Analytical formulation and optimization of Weighted Total Harmonic Distortion in three-phase staircase modulated multilevel inverters

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Staircase Modulation (SCM) is a popular switching strategy for three-phase multilevel inverters (MLI) of higher voltage levels count (N). SCM is often combined with some harmonic mitigation strategy, such as a minimum of Total Harmonic Distortion (THD) in addition to output fundamental component adjustments. While voltage THD-based SCM is desirable in many applications, it is not a panacea for all applications, especially for motor drives and grid-connected MLIs, where minimization of current-THD is preferable. A THD alternative performance parameter, called frequency-Weighted Total Harmonic Distortion (WTHD) can serve as a strong indication of current-THD, especially in inductive loads. For three-phase MLIs with 3-wire loads, line voltage waveform has more significance than phase voltage waveform, as it directly impacts the load's current quality. For this reason, the minimization of Line-voltage WTHD (WLTHD) may be preferred over the minimization of THD, Line-THD (LTHD) or WTHD. Since WLTHD involves complicated expressions, it is typically calculated and expressed using frequency-domain numerical approaches, often resulting in underestimation errors. In this paper, a novel analytical WLTHD formulation for three-phase SCM-based MLIs is revealed and used for subsequent WLTHD minimization with the aim of minimizing load current-THD for 3-wire inductive loads. The proposed WLTHD formulation is a function of all possible Phase Switching Angles (PSA), valid for any MLI topology with arbitrary values of N, either odd or even. Moreover, it is shown that the existing WLTHD minimization approach is a particular case of the proposed generic methodology. The derived formulation, which can produce mathematically exact symbolic WLTHD results, is verified against numerically obtained results in previous works. Controller + Hardware in Loop (C-HIL) based validation is also carried out for 7-level three-phase MLIs feeding an induction motor for current-THD optimization, revealing significant improvements compared to previously obtained methods. The proposed WLTHD minimization possess the highest correlation to current-THD, even for nearly resistive loads. Links for downloading MATLAB and Maple source files of proposed WLTHD formulations as well as pre-calculated optimum PSA results for 5 ≤ N ≤ 16, covering most of the practical cases, are also provided for reader's convenience.

Original languageEnglish
Article number119137
StatePublished - 15 Jan 2021


  • Current THD
  • Line voltage
  • Minimization
  • Optimization
  • Staircase modulation
  • Staircase modulation
  • TermsThree-phase multilevel inverters
  • Three-phase multilevel inverters
  • Total harmonic distortion
  • Weighted THD

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Building and Construction
  • Pollution
  • Mechanical Engineering
  • Industrial and Manufacturing Engineering
  • Electrical and Electronic Engineering


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