# Transmission line dynamics on inverter-dominated grids: analysis and
simulations

Gabriel E. Colón-Reyes⋆, Ruth Kravis⋆, Sunash B. Sharma⋆, Duncan S.
Callaway⋆,† ⋆Department of Electrical Engineering and Computer Sciences,
University of California, Berkeley, United States
†Energy and Resources Group, University of California, Berkeley, United States
{gecolonr, ruth.kravis, sbsharma<EMAIL_ADDRESS>
###### Abstract

In this work, we study fast time scale dynamic interactions between inverters,
synchronous machines, and transmission lines. The overlapping time scales
between inverter controls and electromagnetic phenomena idfentified in recent
years has necessitated a re-evaluation of assumptions made in power system
dynamics studies. We utilize an open-source modeling platform to perform both
small signal stability and dynamic time domain studies of networks containing
inverters, machines, and loads. We use transmission line models of varying
fidelity, including models with multiple segments and frequency dependence.
Our results indicate that, for the cases we study, line dynamics are not
important for characterizing small signal stability. However, while in many
cases the high fidelity line models are unnecessary for dynamic simulations,
there are some cases in which simpler models omit dynamics that could be
important to the operation of inner inverter control loops.

###### Index Terms:

Low-inertia grids, power system dynamics, power system stability, transmission
line dynamics.

Submitted to the 23rd Power Systems Computation Conference (PSCC 2024). This
research was supported by the U.S. Department of Energy’s Solar Energy
Technologies Office through award 38637 (UNIFI Consortium).

##  I Introduction

This paper addresses the question of whether or not electromagnetic (EM)
dynamics should be included in power system dynamic interaction studies. These
dynamics have been largely ignored due to the natural time scale separation
between synchronous machine (SM) states and EM dynamics. However, because
converter-interfaced generation (CIG) sources have feedback control loops
whose time constants overlap with EM dynamics, it is no longer clear whether
fast time scale dynamics can be neglected [1]. This question has become
increasingly prominent with the functionality of grid-forming inverters (GFM)
to regulate frequency and voltage magnitude. Addressing this question has
important compuational implications: whereas phasor-domain modeling tools that
ignore fast time scale dynamics can simulate power system dynamics in a matter
of seconds or less, electromagnetic transient (EMT) software tools can require
many hours to simulate even relatively small systems [2].

The central component in this question – and what determines the difference
between phasor domain simulations and EMT simulations – are the transmission
lines (TLs). Phasor domain simulations typically have detailed load, machine
and inverter models that are coupled via algebraic line models that ignore the
EM dynamics. On the other hand, EMT methods model the detailed physics of TLs
– including wave propagation and frequency dependence – in tandem with other
power system elements. This enables EM time scale phenomena from a variety of
system components to accurately propagate from one network location to
another.

Several recent studies have found that simple models of TL dynamics can impact
small signal stability classifications [3, 4, 5], in some situations showing
that line dynamics can destabilize grids with high shares of CIG, while
stabilizing them in others. In these papers, TLs are modeled with a single RL
branch [3, 5] or dynamic lumped-parameter $\pi$ models [4]. While these models
provide higher fidelity dynamics than algebraic models, they do not capture
transmission line frequency dependence or wave propagation dynamics associated
with distributed parameter or multi-segment line models. However, research on
direct current line models that interface with power electronic converters [6,
7, 8] has shown that frequency dependence and multi-segment dynamics can
produce qualitatively different dynamics relative to single segment frequency
independent models.

This paper aims to contribute to the discussion around TL modeling choice in
system-level dynamics and stability by examining the impact of higher fidelity
TL models on simulation outcomes. In addition to comparing algebraic network
models to dynamic $\pi$ models, we examine multi-segment TL models with and
without frequency dependent dynamics. Our contributions are as follows:

  * •

First, building on the open-source simulation package
PowerSimulationsDynamics.jl [9], we construct a TL modeling package that
allows for comparison among several transmission line models. This facilitates
comparisons between a large variety of TL line modeling choices in a single,
fast-to-simulate, open-source simulation package, rather than relying on
aligning simulation assumptions between commercial phasor-domain simulation
software and EMT simulation software.

  * •

Second, we examine the effect of TL model choice on the small signal stability
of power systems with a mix of grid-forming CIG and SMs. For the GFM-SM cases
we study, adding TL fidelity with more segments and frequency dependence
alters both the real and imaginary parts of the system’s eigenvalues. However,
we find that the loading and line length at which systems become unstable is
independent of TL model. These results are consistent with a subset of results
from earlier studies and provide additional detail with respect to the effects
of including higher fidelity line models.

  * •

Third, we examine the effect of TL model choice on dynamic simulation outcomes
for branch trips across several case studies, loading scenarios, and line
lengths, for systems with a mix of GFM and SMs. We observe various differences
in voltage and current dynamics; however, these differences do not produce
sustained dynamics. For a subset of experiments, higher fidelity models
produce transients of significantly different amplitude and direction than the
lower fidelity models – suggesting that simulations with low order line models
may miss important dynamic events, especially in the study of the action of
CIG inner control loops.

Notation: Dot notation indicates the time derivative of a variable, i.e.,
$\dot{x}=\frac{dx}{dt}$. Bold lower-case symbols are used to represent complex
variables in the $dq$ or $RI$ reference frames, e.g.
$\boldsymbol{x}=x_{R}+jx_{I}$. $\lambda_{A}\in\sigma(A)$ is an eigenvalue of
$A$, where $\sigma(A)$ is the spectrum of $A\in\mathbb{R}^{n\times n}$.
$\lVert\cdot\rVert$ is the Euclidean vector norm.

##  II Modeling

###  II-A EMT and positive sequence simulations

EMT tools such as PSCAD simulate systems via a sequence of coupled nodal
equations in which TL dynamics and bus-level device dynamics are represented
in a discrete time formulation. This allows line wave propagation dynamics and
delays to be captured with high fidelity. In contrast, phasor domain tools
such as PSS/e and PSLF solve the dynamics of devices at each bus and, in order
to accelerate simulation times, represent the network model as an algebraic
system solved in a separate power flow step. Phasor domain tools are further
accelerated relative to EMT tools by using ordinary differential equation
(ODE) solvers that numerically integrate dynamics far faster than discrete
time EMT solvers [10].

These differences complicate simulation-based comparisons of dynamic phenomena
with and without TL dynamics. EMT software tools are slow to solve, and
significant modeling effort (parameter choice, model harmonization) is
required to ensure that simulation outputs without line dynamics match
industry standard phasor domain modeling tools. Moreover, Bergeron-style
discrete-time formulations of high-frequency dynamics for TLs produce delay-
difference equations that cannot be easily deployed in a small signal
stability analysis.

In this paper we leverage the Julia-based modeling and simulation package
PowerSimulationsDynamics.jl (PSID). PSID can simulate power system dynamics in
the phasor domain as well as in a balanced $dq$ form, and it is capable of
precisely reproducing the output of PSS/e in the phasor domain, even on very
large network models, as well as PSCAD EMT with fast inverter and machine
dynamics [9]. In contrast to PSCAD, because PSID preserves a $dq$ ODE
formulation of the system model, it can leverage a suite of numerical
integration solvers that enable significantly faster simulation speeds of
power system dynamics.

However, at the time of writing this artcile, PSID has only been developed to
model dynamic $\pi$ representations of TL dynamics. In this paper we extend
PSID to approximate distributed parameter TL dynamics with and without
frequency dependence. We do so by adapting a multi-segment multi-branch
modeling approach [11, 8] originally developed for modeling direct current
cables, to work in PSID’s multi-machine AC power system simulation setting.

###  II-B Power System Components

####  II-B1 Generator

In PSID, a generator is composed of five main components: a stator, a shaft, a
turbine governor, a power system stabilizer (PSS), and an automatic voltage
regulator (AVR). Our choice of models for each of the components is based on
the time scales over which the dynamics of such component evolves relative to
our interest in fast time scales of inverter controls and TL dynamics.

The models that we choose for each of the components of the generator, along
with the corresponding choices for parameters, can be found in Chapters 15 and
16 of [12]. We adopt the six-state Anderson-Fouad machine stator model. This
model incorporates transient and subtransient EM stator dynamics. We choose a
shaft model given by the swing equations with damping. Further, we opt for a
fixed input turbine governor because its output will be relatively constant in
the time scales of interest. Because of its use for slow time scales, and thus
its irrelevance to our analysis, we choose not to include a PSS. Lastly, we
opt for a Type 1 AVR to capture voltage control dynamics.

####  II-B2 Inverter

In PSID, an inverter is composed of six main components: a DC voltage source,
a model for the switches, an output inverter filter, an outer grid-forming
(GFM) or grid-following (GFL) control loop, an inner control loop, and a
frequency estimator.

The models we choose for these components and its parameters come from [11].
We choose a fixed DC voltage source model, an LCL passive filter, a virtual
synchronous machine GFM model for the outer loops, nested proportional-
integral (PI) loops for the inner control loops, and a phase locked loop for
damping of the virtual frequency. We choose an averaged model for the switches
[13]. For some experiments, we choose to have the inverter be a GFL source,
modeled by active and reactive power PI controllers [14].

(a)

(b)

Figure 1: Line model schematics: (a) $\pi$ model ($statpi$ and $dynpi$), and
(b) Multi-segment model ($MSSB$ and $MSMB$).

###  II-C Transmission lines

We consider TL models that can be expressed as linear time-invariant state-
space models, since these are compatible with differential-algebraic equation
(DAE) representations of other system components.

To model TLs of arbitrary length (denoted $\ell$), we use per unit length
parameters for impedance, $z_{km}=r_{km}+j\omega l_{km}$, and susceptance,
$b_{km}=g_{km}+j\omega c_{km}$. At a particular operating frequency $\omega$,
we compute the lumped parameter equivalent $\pi$ model according to the
following equations, which include hyperbolic correction factors from the
steady state solution to Telegrapher’s equations:

$\displaystyle z_{\pi}$
$\displaystyle=z_{km}\ell\left(\frac{\sinh(\gamma\ell)}{\gamma\ell}\right)$
(1) $\displaystyle y_{\pi}$
$\displaystyle=y_{km}\ell\left(\frac{\tanh(\gamma\ell/2)}{\gamma\ell/2}\right)$
(2) $\displaystyle\gamma$ $\displaystyle=\sqrt{z_{km}y_{km}}$ (3)

where $\ell$ is line length, as distinct from $l$ which refers to inductance.
We assume $g_{km}=0$, which does not imply that
$g_{\pi}=\mathrm{Re}(y_{\pi})=0$. However, we choose to overwrite $g_{\pi}=0$.
From this model, we can compute equivalent $r,l,c$ as
$r_{\pi}=\mathrm{Re}(z_{\pi}),l_{\pi}=x_{\pi}/\omega=\mathrm{Im}(z_{\pi})/\omega,c_{\pi}=b_{\pi}/\omega=\mathrm{Im}(y_{\pi})/\omega$.

####  II-C1 Algebraic $\pi$ model (statpi)

The algebraic $\pi$ model has the form shown in Fig. 1(a). It assumes that any
line dynamics are stable and settle quickly compared to other system dynamics.
Therefore, the differential terms associated with the line capacitance and
inductance are set to zero, giving an algebraic model as follows:

$\displaystyle\boldsymbol{i_{in}}$
$\displaystyle=\left(\frac{1}{z_{\pi}}+y_{\pi}\right)\boldsymbol{v_{1}}-\frac{1}{z_{\pi}}\boldsymbol{v_{2}}$
(4) $\displaystyle\boldsymbol{i_{out}}$
$\displaystyle=-\frac{1}{z_{\pi}}\boldsymbol{v_{1}}-\left(\frac{1}{z_{\pi}}+y_{\pi}\right)\boldsymbol{v_{2}}$
(5)

Since this model is purely algebraic, any dynamics on $\boldsymbol{i_{1}}$,
$\boldsymbol{i_{2}}$, $\boldsymbol{v_{1}}$, and $\boldsymbol{v_{2}}$ that
arise due to the interconnection of the line with dynamic devices will
instantaneously appear at the other end of the line.

####  II-C2 Dynamic $\pi$ model (dynpi)

The dynamic $\pi$ model has the same structure as shown in Fig. 1(a), but it
includes dynamics on line current and voltage states:

$\displaystyle\frac{d\boldsymbol{i}}{dt}$
$\displaystyle=(\boldsymbol{v_{1}}-\boldsymbol{v_{2}})-z_{\pi}\boldsymbol{i}$
(6) $\displaystyle\frac{d\boldsymbol{v_{1}}}{dt}$
$\displaystyle=(\boldsymbol{i_{in}}-\boldsymbol{i})-y_{\pi}\boldsymbol{v_{1}}$
(7) $\displaystyle\frac{d\boldsymbol{v_{2}}}{dt}$
$\displaystyle=(\boldsymbol{i}-\boldsymbol{i_{out}})-y_{\pi}\boldsymbol{v_{2}}$
(8)

Similar to $statpi$, $dynpi$ only captures the effect of the distributed line
parameters in steady state.

####  II-C3 Multi-segment single-branch $\pi$ model ($MSSB$)

In what follows, a line with multiple “segments” is one that is divided into a
discrete set of identical length $\pi$ components connected in series. In
addition, a line with multiple “branches” is one in which each segment is
divided into a set of parallel branches, each with different impedances, to
capture frequency dependent line characteristics.

The multi-segment single branch $\pi$ model ($MSSB$) consists of $N$
$\pi$-shaped segments. The parameters for each segment are given by:

$\displaystyle r_{seg}$ $\displaystyle=r_{km}\ell_{seg}$ (9) $\displaystyle
l_{seg}$ $\displaystyle=l_{km}\ell_{seg}$ (10) $\displaystyle c_{seg}$
$\displaystyle=c_{km}\ell_{seg}$ (11)

where $\ell_{seg}=\frac{\ell}{N}$. Further, $z_{seg}=r_{seg}+j\omega l_{seg}$
and $y_{seg}=j\omega c_{seg}$. This model is seen in Fig. 1(b) with $M=1$,
namely a single RL branch for each segment. The $i^{th}$ segment of an
$N$-segment $MSSB$ model is defined by the following equations:

$\displaystyle\frac{d\boldsymbol{i_{i}}}{dt}$
$\displaystyle=(\boldsymbol{v_{i}}-\boldsymbol{v_{i+1}})-z_{seg}\boldsymbol{i_{i}}$
(12) $\displaystyle\frac{d\boldsymbol{v_{i}}}{dt}$
$\displaystyle=(\boldsymbol{i_{i-1}}-\boldsymbol{i_{i}})-y_{seg}\boldsymbol{v_{i}}$
(13) $\displaystyle\frac{d\boldsymbol{v_{i+1}}}{dt}$
$\displaystyle=(\boldsymbol{i_{i}}-\boldsymbol{i_{i+1}})-y_{seg}\boldsymbol{v_{i+1}}$
(14)

Note that for $i=1$, $i_{i-1}=i_{in}$, and for $i=N$, $i_{i+1}=i_{out}$. As
$N$ is increased, the $MSSB$ model more closely approximates the equivalent
$\pi$ model in steady state frequency response. The advantage of explicitly
representing segments is that unlike the equivalent $\pi$ model, it captures
the distributed nature of the line parameters in both transient and steady
state responses.

####  II-C4 Frequency dependent multi-segment multi-branch $\pi$ model
($MSMB$)

The $MSMB$ model has $M$ branch currents per segment. Let the subscript
$m=1,...,M$ denote the $m^{th}$ parallel branch. Therefore, the equations for
the $i^{th}$ segment are:

$\displaystyle\frac{d\boldsymbol{i_{i,m}}}{dt}$
$\displaystyle=(\boldsymbol{v_{i}}-\boldsymbol{v_{i+1}})-z_{seg,m}\boldsymbol{i_{i,m}}\quad\forall
m$ (15) $\displaystyle\frac{d\boldsymbol{v_{i}}}{dt}$
$\displaystyle=(\boldsymbol{i_{i-1}}-\boldsymbol{i_{i}})-y_{seg}\boldsymbol{v_{i}}$
(16) $\displaystyle\frac{d\boldsymbol{v_{i+1}}}{dt}$
$\displaystyle=(\boldsymbol{i_{i}}-\boldsymbol{i_{i+1}})-y_{seg}\boldsymbol{v_{i+1}}$
(17)

where $z_{seg,m}=r_{seg,m}+j\omega l_{seg,m}$. Note that
$\boldsymbol{i_{i}}=\sum_{m=1}^{M}{\boldsymbol{i_{i,m}}}$ in the voltage
equations, as seen in Fig. 1(b).

The $MSMB$ is considered the highest fidelity line model of those presented,
since it captures dynamics along the line’s length as well as the additional
damping that arises due to the frequency dependence of the line parameters
[8].

###  II-D Line parameters

For the $MSMB$ model, we obtain $r_{km,m}$ and $l_{km,m}$ for each branch from
known line data via vector fitting [15, 11], a form of parameter estimation
based on real transmission line data. In this paper, we use the frequency
dependent line impedance parameters from [16] as a starting point111We use the
data in Table 3.. We derive parameters for all other line models from the
$MSMB$ model by finding the equivalent parallelized branch impedance at
nominal frequency for the $MSMB$ model. This yields the $MSSB$ $z_{km}$, which
we use in Equations 1, 2 and 3 to obtain the lumped parameter model parameters
for $statpi$ and $dynpi$. It proves important to follow a procedure like this
to compute impedances and susceptances for all models to ensure that at the
steady state operating frequency all lines have the same impedance, resulting
in the same network power flow solution.

###  II-E Aggregate system model

All the models described above can be written in DAE form. Linking the devices
and network, we arrive at a mathematical form as follows:

$\begin{bmatrix}\dot{x}\\\ 0\end{bmatrix}=\begin{bmatrix}f(x,y,u)\\\
g(x,y,u)\end{bmatrix}$ (18)

Here, $x\in\mathbb{R}^{n}$ are the system’s dynamic states,
$y\in\mathbb{R}^{m}$ are the system’s algebraic states, $u\in\mathbb{R}^{p}$
are the system inputs.
$f:\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{n}$
and
$g:\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{m}$
are the vector equations associated to the dynamics of the network and the
algebraic constraints.

We are interested in studying the stability of such a system, in particular,
its small signal stability. In general, $f$ is a nonlinear vector field.
Therefore, to study the small signal stability of the system we find an
equilibrium point $(x^{\star},y^{\star},0)$ by setting $\dot{x}=0$ and solving
the nonlinear system of equations, linearize them around that point, and
arrive at a set of linear dynamics that characterize the behavior of the
system in the vicinity of that equilibrium point. The resulting equations will
then be of the following form:

$\Delta\dot{x}=J(x^{\star},y^{\star})\Delta x$ (19)

Here, $J(x^{\star},y^{\star})$ is the reduced system Jacobian matrix. By
studying the eigenvalues of this matrix associated to the linearization, we
can determine if $(x^{\star},y^{\star},0)$ is a stable or unstable operating
condition for the network. See [4] or [9] for further details on the
linearization process in PSID.

##  III Test Cases

To investigate dynamic interactions under the different line models presented,
we choose a simplified two bus test case, and the IEEE 9 Bus test case.

######  Remark.

We choose not to use an infinite bus in any of our studies because it would
instantaneously produce or consume whatever real and reactive power are
necessary to maintain its voltage, which has an unrealistic effect on the
dynamic results. Instead, we choose a SM/GFM inverter as the voltage angle
reference bus to solve the initializing power flow problem and subsequently
simulate dynamics of the multi-machine system.

###  III-A Two Bus test case

The two bus test case, shown in Fig. 2, has two generation sources connected
by two identical transmission lines in parallel. By default, $\ell=100$ km.

Figure 2: Two Bus test system single line diagram.

The test case has a constant impedance load, which can be located at either
one of the two buses. We choose $p_{load}$ and $q_{load}$ according to the
line’s surge impedance loading $(SIL)=\frac{V^{2}}{Z_{c}},$ where $V$ is the
nominal system voltage (230 kV) and $Z_{c}=\sqrt{z_{km}/y_{km}}$ is the line’s
characteristic impedance. Normally, $SIL$ is computed for lossless lines,
however here we consider losses to give nominal $SIL$ values for real and
reactive power. For our parameters of $V$, $z_{km}$, and $y_{km}$, we arrive
at $p_{load}=2.05$ p.u. and $q_{load}=0.08$ p.u.

###  III-B IEEE 9 Bus test case

Figure 3: IEEE 9 Bus test case single line diagram [17]

.

We also study the IEEE 9 Bus test case, shown in Fig. 3. We place two
identical GFM inverters at buses 1 and 3, and a SM at bus 2. The test case
also includes three constant impedance loads at buses 5, 6 and 8. We set
nominal loading and line lengths according to [18].

###  III-C Experiment types

We construct different experiments by varying loading and line lengths. We
vary loading because it is a key variable of interest for a system operator
identifying safe operating limits of a system. We vary line lengths under the
hypothesis that differences between line models may be revealed only for
longer lines.

We vary line length by scaling all nominal line lengths by a line scale. We
vary all loads’ real and reactive power consumption by multiplying nominal
values by a load scale, as well as all generators’ real and reactive power
setpoints.

######  Remark.

We choose the default load for the two bus test case as the $SIL$ for the
line, however the IEEE 9 Bus test case has predefined loads, all of which are
different than the line $SIL$. Therefore, a particular load scale factor
scales a different ‘base’ load in the IEEE 9 Bus test case compared to the Two
Bus test case.

##  IV Small signal analysis

We compare small signal stability under our four different line models.
Following the approach developed in [6], we choose the segment length for
$MSSB$ and $MSMB$ by selecting $N$ such that the highest frequency oscillation
of the line model is at least as large as the highest frequency oscillation
mode of the GFM and SM. For the $MSMB$ model we choose three parallel branches
(i.e., $M=3$), which is consistent with the modelling recommendations in [8].

###  IV-A Two Bus test case

####  IV-A1 Stability boundary

Fig. 4 plots the line scale at which the linearized system becomes unstable
under a range of loading conditions. Four traces are plotted, one for each
line model. Since the traces mostly overlap completely, the key takeaway is
that all line models give very similar predictions of the line length at which
instability arises. The stability ‘boundary’ was calculated using 10 km line
scale increments, so the different models may give slightly different
predictions, but the differences are very small.

(a)

(b)

(c)

(d)

Figure 4: Line length where the system loses stability on the $y$-axis
(measured in terms of $line\ scale$) as a function of system loading on the
$x$-axis (measured in terms of $load\ scale$). Top row is GFM vs SM, subplot
(a) has the load at the GFM bus while subplot (b) has the load at the SM bus.
Bottom row has the load at bus 2, subplot (c) is SM v SM, and subplot (d) is
GFM v GFM.

There are some cases where the models deviate slightly, for example, at high
load scales in Fig. 4(a) (GFM v SM, load at SM), and at low load scales in
Fig. 4(d) (GFM v GFM). However, considering that the total line lengths are
hundreds of kilometers, these deviations are not significant.

The $MSSB$ and $MSMB$ also give almost identical results. This suggests that
the extra damping associated with the $MSMB$ line does not affect the least
stable eigenvalues. The closeness of these models may also reflect the limited
data available for the vector fitting used to construct the $MSMB$ model.

These results are consistent with other GFM-SM simulations [3] which found
that the stability boundary did not differ for algebraic versus dynamic $\pi$
models. Our results extend that finding to much higher order dynamic models.

####  IV-A2 Eigenvalue analysis

Fig. 5 compares the system eigenvalues under different choices of line models
for a system with nominal loading and line scaling. The blue cluster of least
stable eigenvalues near the $j\omega$-axis are common to all line models, the
orange eigenvalues are common to all models except $statpi$, the green to only
$MSSB$, and purple to only $MSMB$. $MSSB$ (green) and $MSMB$ (purple) models
exhibit higher frequency eigenvalues compared to $dynpi$, and the $MSMB$
(purple) high frequency eigenvalues are more damped compared to the $MSSB$
(green) eigenvalues.

Figure 5: System eigenvalues under different line models, for a GFM v SM test
case, with the load at the GFM bus, and $load\ scale=1.0$, $line\ scale=1.0$.

The least stable eigenvalues, in blue, have very low ($<1\mathrm{e}^{-4}$)
participation from line states. This is consistent with our finding that added
line fidelity does not influence small signal stability for the cases we
studied. The high frequency eigenvalues are almost exclusively associated with
internal line states. The only exception is a pair of very high frequency and
very damped eigenvalues, which have some small participation
($<1\mathrm{e}^{-4}$) from GFM filter voltage and SM flux states.

(a)

(b)

Figure 6: $MSMB$ eigenvalues under a sweep of (a) line lengths (with $load\
scale=1.0$) and (b) loading (with $line\ scale=1.0$).

In Fig. 6 we plot how system eigenvalues move in the complex plane under
changing line lengths and loading for the $MSMB$, our highest fidelity model.
As we increase line lengths, we see in Fig. 6(a) that the high frequency
eigenvalues (which are associated with dynamic line states) do not become less
stable overall (although some eigenvalues do move right), and the cluster of
least stable eigenvalues common to all dynamic line models (those with low
line state participation factors) approach the right half of the complex plane
(RHP) first. There is some rightward movement of high frequency line
eigenvalues, and we cannot rule out the possibility that the line dynamics
could destabilize the system with different parameter choices (such as control
loop gains). This merits further investigation.

In Fig. 6(b), we see that that the high frequency line eigenvalues are very
weakly influenced by loading, and instability again arises due to the cluster
of common eigenvalues approaching the RHP.

###  IV-B IEEE 9 Bus test case

The 9 bus system is stable under all load and line scalings chosen. The four
line models gave very similar predictions of the maximum non-zero eigenvalue,
supporting the results demonstrated on the Two Bus test case.

(a)

(b)

(c)

Figure 7: Inverter filter current for the Two Bus test case with branch trip.
The $line\ scale$ is 1.0 on the top row of plots, and 5.0 on the bottom row.
Columns (a), (b), (c) have load scaling factors of 0.5, 1.0, and 2.0,
respectively. Figure 8: Bus 3 inverter filter current for the IEEE 9 Bus test
case with $line\ scale=3.0$, and $load\ scale=1.0$.

##  V Dynamic simulations

In this section, we first justify the variables that we pay close attention
to, we discuss and justify the perturbation we apply to the systems, and we
present the results of our dynamic studies.

###  V-A Variables, disturbance, and parameters of interest

Inverters are known to tolerate $\sim 1.3$ p.u. rated current before risking
damage to device switches. Therefore we will focus our attention on current at
inverter filters to identify whether low order line models could fail to
identify inverter overcurrents. Current dynamics can also serve as a proxy for
broader system dynamics and the potential for qualitatively different
simulation outcomes with different line models. In our case we are modeling
devices with LCL filters, and we therefore capture current closest to the
switches. Note that PSID does not currently model a current saturation block
to protect switches from overcurrent. Therefore our simulation results can be
used to identify events in which overcurrent protection could be activated,
but are not indicative of the dynamics that would ensue following an
overcurrent event.

We chose to perturb the system with a branch trip, because it affects the
system abruptly and at the differential equations closest to the variables of
interest. This is in contrast to a reference change (such as in the GFM
controls or SM generation setpoint), which goes through multiple filtering
levels before reaching the variables of interest. A branch trip also quickly
reroutes power flows throughout the network, causing fast transmission
dynamics and potentially making inverters increase their current outputs to
meet power demand.

As in the small signal analysis above, we test the system under a range of
loading and line length parameters.

###  V-B Two Bus test case

For the two bus case dynamic simulations, we add the load to bus 2. We trip
one of the lines since they are identical and will have the same loading. We
considered several scenarios placing loads and generators in the Two Bus test
case (including GFM-SM systems, GFM-only systems, and SM-only systems), with a
range of loading and line lengths for each. We find that all categories of
dynamic line model produce qualitatively different dynamics following a branch
trip, but that the dynamics decay to nearly the same trajectory for all models
after 10 to 20 milliseconds. Fig. 7 provides examples of these dynamics for
the case with a SM at bus 1 and GFM plus load at bus 2.

For the cases we investigated with the Two Bus model, we did not observe
sustained line dynamics that lead to undesirable interactions between
inverters and/or machines at different buses. These results are for standard
machine and CIG parameter values, but as suggested in the small signal
section, we cannot exclude the possibility that undesirable interactions could
occur at different parameter values (such as different CIG inner control loop
gains). Interestingly, the $MSSB$ and $MSMB$ produce essentially the same
dynamics, suggesting that modeling line frequency dependence is not important
for studies on this scale; this could also be a consequence of limitations of
the data used for obtaining $z_{km}$ and $y_{km}$.

We also see that, although CIG filter current oscillates more quickly with all
three dynamic line models, the amplitude of these oscillations is generally
smaller than, and never greater than, the amplitude of the oscillations
identified with the static model. Though there is clearly more to investigate
here, these results suggest that static line models will not underestimate the
potential for switch current limiting behavior. This is likely because the
distributed nature of the multi-segment lines delay and attenuate the
propagation of disturbances.

###  V-C IEEE 9 Bus test case

In this test case we trip the line connecting buses 4 and 5 because it is the
most heavily loaded line in the pre-fault system. In general, results found
for the IEEE 9 Bus test case align with those of the Two Bus test case: while
dynamics differ, the amplitudes of current excursions are comparable and
dissipate quickly. However, we found several instances in which $statpi$ and
$dynpi$ line models underestimate inverter current magnitude excursions in
important ways. Fig. 8 is one such example. Here, both high-fidelity line
models suggest that the initial deviation in current magnitude is not just in
the opposite direction of $statpi$ and $dynpi$ lines, but its peak is
significantly different than the lowest peak for the $statpi$ model.
Therefore, using the $dynpi$ model may lead one to believe that the
oscillations predicted by $statpi$ are not real. Hoewever, using $MSSB$ and
$MSMB$ reveals that not only are oscillations present, but they occur at a
different time and in a different direction from what $statpi$ predicts. This
is significant in p.u. terms and suggests that high-fidelity line models
should be considered when studying inverter over-current protection schemes.

##  VI Conclusions and recommendations

We find that each line model generates different sets of eigenvalues, but for
the cases we investigated the dynamic line states have very low participation
factors in the least stable eigenvalues, and consequently line model fidelity
does not alter stability conclusions. These results are consistent with and
expand on earlier work [3] that found very simple dynamic line models do not
alter stability assessments in GFM-SM systems. However, we cannot rule out the
possibility that the line dynamics could destabilize the system with different
parameter choices (such as control loop gains or inverter control type).
Indeed, other papers have found that simple line dynamics impact the
assessment of converter gain parameters on stability [4], and that conclusions
may differ with grid following converters [3] or GFM devices using virtual
oscillator control [5]. This merits further investigation.

Dynamic simulation results show that while $dynpi$, $MSSB$, and $MSMB$ models
exhibit higher frequency components than $statpi$, these oscillations are,
mostly, not of significant amplitude to be of concern and dissipate quickly.
This is consistent with our small signal analysis, which indicates that
dynamic line states have very low participation factors associated to
eigenvalues closest to the RHP. However, we did identify cases on the IEEE 9
Bus model in which $statpi$ and $dynpi$ models fail to capture high amplitude
current spikes that may be relevant to studies of inverter inner control loop
performance.

For the cases we investigated, we found no evidence that modeling line
frequency dependency is important. However, different line frequency datasets
could produce differences in dynamic simulation results between $MSSB$ and
$MSMB$, and future work should explore this possibility.

## Acknowledgments

The authors would like to acknowledge R. Henriquez-Auba’s guidance in using
PSID. This research was supported by the U.S. Department of Energy’s Solar
Energy Technologies Office through award 38637 (UNIFI Consortium).

## References

  * [1] “Expert group interaction studies and simulation models (EG-ISSM),” Grid Connection European Stakeholder Committee, Tech. Rep., 2021.
  * [2] R. W. Kenyon, B. Wang, A. Hoke, J. Tan, C. Antonio, and B.-M. Hodge, “Validation of maui pscad model: Motivation, methodology, and lessons learned,” in _2020 52nd North American Power Symposium (NAPS)_. IEEE, 2021, pp. 1–6.
  * [3] U. Markovic, O. Stanojev, P. Aristidou, E. Vrettos, D. S. Callaway, and G. Hug, “Understanding small-signal stability of low-inertia systems,” _IEEE Transactions on Power Systems_ , 2021.
  * [4] R. Henriquez-Auba, J. D. Lara, C. Roberts, and D. S. Callaway, “Grid forming inverter small signal stability: Examining role of line and voltage dynamics,” in _IECON 2020 The 46th Annual Conference of the IEEE Industrial Electronics Society_. IEEE, 2020, pp. 4063–4068.
  * [5] D. Gros, M. Colombino, J.-S. Brouillon, and F. Dorfler, “The effect of transmission-line dynamics on grid-forming dispatchable virtual oscillator control,” _IEEE Transactions on Control of Network Systems_ , vol. 6, no. 3, p. 1148–1160, Sep 2019.
  * [6] S. D’Arco, J. Beerten, and J. Suul, “Cable model order reduction for hvdc systems interoperability analysis,” in _11th IET International Conference on AC and DC Power Transmission_. Birmingham, UK: Institution of Engineering and Technology, 2015, pp. 026 (10 .)–026 (10 .). [Online]. Available: https://digital-library.theiet.org/content/conferences/10.1049/cp.2015.0039
  * [7] S. D’Arco, J. A. Suul, and J. Beerten, “Time-invariant state-space model of an ac cable by $dq$-representation of frequency-dependent $\pi$-sections,” in _2019 IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe)_. Bucharest, Romania: IEEE, Sep 2019, p. 1–5. [Online]. Available: https://ieeexplore.ieee.org/document/8905577/
  * [8] ——, “Configuration and model order selection of frequency-dependent $\pi$ models for representing dc cables in small-signal eigenvalue analysis of hvdc transmission systems,” _IEEE Journal of Emerging and Selected Topics in Power Electronics_ , vol. 9, no. 2, p. 2410–2426, Apr 2021.
  * [9] J. D. Lara, R. Henriquez-Auba, M. Bossart, D. S. Callaway, and C. Barrows, “Powersimulationsdynamics.jl – an open source modeling package for modern power systems with inverter-based resources,” 2023.
  * [10] J. D. Lara, R. Henriquez-Auba, D. Ramasubramanian, S. Dhople, D. S. Callaway, and S. Sanders, “Revisiting power systems time-domain simulation methods and models,” _IEEE Transactions on Power Systems_ , pp. 1–16, 2023.
  * [11] J. Beerten, S. D’Arco, and J. A. Suul, “Frequency‐dependent cable modelling for small‐signal stability analysis of vsc‐hvdc systems,” _IET Generation, Transmission & Distribution_, vol. 10, no. 6, p. 1370–1381, Apr 2016.
  * [12] F. Milano, _Power system modelling and scripting_. Springer Science & Business Media, 2010.
  * [13] A. Yazdani and R. Iravani, _Voltage-sourced converters in power systems: modeling, control, and applications_. John Wiley & Sons, 2010.
  * [14] R. W. Kenyon, A. Sajadi, A. Hoke, and B.-M. Hodge, “Open-source pscad grid-following and grid-forming inverters and a benchmark for zero-inertia power system simulations,” in _2021 IEEE Kansas Power and Energy Conference (KPEC)_. IEEE, 2021, pp. 1–6.
  * [15] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” _IEEE Transactions on power delivery_ , vol. 14, no. 3, pp. 1052–1061, 1999.
  * [16] H. Dommel, “Overhead line parameters from handbook formulas and computer programs,” _IEEE Transactions on Power Apparatus and Systems_ , vol. PAS-104, no. 2, p. 366–372, Feb. 1985.
  * [17] Y. Song, D. Hill, and T. Liu, “Small-disturbance angle stability analysis of microgrids: A graph theory viewpoint,” 09 2015.
  * [18] J. Bérard, “Ieee 9 bus system example,” 2017.