How Are Optimal Pulse Patterns Designed for Multilevel Converters?

Optimal pulse patterns (OPPs) are a technique to control power converters. Basically, the switching angles and levels of a signal are computed offline to minimize load current harmonic distortion. Besides limiting harmonics, extra constraints can be added to cut converter losses. The problem is tough: the objective function is trigonometric and involves both continuous and integer variables, making it non‑convex. The authors treat OPP synthesis as an optimal control problem for a hybrid system. They then lift it into a convex, infinite‑dimensional conic program using occupation measures. This allows them to obtain lower bounds on the minimal achievable harmonic distortion by solving a sequence of semidefinite programs (SDPs) in the moment‑sum‑of‑squares hierarchy. Each SDP scales linearly with the allowed switching transitions and the converter voltage levels, making the approach scalable for real‑world use.

In the realm of multilevel converters, OPPs represent an advanced multiple‑switching modulation strategy. The original synthesis problem is a non‑convex optimization with mixed variables: switching angles (continuous) and voltage levels (integer). To overcome this, the authors employ occupation measure theory, which describes a dynamic system's behavior via a probability distribution over its trajectory space. This transformation yields a conic moment program, where the variables are the moments of the occupation measures. The moment‑sum‑of‑squares (SOS) hierarchy provides a sequence of increasingly tight semidefinite relaxations, guaranteeing convergence to the global optimum under regularity conditions. Each hierarchy level requires solving an SDP whose size scales with the number of monomials of degree ≤2d, d being the hierarchy order. Since the number of switching transitions and voltage levels are discrete parameters, the SDP complexity grows linearly with them, making the method feasible for converters with many levels. Moreover, the hybrid nature of the system allows direct inclusion of on‑off constraints in the formulation, preserving solution feasibility. This approach paves the way for systematic OPP design with theoretical optimality guarantees, surpassing traditional heuristic or local search methods.