Inertia effect and load sharing capability of grid forming
converters connected to a transmission grid
T. Qoria*, F. Gruson*, F. Colas*, G. Denis †, T. Prevost †, X. Guillaud*
* Univ. Lille, Centrale Lille, Arts et Métiers Paris Tech, HEI,
EA 2697 - L2EP - Laboratoire d’Electrotechnique et d’Electronique de Puissance, F-59000 Lille, France
†Réseau de Transport d’Electricité, Versailles, France
Keywords: Grid-forming converter, improved droop control,
inertia emulation, active power dynamic, power sharing
The virtual synchronous machine concept (VSM) has been
developed initially to reproduce the synchronous machine
stabilizing effect by providing inertia with the emulation of
swing equation, whereas droop control is developed initially
to ensure load sharing and has no inertia. An introduction of a
low pass filter to droop control has been motivated to filter
the active power measurement and ensures a time decoupling
with the inner control loops, whereas, this low-pass filter can
also provide inertia to the system. This functionality is limited
due to its negative impact on the active power dynamic. This
paper proposes an analysis of the conventional droop control
by showing its limitations and proposes an improved inertial
droop control that allows providing the inertia to the system
and ensures a good dynamic behavior of the active power at
once in simple manner, and without modifying the load
sharing capability. The results obtained are compared to the
conventional method (Droop control and VSM) in various
topologies in order to show the relevance of the proposed
Electrical machines have the physical property of storing
kinetic energy in their mechanical part; this energy
contributes essentially to the stability of the electrical grid.
In case of failure, the inertia is able to compensate
immediately the power imbalance and limits the frequency
variations which help the system to remain stable.
In recent years, renewable energy sources have been steadily
increasing, these latter are interfaced to the AC system
through power converters. Currently these converters are
controlled to inject active and reactive power by relying on
the voltage formed by the AC grid; this control strategy is
well-known as grid-following control.
This project has received funding from the European Union’s Horizon 2020
research and innovation program under grant agreement No 691800. This
paper reflects only the author’s views and the European Commission is not
responsible for any use that may be made of the information it contains.
A rise of renewable energy sources based on grid-following
control causes a significant reduction of the total electrical
grid inertia . It induces a faster dynamic response of the
frequency since they do not naturally bring this inertia effect
. Therefore the reaction time of the primary control
becomes slower than the frequency response time which may
lead to an unstable operation. This low-Inertia phenomenon is
already noticed in several areas, such as Ireland and UK.
More details about those data are published by ENTSOE .
It is possible to modify the active power with respect to the
derivative of the frequency to mimic the inertial effect. But
this supposes to measure the frequency and operate a
derivation which may induce some time delay and frequency
oscillations , .
Because of these limitations, new control laws are needed in
order to emulate the inertia effect. This subject has been
widely discussed in the literature, which has led to the
development of the grid forming control-  that induces
more natural inertia effect linked with the way to synchronize
the converter to the grid. This has been largely documented
on various publications dealing with virtual synchronous
machine, synchronverter, virtual synchronous generator
(VSG) or VISMA –. Some of these concepts emulate
the electrical behavior of the real synchronous machine in
order to enhance the load sharing transient; others imitate the
stabilizing effect by reproducing only the swing equation
effect. The virtual synchronous machine concept has been
developed also to ensure a self-synchronization to the main
AC grid .
From another side, the principle of the droop control is also
linked with the synchronous behavior of synchronous
generators. Lots of classical synchronous generators have a
static droop control that adjusts their mechanical power
injection with respect to the frequency variation, and thus
compensate for power unbalance. The static proportional
relationship between power and frequency creates a
mechanism of load sharing between generation units. This
effect can be reproduced by the droop control applied to the
grid forming converter, this concept has been widely
discussed in the context of micro-grids – and
uninterruptible power supply , . The droop control is
also known in the literature as Power Synchronization
Method (PSM) . The droop control principle has been
developed for a variety of applications. Then, it has been
improved in order to achieve a good decoupling between
active and reactive power, to guarantee an accurate reactive
power sharing by introducing virtual impedances, and also to
ensure a good transient behavior by filtering the oscillations
around the grid frequency .
Although conceptually these two concepts are used for
different reasons .i.e. droop control ensures the power sharing
between units in parallel operation, while VSM provides
inertia and damping; it has been shown in this context that the
two approaches are mathematically equivalent . In some
papers the load sharing using the VSM concept is ensured by
an extra loops, thus the model become more complex and
require additional loops , –.
This paper aims to combine the inertial effect of a VSM and
the load sharing capability of a droop control in a single
algorithm without adding any more loops or complexities.
In this paper, small-signal model for conventional droop
control is presented in order to show its limitations and
propose a new simple control strategy that provides inertia;
ensure a better active power transient and load sharing.
In order to simplify the study, only the average VSC model is
used in the development and the DC bus is considered as an
Ideal DC voltage source.
The inertia effect will be analyzed, improved and simulated in
various grid topologies (e.g. grid connected mode and parallel
The reminder of this paper is structured as follows. In
Section.2, a mathematical comparison between droop control
and VSM is presented with reference to the shortcomings of
droop control. Section.3 presents the proposed inertial droop
control with its design method. Section.4 shows a
comparative synthesis between the proposed method and
conventional control techniques.
2 Droop control and VSM comparison
2.1 Recall on classical algorithms
The choice of the droop control is motivated by its capacity to
interact with other units without dedicated communication
link. The architecture is presented in Figure 1. This controller
is responsible for power regulation and load sharing through
the parameter which defines the maximum variation of
the frequency for a nominal power of the AC source. Usually
the droop control gain is set to 5%. This means that for a
maximum variation of the active power, the frequency will
decrease by 5%.
Figure 1: Droop control.
A low-pass filter is often added on the power measurement or
on the frequency derivation as shown in figure 1 for dynamic
decoupling with inner loops, active power noises filtration,
and to avoid frequency jump.
The architecture of the VSM is presented in figure 2. This
concept is mainly chosen because it reproduces the
synchronous machine stabilizing effect by introducing inertial
effect of the synchronous machine. More complex models
have been also developed to mimic also the electrical
behavior of the synchronous machine as mentioned in the
Figure 2: VSM for virtual inertia and damping
2.2 Mathematical equivalence
The transfer function of the droop control is expressed in per-
unit as follow:
Where , and are respectivey the set-point active
power, the set-point frequency and the output converter
Note that the red expression in the equation above can be
neglected as the set-point is constant.
The transfer function of the VSM is expressed in per-unit as
Whereand are respectively the inertia constant and
damping coefficient, while can be a constant frequency
set-point or a grid frequency estimation depending on the
mode operation (i.e. power regulator of frequency regulator)
Equivalence between both approaches exists and can be
However, the aim of these two approaches is slightly
different. The VSM is mainly focused on bringing an inertial
effect to the grid. The coefficient is chosen first (e.g. =
5s) and then the coefficient is choosen to give a stable
From the analogy between equations (1) and (2), it is clear
that the droop control can provide inertia such as VSM thanks
to the low-pass filter dynamic where the inertia constant can
be tuned through the decrease of the filter cut-frequency.
While the decrease of induces oscillations on the active
power. In , must fulfills the following condition to
maintain a stable operation:
Hence, with the classical parameters, the inertia provided by
this control is limited to . If more inertia is needed,
these approaches need to be improved.
3 Proposed inertial droop control
In control theory, a derivative action aims to stabilize the
system and to control the undesirable overshoot; however a
lead-lag filter is generally used to overcome the numerical
issue linked to the derivative computation.
The proposed idea is to add a lead-lag controller on the active
power measurement in order to damp the active power
oscillations. The expression of this lead-lag action is
presented in (5).
The interval of should not be chosen too high in order to
ensure a proper operation in practice and to avoid soliciting
much derivative action.
Figure 3: The proposed inertial droop control
To tune the lead controller, the root-locus method based on
sensitivity analysis is used. A linear model in equation 6 has
been developed for the system presented below:
Figure 4: Grid-forming VSC connected to an infinite bus.
The system contains 5 state variables which are respectively
the grid currents dynamics, the converter output frequency,
the converter angle and the filter active power measurement.
The analysis will be interested only on dominant oscillatory
modes shown in Figure 5 for that corresponds
Figure 5: Lead-lag controller design
Using an iterative variation of the controller parameters, the
obtained results in figure 5 show that for a given lead-lag
controller frequency variation
[1 60] rad/s range, the
active power modes become more damped when the value of
increases. It means that the proposed control improves
widely the active power dynamic for large inertia constant
H=5s. A comparison between modes related to active power
dynamic with and without lead-lag controller is presented in
Table 1: The active power dynamic improvement
Inertia constant H = 5s
Droop with LP filter
Proposed Droop control
The improvements brought by the developed control are
checked with time domain simulations in figure 6. A step on
the active power ( ) is applied at t =1s.
Figure 6: comparison of active power dynamics between
conventional droop control and the proposed droop control.
(a) VSC Frequency, (b) Active Power
The next section presents a modified AC grid model with
variable frequency that behaves similarly to the traditional
synchronous machine response in order to show the active
power dynamic, load sharing and the inertia contribution of
the proposed control.
4 Comparison with other control techniques
Till now, the frequency of the voltage source modelling the
grid has been considered as fixed. In this section, a variable
frequency is considered in order to assess the behavior of this
source with respect to frequency variation. The studied
system is presented in figure 7. The rated power of the
converter and the grid are considered identical ( Power
Electronics integration). The system parameters are described
in Table 2.
The objective of this test case is to validate the proposed
method and to compare it with other control strategies by
showing the frequency variation; load sharing and the active
power dynamic (e.g. load variation for different nominal
power of the converter and the equivalent AC grid).
The AC grid frequency is driven by a model representing a
kind of simplified equivalent synchronous machine with a
droop control. Where , , and are respectively the
droop gain, the inertia constant, the lead time constant and the
lag time constant. The lead-lag component aims to reproduce
the synchronous machine frequency dip behavior.
Figure 7: Controlled voltage source connected to an infinite
AC grid with variable frequency
(a) (b) (c)
(a) (b) (c)
Figure 8: Comparison of the proposed method to conventional ones (a) Conventional droop control with H=5s, (b) VSM
with high damping coefficient , (c) VSM with a small damping coefficient equivalent to the droop gain.
Table 2: System parameters
The developed control is compared to the control strategies of
figures 1 and figure 2. In simulations of figure 8, a load
variation is applied at t=5s.
In figure (8.a), the conventional droop is parametrically
adapted to obtain an inertia constant of =5s. In these
conditions, the conventional droop control shows a good
frequency response and power sharing in steady state,
however, the active power dynamic remains very oscillatory.
Comparing to the proposed method, this latter ensures at once
a good active power dynamic, power-sharing capability and
Comparing the proposed method to the VSM and according
to the function that VSM should ensure, the value of can be
adapted to obtain a good damping , in this case, the VSM
cannot ensure any more the power sharing capability since the
droop gain of the two AC source is completely different as
shown in figure (8.b), this operation mode is not acceptable in
large power transmission grid. Or, the parameter is adapted
to ensure power sharing and in this situation we fall back on
the same problematic of the conventional droop as shown in
figure (8.c) (oscillatory active power). In both cases the
developed control remains advantageous since it fulfills the
requested specifications (.i.e. Load sharing capability, good
active power dynamic and inertia providing).
It is possible to add two additional loops to the VSM (i.e. PLL
and classic frequency droop control) in order to ensure
separately the functionalities needed, where the PLL is
required for frequency measurement and the classical droop
for frequency regulation. The principle of this control is
similar conceptually to the one of the real synchronous
machine (Figure.8), while the implementation remains
Figure 9: VSM with frequency droop
Taking into account these changes, the results obtained by
VSM in figure 9 remains nearly the same to the proposed
method as shown in the figure 10. In the following simulation
the response time of the PLL is 50ms.
If the PLL dynamic is chosen slower (e.g. ), the
frequency response will change and the active power dynamic
also (Figure 11), therefore, the control parameters require a
new tuning in order to get acceptable performances.
Moreover, PLLs introduce a non-negligible delay in practice
which can limit the performance of the controllers that
depend on the frequency estimation of the PLL. Recent
publications have recognized the impact of PLLs in the
regulation provided by non-synchronous devices, but also the
potential instabilities that these devices can cause to power
converter , hence the advantage of the proposed method
which remains very simple to implement without additional
measurement or control complexity.
Figure 10: Comparison of the proposed method to the VSM
with frequency droop. (a) The frequency. (b) The active
power of the VSC and the AC grid
Figure 11: Comparison of the proposed method to the VSM in
case of PLL dynamic change. (a) The frequency. (b) The
active power of the VSC and the AC grid
In this paper, a mathematical comparison between
conventional droop and VSM is presented to show the
similarity of the two control strategies. Subsequently the
limitations of the conventional droop in terms of inertia
providing is highlighted, which led to the development of a
new control law based on droop control principle that ensure
a good dynamic behavior (Frequency and active power) and
also static one (power sharing capability).
In order to show the relevance of the proposed method, this
latter has been compared with conventional methods such as
droop control, VSM and VSM + Frequency Droop.
From a perspective point of view, this technique will be
analyzed in case of several converters to show its impact on
the system stability taking into account all converter
dynamics (i.e. LC filters).
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