A Comparative Study of Grid-Forming Controls and their Effects
on Small-Signal Stability
This paper was downloaded from TechRxiv (https://www.techrxiv.org).
LICENSE
CC BY-NC-SA 4.0
SUBMISSION DATE / POSTED DATE
03-02-2023 / 05-09-2023
CITATION
Lamrani, Yahya; Colas, Frédéric; Van Cutsem, Thierry; Cardozo, Carmen; Prevost, Thibault; Guillaud, Xavier
(2023). A Comparative Study of Grid-Forming Controls and their Effects on Small-Signal Stability. TechRxiv.
Preprint. https://doi.org/10.36227/techrxiv.22006310.v2
DOI
10.36227/techrxiv.22006310.v2
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. XX, NO. Y, MONTH 2023
1
A Comparative Study of Grid-Forming Controls and
their Effects on Small-Signal Stability
Yahya Lamrani, Student Member, IEEE, Frédéric Colas, Member, IEEE, Thierry Van Cutsem, Fellow
Member, IEEE, Carmen Cardozo, Member, IEEE, Thibault Prevost, Member, IEEE, Xavier
Guillaud, Member, IEEE
Abstract—The increasing penetration of power-electronics interfaced resources brings new challenges regarding the smallsignal stability of power systems. To address this issue, gridforming controlled converters have emerged as an alternative
to their conventional grid-following counterparts. This paper
investigates the mechanisms behind converters driven stability
and quantifies the stabilizing effect of grid-forming controls. The
linearized state space model of different combinations of control
strategies is analyzed in a multi-infeed system considering various
operating points. Through a parametric sensitivity study and an
examination of the participation factors of key eigenvalues of
the linearized models, it is confirmed that grid-forming controls
contribute to system stabilization. Moreover, this paper demonstrates that this stabilizing effect varies significantly depending on
the specific grid-forming control implemented: whether a current
control loop is used or not, notably impacts stability.
Index Terms—Small-signal stability, grid-following, gridforming, current control, interaction phenomena.
I. I NTRODUCTION
D
UE to the increasing efforts to limit the effects of climate
change, the energy transition has been accelerated to
meet the growing electricity consumption with carbon-neutral
energy sources. Most of these sources, such as photovoltaic or
wind, use converters to connect to the grid. This paradigm shift
from an electrical grid dominated by synchronous generators
to a system with a high rate of heterogeneously distributed
non-synchronous generation brings new challenges to the
power system. Non-synchronous generation behaves differently; for instance, it does not inherently provide inertia, system strength and it has a significantly lower current overload
capacity. This combination of properties results into grids that
are characterized as weaker than the traditional grids [1].
Today, the majority of the deployed non-synchronous generation is Grid-Following (GFL) controlled. This solution is well
established in both academia and industry, reaching a certain
level of harmonization. It always refers to an almost identical
control structure including outer loops in cascade with a
current control loop, and a Phase-Locked Loop (PLL) for
synchronization [2]. However, GFL controls are known to be
sensitive to the grid strength and can become unstable in weak
network conditions [3]. Therefore, the European Network of
Transmission System Operators for Electricity (ENTSO-E)
recognizes both the decrease of synchronous generation and
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Copyright may be transferred without notice, after which this version may
no longer be accessible
the increase of GFL-controlled sources as destabilizing factors
of the power system.
More recently, Grid-Forming (GFM) control has been acknowledged as a promising alternative to overcome this potential barrier to the massive deployment of converter interfaced
resources [1], [4]. However, many control schemes fall under
the umbrella of GFM control.
The common trait of all GFM schemes is that the controls
set the references for the angle and the amplitude of the converter modulated voltage. The angle reference can be obtained
with different solutions: droop control, power-synchronization
control, synchroverters, Virtual Synchronous Machine (VSM),
virtual synchronous generator or IP control [2], [5]–[8]. It has
been shown that these schemes, when properly tuned, lead to
an equivalent dynamic behavior [9], with the VSM scheme
being the most adopted in literature for transmission systems
applications as it offers a more familiar transition from the
traditional synchronous generation [10]. As for the voltage
amplitude reference, it can be set directly or via a current
control loop. The use of a current loop has been privileged so
far as it allows a straightforward saturation of the currents
during faults and the re-use of extensively studied current
loops implemented in the GFL control [11]. Multiple solutions
have been proposed to generate the current references under
the GFM control scheme such as the conventional dual loop,
the virtual impedance or the use of a Quasi-Static Electrical
Model (QSEM) (also referred to as the virtual admittance) with
the use of the QSEM proving to be the most stable solution
[11], [12]. Moreover, recent works have shown that it is
possible for GFM controls to handle large disturbances events
without requiring a current loop [13]. Therefore, investigating
the necessity and the impact of the current control on GFM
schemes becomes a valid question, which, to the knowledge of
the authors, hasn’t been sufficiently addressed in the literature.
Past the controls nomenclature, the comparative stability
studies between GFL and GFM controls have mostly used a
setup consisting of a single converter connected to a Thévenin
equivalent, where the equivalent impedance represents the
grid strength expressed by the Short-Circuit Ratio (SCR).
Such a setup is quickly limited by the static power transfer
limit (regardless of the controls) when considering realistic
reactive power limits of the units. It also fails to highlight the
interaction phenomena encountered in the real power system
due to the presence of multiple converters. In [14], a multiinfeed system was studied to determine the maximal hosting
capacity of GFL-controlled converters of a given system.
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However, the many simplifications and assumptions (such as
identical control for all converters) reduces the model to a
single converter to infinite bus configuration that overlooks
interaction phenomena among converters. When interaction
phenomena are studied [15], they are presented among GFLcontrolled converters only to highlight the impact of the
bandwidth of the PLL and of the power injection on the smallsignal stability of the system. A setup was proposed in [16]
to study the small-signal stability of multiple converters while
overcoming the static power transfer limit and without neglecting interaction phenomena. However, its scope was limited to
GFL-controlled converters. Additionally, when both GFL and
GFM controls are considered [17], [18], the analysis either
provides tuning recommendations rather than an explanation of
the stabilizing impact of the GFM control, or it compares GFM
and GFL based on a voltage source/current source behavior,
which is an unfair comparison as the distinction stems from a
circuit difference rather than a control difference (LC filter for
GFM-controlled converters and LCL filter for GFL-controlled
converters). Finally, even in studies directly targeting the
improvement brought by GFM controls, little attention is paid
to their specific type. In fact, it is often assumed that all GFM
controls yield similar stability improvements, provided they
are properly tuned and fulfill the primary functions of synthetic
inertia and/or frequency droop [19].
In conclusion, although several studies have analyzed multiinfeed systems, no demonstration of the stabilizing effect of
different types of GFM controls has been provided. To fill
these gaps, this paper proposes the following contributions:
• Investigation of the GFM control stabilizing effect: it is
shown that the stabilizing properties of a GFM-controlled
converter vary significantly depending on the presence of
current control for an otherwise identical GFM control.
• Proposal of a dedicated test setup suitable for studying
interaction phenomena among converters in a weak grid.
The setup is voluntarily kept as simple as possible to
allow a physical interpretation of the analysis of the
underlying instability mechanisms.
This paper is organized as follows. Section II describes
the building blocks of the studied systems by presenting the
control structures, their tuning, as well as their state-space
models. Section III studies the parametric sensitivity of the
proposed 2-converter setup with respect to the grid strength,
in order to showcase the variation on the stability limits
resulting from different controls combinations. The underlying
interaction phenomena and their impact on the stability limits
is also analyzed. Section IV utilizes the findings of Section III
to determine the minimal proportion of GFM-controlled converters required to stabilize the system by mitigating GFLrelated interactions. This analysis is carried out on a threeconverter system operating close to the previously identified
stability limit. Finally, Section V concludes the paper.
II. P RESENTATION OF DIFFERENT CONTROLS
In the following subsections, three controls are presented: GFL, voltage-controlled GFM (vcGFM) and currentcontrolled (ccGFM), respectively. In addition to the control
2
structure, the tuning of its parameters and the equations
describing the dynamic behavior of the controls are also
detailed. The linearized state-space model of each control is
then derived considering the building blocks shown in Fig. 1,
where the inputs are the Point of Common Coupling (PCC)
voltages, the power and voltage references and the outputs are
the injected currents.
Fig. 1: Converter module
A. Grid-following control
The GFL control structure is recalled in Fig. 2. The smallsignal model is established in the Park reference, with the
assumption of a constant DC link voltage. The control operates
in a local Park reference frame denoted by the subscript ”dq ”,
while the subscript ”xy ” denotes the grid-side Park reference
frame. The PLL synchronizes the local d-q frame to the
PCC voltage vg by providing θ̃g , an estimate of the phase
angle θg of that voltage. The estimated angle is used in the
Park transformation and in its inverse, respectively denoted as
P (θ̃g ) and P (θ̃g )−1 [2]. The currents and voltages used by
the control are measured at the PCC and are filtered to reflect
the bandwidth of the measurement devices.
Fig. 2: Grid-following control structure
Naturally, as the voltages and currents are filtered in the
three-phase frame, the differential equations for vgfxy and ifgxy
accounts for the axis cross coupling.
The control is set to track the active power injection P to
P ∗ and to regulate the PCC voltage to V ∗ , where V ∗ is set
to provide a Q-V droop shown in Eq. (1) [20]:
V ∗ = V0 + kQ (Q∗ − Q)
(1)
where kQ is the droop gain, Q is the measured reactive
power produced by the converter, and V0 and Q∗ are the
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voltage and the reactive power references, respectively, such
that the voltage setpoint V ∗ is equal to V0 when the reactive
power produced is equal to Q∗ . The outer loops controlling
the active power and the PCC voltage generate the reference
currents i∗dq used by the current controller. In turn, the current
∗
control sets the reference for the modulated voltages vm
dq
using the Proportional Integral (PI) controllers. Finally, the
∗
modulated voltage vm
is affected by a delay to account for
dq
the effect of the Pulse Width Modulation (PWM), reflected by
the switching frequency fsw .
Using various design studies and recommendations for
stable GFL control under weak grid conditions [21]–[23], the
above GFL is tuned as shown in Table I. In this table, ωp ,
ωv , ωLP F and ωcc denote the bandwidths of active power
control, voltage control, low-pass filter and current control,
respectively. Besides the small-signal stability considerations,
ωcc is also adjusted to account for the switching frequency
fsw , in order to avoid undesired interactions with the low level
control. The PLL PI controller gains are calculated based on
the PLL natural frequency ωn and its damping ratio ξ.
TABLE I: GFL parameters
Control
Inputs filtering
Outer loops
PLL
Current control
PWM
Parameter
finputs
ωp
ωv
kQ
ωLP F
ωn
ξ
ωcc
fsw
Value
5 kHz
10 rad/s
50 rad/s
0.15
300 rad/s
50 rad/s
1
1200 rad/s
2 kHz
The relationship between the ”dq ” and the ”xy ” sets of
variables is detailed in Eqs. (2) involving the phase angle
estimate provided by the PLL.
cos(θ̃g − θg ) sin(θ̃g − θg )
vgfdq =
vgfxy
−sin(θ̃g − θg ) cos(θ̃g − θg )
cos(θ̃g − θg ) sin(θ̃g − θg )
ifgdq =
ifgxy
(2)
−sin(θ̃g − θg ) cos(θ̃g − θg )
cos(θ̃g − θg ) −sin(θ̃g − θg )
∗
∗
vm
=
vm
xy
dq
sin(θ̃g − θg ) cos(θ̃g − θg )
The PWM-induced delay is modeled using a first-order Padé
approximation as follows:
−s + α ∗
v
(s)
(3)
vmabc (s) =
s + α mabc
where α depends on the switching frequency.
The state-space model inputs, state variables and outputs
are defined as follows:
UGF L = [vgx , vgy , P ∗ , Q∗ ]
XGF L = [igx , igy , P f , Qf , V f , ζ P , ζ V , ζ id ,
ζ iq , ζ P LL , θ̃g , Px , Py , vgfx , vgfy , ifgx , ifgy ]
YGF L = [igx , igy ]
(4)
(5)
(6)
where igx and igy are the currents injected into the grid
by the converter. P f , Qf , V f are respectively the filtered
3
active power, reactive power and PCC voltage. The five ζ i
state variables are related to the integrators of the various PI
controllers and Pxy are the state variables associated with the
Padé approximation of the delay in the Park reference frame.
The differential equations converted in Laplace domain, in
per-unit (pu), can be found in the Appendix.
B. Voltage-controlled GFM
The vcGFM control structure, shown in Fig. 3 uses the VSM
∗
scheme proposed in [9] to generate the reference angle θm
and
track the active power reference, while providing an inertial
and damping effect, denoted H and ξ, respectively. The VSM
scheme utilizes a PLL, identical to the one previously chosen
for the GFL scheme, to obtain an estimation of the grid angular
frequency ω̃g . Furthermore, the current dynamics are actively
∗
damped by modifying the voltage references vm
using a
dq 0
Transient Virtual Resistor (TVR) [24], [25]. The modulated
voltage is controlled by directly setting the references of vmdq 0
to (V ∗ , 0), which are then rotated by the generated reference
∗
angle θm
. By controlling the magnitude of the modulated
voltage behind the connection transformer, the control scheme
yields a QV droop whose value corresponds to the leakage
inductance Lc of the transformer. Incidentally, this is common
practice in conventional power plants.
Fig. 3: Voltage-controlled GFM control structure
The delay, seen in Fig. 3, is identical to the one considered
for the GFL control. Similarly to the GFL control, as well,
the currents and voltages measured at the PCC are filtered
to reflect the measurement devices bandwidths. The TVR
parameters are chosen to cover the possible resonances of the
AC system [26]. The full control parameters are provided in
Table II.
TABLE II: vcGFM parameters
Control
Angle control
TVR
Parameter
H
ξ
ωf
Rv
Value
5s
1
60 rad/s
0.09
The linearized state-space model of the vcGFM-controlled
converter is built similarly to that of the GFL-controlled
converter with the local Park reference frame being rotated
∗
by θm
instead of θ̃g . The state-space model inputs and state
variables are listed in Eqs. (7) and (8), respectively, while the
output variables are those already defined in Eq. (6).
UvcGF M = [vgx , vgy , P ∗ , V ∗ ]
(7)
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∗
Pxy , ζ P LL , θ̃g , vgfxy , ifgxy , igxy and ζ θm variables obey the
same differential equations as for the vcGFM-controlled converter. ζ idq are the state variables associated with the current PI
controllers. The differential equations describing the dynamic
behavior of the newly introduced state variables are detailed
in Eqs. (A3).
III. S TABILITY LIMIT OF A 2- CONVERTER SYSTEM
As shown in [16], the classical small-signal studies using a
Thévenin equivalent are not sufficient to determine the stability
limits of different controls, for various grid strengths. In this
section, precisely, the stability limits are investigated in a
completely different setup where no power flows into the
Thévenin equivalent representing the external grid. To that
purpose, the 2-converter system proposed in [16] to improve
GFL controls is used here to determine the small-signal
stability limit of the previously presented controls for various
levels of grid strength. First, the stability limit of two GFLcontrolled converters is assessed. One of the two converters is
then changed into vcGFM and ccGFM control, respectively, to
evaluate the contribution to stability of each GFM VSC type.
Fig. 4: Current-controlled GFM control structure
TABLE III: ccGFM parameters
Control
QSEM filter
Current control
Parameter
QSEM
ωLF
P
ωcc
Value
62.5 rad/s
1200 rad/s
∗
∗
XvcGF M = [igx , igy , ζ θm , θm
, ζdT V R , ζqT V R ,
Px , Py , ζ P LL , θ̃g , vgfx , vgfy , ifgx , ifgy ]
(8)
A. Setup description
Pxy , ζ P LL , θ̃g , vgfxy , ifgxy and igxy variables obey the differential equations already detailed in the Appendix for the GFL
∗
control. ζ θm is the state variable related to the VSM controller
TV R
of the power loop and ζdq
are the state variables related to
the TVR washout filter. The differential equations describing
the dynamic behavior of the newly introduced state variables
are detailed in Eqs. (A2).
C. Current-controlled GFM
Fig. 5: Description of the 2-converter setup
The ccGFM control structure, presented in Fig. 4, involves
∗
.Similarly to the vcGFM
the same VSM scheme to generate θm
control, the voltage references are set for the modulated voltage.However, a current loop is used to generate the modulated
voltage references, as presented in [11]. The QSEM of the
circuit between the modulated voltage and the PCC voltage
is used to determine the reference values of the currents i∗dq ,
according to Eq. (9).
1
Rc
ω̃Lc
f
∗
∗
(vm
− vdq
) (9)
idq = 2
dq
Rc + (ω̃Lc )2 −ω̃Lc Rc
where ω̃ is the estimated converter angular frequency obtained
f
∗
as the time derivative of θm
and vdq
is the filtered PCC voltage.
It must be noted that the PCC voltage used for the QSEM is
filtered to cover only the frequency range appropriate to the
above-mentioned quasi-static model. For a fair comparison,
the current control loop used here is identical to that of the
GFL scheme. The set of parameters used is given in Table III.
The linearized state-space model of the ccGFM-controlled
converter is built similarly to that of the vcGFM-controlled
converter, with identical inputs and outputs. The state variables
are as follows:
∗
∗
XccGF M = [igx , igy , ζ θm , θm
, vdf , vqf , ζ id , ζ iq ,
Px , Py , ζ P LL , θ̃g , vgfx , vgfy , ifgx , ifgy ]
(10)
TABLE IV: 2-converter setup parameters
Circuit
VSC
Filter
OHL
Grid
Parameter
Snom , Sb
Unom , Ub
Pnom
Qmax
Lc
Rc
Ll
Rl
Lg
Rg
Value
1.044 GVA
400 kV
1 GW
300 MVAr
0.15 pu
0.005 pu
0.144 pu
0.0072 pu
0.5 pu
0.05 pu
The setup used to study the small-signal stability is shown
in Fig. 5. The system consists of two converters: V SC1
and V SC2 , where V SC1 is GFL-controlled while V SC2 is
controlled in GFL, vcGFM and ccGFM mode, successively.
Each converter is connected via a 30-km long Overhead Line
(OHL) to a Thévenin equivalent, denoted with subscript ”g ”.
The Thévenin impedance (Rg + ωb Lg ) is used as a metric of
the grid strength.The circuit parameters are detailed in Table
IV.
The operating point is set so that V SC1 is absorbing its
nominal active power while V SC2 is injecting its nominal
active power, thus limiting the power flowing through the grid
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5
1.08
VSC2 in GFL mode: Non-linear model
VSC2 in GFL mode: Linear model
VSC2 in ccGFM mode: Non-linear model
VSC2 in ccGFM mode: Linear model
VSC2 in vcGFM mode: Non-linear model
VSC2 in vcGFM mode: Linear model
1.06
1.04
1.02
1
0.98
0.96
0.94
0.92
1
1.05
1.1
1.15
1.2
1.25
1.3
Time (s)
Fig. 6: V SC1 PCC voltage response to a grid side phase jump
impedance. This way, the system can operate for increasing
values of the grid impedance with limited reactive power
requirements.
The state-space model of the setup is developed by associating the previously established state-space models to the
state-space model of the network, which has been detailed in
[16]. The state-space variables of the network are expressed
as:
Xsys = [ig0x , ig0y , ig1x , ig1y , ig2x , ig2y ]
(11)
Therefore, the full system state variables are represented by
the vector [Xsys , XV SC1 , XV SC2 ].
Fig. 7: System eigenvalues for varying grid impedance
TABLE V: Dominant modes for Zg = 0.7 pu
Setup
Eigenvalue
Damping ratio
GFL-GFL
3.67 ± 97.13i
3.7%
GFL-vcGFM
−54.7 ± 312.5i
17.3%
GFL-ccGFM
−76.4 ± 515i
14.6%
B. Validation of the linearized model
Using the state-space models of the network and the VSC
blocks under different controls, the 2-converter system statespace model is assembled and linearized at the following
operating point:
 ∗
∗

PV SC1 = −PV SC2 = −Pnom


VV∗SC1 = VV∗SC2 = 1 pu
egxy = [1, 0]
(12)
Before proceeding to stability studies, the linearized model has
been validated in time-domain by comparing it to the full nonlinear model implemented in Matlab-Simulink. The validation
consisted in comparing the responses to the non-linear model
in case of a π/40 phase jump of the voltage source eg . The
results are shown in Fig 6. It is clearly seen that the linearized
model accurately replicates the non-linear model’s dynamics
and is used henceforth for the system stability analysis.
C. Sensitivity to the grid strength
The sensitivity of the system to the grid strength is studied
by assessing its small-signal stability for increasing values of
the grid impedance. Thus, the state-space model of the full
2-converter setup is linearized for various values of the grid
impedance and the eigenvalues of the linearized system are
analyzed. The parametric study is conducted for values of
Zg ranging from 0.5 pu to 3 pu, which would characterize
an extremely weak grid. Lg and Rg are accordingly varied
to keep a X/R ratio of 10.Figure 7 shows a sample of the
parametric sensitivity of the three controls combinations. The
stability limit is defined as the first operating point for which
at least one of the eigenvalues of the linearized system has
a positive real part, which is also confirmed via EMT timedomain simulations.
The figures show a clear stability advantage of the GFM
controls with the vcGFM outperforming the ccGFM. The
GFL-GFL setup is the first to become unstable, at Zg = 0.8 pu
while the GFL-GFM combinations remain stable for higher
values of Zg . In fact, the GFL-vcGFM shows very little
sensitivity to the grid impedance and no stability limit for the
studied range of Zg values. The GFL-ccGFM combination
has a significantly higher stability limit than the GFL-GFL
setup as it is stable up to Zg = 1.38 pu. Beyond the simple
comparison of the stability limits of the setups, the parametric
sensitivity also allows analyzing the dynamic behavior of
the three setups as they approach their stability limits. For
example, at Zg = 0.7 pu, the three configurations can be
compared in terms of the eigenvalues that are most sensitive
to the grid strength. This comparison is presented in Table
V, where both GFL-GFM setups exhibit significantly better
damping than the GFL-GFL counterpart.
D. Interaction phenomena interpretation through participation factors
For a deeper understanding of the observed differences
between the three setups, it is of interest to study how
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Grid
VSC1
𝑋
𝑋
6
VSC2
𝑋
(a) VSC2 in GFL mode
Grid
VSC1
𝑋
𝑋
VSC2
𝑋
(b) VSC2 in vcGFM mode
Grid
𝑋
VSC1
𝑋
VSC2
𝑋
(c) VSC2 in ccGFM mode
Fig. 8: Participation factors of the three setups in their respective dominant modes
the various controls and system parameters contribute to the
instability phenomenon, via the state variables governed by
the said control and system parameters [27]. To that purpose,
for the previously identified dominant, or eventually unstable
eigenvalue, the participation factors of the state variables are
analyzed. The results, corresponding to the eigenvalues from
Table V and Zg = 0.7 pu, are shown in Fig. 8.
For the GFL-GFL setup, the results definitely confirm the
converter interaction and instability mechanisms previously reported in the literature. The states of both converters contribute
to the critical eigenvalue via the voltage regulation and the
PLL state variables (ζ V and θ̃g , grayed out in Fig. 8a), which
reflects the electrical proximity of the two converters PCCs.
The PLL and the voltage control are closely correlated: the
PLL uses the PCC voltage to synchronize the control frame
in which the current references are generated to regulate the
voltage. The injection of these currents causes the volatility of
Vpcc which is accentuated in weaker grids, eventually leading
to unstable operation, as reported in multiple references (e.g.
[15], [28]). Note that the observed asymmetry between the
contributions of the two converters state variables is due to the
higher stability of GFL control for a rectifier mode operating
point. In fact, a GFL-controlled converter displays a higher
stability margin when, all things being equal, the active power
Fig. 9: Description of the multi-infeed setup
reference point is negative (rectifier mode) than when it is
positive (inverter mode) [29].
When V SC2 is in vcGFM mode, the noticeable difference
is the absence of contribution of the GFL state variables to
the dominant mode, which further confirms that this setup
remains stable even for extremely weak grids. In fact, in this
system, the vcGFM provides a full decoupling between the
GFL converter and the weak grid, thus mitigating the potential
GFL-weak grid interactions and significantly improving the
system dynamic behavior and stability limit.
Finally, when V SC2 is controlled in ccGFM mode, the
contributions of the GFL control are significantly limited in
contrast to the GFL-GFL setup. This mitigation of the GFLweak grid interaction can be attributed to the improvement
brought by the ccGFM control. However, the GFL contribution
to the critical eigenvalue is not completely neutralized, which
explains why this setup becomes unstable before its GFLvcGFM counterpart. Furthermore, two ccGFM state variables
f
are contributing to the eigenvalue: vdq
, grayed out in Fig. 8c.
These state variables are used by the QSEM to generate the
current references used by the current control (see Eq. (9)).
This explains why vcGFM control shows better stability than
the ccGFM control: the ccGFM control relies on the PCC
voltage measurements for its current control, which makes it
more vulnerable to the grid strength: indeed, a less stiff grid
leads to a more volatile Vpcc and consequently a less stable
operation of the ccGFM.
IV. S TABILIZING PROPERTIES OF GFM CONTROLS IN A
MULTI - INFEED SYSTEM
This section aims at showing the stability improvement
brought by adding a GFM-controlled converter to multiple
GFL-controlled converters. As GFM control is more recent
than GFL control and not all existing GFL-controlled installations can be retro-fitted to GFM control, it is interesting to
determine the minimal additional capacity of GFM-controlled
converters required to stabilize a given system hosting multiple
GFL-controlled converters.
A. Setup description
The setup used in the previous section is modified such
as V SC2 is split into two converters: V SC21 and V SC22 ,
respectively, where V SC21 operates in GFL-mode and V SC22
in vcGFM or ccGFM mode, while obeying the constraint:
V SC21
V SC22
V SC2
Snom
+ Snom
= Snom
= Sb
(13)
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7
Grid
𝜆
VSC1
VSC22
VSC21
𝜆
(a) VSC22 in vcGFM
𝑋
𝑋
𝑋
𝑋
Fig. 11: Participation factors with VSC22 in ccGFM mode
Grid
VSC21
VSC1
VSC22
(b) VSC22 in ccGFM
Fig. 10: System eigenvalues for varying VSC22 proportion
This model is representative for example, of a wind farm, in
which a subset of generators are controlled in GFM mode and
the rest in GFL mode, each subset being represented by an
aggregated model. The setup is shown in Fig. 9. The network
parameters remain identical to those in Table IV, with Rc
and Lc keeping their values in pu but each referred to the
V SC22
V SC21
) of their respective converter.
or Snom
MVA base (Snom
Furthermore, Zg is set to 1 pu. This leads to an operating
point far beyond the stability limit of the GFL-GFL setup
(Zg = 0.8 pu), and, hence, allows assessing the stabilizing
impact of GFM-controlled converters.
Similarly to the setup of Section III, the state space model
of the network is built with the state variables Xsys shown
in Eq. (11). The full system state variables are represented by
the vector [Xsys , XV SC1 , XV SC21 , XV SC22 ].
In accordance with the 2-converters setup, the operating
point of the whole system is chosen as follows:
 ∗
∗
∗

PV SC1 = −(PV SC21 + PV SC22 ) = −Pnom
VV∗SC1 = VV∗SC2 = 1pu
(14)


egxy = [1, 0]
𝑋
𝑋
Grid
𝑋
VSC21
VSC1
𝑋
𝑋
𝑋
𝑋
VSC22
𝑋
Fig. 12: Participation factors with VSC22 in vcGFM mode
with the type of the GFM control: the ccGFM control has to
be 15 times bigger than its vcGFM counterpart. Indeed, while
the 3% proportion is enough with a vcGFM control for the
system to reach small-signal stability, the proportion climbs
to 45% with the ccGFM control.
Past the strict definition of small-signal stability, the minimal proportion required by both GFM controls to reach a
satisfactory dynamic performance also varies significantly. As
shown by the 10% damping ratio cone in Fig. 10, a 8%
proportion with vcGFM control is enough to guarantee an
acceptable dynamic performance of the system while the
ccGFM control required for an equal performance is only
achieved for a 99% proportion.
B. Comparison of the respective contributions of the ccGFM
and vcGFM to system stability
C. Mitigation of interactions phenomena
In the same manner as presented in the previous section, the
state-space model has been linearized and validated through
comparisons with the non-linear model in time-domain.
The stabilizing effect of the GFM controls is evaluated
V SC22
by determining the minimal proportion (Snom
/Sb ) of the
GFM-controlled converter required to stabilize the system at
the operating pointFigure 10 shows the root loci relative to
the vcGFM and ccGFM control, respectively, when varying the
above mentioned proportion. It is remarkable that the required
converter proportion can be as low as 3%, further highlighting
the stabilizing impact of the GFM control.
V SC22
On the other hand, the minimal Snom
/Sb proportion
required to restore small-signal stability varies significantly
In Section III, the stabilizing effect of the GFM controls
has been traced back to the mitigation of the interactions
between the GFL-controlled converters through the weak grid,
brought by GFM controls decoupling the GFL-controlled
converters from the network variables.Now, the participation
factor analysis is repeated to shed light on the underlying
mechanisms explaining these latter results.
For the case with ccGFM control, the parametric sensitivity
allows identifying the eigenvalue that first becomes unstable,
and, consequently, dominates the system dynamics. Figure 11
shows the participation factors of the various state variables
when V SC22 is in ccGFM mode. The contributions of the
two GFL-controlled converters, previously identified in Fig. 8,
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. XX, NO. Y, MONTH 2023
8
the same power flow pattern as before. For this study, the
passive load is chosen as a purely resistive load chosen such
as its power consumption is 10% of Sb , it follows that:
V SC1
Snom
= 0.9Sb
(a) VSC22 in vcGFM
(b) VSC22 in ccGFM
Fig. 13: Parametric sensitivity to VSC22 size with a load
are reaffirmed, though with the additional contributions of
the ccGFM filters used for the V SC22 QSEM. The ccGFMcontrolled converter fails to sufficiently limit the interactions
between the GFL-controlled converters and the weak grid,
which explains why a larger ccGFM-controlled converter
V SC22 (and a correspondingly smaller GFL-controlled converter V SC21 in this setup) is required to achieve system
stability.
The parametric sensitivity with regard to the proportion
of the vcGFM-controlled converter highlights two different
pairs of eigenvalues, denoted λ1 , λ2 in Fig. 12. For a low
GFM proportion, λ1 is the dominant eigenvalue. The participation factors reveal that all three converters contribute to the
eigenvalue λ1 with the PLL and the voltage regulation of the
GFL converters participating the most. However, the system
dynamics experience a change as the proportion of GFM
converter grows: beyond a 30% proportion, λ2 becomes the
dominant eigenvalue of the system. The participation factors
show that the GFL-controlled converters have a very limited
contribution to λ2 , thus highlighting again the decoupling of
the GFL-controlled converters from the weak grid, and hence
the faster (with regard to GFM proportion) stability restoration.
D. Effect of load type on previous findings
The previous studies have shown the interest of the proposed
test setup. Interaction phenomena and small-signal stability
limits have been highlighted without the constraints previously
encountered in a Thévenin equivalent setup.
However, it has been reported in literature that the type of
load plays a key role in the dynamic behavior and the stability
of the system [30]. Therefore, it is prudent to assess the
validity of the previously observed trends beyond the specific
case of a 100% active power-controlled power.
In this subsection, the previous setup is modified to include
a passive load in parallel with VSC1 .Accordingly, the nominal
power of VSC1 (here, the active load) is reduced to maintain
(15)
It is first verified, by time-domain EMT simulation and eigenvalues analysis, that if Zg = 1 pu and VSC2 is fully controlled
in GFL mode, the system would remain unstable despite the
change in the load type. Therefore, the previous subsection
study can be reconducted to evaluate the stabilizing effect of
both GFM control schemes. After linearizing and validating
the state-space model against time-domain EMT simulations,
a parametric sensitivity study with regard to the size of
the GFM-controlled VSC22 is carried on. Figure 13 shows
the evolution of the system eigenvalues for an increasing
proportion of the GFM-controlled converter.
When VSC22 is in vcGFM mode, the system becomes
small-signal stable for as little as a 2% proportion and reaches
a satisfactory dynamic performance for a 5% proportion.
The required minimal proportion is only slightly improved
compared to the test case with no passive load, showing that
the vcGFM control scheme is quite robust with regards to the
type of load.
However, when VSC22 is in ccGFM mode, the minimal
proportions to reach small-signal stability and satisfactory
dynamic behavior are significantly reduced from 45% to 6%
and from 99% to 89%, respectively. This finding shows that
the ccGFM control scheme provides a better stabilizing effect
for a smaller active load and with the presence of a passive
load. The passive load being resistive also contributes to the
system damping, making the stabilizing task easier for the
GFM-controlled converter.
It is also worth noting that the differences between the
two GFM schemes, while reduced, are still significant. It can
be shown that the gap between both controls can be further
reduced by increasing the size of the passive load (while
decreasing the active load) but it remains unbridgeable.
V. C ONCLUSION
In this paper, the small-signal stability of different controls
has been studied by analyzing the linearized state space model
of different combinations of controls under various operating
conditions. Using a dedicated test setup, the stability limits of
the controls are investigated. It is confirmed, thanks to the
analysis of participation factors, that the GFL controls are
more sensitive to the grid strength: they are more vulnerable
to interaction phenomena, and they rely on the stiffness of
the PCC voltage to synchronize. It is also found that the GFM
controls improve the stability of the GFL-controlled converters
by limiting their interactions with the weak grid. This smallsignal stability enhancement depends significantly on the type
of GFM control. In fact, while both studied GFM controls
are set to identically provide inertia, the vcGFM-controlled
converter fully decouples the GFL-controlled converter from
the weak grid, while the ccGFM-controlled converter only
limits the interactions of the GFL-controlled converter with
the weak grid and presents new interaction phenomena as it
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. XX, NO. Y, MONTH 2023
also relies on the PCC voltage for current reference generation,
thus being itself vulnerable to less stiff grids. This finding is
further extended by studying the stabilizing impact brought
by a GFM-controlled converter to a multi-infeed system and
under different types of load.
More importantly, while this paper has confirmed that using
GFM-controlled converters indeed improves the small-signal
stability of multi-infeed systems, it has shown that not all GFM
controls are equally effective, mainly due to current control.
Furthermore, considering the uncertainties and the lack of
knowledge over the load type and its distribution in future
power systems, the present study should not be considered as
a simple comparative analysis of two GFM control schemes
only, but also as a robustness study showing that when the
GFM controls account for a current control loop, they become
more dependent on the voltage stiffness provided by the rest of
the system due to various grid strength levels and load types.
Going forward, these findings are currently being validated
on complex systems more representative of realistic transmission grids and converters topologies (MMC, LCL-connected
VSC...). Finally, time domain studies with full EMT nonlinear models will ultimately confirm the validity of the results
discussed here for unbalanced conditions and large disturbance
stability.
A PPENDIX
The state variables of the GFL control are governed by the
following differential equations:

ωb

s.igx =
(vmx − vgx − Rc igx + ωLc igy )


L
c



ω
b


s.igy =
(vmy − vgy − Rc igy − ωLc igx )


L

c




s.P f = ωLP F (P − P f )





s.V f = ωLP F (Vg − V f )





s.Qf = ωLP F (Q − Qf )






s.ζ P = Kip (P ∗ − P f )





s.ζ V = Kiv (V ∗ − V f )




 s.ζ id = Kicc (i∗d − igd )
s.ζ iq





s.ζ P LL





s.θ̃g





s.Px




 s.Py





s.vgfx






s.vgfy





s.ifgx




s.ifgy
= Kicc (i∗q − igq )
(A1)
= KiP LL vgq
= ωb (ζ P LL + KpP LL vgq )
∗
= α(vm
− vmx ) + ωPy
x
∗
= α(vm
− vmy ) − ωPx
y
= ωinputs (vgx − vgfx ) + ωvgfy
= ωinputs (vgy − vgfy ) − ωvgfx
= ωinputs (igx − ifgx ) + ωifgy
= ωinputs (igy − ifgy ) − ωifgx
where ωb is the base frequency of the system in rad/s, ω is the
grid frequency in pu and the Ki , Kp parameters refer to the
integral and proportional gains of the various PI controllers.
9
The state variables of the vcGFM control are governed by
the following differential equations:

∗
1


(P ∗ − P + kd (ω̃g − ω̃))
s.ζ θm =


2H


∗
∗
s.θm
= ωb (ζ θm − ω ∗ )
(A2)

TV R
TV R

s.ζ
=
ω
(R
i
−
ζ
)

T
V
R
v
g
d
d
d


 TV R
s.ζq
= ωT V R (Rv igq − ζqT V R )
where ω ∗ is the frequency setpoint in pu, and kd is the VSM
gain calculated for the desired GFM damping ξ.
The state variables of the ccGFM control are governed by
the following differential equations:
 f
QSEM
f

s.vd = ωLP

F (vgd − vd )


 s.v f = ω QSEM (v − v f )
gq
q
q
LP F
(A3)
id
cc ∗

s.ζ
=
K
(i
−
i
)

g
i
d
d


 iq
s.ζ = Kicc (i∗q − igq )
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B IOGRAPHY S ECTION
Yahya Lamrani (Student Member, IEEE) received the M.Eng degree in
2021 from CentraleSupélec, Paris-Saclay, France. He also received the M.Sc.
degree in 2021 in electrical engineering, information technology and computer
engineering from RWTH Aachen, Germany. He is currently pursuing the
Ph.D. degree with the Centrale Lille Institute in the Laboratory of Electrical
Engineering and the Power Electronics (L2EP), Lille, France.
Frédéric Colas (Member, IEEE) received a Ph.D. in control system in 2007
from Ecole Centrale de Lille (France). Frédéric Colas is a member of the
Laboratory of Electrical Engineering (L2EP) in Lille and is a Research
Engineer at Arts et Métiers. His field of interest includes the integration of
dispersed generation systems in electrical grids, advanced control techniques
for power system, integration of power electronic converters in power systems
and hardware-in-the-loop simulation.
Thierry Van Cutsem (Fellow, IEEE) received the M.Sc. and Ph.D. degrees
from the University of Liège, Belgium. When retiring in 2021, he was a
Research Director of the Fund for Scientific Research (FNRS) and an Adjunct
Professor with the Department of Electrical Engineering and Computer
Science, University of Liège. He is currently active as a Consultant and
Adviser. His interests included power system dynamics, security, monitoring,
control, and simulation. He has been involved in collaborations with several
TSOs in Europe and Canada in the field of voltage control and stability. He
also served as the Chair of the IEEE PES Power System Dynamic Performance
Committee.
Carmen Cardozo (Member, IEEE) received a M.Eng in electrical engineering
from Universidad Simón Bolı́var, Miranda, Venezuela in 2008 and a M.Sc. in
Energy Physics from ENS Cachan, France in 2012. She has also received
a Ph.D. in 2016 from Université Paris-Saclay. Since then, she has been
working for the research and development department of Réseau de Transport
d’Electricité (RTE), the french TSO. Her research topics include the modeling
and control of power electronic interfaced resources, including HVDC crossborder inter-connector, for power system stability assessment.
Thibault Prevost (Member, IEEE) received his M.Eng from Supélec, Paris,
France. He has been working for research and development of the Réseau de
Transport d’Electricité (RTE) since 2007. He has worked on grid connection
studies for renewable and European grid codes for generators. He has been
working on European Projects Migrate and Osmose, still focus on the
operation of a system dominated with power electronic interfaced generation.
Xavier Guillaud (Member, IEEE) received a Ph.D. from University of Lille
in 1992 and joined the Laboratory of Electrical Engineering and Power
Electronics (L2EP) in 1993. He has been professor in Ecole Centrale of Lille
since 2002. First, he worked on modeling and control of power electronic
systems. Then, he studied the integration of distributed generation and
especially renewable energy in the power system. Nowadays, he is focused on
the integration of high voltage power electronic converters in the transmission
system. He is involved on several projects about power electronics on the grid
within European projects and a large number of projects with French electrical
utilities.