Comparative Analysis of Single-Branch, Two-Branch, and Three-Branch RC Equivalent Circuit Models for Supercapacitor Simulation in OpenModelica

Energy storage is the silent engine of modern technology, powering everything from the smartphone in your pocket to the electric bus navigating city streets. While batteries have long held the crown for storing large amounts of energy, they often struggle when asked to release that energy quickly or to absorb a sudden surge of power. This is where supercapacitors step in. Unlike batteries, which store energy through chemical reactions, supercapacitors store energy electrostatically, much like a bucket holding water. They can charge and discharge in seconds, handle millions of cycles without wearing out, and deliver massive bursts of power. Because of these traits, they are becoming essential for regenerative braking in vehicles, stabilizing power grids, and smoothing out energy fluctuations. However, to use them effectively, engineers must be able to predict exactly how they will behave under different conditions. This requires a mathematical map, or a model, that simulates the device's internal physics. If the map is too simple, it leads engineers astray; if it is too complex, it becomes too slow to use in real-time design.

A researcher at the Air Force Institute of Technology in Kaduna, Nigeria, set out to test three different ways of drawing this map for a specific, commercially available supercapacitor. They focused on a device rated at 310 farads and 2.7 volts, a common component in the industry. The researcher used a free, open-source software environment called OpenModelica to run their experiments. This software allows engineers to write equations that describe how physical systems behave and then watch those systems run on a computer screen. The researcher compared three distinct models: a single-branch model, a two-branch model, and a three-branch model. In the simplest version, the single-branch model, the supercapacitor is treated as a single capacitor connected to a single resistor. It is a straightforward approach, but it assumes the device reacts instantly to changes in current. The two-branch model adds a second layer of complexity, introducing a second capacitor and resistor that react more slowly, mimicking the way charge moves between different layers inside the device. The most detailed version, the three-branch model, adds a third layer to account for very slow processes that happen over minutes or hours, such as the gradual leakage of energy or the slow redistribution of charge deep within the material.

To see which model worked best, the researcher subjected all three versions to the exact same test: a rigorous cycle of charging, resting, and discharging. They pushed a steady current of 10 amps into the device for two and a half minutes, let it sit idle for thirty seconds, and then drew the power out at the same rate for another two and a half minutes. This specific pattern is a standard way to test supercapacitors because the rest period reveals a crucial behavior known as voltage relaxation. When the current stops, the voltage in a real supercapacitor does not stay flat; it shifts as the internal charge settles. The researcher measured how closely each model's predicted voltage matched the reference three-branch model, which they treated as the most accurate representation of reality. They also tracked how much energy was stored and how much heat was generated during the process.

The results were stark. The simplest, single-branch model failed to capture the reality of the device's behavior. During the charging phase, it predicted a peak voltage that was nearly 0.8 volts higher than the more complex models. More importantly, during the thirty-second rest period, the single-branch model showed a flat line, completely missing the voltage relaxation that occurs in the real device. Because it ignored the internal delays, it also calculated that the device generated almost no heat, underestimating the peak heat generation by about 84 percent compared to the reference model. This is a critical finding, as using such a simplified model for thermal design could lead to dangerous overheating in real-world applications. The single-branch model also suggested that the device stored almost no energy after a full discharge, whereas the more complex models showed that a significant amount of energy remained trapped in the slower-reacting parts of the system.

In contrast, the two-branch model proved to be a remarkably effective middle ground. By adding just one extra state variable and a few more parameters, it reduced the voltage prediction error by a factor of twenty compared to the simple model. Its peak error dropped to just 0.038 volts, and it successfully reproduced the voltage relaxation during the rest period. The researcher found that this model captured the dominant behavior of the supercapacitor with high accuracy while remaining simple enough for practical use in energy management systems. The three-branch model, while the most accurate, offered only marginal improvements over the two-branch version for this specific test cycle, though it did account for the very slow, long-term energy storage that the two-branch model missed.

The study concludes that for most simulation purposes, the two-branch model offers the best balance between simplicity and precision. It provides a level of accuracy that is sufficient for designing control systems and managing energy, without the computational burden of the most complex model. The researcher made all their code and simulation scripts available to the public, allowing other engineers to replicate the work or build upon it. Their next step involves taking the most accurate three-branch model and coupling it with a thermal simulation tool to create a complete picture of how heat and electricity interact inside the device. This work highlights that in the world of energy storage, the right level of detail matters; a model that is too simple can lead to costly errors, while one that is just right can unlock safer and more efficient designs.