1
Frequency Dynamics with Grid Forming Inverters:
A New Stability Paradigm
arXiv:2102.12332v1 [eess.SY] 24 Feb 2021
R. W. Kenyon, Student Member, IEEE, A. Sajadi, Senior Member, IEEE, B. M. Hodge, Senior Member, IEEE
Abstract—Traditional power system frequency dynamics are
driven by Newtonian physics, where a synchronous generator
(SG), the historical primary source of power, follows a deceleration frequency trajectory upon power imbalances according
to the swing equation. Subsequent to a disturbance, an SG will
modify pre-converter, mechanical power as a function of frequency; these are reactive, second order devices. The integration
of renewable energies is primarily accomplished with inverters
that convert DC power into AC power, and which hitherto have
employed grid-following control strategies that require other
devices, typically SGs, to establish a voltage waveform and elicit
power imbalance frequency dynamics. A 100% integration of this
particular control strategy is untenable and attention has recently
shifted to grid-forming (GFM) control, where the inverter directly
regulates frequency; direct frequency control implies that a GFM
can serve power proactively by simply not changing frequency.
With analysis and electromagnetic transient domain simulations,
it is shown that GFM pre-converter power has a first order
relation to electrical power as compared to SGs. It is shown that
the traditional frequency dynamics are dramatically altered with
GFM control, and traditional second-order frequency trajectories
transition to first-order, with an accompanying decoupling of the
nadir and rate of change of frequency.
Index Terms—synchronous generators, grid forming inverters,
frequency response, nadir, rate of change of frequency
I. I NTRODUCTION
Frequency dynamics in AC power systems have forever
been govnerned by Newtonian physics, where a synchronous
generator (SG), the conventional primary source of power,
follows a deceleration frequency trajectory upon mechanical–
electrical power imbalances. The integration of variable renewable energy generation into power systems is primarily
accomplished with inverters (i.e., inverter based resources
(IBRs)), which hitherto have employed grid-following (GFL)
control strategies that rely on other devices to establish the
voltage profile [1]. As renewable shares continue to grow at an
accelerating pace, small to medium sized power systems with
significant instantaneous penetrations of IBRs have become
commonplace in many systems across the globe [2]–[4]. As
the penetration of GFL devices increases, overall system
inertia decreases due to the supplanting of SGs [5]; it is
R. W. Kenyon and B. M. Hodge are with the Electrical Computer and Energy Engineering (ECEE) and the Renewable and Sustainable Energy Institute (RASEI) at the University of Colorado Boulder, Boulder, CO 80309, USA, and the Power Systems Engineering
Center, National Renewable Energy Laboratory (NREL), Golden, CO
80401, USA, email: {richard.kenyonjr,BriMathias.Hodge}@colorado.edu,
{richard.kenyon,Bri-Mathias.Hodge}@nrel.gov
A. Sajadi is with the Renewable and Sustainable Energy Institute (RASEI)
at the University of Colorado Boulder, Boulder, CO 80309, USA, email:
Amir.Sajadi@colorado.edu
well documented that instability can occur for high/complete
penetrations of GFL inverters (i.e., low–zero inertia systems)
[5]–[8]. As a result, attention has shifted to grid-forming
(GFM) control, where instead of regulating to active power
set points as a current source, the IBR establishes a voltage
and frequency at the point of interconnection to the grid.
A primary challenge concerning the operation of these lowinertia power systems is the maintenance of system stability, in
particular the frequency response when SGs are displaced by
GFL IBRs; recent work has pointed towards the potential of
GFM inverters to mitigate these stability challenges [9]–[16].
The authors of [9]–[12] have extensively studied the smallsignal stability of power systems with integrated GFMs by
developing high-fidelity differential-algebraic models; often,
non-zero, minimum SG quantities are declared in order to
preserve system stability. Conversely, the feasibility of operating bulk power systems with 100% GFM-based generation has been computationally demonstrated in the electromagnetic transient domain [13], and positive sequence [14].
[13] investigated the dynamic interactions between GFMs and
SGs and have identified that the integration of GFMs improves
the system frequency response; however, these studies fail to
link the fundamental shift in power conversion order to these
improved dynamics. Other studies have started to identify the
damping-like contribution of droop controlled GFM inverters
to frequency dynamics [13]–[15], but cast these conclusions
on the basis of similar frequency trajectories across an entire
system.
This paper investigates the power conversion dynamics of
SGs and GFMs (i.e., converters) and the associated frequency
dynamics in bulk electric power systems along a shifting
generation trajectory; namely, from a 100% SG, standard
inertia system, to a 100% GFM, inertia-free system. The main
conclusions of this paper are summarized:
• It is analytically derived that, with respect to preconverter power, the primary frequency dynamics of
a GFM are first order, and second order for an SG.
Frequency-power portrait analyses of test system load
step responses with varied GFM and SG quantities confirm these lower order dynamics.
• The standard assertion of the fast frequency response of
IBRs, as applicable to pre-converter power of a GFM, is
refuted. A GFM serves power simply by not changing
the local frequency.
• Common average frequency metrics, typically used to
approximate an overall system frequency response, are
demonstrated as inadequate with GFM devices, and the
decoupling of traditional nadir and rate of change of
2
frequency relations with GFMs is demonstrated.
II. M ATHEMATICAL M ODELS OF C ONVERTERS
This section focuses on the mathematical models that describe the power converter devices of interest; namely, the
mechanics and associated time scales in which a converter
can modulate the power throughput. To facilitate the comparison of frequency dynamics between GFMs and SGs, the
notation relating the flow of power through either device is
first summarized in Fig. 1. As shown, both devices have a
pre-converter power (pm ) flowing into the device, which is
analogous to the mechanical torque applied to the shaft of
an SG, or the power supplied by an energy storage system
to the GFM1 . Each type of converter stores a quantity of
2
energy, either as kinetic energy for the SG Eint,G = 21 Iωmech
with I and ω being the moment of inertia and shaft rotation
rotational speed, respectively, or within the GFM primarily
2
, with
as electrical capacitive storage, Eint,I = 12 CDC VDC
CDC and VDC being the DC link capacitance and capacitor
voltage, respectively. Generally, Eint,G >> Eint,I [17], [18].
Electrical power is delivered to the grid by the converter,
represented as pe . The distinction between GFM pre-converter
power, pm,I , and electrical power, pe,I , is made to create the
comparison analogy; these values are only distinguished by a
low-pass filter in this particular GFM control design (pm,I is
often referred to as filtered power, pavg , in the literature [10],
[19]). Here, it is assumed that pm,I is readily available within
the low-pass filter rise time on account of standard energy
storage/inverter response times [20], [21]. Neglecting losses,
conservation of energy requires that Ė 6= 0 if pm 6= pe 2 .
export relationship. For this work, we focus on the frequency
dynamics of the multi-loop droop control GFM, with the
simplified control block diagram shown in Fig. 2. Henceforth,
GFM implies this particular control strategy. In this diagram,
from the left, pm is injected into the converter via the DC
voltage source, where it is modified with a pulse width
modulation (PWM) controller that outputs a series of modified
amplitude square waves that are filtered and interfaced with
the power system. The LC output filter meets with a coupling
inductor to create the LCL topology shown in Fig. 2. The
current (if ) and capacitor voltage (EI ) are regulated by
proportional-integral (PI) controllers operated in the directquadrature reference frame. Additional detail on the controller
design and implementation can be found in [25].
Lf
LI
RI
io
[vo , ωI]
vo
EI
vi
+
-
if
Rf
Cf
PWM
Current
Controller
Voltage
Controller
e[v , ω]
+-
[vset ,ωset]
Fig. 2: Control scheme of multi-loop droop grid-forming
inverter.
In this paper, where the interest is in frequency dynamics,
the voltage can be assumed to be stiffly regulated and therefore
constant within the subsequent mathematical formulations.
This complements the SG model approximation to be discussed in Section II-B. The low pass filter dynamical relation
between pm,I and pe,I is provided in (1).
2π (pe,I − pm,I )
(1)
τI
where τI,f il is the filter time constant; as cutoff frequency,
ωI,f il = 2π/τI . The work in [10] found a limit of 5ωI,f il >
ωn , which indicates that ωI,f il should be greater than 75 rad/s
in a 60 Hz system, and therefore τI ≤ 0.08 s; i.e., on the order
of rapid energy system response times. By (1), pm,I has a first
order relation to pe,I . The active power–frequency governing
equations, which are the control basis of the multi-loop droop
relation, are shown in (2) and (3):
ṗm,I =
Fig. 1: Converter topology showing the relation between the
device internal energy (Eint ), pre-converter power (pm ), and
electrical power (pe ).
A. Grid Forming Inverter
Although a variety of GFM control strategies have been
presented in the literature, such as the droop [22], multiloop droop [15], virtual synchronous machine [23], and virtual
oscillator control [24], they all approximate a similar objective;
namely, the construction of a voltage and frequency at the
point of interconnection with dynamics associated via a power
1 other sources of power exist for the GFM, such as curtailed photovoltaic
output, although here an energy storage system with available energy is
assumed for simplicity
2 dot notation indicates time derivative; ẋ = d x
dt
δ̇I = MP (pm,I,o − pm,I )
(2)
2πMP (pm,I − pe,I )
ω̇I =
(3)
τI
where δI is the GFM phase angle, MP is the droop gain,
pm,I,O is the pre-converter power set point, pm,I is the preconverter power, ωI is the GFM frequency, and pe,I is the
power transferred to the grid; (3) is formulated by combining
(2) and (1). Note that (2) is expressed in relative form (i.e.,
the steady state value is zero).
Based on (1), the pm,I evolves based only on pe,I . Following, δI evolves according to (2). It can therefore be stated that
the GFM frequency is a function of the pre-converter power;
the device meets the increase in power demand (pe,I ), and then
3
adjusts the frequency according to the droop relation. This
distinction indicates that a GFM does not require frequency
deviations in order for pm,I to evolve. In this control scheme,
frequency is changed to accomplish power sharing, but the
underlying throughput power mechanics indicate an entirely
different, and inverted, relationship with power differentials
as compared to the SG.
B. Synchronous Generator
The SG is a well understood and documented device;
see [17], [26] for in-depth discussions. Here, ideal voltage
regulation is assumed and the machine/voltage dynamics are
neglected. While governor models are generally non-linear
with higher order filtering, deadbands, and saturation, a valid
approximation for frequency dynamics can be made with a first
order system [17]. The result is a 3rd order dynamical system
consisting of the swing equation (shown in (4) and (5)) and
the first order governor dynamics (6). The swing equation,
i.e., Newton’s second law in rotational form, relates the SG
rotation speed with pm,G – pe,G power imbalance.
δ̇G = ωG − ωs
1 pm,G − pe,G − Dδ̇G
ω̇G =
M
(4)
(5)
where δG is the SG phase angle, ωG is the SG rotor speed,
ωs is the synchronous speed (equivalent to ω0 for a two
pole machine), M = 2H
ωs where H is the inertia of the
machine, pm,G is the pre-converter power, pe,G is the electrical
power, and D is the damper winding component. The governor
dynamics of the SG can be approximated by the slowest action
as:
RD −1 (ωo − ωG ) − (pm,G − pm,G,o )
ṗm,G =
(6)
τG
where RD is the droop setting of the device (it includes a
factor of 2π), pm,G,o is the pre-converter power set point, and
τG is the governor response time. The value of τG can vary
substantially depending on type and model, but is generally
not less than 0.5 s [17]. Taking (5) within the derivative of
(6), it can be concluded that pm,G has a second order relation
to pe,G :
p̈m,G
−RD −1 ω̇G − ṗm,G
=
τG
−1
−(RD M ) (pm,G − pe,G ) − ṗm,G
=
τG
(7)
(8)
Therefore, pm,G has a second order relation to pe,G , as
compared to the first order relation seen with the GFM device.
Furthermore, it can be stated that pm,G is a function of
frequency, the inverse of the GFM pre-converter–frequency
relationship. More simply, it can be said that whereas the SG is
a reactive device, the GFM is a proactive device, with respect
to frequency.
III. S INGULAR P ERTURBATION A NALYSIS
A common practice in the mathematical formulation and
analysis of power systems is the utility of singular perturbation
theory, wherein dynamical equations are reduced to algebraic
expressions because the associated dynamics are too fast to be
of interest [27]. Within the context of power system dynamics
analysis, this is most transparent in the algebraic treatment
of the transmission system. Here, we apply this tool to the
frequency dynamic analysis of the GFM and SG. Note that
both (1) and (6) are first order differential equations, with
time constants τI and τG . There is, in general, an order of
magnitude of separation between these values; i.e. τI << τG .
Therefore, where the time scale of interest is the settling
of frequency dynamics associated with the SG, using the
foundations of singular perturbation analysis, ṗm,I can be
assumed 0; thus, according to (1), pm,I ≈ pe,I .
Due to the relatively slower response of the SG governor,
immediately following a disturbance the difference between
pm,G,o and pm,G is negligible; i.e., pm,G,o ≈ pm,G . Applying
these approximations, the frequency dynamics relevant before
substantial SG governor action of the GFM ((2) and (3)) and
SG ((4) and (5)) can be reformulated. The approximated GFM
frequency dynamics are given in (9):
δ̇I = MP (pm,I,o − pe,I )
∝ −MP pe,I
(9)
The SG dynamics are given in (10) and (11):
δ̇G = ωG − ωo
(10)
1
(pm,G,o − pe,G )
ω̇G =
M
(11)
pe,G
∝−
M
From (9), it can be concluded that GFM frequency is algebraically related to pe,I ; therefore, considering the first
order relation of pe,I and pm,I in (1), GFM frequency has
a first order relation with pre-converter power. From (10) and
(11), with respect to pe,G the frequency dynamics of the SG
follow a first order response; therefore, with (6), the frequency
dynamics of the SG have a second order relation with preconverter power.
A. Device Step Response
The conclusion that the SG manifests a second-order frequency dynamic is extensively studied and well understood
[17], [26]. However, our analytical model suggests that the
GFM exhibits a first order response. For verification, we
simulate the frequency response of the SG and GFM following
a load. All simulations are performed in the power system
computer aided design (PSCAD) [28] simulation platform.
Frequency, rate of change of frequency (ROCOF), and inertia
metrics are defined as in the Appendix.
The GFM model is a full order, averaged model that
was created based on the state of the art [10], [29] and is
made available open-source at [30]. A full text on the model
implementation is available at [25], with all default parameters
listed. Parameters of note are MP = 0.05(Hz, pu/S, pu)
4
and τI = 0.05s. The SG model uses the standard PSCAD
machine model with default prime–double prime reactance
and time constants, two damper windings sans saturation and
resistive windage losses. The exciter is an AC7B model with
default parameters. All instances of SGs share a common
set of parameters; for all 9 and 39 bus system simulations,
RD = 0.05(Hz/Sbase ), H = 4s (inertia seconds), and
τG = 0.5s. The single device tests examine varied H values
to exhibit the traditional nadir (lowest frequency excursion)
and rate of change of frequency relationship.
The frequency and power step response of each device for
a 10% load step are presented in Figs. 3 and 4. In this model,
each device is operated in standalone fashion, dispatched at
50% with a constant power load connected directly to the
terminals. The SG frequency trace in Fig. 3 shows the second
order negative step response following the load step with
overshoot and subsequent damped oscillations that settle to
the droop determined steady state. The GFM frequency traces
follows a standard first order response; steady state is achieved
with no overshoot and a far smaller response time as compared
to the SG. A summary of the nadir, and ROCOF are presented
in Table I. Note the inverse proportionality between inertia and
ROCOF and the correlation between a lower nadir and reduced
inertia. Conversely, note the larger ROCOF of the GFM device
but a resultant higher nadir.
TABLE I: Single Device Step Response Results
Device
SG (H = 4s)
SG (H = 3s)
SG (H = 2s)
SG (H = 1s)
GFM
ROCOF
(Hz/s)
0.48
0.63
0.95
1.90
1.50
Nadir
(Hz)
59.77
59.73
59.67
59.52
59.85
Settling Frequency
(Hz)
59.85
59.85
59.85
59.85
59.85
singular perturbation application. The pm,G follows a second
order response complete with overshoot and oscillations, as
well as the initial acceleration and inflection change.
Fig. 4: Power response of isolated GFM and SG devices to
a 10% load step at a 50% initial loading. pe is identical for
each device. The acceleration period of pm,G , a second order
characteristic, is evident in the magnified window.
The results of two standard benchmarks, the IEEE 9 and 39
bus test systems, that were used to validate our analysis are
presented in the next two sections.
IV. T EST C ASE I: IEEE 9 B US S YSTEM
Simulations on the IEEE 9 bus test system were performed
in the PSCAD simulation environment. The network and all
associated dynamic elements are available open source at [30],
[31]. Buses 4-9 are 230 kV, buses 1-3 are 16.5 kV, 18.0 kV,
and 13.8 kV, respectively. All SG or GFM devices are rated
at 200 MVA, with other pertinent parameters as established
in Section III. The load is modelled as constant power with
no frequency or voltage dependence. A 10% load step (31.5
MW, 11.5 MVar) occurs at bus 6. Prior to the perturbation, the
system is brought to steady state by initiating all devices as
ideal sources, and then systematically releasing the associated
dynamics in a manner conducive to maintaining steady state
stability. Greater detail on this startup process can be found
in [32]. Different interconnection scenarios of SG and GFMs
into the system are created by systematically supplanting an
SG with a GFM, as shown in Table II.
TABLE II: 9 Bus Configuration and Results
Fig. 3: Frequency response of isolated GFM and SG devices
to a 10% load step at a 50% initial loading.
The pe and pm responses of each device are presented in
Fig. 4. Note that the pe response is identical for each device;
because the voltage dynamics are near ideal, the electrical
power is primarily determined by network changes and not
the device dynamics. The slight pe overshoot is a relic of
the constant power load modeling in PSCAD load impedance
values are modulated every half cycle. The first order relation
of pm,I and pe,G is obvious in the GFM traces, and at a
much faster rate than the SG response, which corroborates the
Scenario
A
B
C
D
Device at Bus
1
2
3
SG
SG
SG
GFM
SG
SG
GFM
GFM
SG
GFM
GFM
GFM
Inertia
(s)
4.0
2.6
1.3
0.0
ROCOF
(Hz/s)
0.50
0.73
1.12
1.61
Nadir
(Hz)
59.72
59.76
59.79
59.83
The results in Fig. 5 show the results for four simulations
(scenarios A, B, C, and D - explained in Table II) where the
the SGs are systematically replaced by the GFMs, resulting
in a incremental decrease in H. These reduced values are
captured in Table II, along with the the resultant nadir and
ROCOF statistics for each scenario. It is evident that as the
mechanical inertia is decreased, the nadir increases, which is
indicative of the dominant first order response of the additional
GFMs. Additionally, although the ROCOF increases as inertia
5
is reduced, it does not correlate with a lower nadir, as would
be expected in a second order system.
trajectories [15], [17]; these results show this assumption is
no longer valid with GFM devices.
The Scenario C frequency exhibits overshoot, but a
smoother recovery; i.e. the concavity in the green trace from t
= 1.5–2.5s is the inverse of the expected second order recovery.
This system frequency trace is not representative of an overdamped second order system, nor a critically damped system
which would follow a far more gradual trajectory to the settling
frequency. The GFM devices are acting too quickly for all
of the devices to remain synchronized during the first 0.5
s following the perturbation. In Scenario D, with all GFM
devices, the system frequency follows a typical first order
response where overshoot and similar frequency oscillations
are absent. The device frequencies show that with all GFM,
the devices maintain a general synchronization throughout the
recovery.
Fig. 5: Average system frequency response for varied quantities of GFM/SGs. Although the peak ROCOF grows with
fewer online SGs, the nadir is simultaneously reduced.
The system frequency oscillation period with all SGs (Scenario A) matches the single machine step response; i.e.,
0.4 Hz oscillations. Individual device frequencies for each
Scenario are presented in Fig. 6, where it is evident that for
Scenario A, all three SGs have similar frequency trajectories;
the three devices maintain broad synchronization following the
perturbation. Herein lies the motivation for center of inertia
and average frequency metrics.
Fig. 7: Comparison of pre-converter and electrical powers of
each device for the four 9 bus simulation scenarios. Note that
Pe and GFM Pm are overlapped at these resolutions.
Fig. 6: Initial frequency response of each device for the four
simulated scenarios on the 9 bus system. Note the contrary
motion present with mixed systems (scenario B and C),
indicating an initial lack of broad synchronization.
The Scenario B system frequency follows a standard second
order step response with a damping value around 50%, with
a brief inversion between 0.5–0.75s; this corroborates the
assertion that droop controlled GFMs add to the damping
of the system [15]. From Fig. 6, it is obvious that the three
devices are not broadly synchronized immediately following
the disturbance. The large GFM frequency changes are the
cause of the average frequency inversion (Fig. 5) just after
the perturbation. The bedrock assumption of average system
frequency metrics is that all devices have similar frequency
The pe and pm response for each scenario are presented
in Fig. 7. The electrical powers show inter-area oscillations
(f = 1.6Hz) between SG 1 and SG 2 in Scenario A. Note
the time separation between these pe,G oscillations and pm,G .
Scenario B shows the very rapid changes in pm,I –pe,I (these
are indistinguishable at this resolution) of GFM 1, which
exacerbates a larger peak pe,G of SG 2 and 3, while the subsequent oscillations are more damped. The peak pe,G output
of SG 3 in Scenario C is further increased, although there
are no oscillations following this overshoot. The conclusion
is that the rapid frequency changes of the GFMs, evident
in Fig. 6, cause the network conditions to change rapidly,
while the SG frequency follows the slower second order
response and resulting in a larger pe,G extraction. Therefore,
the fast frequency change of the GFM forces the SG into
larger oscillations because of its slow response and exacerbates
the dynamic excursions. Scenario D power outputs show a
6
large reduction in power oscillations (including interarea), with
minimal overshoot and a relatively rapid arrival to settling
outputs. Frequency–power portraits are omitted for the 9 bus,
due to strong similarities with the 39 bus results subsequently
presented (Fig. 9).
These 9 bus system results depict that the presence of
GFM inverters reduces the average system frequency nadir,
while increasing ROCOF. In a second order system, these two
directional changes would not be correlated; the cause is due
to the first order response of the GFM devices. The traditional
average frequency determination methods therefore do not
produce an appropriate metric with GFMs because the rapid
changes of the GFM frequency result in device frequencies that
are at times contrary and divergent to adjacent SGs, negating
the fundamental assumption of these metrics.
of the 39 bus test system in PSCAD with only inverters is a
significant testament to the viability of zero inertia systems.
V. T EST C ASE II: IEEE 39 B US S YSTEM
Fig. 8: Average frequency response of 39 bus system for varied
quantities of GFMs and SGs and a 10% load step at bus 15.
The IEEE 39 bus test system [33] is also presented as
a larger case study, with the entire system as simulated in
PSCAD and supporting Python code available open-source
at [30]. The network has been partitioned into 6 subsystems
using the Bergeron parallelization components; a valuable
contribution to future research. The network elements are
unchanged. All buses operate at 230 kV, with a 230/18 kV
generator step–up unit installed to connect all generation
elements at 18 kV. All generation elements are rated at 1000
MVA. Dispatch and voltage set points are unchanged from
the test system configuration. All ten initial SG devices are
systematically replaced by GFMs, with a scenario defining
each iteration; i.e. scenarios 0–10 in Table III. The 10% load
step (600 MW/141 Mvar) occurs at bus 15.
TABLE III: 39 Bus Configuration and Results
Scenario
0
1
2
3
4
5
6
7
8
9
10
GFMs at Buses
n/a
30
30–31
30–32
30–33
30–34
30–35
30–36
30–37
30–38
All GFM
Inertia
(s)
4.0
3.6
3.2
2.8
2.4
2.0
1.6
1.2
0.8
0.4
0.0
ROCOF
(Hz/s)
0.567
0.587
0.669
0.808
0.930
1.071
1.225
1.396
1.525
1.648
1.852
The f –pm,I portraits for a selection of 39 bus system simulations are presented in Fig. 9, where the subtitles correspond
with the Table III entry. With no GFMs, the SGs follow
the trajectory of an initial frequency deviation prior to preconverter changes. Prior to convergence on the steady state
values, the trajectories exhibit oscillations in the form of converging spirals. With half GFMs in scenario 5, the first order
relation between frequency and pre-converter power is evident,
while the SG trajectories are shortened with less overshoot.
With only a single SG online in scenario 9, the SG exhibits
no pre-converter power overshoot. This is due to the faster
frequency oscillations, where the large governor response time
does not permit a reaction. Scenario 10, with only GFMs
online, exhibits the first order/quasi linear relationship between
frequency and pm,I .
Nadir
(Hz)
59.690
59.712
59.717
59.724
59.730
59.738
59.748
59.748
59.756
59.772
59.808
Figure 8 shows the average frequency for each of the 11
scenarios simulated on the 39 bus system. Along with the
frequency statistics presented in Table III, it is concluded that
while ROCOF increases with larger quantities of GFMs and
a resultant decrease in system mechanical inertia, the nadir is
raised. Additionally, the the nadir occurs sooner after the load
step. The damping of the frequency oscillations increases with
a larger quantity of GFMs. The average frequency shows more
variance immediately following the disturbance with larger
quantities of GFM. These unusual frequency traces are the
result of the at times contrary GFM frequency as compared to
the SG units, further diminishing the bedrock assumptions of
average frequency metrics. Here, it is noted that the simulation
Fig. 9: Frequency–power portraits of each device for scenario
0, 5, 9, and 10 for the 10% load step on the 39 bus test system.
VI. D ISCUSSION
With only reactive devices such as SGs (see Section II-B)
matching system aggregate pm to pe and the associated slow
7
governor response (6), a larger ROCOF generally yields a
deeper nadir for the same magnitude power imbalance [34].
Consider the following approximating equation:
∆fprior = αROCOF × tresponse
(12)
where ∆fprior is the frequency deviation prior to substantive
pm,G changes, αROCOF is the ROCOF for a particular system,
and tresponse is the pre-converter power response time (as used
in (1) and (6)). Evidently, a relatively larger ROCOF for the
same tresponse (such as τG , in (6)), yields a larger ∆fprior
before substantial pm changes take place. This is the wellknown inertial response period when rotational kinetic energy
(Eint,G ) is extracted from the SGs; for SG frequency response
dominated systems, less inertia yields larger αROCOF values,
which are susceptible to lower nadirs potentially triggering
frequency load shedding [35]. This is corroborated by the
simulation results presented in Table I for a single device.
A defining feature in this relationship is the reactive nature
of SGs to frequency; the frequency must change prior to a
change in pm,G . This relation is inverted with a GFM device;
when pe,I changes due to varying network conditions, pm,G is
directly impacted prior to any change in frequency. However
a GFM changing frequency is a control response, ostensibly
designed to achieve some type of load sharing, and not the
result of a necessary chain of events to match pe,I and pm,I
as for an SG. Therefore and as derived in Section III, the preconverter–frequency relationship is of lower order with GFMs.
Resulting from this lower order relationship is a reduction in
frequency dynamics in the presence of GFMs as presented in
the simulation results of Sections IV and V, where it is evident
that the standard inertial frequency response of SG dominated
systems is no longer to be expected with GFM devices.
Fig. 10: Nadir and ROCOF as a function of mechanical inertia
for the single device, 9 bus, and 39 bus systems following a
10% load step.
Figure 10 summarizes the relation between ROCOF and
Nadir of the single SG device, 9 bus, and 39 bus test systems
as a function of mechanical inertia. The data presented is from
Tables I II and III. Evident is the standard anti-correlation
between larger ROCOF and lower nadirs in the traditional,
SG dominated power systems; these are approximated with the
single device system, which is de facto the center of inertia
response. In the presence of GFM devices, this relation is
no longer present, as shown by the now correlated nadir and
ROCOF data for the 9 and 39 bus systems. Here, it is noted
that there are potential broader issues due to larger ROCOFs
such as relay tripping, device disconnection, and machine shaft
strain, but the analysis of these considerations is beyond the
scope of this work. However, the ability to greatly reduced
the severity in nadirs with GFM may out weigh these other
potential issues.
VII. C ONCLUSION
This work investigated the power transfer dynamics of
synchronous generators and grid forming inverters and the
driving factors associated with these dynamics. It was shown
analytically that grid forming devices have a lower order relation between electrical and pre-converter power as compared
to the synchronous generator, which results in a reduction
in device frequency dynamics. From these relations, it was
demonstrated that the multi-loop droop grid forming inverter is
a proactive device with respect to frequency and pre-converter
power, contrasting with the reactive nature of the synchronous
generator. As implemented in the 9 and 39 bus test systems,
it was shown that the traditional frequency metric relations of
larger rate of change of frequency and nadir are decoupled in
the presence of these grid forming devices.
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A PPENDIX
A. Average System Frequency
The frequency at a generation bus is defined as the shaft
rotation speed for an SG, or the control system derived
frequency for a GFM. The center of inertia system frequency
formulation, where an aggregate system frequency response
can be derived based on the summation of SG parameters
system wide [15], [17], is not used because the method will
not capture the frequency regulation of the zero inertia GFMs;
i.e. this metric lacks applicability to these zero inertia devices.
To arrive at a system average frequency, f (t), each device
frequency is weighted according to the device rating, as shown
in (13):
Pn
(M V Ai ∗ fi (t))
Pn
f (t) = i=1
(13)
i=1 M V Ai
where fi (t) is the frequency of device i at time t, M V Ai is
the device i rating, and n is the number of devices.
B. Rate of Change of Frequency
ROCOF, with respect to the rotation speed of a SG, is
a continuous function; however, for practical purposes such
as device action (i.e., protection, inverter response, etc.) it is
calculated with a sliding window averaging method, as shown
in (14):
f (t + TR ) − f (t)
f˙(t) :=
(14)
TR
where f is the frequency, and TR is the size, in seconds,
of the sliding window. A TR = 100 ms window is used,
in accordance with [34]. The largest absolute ROCOF value
during a particular event is the result presented.
C. Mechanical Inertia
An aggregate mechanical inertia value is calculated as: (15).
This serves the purpose of quantifying the changeover from
inertial devices, SGs, to non-inertial devices, GFMs. Lower
inertia here implies a greater quantity of generation supplanted
by GFMs.
Pn
i=1 Hi SB,i
H= P
(15)
n
i=1 SB,i
where Hi is the inertia rating (in s) of device i, SB,i is the
MVA rating of device i, and n is the number of devices.