# Application of the Waveform Relaxation Technique to the Co-Simulation of Power Converter Controller and Electrical Circuit Models

## TL;DR

This paper introduces a waveform relaxation-based co-simulation method for power converter controllers and electrical circuits, enabling separate yet coupled simulation with mathematical analysis of iteration bounds.

## Contribution

It presents a novel co-simulation approach combining fixed and adaptive time stepping schemes using waveform relaxation, with mathematical analysis and practical application example.

## Key findings

- Effective coupling of controller and circuit models demonstrated
- Mathematical bounds for algorithm iterations established
- Application to Large Hadron Collider circuit validated

## Abstract

In this paper we present the co-simulation of a PID class power converter controller and an electrical circuit by means of the waveform relaxation technique. The simulation of the controller model is characterized by a fixed-time stepping scheme reflecting its digital implementation, whereas a circuit simulation usually employs an adaptive time stepping scheme in order to account for a wide range of time constants within the circuit model. In order to maintain the characteristic of both models as well as to facilitate model replacement, we treat them separately by means of input/output relations and propose an application of a waveform relaxation algorithm. Furthermore, the maximum and minimum number of iterations of the proposed algorithm are mathematically analyzed. The concept of controller/circuit coupling is illustrated by an example of the co-simulation of a PI power converter controller and a model of the main dipole circuit of the Large Hadron Collider.