Dimensionally Consistent Fractional-Order Averaged Modeling and Stability of an Interleaved Bidirectional DC–DC Converter for More-Electric Aircraft

Modern aircraft are undergoing a quiet revolution. The heavy, noisy systems that once relied on hydraulic fluid and pneumatic air pressure are being replaced by electric motors and electronic controls. This shift, known as the "more-electric aircraft," demands a power grid that is lighter, more efficient, and incredibly reliable. At the heart of this new system are converters, devices that act like traffic directors for electricity, stepping voltage levels up or down to match the needs of different components. In these aircraft, a critical task is managing the flow of power between a high-voltage bus, typically around 270 volts, and lower-voltage systems that run at 90 volts. These converters must handle power in both directions, moving energy from the main bus to the loads and back again, all while keeping the current perfectly balanced between parallel paths to prevent overheating.

For engineers, predicting how these converters behave during sudden changes is a constant challenge. When a pilot demands more power or the system switches modes, the voltage and current do not settle instantly; they ripple, overshoot, and slowly decay. Traditional mathematical models treat these systems as if they have no memory of the past, assuming that once a disturbance stops, the system snaps back to normal in a predictable, exponential curve. However, real-world electronics often exhibit "long-tail" behavior, where the system remembers past events for a surprisingly long time, causing slow, lingering adjustments that standard models struggle to capture. This is where the work of researchers Taylan Demir and Atakan Koçyiğit comes in. They have developed a new way to model these aircraft power converters that accounts for this memory without breaking the fundamental laws of physics.

The researchers focused on a specific type of converter used in aircraft: a two-phase interleaved bidirectional DC-DC converter. "Interleaved" means the device uses two parallel lanes to share the work, which reduces stress on the components and smooths out the current. "Bidirectional" means it can push power forward or pull it back. To understand how this machine behaves, the team created a mathematical description that is "dimensionally consistent." In simpler terms, they ensured that the units of measurement—like seconds, volts, and amps—remained correct throughout their equations. Previous attempts to add "memory" to these models often resulted in equations where the units no longer matched, making the results physically meaningless. The team solved this by introducing a reference time, a fixed scale that allows them to adjust the model's memory without changing the physical reality of the inductors and capacitors inside the hardware.

Using this new framework, the researchers explored what happens when the fractional order of the model, a number that controls how much memory the system has, is changed. In their simulations, they tested values ranging from the standard integer value of one down to lower numbers like 0.9 and 0.8. They found that while the final, steady state of the system—the voltage it settles at after everything calms down—remained exactly the same regardless of the memory setting, the journey to get there changed significantly. When the memory was increased (by lowering the fractional order), the system became more cautious. It avoided the sharp, jarring spikes in voltage that the standard model predicted. Instead of a quick, oscillating bounce, the voltage rose more gently and settled with a slower, more gradual decay. This "long tail" of behavior, which the researchers describe using a specific mathematical function known as the Mittag-Leffler function, suggests that the system retains a history of its past movements, smoothing out the transition.

Crucially, the team also discovered that this memory effect behaves differently depending on whether the two parallel lanes of the converter are working together or fighting against each other. When the lanes are balanced, the system responds smoothly. However, if one lane carries more current than the other, the imbalance does not disappear instantly. In the standard model, this mismatch would vanish quickly and exponentially. In the new fractional model, the mismatch fades away much more slowly, following a different, more complex curve. This finding is vital for aircraft safety because it implies that even if the control system sends identical commands to both lanes, small differences in current can linger longer than previously thought, requiring different strategies for monitoring and fault detection.

The researchers were careful to clarify what their work does and does not prove. They did not claim that a specific fractional number is a hidden constant of nature inside every aircraft converter. Instead, they demonstrated that their new mathematical tool is a valid, consistent way to represent the complex, memory-filled transients seen in real-world data. They ran their simulations on a representative 270-volt to 90-volt system, using parameters typical for aerospace applications, such as inductors of 250 microhenries and capacitors of 1 millifarad. The results showed that by adjusting the fractional order, engineers could tune the model to match the specific "tail" behavior observed in experiments without altering the basic design of the converter.

This work serves as a bridge between simple, traditional models and the messy reality of high-performance electronics. It offers a reduced-order tool that engineers can use to study transient phenomena—those fleeting moments of change that happen when an aircraft powers up or shifts loads. By separating the memory effects from the core hardware design, the researchers provide a way to study how these systems behave under stress without having to build a new physical prototype for every test. The next step, as the authors note, is to take this theoretical framework and test it against real hardware, measuring the actual currents and voltages to see if the fractional model truly captures the behavior of a physical aircraft converter. Until then, this study stands as a rigorous, mathematically sound method for understanding how the memory of the past shapes the power of the future in the skies.