  Impedance models of power systems are useful when state-space models of
apparatus such as inverter-based resources (IBRs) have not been made available
and instead only black-box impedance models are available. For tracing the root
causes of poor damping and tuning modes of the system, the sensitivity of the
modes to components and parameters are needed. The so-called critical
admittance-eigenvalue sensitivity based on nodal admittance model has provided
a partial solution but omits meaningful directional information. The
alternative whole-system impedance model yields participation factors of
shunt-connected apparatus with directional information that allows separate
tuning for damping and frequency, yet do not cover series-connected components.
This paper formalises the relationships between the two forms of impedance
models and between the two forms of root-cause analysis. The calculation of
system eigenvalue sensitivity in impedance models is further developed, which
fills the gaps of previous research and establishes a complete theory of
impedance-based root-cause analysis. The theoretical relationships and the
tuning of parameters have been illustrated with a three-node passive network, a
modified IEEE 14-bus network and a modified NETS-NYPS 68-bus network, showing
that tools can be developed for tuning of IBR-rich power systems where only
black-box impedance models are available.
