Approximate Kron Reduction Methods for Electrical Networks With Applications to Plug-and-Play Control of AC Islanded Microgrids

Kron reduction (KR) is a methodology for analyzing an electrical network by replacing it with a simpler circuit having less nodes but the same terminal behavior of voltages and currents at target vertices. Existing approaches to instantaneous KR, however, can fail in preserving the structure of transfer functions representing power lines. Therefore, even if the original lines are passive $RLC$ circuits, reduced lines might have transfer functions that do not correspond to a physically realizable passive system. To overcome this drawback, in this paper we focus on $RL$ line models and propose two approximate KR algorithms producing reduced lines with the first-order transfer functions and capable of representing exactly the asymptotic behavior of electric signals, even if they are unbalanced. Then, we show how to apply these KR methods to the design of decentralized voltage and frequency controllers for AC islanded microgrids. Notably, we focus on the plug-and-play algorithm previously proposed by some of the authors, which assumes loads connected to the inverter outputs, and generalize it to networks where loads appear in arbitrary positions. Theoretical results are validated with numerical examples and the application of KR for designing PnP controllers is assessed through a simulation on a 21-bus microgrid.