Circuit-to-Circuit Cosimulation for Closed-Loop Electrical Systems Using Waveform Relaxation

A computational time saving approach is proposed for the cosimulation of a buck–boost converter using waveform relaxation (WR) and time windowing. This study implements the Gauss–Seidel (GS) method, an iterative process that is employed to solve systems of linear and nonlinear equations using the WR technique, improving the convergence speed and accuracy of closed-loop system dynamic simulations. The proposed technique is applied to a 3.3-V buck–boost converter working both in buck and boost modes, therefore, simulations and measurements are carried out with a 1.8- to 5.5-V input voltage range. A comparative analysis between GS-WR without and with time windowing demonstrates that with time windowing, the number of iterations increases by 60<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> and 67.8<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula>, while the elapsed time is reduced by 22.3<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> and 26.6<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> with respect to a single time windows in buck and boost modes, respectively. Moreover, the optimal number of time windows (i.e., 4), computed by simulation, yields a 8.1<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> and 18.1<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> faster simulation time compared to the highest considered value in the study (i.e., 10). The outcome of this comparison reveals that a higher number of time windows does not necessarily result in a quicker computation compared to a lower value. More specifically, the GS-WR technique was found to be the main contributor to the acceleration, while time windowing ensures the convergence to the same results as the full system simulation.