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Optimal Pulse Patterns through a Hybrid Optimal Control Perspective

Jared Miller, Petros Karamanakos

TL;DR

The paper addresses designing optimal pulse patterns (OPPs) for multilevel converters to minimize harmonic distortion, a nonconvex problem due to trigonometric terms and mixed-integer decisions. It reframes OPP synthesis as a hybrid optimal-control problem and lifts it to an infinite-dimensional linear program over occupation and jump measures, enabling convex relaxations via the moment-SOS hierarchy. This framework yields computable lower bounds on distortion and a practical recovery path to feasible switching sequences, with complexity scaling linearly in the number of levels and allowed transitions. Numerical results on a five-level converter demonstrate tight SDP bounds and feasible patterns, illustrating a scalable approach to globally informative OPP design that can accommodate additional constraints and objectives.

Abstract

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.

Optimal Pulse Patterns through a Hybrid Optimal Control Perspective

TL;DR

The paper addresses designing optimal pulse patterns (OPPs) for multilevel converters to minimize harmonic distortion, a nonconvex problem due to trigonometric terms and mixed-integer decisions. It reframes OPP synthesis as a hybrid optimal-control problem and lifts it to an infinite-dimensional linear program over occupation and jump measures, enabling convex relaxations via the moment-SOS hierarchy. This framework yields computable lower bounds on distortion and a practical recovery path to feasible switching sequences, with complexity scaling linearly in the number of levels and allowed transitions. Numerical results on a five-level converter demonstrate tight SDP bounds and feasible patterns, illustrating a scalable approach to globally informative OPP design that can accommodate additional constraints and objectives.

Abstract

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.

Paper Structure

This paper contains 24 sections, 2 theorems, 36 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Optimal Pulse Patterns
  5. OPPs as Hybrid Control
  6. Assumptions
  7. Hybrid System Description
  8. Mode and Switch Dynamics
  9. Optimal Control Problem
  10. Convex Linear Measure Program
  11. Measure Preliminaries
  12. Involved Measures
  13. Construction Procedure
  14. Convex Reformulation
  15. Objective
  16. ...and 9 more sections

Key Result

Proposition III.6

Problems prob:tdd_orig and prob:tdd_hy have the same objective $(J^* = J^*_{hy})$.

Figures (8)

  • Figure 1: QaHW signals $u(\theta)$ and $I(\theta)$ with $L = \{-1, 0, 1\}$
  • Figure 2: Transition graphs for $d=8, N=7$
  • Figure 3: Valid regions for $(c, s)$ under $d=4, \ \Theta = \pi/18$
  • Figure 4: Continuity relation between measures
  • Figure 5: $Q$ bounds and timing for $d=8$, $M=0.9$
  • ...and 3 more figures

Theorems & Definitions (6)

  • Proposition III.6
  • proof
  • Remark 1
  • Theorem IV.2
  • proof
  • Remark 2