Optimal Pulse Patterns through a Hybrid Optimal Control Perspective

Optimal pulse patterns (OPPs) are a modulation method in which the switching angles and levels of a switching signal are computed via an offline optimization procedure to minimize a performance metric, typically the harmonic distortions of the load current. Additional constraints can be incorporated into the optimization problem to achieve secondary objectives, such as the limitation of specific harmonics or the reduction of power converter losses. The resulting optimization problem, however, is highly nonconvex, featuring a trigonometric objective function and constraints as well as both real- and integer-valued optimization variables. This work casts the task of OPP synthesis for a multilevel converter as an optimal control problem of a hybrid system. This problem is in turn lifted into a convex but infinite-dimensional conic program of occupation measures using established methods in convex relaxations of optimal control. Lower bounds on the minimum achievable harmonic distortion are acquired by solving a sequence of semidefinite programs via the moment-sum-of-squares hierarchy, where each semidefinite program scales in a jointly linear manner with the numbers of permitted switching transitions and converter voltage levels.