Closed-Form Solutions for Grid-Forming Converters: A Design-Oriented Study

Fangzhou Zhao, Tianhua Zhu, Lennart Harnefors, Bo Fan, Heng Wu, Zichao Zhou, Yin Sun, Xiongfei Wang

Research output: Contribution to journalJournal articleResearchpeer-review


This paper derives closed-form solutions for gridforming converters with power synchronization control (PSC) by subtly simplifying and factorizing the complex closed-loop models. The solutions can offer clear analytical insights into control-loop interactions, enabling guidelines for robust controller design. It is proved that 1) the proportional gains of PSC and alternating voltage control (AVC) can introduce negative resistance, which aggravates synchronous resonance (SR) of power control, 2) the integral gain of AVC is the cause of sub-synchronous resonance (SSR) in stiff-grid interconnections, albeit the proportional gain of AVC can help dampen the SSR, and 3) surprisingly, the current controller that dampens SR actually exacerbates SSR. Controller design guidelines are given based on analytical insights. The findings are verified by simulations and experimental results.

Original languageEnglish
JournalIEEE Open Journal of Power Electronics
Pages (from-to)186-200
Number of pages15
Publication statusPublished - 2024


  • Automatic voltage control
  • Closed-form solutions
  • Converters
  • Damping
  • Grid-connected converter
  • Integrated circuit modeling
  • Power conversion
  • Resistance
  • grid-forming control
  • stability
  • sub-synchronous resonance
  • synchronous resonance


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