Supercapacitors are becoming increasingly important storage system components. To effectively control their terminal voltage, even in real time, numerous circuit models capable of faithfully simulating their behavior in energy systems and various applications are being explored. The three-branch supercapacitor model appears to be a good compromise between simplicity and accuracy. Typically, this model lacks accuracy in dynamic cycling and long stand-by periods. In this study, a new model identification method based on the state equations of the circuit is described and tested on a 400 F supercapacitor, and the obtained results are validated by measurements. Such an approach, suitably optimized, provides good agreement with the measurements, with discrepancies below 50 mV even in repeated cycles. In the static identification, after 90 minutes of self-discharge, the discrepancy was approximately 5 mV. The study also discusses the sensitivity of the model output to the circuit parameters, which is useful for choosing the appropriate timespan for parameter optimization and introduces variable leakage resistance and a method for its determination. Through this parameter, good agreement with the measurements is observed during the long self-discharging phases. A discrepancy of less than 50 mV between the measured and computed results is observed after one week. The union of the circuit state equations based model and the nonlinear leakage resistance determination allows the three-branch circuit model to achieve a high accuracy both in real-time simulation and in the presence of long stand-by phases.

The accurate simulation of the behavior of a supercapacitor (SC) and its control in an electronic system, cannot be achieved by a simple one-branch circuit. A correct simulation requires the use of more complex equivalent circuits, with at least two or three branches. These equivalent circuits guarantee a good reproduction of the device’s behavior. The effectiveness of an equivalent circuit is linked to the limits of the circuit parameter identification, which is commonly achieved by means of the voltage and current measurement of charge and self-discharge cycles. The uncertainty in the identification of these circuit parameters, is dependent on the accuracy of the measurement instrumentation and on the repeatability of the SC. A cycle for determining the parameters can extend over a few hours for larger SCs, also considering the time needed by the software algorithm for the parameter identification. Therefore, having a large set of cycles including the parameters determination is a time-consuming procedure. In this study, an efficient method for the repeatability and uncertainty assessment of the equivalent circuit parameters is proposed; this approach relies on a limited set of experimental data and on a single parameter identification process. The analysis presented in this paper highlights how the limited repeatability of the device is an important source of uncertainty for the identification of the equivalent circuit parameters, but it is not the main one for all parameters.

This study demonstrates the response of the three-branch equivalent circuit model in reproducing the terminal voltage of a supercapacitor that undergoes dynamic galvanostatic pulse trains for frequencies from 0.1 Hz to 100 Hz and signal amplitudes of 1 A, significantly wider than the sinusoidal signals used in impedance spectroscopy. Clean and accurate pulse trains were generated at different frequencies using a transconductance power amplifier, and the voltages at the supercapacitor terminals of three different sizes of supercapacitors were successfully compared with the model. The latter was also utilized to check its ability to accurately reproduce charging and self-discharging processes and the ability to simulate voltammetric cycles. This was verified for six supercapacitor sizes from 1 F (rated 1 mWh, 10 W) up to a large 130 F module (rated 70 Wh, 144 kW). In all cases, the relative difference, with respect to the rated voltage, between the terminal voltage provided by the measurements and that yielded by the model simulations did not exceed 3.5%. This extensive multi-level validation demonstrates how the identified three-branch model based on state equations can provide an accurate electrical design tool and a reference for standards.