lithium ions diffuse in the electrode material, and R2 and C2 represent the concentration polarization internal resistance and concentration polarization capacitance, respectively. U1 and U2 represent the terminal voltages of the two RC resistance-capacitance networks; Uoc represents the open circuit voltage (OCV) of the battery, which has a one-to-one nonlinear function relationship with the SOC of the battery; R0 represents the ohmic internal resistance of the battery. I is the terminal current of the battery, and the discharge current direction is defined as positive; Ut represents the terminal voltage of the battery.
1.2 Parameter Identification
The parameters to be identified in the model are Uoc, R1, C1, R2, C2, and R0. In this paper, the open circuit voltage experiment and the cyclic pulse discharge experiment are used to identify the model parameters. Considering that the model parameters of the battery are functions of three factors: SOC, temperature, and current rate, and the model accuracy is easily affected by these factors. To improve the accuracy of the basic model and avoid the deepening of the aging degree of the retired battery caused by the traditional offline modeling method, only the influence of SOC on the model parameters is considered in the parameter identification experiment, and then the model is compensated by the model terminal voltage error under different temperatures and current rates to further improve the accuracy of the basic model. The battery experimental object used in this paper is a certain 18650-type lithium cobalt oxide battery, with a nominal voltage of 3.62 V, a maximum voltage of 4.2 V, a cut-off voltage of 2.75 V, and a nominal capacity of 2.15 Ah.
1.2.1 Open Circuit Voltage Identification
The calibration experiment steps for the open circuit voltage and SOC relationship curve are as follows:
At room temperature, the battery is charged to full power using the standard charging method of constant current followed by constant voltage, and then left to stand for 2 hours. After that, its open circuit voltage value is measured and recorded, and the SOC corresponding to this voltage value is calibrated as 1.
The battery is discharged at a constant current of 1.0C (the current when the battery is completely discharged in 1 hour) for 6 minutes, that is, 10% of the capacity is discharged, and then left to stand for 2 hours. At this time, the open circuit voltage value is recorded, and the corresponding SOC is calibrated as 0.9.
Repeat step 2) until the SOC drops to 0 or the battery terminal voltage drops to 2.75 V.
Using the Cftool toolbox in Matlab to curve fit the experimental data, after considering the computation amount and fitting accuracy, an 8th-order polynomial function is selected as the fitting result, and the OCV-SOC relationship equation and curve are obtained as shown in Equation (2) and Figure 2.
1.2.2 Resistance-Capacitance Parameter Identification
For the parameters R1, C1, R2, C2, and R0 in the battery model, the cyclic pulse discharge experiment is used for identification. The complex characteristic changes inside the battery are expressed by the measurable voltage and current outside the battery. The specific experimental steps are as follows:
At room temperature, the battery is charged to full power using the standard charging method of constant current followed by constant voltage, and then left to stand for 2 hours.
The battery is discharged at a constant current of 1.0C for 3 minutes, that is, 5% of the capacity is discharged, and then left to stand for 5 minutes. During the whole process, the battery terminal voltage change data is recorded.
Repeat step 2) until the SOC drops to 0 or the battery terminal voltage drops to 2.75 V.
Figure 3 shows the current change curve of the cyclic pulse discharge experiment. Figure 4 shows the terminal voltage response curve within one pulse cycle in step 2). This response curve can be divided into four stages: stage AB in the first stage, the battery jumps down in terminal voltage when the current is suddenly applied from the standby state. For the second-order RC equivalent circuit model, the voltages on the capacitors C1 and C2 cannot change abruptly, so this jump voltage is mainly determined by the ohmic internal resistance R0. In the second stage BC, as the discharge time increases, affected by the two RC resistance-capacitance networks, the battery terminal voltage decreases exponentially. In the third stage CD and the fourth stage DE, when the battery changes from loaded discharge to no-load standby, the voltage response also shows a jump-up and exponential increase process, and the principle is the same as that in stages AB and BC.
Taking Figure 4 as an example, the specific derivation process of the internal resistance and capacitance parameter identification in one pulse cycle is given. Both stages AB and CD show strong internal resistance characteristics, so the average value of the voltage difference in these two stages can be selected to calculate the ohmic internal resistance, and the calculation formula is:
Stages BC and DE are the results of the action of the RC resistance-capacitance network. The values of the two polarization resistances and polarization capacitors can be obtained by performing exponential fitting on these two stages of curves.
Stage DE is the zero-input response stage when the pulse current is removed and the RC network loses excitation. Taking point D as the moment t = 0, the zero-input response expressions of the two RC networks are:
At this time, the terminal voltage output equation of the battery is as shown in Equation (5). Using the Cftool toolbox in Matlab to fit the curve in stage DE, the values of the two time constants τ1 and τ2 can be obtained.
Stage BC is the response process after a period of standby and then loaded discharge. At this time, the voltages at both ends of C1 and C2 are approximately 0, so stage BC can be regarded as the zero-state response stage of the RC network. Taking point B as the moment t = 0, the zero-state response expressions of the two RC networks are:
At this time, the terminal voltage output equation of the battery is:
Substituting the two time constants obtained in stage DE into Equation (7), after Matlab exponential fitting, the values of R1 and R2 can be obtained. Then according to τ1 = R1C1 and τ2 = R2C2, the values of C1 and C2 can be obtained.
After fitting the voltage curves in all pulse cycles according to the above steps, a dynamic model can be obtained to simulate the complex electrochemical reaction process inside the battery when the SOC changes.
2.Battery Model Error Analysis and Improvement
2.1 Model Error Analysis at Different Temperatures and Discharge Rates
Based on the battery model established above that only considers the influence of SOC, the model error at different temperatures and different discharge rates is analyzed. According to the working temperature range of the battery selected in this paper, at constant temperatures of -10, 0, 25, and 30 °C, the battery is subjected to constant current discharge experiments at rates of 0.3C, 0.5C, 1.0C, and 1.2C, and the terminal voltage change data of the battery is recorded. The specific experimental steps are as follows:
- At room temperature, the battery is charged to full power using the standard charging method of constant current followed by constant voltage, and then left to stand for 2 hours.
- The fully charged battery is placed in a constant temperature box set at the temperature and left to stand for more than 30 minutes.
- At different temperatures, the battery is discharged at different discharge rates until the battery terminal voltage drops to 2.75 V. During the whole process, the terminal voltage change data of the battery is recorded.
Using the Lookup-Table module in Matlab/Simulink to import the identified model parameters into the simulation model, taking the current as the input of the model, and comparing the output terminal voltage of the model with the measured terminal voltage under the same excitation current. The comparison waveform diagrams of the model terminal voltage and the measured terminal voltage at different temperatures and discharge rates are shown in Figure 5.
It can be seen from Figure 5 that when the discharge rate is the same, the lower the temperature, the greater the model error; when the temperature is the same, the greater the discharge rate, the greater the model error. This indicates that the second-order RC equivalent circuit model considering only the influence of SOC still has a certain simulation error compared to the real battery.
By processing the terminal voltage error data under different working conditions, the model error table shown in Table 1 is obtained, where the definitions of the two errors are as shown in Equation (8).
It can be seen from Table 1 that under the same temperature condition, the model error is basically positively correlated with the discharge rate; under the same discharge rate condition, the model error is negatively correlated with the temperature, and the data in the table is consistent with the conclusion drawn from Figure 5. The reason for the analysis is that as the discharge rate increases, the internal chemical reactions of the battery become more intense, especially in the final stage of discharge, the internal reactions are even more complex and changeable. However, the model parameters are fitted based on a fixed current rate, and the description effect of the model on the battery will decrease at this time. At the same time, the model parameters are also fitted based on the room temperature condition. As the temperature decreases, the temperature difference from the room temperature becomes larger, which will also lead to a worse description effect of the model on the battery.
It is worth noting that at a constant temperature of 30 °C, only the model error of constant current discharge at a rate of 1.0C is given in this paper. This design is to pave the way for the construction and analysis of the BP neural network in the following.
2.2 Construction and Analysis of BP Neural Network Based on Small Samples
Through the above analysis, the average error of the model is affected by the changes in temperature and load current, which is equivalent to a controlled voltage source. Therefore, this voltage source can be compensated into the circuit model to improve the model accuracy. However, to obtain the model error under a wide range of temperatures and current conditions, if the retired battery is repeatedly subjected to the experimental process described above, it will not only consume a lot of time but also cause the reduction of the cycle life of the battery, thereby reducing the cascade utilization rate of the retired battery.
Based on the above problems, this paper proposes a method to predict large sample data by expanding small sample data. First, the average error data of the model under the existing temperature and discharge rate conditions is fitted into an expression, and then the BP neural network is trained based on the expression. The neural network is used to predict the average model error under a wide range of temperatures and current conditions, and compensate it into the circuit model to improve the accuracy of the model.
Using the Cftool toolbox in Matlab to fit the temperature, load current, and average error data in Table 1, the expressions of each order regarding the model error can be obtained, and the output values of each expression corresponding to different discharge rates and temperatures are shown in Table 2. Among them, the independent variable x represents the temperature, and the independent variable y represents the load current.
By comparing the output values of each order expression, it can be observed that when both x and y are of the 3rd order, this expression has the smallest difference from the average model error compared to the other expressions in Table 2, and this expression is:
Training the BP neural network based on Equation (9), the input layer is set to 1 layer with 2 nodes, and the input quantities are the load current and temperature; the hidden layer is set to 1 layer with 5 nodes; the output layer is set to 1 layer with 1 node, and the output quantity is f(x^3, y^3). The training set of the BP neural network is the changing current and temperature and the corresponding average model error. The current starts from 0.6 A and increases to 2.6 A in increments of 0.05 A; the temperature starts from -10 °C and increases to 30 °C in increments of 1 °C. The transfer function of the hidden layer is set to tansig, the transfer function of the output layer is set to purelin, and the training function is trainlm using the Levenberg_Marquardt algorithm.
Figure 6 shows the regression curve of the BP neural network, where R = 0.9982, close to 1. Figure 7 shows the comparison curve and error curve of the predicted value and expected value of the BP neural network, where the x-axis represents the temperature, the y-axis represents the load current, and the z-axis represents the average model error output by the BP neural network. It can be seen that the predicted value and the expected value basically completely overlap, and the error is basically zero.
Table 3 shows the model error value of the BP neural network training. From the data in the table, it can be seen that the output value of the BP neural network is basically equal to the output value of the expression, which once again verifies the conclusion drawn in Figure 7, indicating that the BP neural network achieves the expected effect.
To verify that the BP neural network has a better voltage compensation effect than each order expression, this paper sets the operating conditions of the battery at different charge and discharge rates and temperatures, and compares the output values of the BP neural network and each order expression respectively, as shown in Tables 4 and 5.
Table 4 shows the model error table corresponding to the BP neural network and each order expression at different temperatures and discharge rates. The data in the table can be analyzed from the following three aspects: - The set temperature is 5 °C, the discharge rate is 1C, the temperature is 20 °C, the discharge rate is 1.0C, and the temperature is 25 °C, and the discharge rate is 0.7C. These three working conditions verify the discharge rate and temperature within the training set of the BP neural network. It can be found that the output value of the BP neural network is basically equal to the output value of the expression f(x^3, y^3), once again indicating that the BP neural network has achieved the expected effect.
- The set temperature is 25 °C, and the discharge rates are 2.5C and 4.0C, which verify the discharge rates outside the training set of the BP neural network. It can be found that when the discharge rate exceeds the range of the training set, the output value of the BP neural network has a smaller difference from the average model error compared to the expression f(x^3, y^3), indicating that the BP neural network has better predictability.
- The set temperature is 40 °C, and the discharge rates are 0.5C and 1.0C, which verify the temperature outside the training set of the BP neural network. It can be found that even when only the error data of the discharge rate of 1.0C is fitted at 30 °C, the BP neural network can still make an accurate prediction of the model error when the battery is discharged at a rate of 0.5C at a constant temperature of 40 °C, and the prediction effect is better than that of the expression, verifying the feasibility of the constructed BP neural network, which is also consistent with the data in Table 1.
Table 5 shows the model error table corresponding to the BP neural network and each order expression at different charging rates. The temperature is set at 25 °C, and the model errors at charging rates of 0.6C and 1.0C are verified. It can also be found that the output value of the BP neural network during charging has a better prediction effect than the expression f(x^3, y^3), once again verifying the feasibility of the constructed BP neural network.
By analyzing the data in Tables 4 and 5, the following conclusions can be drawn:
- The neural network trained under small-range temperature and small discharge rate conditions can make a relatively accurate prediction of the model error under conditions beyond the range of temperature and large discharge rate.
- The neural network trained under small-range discharge rate conditions can also make a good prediction of the model error under the charging rate. Although the output value of the neural network is slightly worse than the effect of other expressions in some working conditions, overall, the BP neural network has better predictability and adaptability than each order expression, indicating that the proposed method of small sample data expansion has certain feasibility.
2.3 Model Error Analysis with BP Neural Network Compensation
The controlled voltage source generated by the BP neural network is connected in series into the second-order RC equivalent circuit model to achieve dynamic compensation of the terminal voltage of the battery model. The improved second-order RC equivalent circuit model with the additional controlled voltage source is shown in Figure 8, where VBP(I, T) is the controlled voltage source, which is affected by the two independent variables of current and temperature.
To verify the accuracy of the improved model, the improved model terminal voltage and the measured terminal voltage under different operating conditions in Figure 5 are compared. Taking the discharge rate of 1.2C as an example, the comparison waveform diagrams of the terminal voltage before and after the model improvement at different temperatures are shown in Figure 9.
It can be seen from Figure 9 that under different temperature operating conditions, the terminal voltage curve of the improved model can better approach the measured terminal voltage curve, with a smaller terminal voltage error, indicating that the accuracy of the model after adding the BP neural network compensation is significantly improved. The error comparison before and after the model improvement is shown in Table 6. It can be seen from the data in Table 6 that the average error value and the average relative error value of the improved model are significantly reduced, indicating that the adaptability of the improved model under different temperatures and current rates is improved, and the simulation effect on the battery is better.
- Application of Energy Storage System Based on the Model
Using the model proposed in this paper, the battery cells in the energy storage system can be sorted and simplified for consistency to form an energy storage system for new energy power generation. At the same time, based on the model, the prediction analysis of the heating of the battery cells and the application of battery thermal management can be carried out, which is beneficial to the safety assessment of the energy storage system. Based on this model, the prediction of the terminal voltage and SOC of the battery in the energy storage system can be carried out, which can be used to evaluate the operational energy efficiency simulation of the energy storage system, as shown in Figure 10.
4. Conclusion
Based on the second-order RC equivalent circuit model, this paper proposes a model improvement method with an additional controlled voltage source. Through theoretical analysis and experimental results, the effectiveness of the proposed method and the improved model is verified, and the following main conclusions are obtained:
- The BP neural network has better predictability than the expression, and the proposed method of predicting large sample data with small sample data expansion has certain feasibility and has certain application value in practical engineering.
- The terminal voltage curve of the improved model can better follow the actually measured battery terminal voltage curve, and the simulation effect on the battery is better at different temperatures and current rates, and the accuracy of the model is improved.
- This model can be used for the sorting of cascade batteries and the monitoring and evaluation of the operating status of the energy storage system, which can promote the application of lithium-ion batteries in new energy power generation and energy storage systems.