  Stability of power grids with synchronous generators (SGs) and renewable
generation interfaced with grid-forming converters (GFCs) under dc-side current
limitation is studied. To that end, we first consider a simple 2-bus test
system and reduced-order models to highlight the fundamental difference between
two classes of GFC controls -- (A) droop, dispatchable virtual oscillator
control (dVOC) and virtual synchronous machine (VSM), and (B) matching control.
Next, we study Lyapunov stability and input-output stability of the dc voltage
dynamics of class-A GFCs for the simple system and extend it to a generic
system. Next, we provide a sufficiency condition for input-to-state stability
of the 2-bus system with a class-B GFC and extend it for a generic system.
Finally, time-domain simulations from a reduced-order averaged model of the
simple test system and a detailed switched model of the GFC validate the
proposed conditions.
