Analytical Formulation and Minimization of Voltage THD in Staircase Modulated Multilevel Inverters with Variable DC Ratios

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Staircase Modulation (SCM) is a commonly employed switching strategy in Multilevel Inverters (MLI) with a high number of voltage levels (N). The main challenges in SCM are adjusting the Modulation Index (MI) to the desired value while minimizing the Total Harmonic Distortion (THD). This is achieved by solving an objective function for its optimum variables. These variables are the Switching Angles (SA) in the case of an MLI with Equal DC voltage Supply (EDCS), or both the SAs and the DC voltage supply Ratios (DCR), in the case of an MLI with Unequal DC voltage Supply (UDCS). Such an approach, which is referred to as Optimum Minimization of THD (OMTHD), relies on the accuracy of the THD expression used for the objective function. This paper reveals generic analytical closed-form formulations of both Phase-voltage THD (PTHD) for single-phase (1φ) MLIs and Line-voltage THD (LTHD) for three-phase (3φ) MLIs. The revealed THD expressions apply to SCM based MLIs of any topology, with either EDCS or UDCS voltage source configurations, which may be fixed or varying, and arbitrary value of N (odd and even). Given an arbitrary N value, the proposed unified formulations can be used to generate symbolic N-level PTHD or LTHD expressions, which are analytical functions of the SA and (optionally) DCR variables. The proposed THD expressions were used to form a generic OMTHD problem and solve for optimum SA and DCR variables, ensuring PTHD or LTHD minimization, subject to a set of optional constraints, such as the target MI, tolerable Modulation Error (ME), and the Maximum DCR (MDCR). Outcomes of the proposed OMTHD algorithm were successfully verified by Controller + Hardware-in-Loop (C-HIL) based experiments and compared against results of previous works, raveling significant improvement in THD accuracy and MLI's performance. A downloadable supplemental file containing Maple and MATLAB functions of the proposed THD expressions, as well as pre-calculated sets of optimum SAs and DCRs tables for 13 different values of N (4≤ N ≤16), is provided for readers' convenience.

Original languageEnglish
Article number9261364
Pages (from-to)208861-208878
Number of pages18
JournalIEEE Access
StatePublished - 1 Jan 2020


  • Multilevel inverters
  • optimization
  • staircase modulation
  • total harmonic distortion

ASJC Scopus subject areas

  • Computer Science (all)
  • Materials Science (all)
  • Engineering (all)


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