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1

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

taoufik.qoria@ensam.eu

Keywords: Grid-forming converter, improved droop control,

inertia emulation, active power dynamic, power sharing

capability.

Abstract

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

method.

1 Introduction

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 [1]. It induces a faster dynamic response of the

frequency since they do not naturally bring this inertia effect

[2]. 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 [3].

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 [2], [4].

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[5]- [6] 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 [7]–[13]. 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 [11].

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 [14]–[17] and

uninterruptible power supply [18], [19]. The droop control is

also known in the literature as Power Synchronization

Method (PSM) [20]. 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

2

power sharing by introducing virtual impedances, and also to

ensure a good transient behavior by filtering the oscillations

around the grid frequency [20].

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 [21]. 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 [8], [22]–[24].

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

mode).

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

introduction.

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:

(1)

Where , and are respectivey the set-point active

power, the set-point frequency and the output converter

frequency.

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

follow:

(2)

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

expressed by:

,

(3)

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

behavior.

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 [25], must fulfills the following condition to

maintain a stable operation:

(4)

3

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).

(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.

(6)

The analysis will be interested only on dominant oscillatory

modes shown in Figure 5 for that corresponds

to .

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

the table.1.

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.

𝑇𝑟𝑎𝑑𝑠

𝑇𝑟𝑎𝑑𝑠

𝑇𝑟𝑎𝑑𝑠

4

Figure 6: comparison of active power dynamics between

conventional droop control and the proposed droop control.

(a) VSC Frequency, (b) Active Power

System parameters:

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.

𝑎

𝑏

5

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

inertia providing.

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 [26], 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

different.

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 [27], hence the advantage of the proposed method

which remains very simple to implement without additional

measurement or control complexity.

(a)

(b)

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

(a)

(b)

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

6

5 Conclusion

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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