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Bounding the Minimal Current Harmonic Distortion in Optimal Modulation of Single-Phase Power Converters

Jared Miller, Petros Karamanakos, Tobias Geyer

arXiv:2512.08201v1eess.SYcs.SYmath.OC

TL;DR

OPP design seeks switching signals that minimize current harmonic distortion, but its mixed-integer nonconvex optimization is difficult and existing numerical methods cannot guarantee global optimality. The paper formulates OPPs as periodic mode-selecting hybrid-system optimal control and applies convex relaxation. The resulting semidefinite programs lower-bound minimal distortion, with jointly subquadratic scaling in converter levels and switching transitions.

  • Problem

    OPP optimization combines integer-valued pulse-pattern choices with trigonometric nonconvexity, and numerical methods cannot guarantee global optimality.

    Source: §I · pp. 1–2

  • Method

    The paper casts OPP design as periodic mode-selecting hybrid-system optimal control and applies convex relaxation to obtain semidefinite-program lower bounds.

    Source: §I · p. 2, C. Hybrid Optimal Control Problem · p. 7, §V · p. 10

  • Results

    The semidefinite-program time complexity has jointly subquadratic scaling in converter levels N and switching transitions k.

    Source: D. Convex Relaxation · p. 8

  • Takeaways & Limitations

    The resulting lower bounds provide a benchmark for assessing the optimality of a given pulse pattern.

    Source: §V · p. 10

Abstract

from arXiv · show

Optimal pulse patterns (OPPs) are a modulation technique in which a switching signal is computed offline through an optimization process that accounts for selected performance criteria, such as current harmonic distortion. The optimization determines both the switching angles (i.e., switching times) and the pattern structure (i.e., the sequence of voltage levels). This optimization task is a challenging mixed-integer nonconvex problem, involving integer-valued voltage levels and trigono metric nonlinearities in both the objective and the constraints. We address this challenge by reinterpreting OPP design as a periodic mode-selecting optimal control problem of a hybrid system, where selecting angles and levels corresponds to choosing jump times in a transition graph. This time-domain formulation enables the direct use of convex-relaxation techniques from optimal control, producing a hierarchy of semidefinite programs that lower-bound the minimal achievable harmonic distortion and scale subquadratically with the number of converter levels and switching angles. Numerical results demonstrate the effectiveness of the proposed approachs

I. INTRODUCTION

Optimal pulse patterns (OPPs) allow switching instants and pattern structure to be optimized for harmonic distortion, but their mixed-integer, nonconvex design problem has no globally optimality guarantee from existing numerical methods. This paper recasts OPP design as hybrid-system optimal control, enabling convex lower bounds on minimal current distortion.

I. INTRODUCTION

  • Programmed PWM methods such as SHE and OPP optimize switching instants offline, unlike conventional PWM methods with deterministic switching instants.SHE solves nonlinear equations for a fixed switching sequence; OPPs optimize a criterion such as current TDD and can relax symmetry assumptions.Source: §I · pp. 1–2
  • OPP computation is mixed-integer and nonconvex, and explored numerical methods cannot guarantee a globally optimal solution.The integer variables encode pulse sequence, while nonconvexity stems from switching-angle trigonometry.Source: §I · p. 2
  • The moment-SOS hierarchy provides a systematic sequence of semidefinite programs whose optimal values form monotonically nondecreasing lower bounds.This hierarchy applies when problem data are polynomial; a prior three-phase OPP approach used Taylor and minimax interpolation approximations.Source: §I · p. 2
  • The paper interprets OPP design as mode-selecting hybrid-system optimal control and applies convex relaxation to lower-bound minimal TDD.The time-domain formulation selects transition times and arcs and applies moment-SOS without Taylor or interpolation approximations.Source: §I · p. 2

B. Model of the Power Electronic System

  • The modeled system uses a converter voltage source supplying a resistive-inductive load, with current tracking as a low-ripple control objective.The load can represent a grid or electrical machine, and the converter is assumed to produce N discrete voltage levels.Source: B. Model of the Power Electronic System · p. 3

C. Optimal Pulse Patterns

Optimal pulse patterns choose switching angles and voltage levels to minimize load-current total demand distortion (TDD). Symmetries can constrain harmonics and reduce the number of switching angles needed to specify a pattern.

C. Optimal Pulse Patterns

  • OPPs minimize load-current TDD by choosing switching angles and voltage levels; for three-level converters, the pulse number is d = k/4 and average device switching frequency is fsw = kf1.The switching signal is 2π-periodic, with an even number k of transitions per fundamental period.Source: C. Optimal Pulse Patterns · p. 3
  • The Fourier coefficients depend nonlinearly on the switching signal and angles, while the pulse-driven current-energy expression is piecewise cubic when τ = 0 and A = 0.For other values of τ or A, the expression is piecewise nonlinear; the stated formula assumes a fixed switching sequence and neglects the load-source contribution.Source: C. Optimal Pulse Patterns · pp. 3–4
  • Minimizing normalized load-current energy ∥I∥2 2 is equivalent to minimizing TDD when the fundamental coefficients are fixed.The equivalence follows from Parseval’s relation and the fixed fundamental coefficients.Source: C. Optimal Pulse Patterns · p. 4

D. Symmetries

  • Half-wave symmetry removes even-order harmonics, while quarter-and-half-wave symmetry sets all cosine coefficients to zero; both enforce zero mean.Half-wave- and quarter-and-half-wave-symmetric patterns are determined by the first k/2 and k/4 switching angles, respectively.Source: D. Symmetries · p. 4, Figure 3 · p. 4

E. Constraints in Optimal Pulse Patterns

The OPP formulation selects switching angles and level indices to minimize normalized load-current energy subject to design and system requirements. Its hybrid-system mapping is developed for full-wave symmetry, with other symmetry cases detailed in Appendix B.

E. Constraints in Optimal Pulse Patterns

  • The objective uses normalized load-current energy because minimizing it is equivalent to minimizing load-current TDD.The equivalence is stated through Parseval’s relation.Source: E. Constraints in Optimal Pulse Patterns · p. 4
  • Users may choose an even switching-angle count between kmin and kmax, select FW, HW, or QaHW symmetry, and impose unipolarity for HW or QaHW patterns.Unipolarity requires u(θ) ≥ 0 over [0, π] and can be enforced by restricting allowable transitions.Source: E. Constraints in Optimal Pulse Patterns · p. 4
  • Harmonic constraints apply to selected Fourier coefficients, while the interlocking angle derives from the device’s minimum switching interval Ts as Θ = Tsf1.The interlocking interval is specified to prevent short circuits in dc-link capacitors.Source: E. Constraints in Optimal Pulse Patterns · pp. 4–5

F. Optimal Pulse Pattern Problem

  • For fixed k, the OPP problem chooses switching angles and level indices to minimize normalized load-current energy, with integer levels, transcendental harmonic constraints, and nonlinear objective terms making it nonconvex.Level indices permit searches over inhomogeneous converter levels.Source: F. Optimal Pulse Pattern Problem · p. 5, Symmetry and Unipolarity · p. 5

Symmetry and Unipolarity.

  • When fundamental coefficients are fixed, the optimal pattern for zero external signal also applies to a sinusoidal load-source signal because TDD ignores the fundamental component.This claim is stated for the symmetry and unipolarity setting described in the passage.Source: Symmetry and Unipolarity · p. 5

III. OPP AS MODE-SELECTING OPTIMAL CONTROL

  • The hybrid-system mapping targets FW symmetry; adaptations for HW or QaHW symmetry, including possible unipolarity, are given in Appendix B.Assumptions require even kmax, positive interlocking angle, finitely many sorted converter levels, and finite-dimensional bounded harmonic constraints; the first two prevent Zeno executions.Source: §III · p. 5, A. Assumptions · p. 5

B. Hybrid System Description

The hybrid formulation represents each voltage-level choice and switching transition as a mode in a graph, with switching angles encoded as dwell times. Trajectory integrals then express the distortion objective and harmonic constraints.

B. Hybrid System Description

  • A switching angle-level pair corresponds to the dwell time spent in a mode, and representing angles through cosine and sine states enables polynomial-optimization convex relaxations.The hybrid modes are indexed by converter level and elapsed transition count, with linear dynamics differing by applied voltage.Source: B. Hybrid System Description · p. 5

1) Transition Graph :

  • The transition graph has N(k + 1) vertices and 2(N − 1)k edges, with vertices indexing level and elapsed transitions and edges representing step-up or step-down moves.A periodic path starts and ends at the same level; its level sequence encodes the switching pattern, while dwell times encode switching angles.Source: 1) Transition Graph · p. 5

2) Mode and Switch Dynamics:

  • Within each mode, cosine and sine states rotate with the angle, the clock advances, and current evolves according to the applied voltage; switches reset the clock while preserving the other states.The interlocking-time guard requires the clock variable ϕ to be at least Θ before a transition.Source: 2) Mode and Switch Dynamics · p. 6

3) Integration:

  • The distortion objective accumulates current-squared energy across traversed modes, while harmonic constraints become trajectory integrals expressed using Chebyshev polynomials.The mode and edge maps identify which modes and transitions are occupied at each angle.Source: 3) Integration · pp. 6–7

C. Hybrid Optimal Control Problem

Problem 3.5 casts OPP design as an optimal-control task over an initial condition and a sequence of hybrid-system transitions. For fixed k, it has the same objective as Problem 2.1, and its interlocking constraint ensures feasible timing.

C. Hybrid Optimal Control Problem

  • Problem 3.5 formulates OPP design as an optimal-control task that selects an initial condition and sequence T under device and design requirements.The formulation includes symmetry and unipolarity restrictions on the path.Source: C. Hybrid Optimal Control Problem · p. 7
  • For fixed k, Problems 2.1 and 3.5 have the same objective, and every feasible Problem 2.1 pattern has a representation through T.The correspondence maps the switching signal to a path P.Source: C. Hybrid Optimal Control Problem · p. 7
  • A generic sequence T may violate interlocking timing, but constraint (29e) ensures the required interlocking time is obeyed.Source: C. Hybrid Optimal Control Problem · p. 7

D. Convex Relaxation

The relaxation represents pulse patterns through convex dwell tables and moment-based semidefinite programs, whose bounds tighten with polynomial degree. Numerical examples illustrate how these bounds compare with recovered patterns and SHE solutions.

D. Convex Relaxation

  • The convex relaxation uses moment-based techniques to produce semidefinite-program lower bounds on the TDD objective.The programs use pseudo-moments; increasing polynomial degree β tightens their necessary conditions and yields a rising sequence of lower bounds.Source: D. Convex Relaxation · pp. 7–8
  • A dwell table records mode occupancy, while a mixed dwell table is a convex combination that need not correspond to one pulse pattern.Pure dwell tables map one-to-one to pulse patterns; approximate extraction from a mixed table can yield a pattern that violates harmonics or interlocking constraints.Source: D. Convex Relaxation · p. 7
  • Under FW symmetry, the semidefinite-program solve has worst-case complexity O((Nk)^(3/2)β^16), jointly subquadratic in converter levels N and switching angles k.This scaling follows from |V| = N(k + 1) and |E| = 2Nk.Source: D. Convex Relaxation · p. 8

A. Single Pattern

  • The order-3 relaxation gives a current-energy lower bound of 1.6092, while the candidate pattern’s energy exceeds it by 2.151×10^-5.Direct pattern extraction was infeasible for the required b1 = 0.8; the candidate was obtained by warm-starting fmincon and IPOPT at a fixed level sequence.Source: A. Single Pattern · p. 8
  • For the five-level, 24-angle single-pattern example, the TDD lower-bound gap is 2.2799×10^-4.The pattern uses QaHW symmetry, b1 = 0.8, and load ratio τ = 0.5.Source: A. Single Pattern · p. 8
  • For k = 28, the recovered pattern has energy 1+1.5873×10^-3, below the SHE pattern’s 1+5.950×10^-3 and the order-3 lower bound of 1+5.812×10^-3.The recovered pattern follows Algorithm 1 and fmincon local search; its modulation sequence matches SHE.Source: A. Single Pattern · p. 9

B. Parameter Sweep: Length vs. Degree

The parameter sweeps examine how relaxation bounds and computation vary with pattern length, degree, load ratio, and modulation index. The conclusion frames the bounds as benchmarks for assessing pulse-pattern optimality and identifies extensions and computational improvements as future work.

B. Parameter Sweep: Length vs. Degree

C. Parameter Sweep: Load Ratio vs. Degree

D. Parameter Sweep: Modulation Index vs. Length

V. CONCLUSION

  • The moment-SOS hierarchy provides lower bounds on minimal achievable distortion under device and design constraints, which can benchmark pulse-pattern optimality.Future work includes three-phase extensions, additional device-loss constraints, and reducing semidefinite-program complexity.Source: §V · pp. 10–11

APPENDIX A SIGNAL ENERGY EXPLICIT EXPRESSION

The appendix gives expressions for signal energy across load cases and describes how symmetry and unipolarity restrictions are encoded in transition graphs and support sets.

APPENDIX A SIGNAL ENERGY EXPLICIT EXPRESSION

  • The signal-energy formulation distinguishes current responses from the pulse pattern and external signal, and assumes the initial pattern current is given.For a zero-mean pattern, the initial current can be chosen to produce a zero-mean, minimal-energy waveform.Source: Appendix A · p. 11

A. Pure Reactance

  • For τ = 0, the pure-reactance case defines current values at switching angles and gives energy contributions by interval.The current at each switching angle is denoted Ii p = Ip(αi).Source: A. Pure Reactance · p. 11

B. Mixed Load Characteristic

  • For τ > 0, the mixed-load case uses angle ψ = tan^-1(τ) and sums energy contributions from the pulse-pattern signal and its mixture with the external signal.The component energies are summed as in equation (45).Source: B. Mixed Load Characteristic · p. 11

A. Transition Graph

  • Symmetry and unipolarity are imposed by restricting the transition graph, including which vertices can be initial or reachable.HW and QaHW symmetry reduce the vertices that need consideration; unipolarity removes vertices below (N + 1)/2 in the first half-period.Source: A. Transition Graph · p. 12, Appendix B · p. 12

B. Support Sets

  • The symmetry-restricted formulation tracks only the first π for HW or π/2 for QW, then completes the signal by symmetry.The specialized QaHW support construction applies only when τ = 0 and ϕ = 0; otherwise the HW construction is used with an HW-compatible graph.Source: B. Support Sets · p. 12
  • Symmetry also constrains path endpoints: an HW path starting at (n, 0) terminates at (N−n, k/2), while a QW path starts at ((N + 1)/2, 0).The QW endpoint may be any (n, k/4) for which n + k/4 is even.Source: B. Support Sets · p. 12

C. Integration

The appendix defines integration operators for symmetry-restricted patterns and reviews the measure concepts used to form convex relaxations. Occupation measures express trajectories as measures, while a Liouville conservation relation characterizes admissible initial, terminal, and occupation measures.

C. Integration

  • HW and QW integration operators use symmetry-specific intervals and averaging operators, with simplified forms for functions matching the pattern’s symmetry.For HW-symmetric functions, averaging becomes a factor of 2; for QW-symmetric or antisymmetric functions, it becomes a factor of 4.Source: C. Integration · pp. 12–13

APPENDIX C MEASURE THEORY AND OCCUPATION MEASURES

  • The appendix introduces Borel measures and their integration pairing, then defines moments as integrals of monomials against a measure.These measure constructions are used to construct convex relaxations to Problem 3.5.Source: Appendix C · p. 13, A. Analysis · p. 13

B. Standard and Occupation Measures

  • An occupation measure represents time spent by a trajectory across its state-space curve and is the pushforward of time measure along the curve graph.The text also expresses trajectory evolution through a conservation law using the occupation measure.Source: B. Standard and Occupation Measures · p. 13
  • The Liouville relation generalizes the conservation law to initial, terminal, and occupation measures; under stated compactness and Lipschitz conditions, satisfying tuples correspond to convex combinations of trajectories.The stated result allows possibly infinite convex combinations.Source: B. Standard and Occupation Measures · p. 13
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