  Low inertia, nonlinearity and a high level of uncertainty (varying topologies
and operating conditions) pose challenges to microgrid (MG) systemwide
operation. This paper proposes an online adaptive Koopman operator optimal
control (AKOOC) method for MG secondary voltage and frequency control. Unlike
typical data-driven methods that are data-hungry and lack guaranteed stability,
the proposed AKOOC requires no warm-up training yet with guaranteed
bounded-input-bounded-output (BIBO) stability and even asymptotical stability
under some mild conditions. The proposed AKOOC is developed based on an
ensemble Koopman state space modeling with full basis functions that combines
both linear and nonlinear bases without the need of event detection or
switching. An iterative learning method is also developed to exploit model
parameters, ensuring the effectiveness and the adaptiveness of the designed
control. Simulation studies in the 4-bus (with detailed inner-loop control) MG
system and the 34-bus MG system showed improved modeling accuracy and control,
verifying the effectiveness of the proposed method subject to various changes
of operating conditions even with time delay, measurement noise, and missing
measurements.
