IEEE Electrification - June 2021 - 81

model: GENROU. Additionally, we use an IEEE AC1A automatic
voltage regulator (AVR) system and a gas steam turbine
model turbine governor in the slack bus. The ability to
quickly switch devices and models is another core feature of
PSID. In this example, we conduct two simulations: one with
synchronous machines only and a second case in which we
replace the generator located at bus 6 for an aggregated
model of a VSM grid-supporting inverter with 19 states, the
same model used in example case A. This example allows
us to explore the effects of substituting synchronous
machines with converter-interfaced generation.
1.001
1.002
1.003
1.004
1
0.999
01 2
Time (s)
3
ωvsm Fourth-Order Case: Static Lines
ωvsm Sixth-Order Case: Static Lines
ωvsm Sixth-Order Case: Dynamic Lines
ωvsm Eighth-Order Case: Dynamic Lines
Figure 8. The virtual speed ~vsm using different machine and network
models for example case A. Continuing from the results in Figure
7, the machine's and lines' modeling assumptions do not affect
the evolution of inverter's virtual speed.
After the system's components are defined, all the
devices are automatically initialized based on the power
flow results. If a feasible point exists, then the system is
appropriately initialized and can be used for simulation.
Using AD, PSID provides small-signal information for
the operating point, such as eigenvalues, damping of
each mode, and participation factors. In this example
simulation, we use an admittance matrix formulation
for the network model representation.
Figure 9 presents a comparison of the eigenvalues (at
the operating point) for both systems with and without
the inverter, separated into " fast " and " slow " dynamics.
The states are assigned to eigenvalues via their participation
factors. This comparison is crucial for determining, in
each simulation, the fastest dynamics that could be
approximated via an algebraic representation.
As shown in Figure 9, the electromagnetic dynamics
of the RLC filter in the inverter are the fastest dynamics,
and for this operating point could reasonably be
neglected in our study case because they are, effectively,
on different time scales.
Similarly, the low-pass filter model used to measure
the bus voltage of the excitation systems could also be
ignored and considered an ideal measurement. However,
based on this analysis, the inverter eigenvalue associated
with the virtual speed and the PLL dynamics are sufficiently
close to the AVR exciter time scale and should be
considered in a time-domain simulation. These results
highlight the complexity of defining modeling strategies
to study converter and system dynamics interactions.
We perform a time-domain simulation to explore the
behavior of the system under a perturbation. Figure 10
depicts the rotor speed of the generator and the virtual
speed of the inverter after an ideal trip of the line
10
-5,000
-2,500
-5,000
-2,500
-2,600
-2,000
-1,400
Re (λ)
(a)
Original 14-Bus Eigenvalues
Generators and Inverter: 14-Bus Eigenvalues
Figure 9. The eigenvalues for machines based on example case B: a 14-bus system (blue circles) and a modified 14-bus system (red crosses).
(a) " Fast " dynamics and (b) " slow " dynamics. The inverter model introduces new fast dynamics that interact with the dynamics of synchronous
machines and cannot be neglected when performing dynamic simulations.
IEEE Electrification Magazine / JUNE 2021
81
PLL
Eigenvalues
-800
-200
RLC Filter
Eigenvalues
AVR-Voltage
Measurement
Eigenvalues
VirtualSpeed
Eigenvalue
5
-10
-5
-60
-50
Eigenvalues:
Generator
Damping Flux Linkages States
and Inverter Current Controller
-40
-30
Re (λ)
(b)
-20
-10
Eigenvalues: AVR States and
Inverter Voltage Controller
lm (λ)
Virtual Speed (p.u.)
lm (λ)

IEEE Electrification - June 2021

Table of Contents for the Digital Edition of IEEE Electrification - June 2021