While being part of the Formula Student team of NTUA, I worked on building an accurate electrical model of the accumulator. Our final goal was to create a Simulink model that can accurately describe the cell voltage dynamics in the whole State of Charge (SoC) range. When current is drawn from the cells, the terminal voltage drops in a way that is strongly dependent on the state of charge. That means that the output impedance of the cell is a function of the state of charge. Other variables affecting the output impedance are the temperature and the magnitude of the current. In this work those parameters were neglected and we assumed only a SoC dependence.
Embedded Files
A model commonly used to describe Li-ion cells is a voltage source, in series with a resistance and several serially connected RC branches (Fig. 1). The more the branches, the higher the accuracy and complexity of the model. Accuracy is particularly important at low SoC values where the behavior of the cell changes abruptly. We decided to use a 2-RC branch model. This choice appears to be a good compromise between complexity and accuracy. All parameters of the circuit depend on the SoC value of the cell. The aim of this work was to determine this dependence and incorporate the results into lookup tables.
Fig. 1: Equivalent circuit with 2 RC branches
The current drawn from the cells can be considered the excitation of the dynamical system while the terminal voltage of the cell is the response of the system. Current pulses were used since the cell response to a current pulse can reveal critical parameters of the cell in relation to the equivalent circuit used. In Fig. 2, a 4 A current pulse and the respective voltage response are shown.
The voltage response can be divided into two regions: the pulse region that coincides with the time interval during which current is drawn from the cell and the relaxation region that corresponds to the time interval between the current pulses. The topology of the equivalent circuit is in agreement with the form of the voltage response. Initially, we assume the circuit to be at rest. That means that the capacitors have been fully discharged through the resistors and the open circuit terminal voltage is equal to Vcell. During the first moments when the current pulse arrives, the current passes through the resistor R0 and the capacitors C1 and C2 . The terminal voltage is thus the internal voltage of the cell minus the voltage drop at R0 . This voltage drop corresponds to the step decrease in voltage depicted in the measured data.
Fig. 2: Current pulse and voltage response.
Then, the capacitors C1 and C2 start to get charged. The voltage built across them causes a current to flow through the resistors R1 and R2. This current leads to a further voltage drop that builds gradually as the voltage across the capacitors gets higher. The above-mentioned behavior corresponds to the decreasing voltage curve seen immediately after the step voltage drop. Finally, when the current pulse ends the voltage drop at R0 will go to zero. That causes the small step increase in the terminal voltage. As time passes, the capacitors get discharged through the resistors R1 and R2 and the terminal voltage slowly increases until it reaches a constant value that is, however, somewhat lower than the initial open circuit voltage value.
Fig. 3: Pulse and Relaxation Regions
In order to draw current, a current source capable of drawing currents up to 20 A from a 4.2 V battery cell was designed and built. The main elements of the current source are a power MOSFET connected in series with heat dissipation resistors (Fig. 4). A negative feedback scheme was implemented to control the current. A microcontroller was used to apply an analog voltage signal Vsig to the positive input of the opamp. Through the feedback mechanism the same voltage appears across the resistors, producing a proportional current. The voltage response and current drawn were sampled with a frequency of 50 Hz using an Arduino microcontroller with a 10-bit resolution. The data were logged in real time using Simulink. Before each discharging event, the cell was subjected to a slow charging process. The final setup is shown in Fig. 5.
Fig. 4: Current source schematic.
The cell was subjected to current pulses until the minimum allowable voltage was reached. Enough relaxation time was given between the pulses in order to ensure that the open circuit voltage is reached before the next pulse. (i.e., the capacitors in the equivalent circuit are fully discharged). During each pulse the SoC drops linearly. The SoC was measured with the coulomb counting method, that is by simply integrating the current pulses over time and dividing by the total cell capacity. The open circuit voltage Vcell is determined by measuring the voltage after the relaxation period has ended. The R0 resistance is calculated by measuring the step voltage drop at the edges of the current pulses. The parameters can then be converted to the equivalent parameters for the whole accumulator arrangement (140s2p).
Fig. 5: Implemented setup.
Fig. 6: Cell terminal voltage during a complete discharge cycle.
Fig. 7: SoC, cell terminal voltage and to Vcell estimation.
In order to estimate the other RC parameters the measured voltage response data was fitted to the analytical response solutions. The relaxation regions were fitted first, in order to calculate the respective RC time constants. Then, the the data from the pulse regions were fitted and the exact R1 , C1 and R2 , C2 values were estimated. The results are shown in Figs. 8-9.
Fig. 8: Open-circuit voltage and R0 estimation for the whole accumulator (140s2p).
Fig. 9: Estimation of the remaining parameters.
In order to automate the procedure, a script was developed in Matlab that recognizes the respective regions and performs the fitting and estimation method automatically. The parameter data was incorporated into lookup tables that allowed us to build the final accumulator model.
Fig. 10: Simulink equivalent model with lookup tables.
Page updated
Google Sites
Report abuse