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Characterization of the Grid-forming function of a

power source based on its external frequency

smoothing capability

Debry Marie-Sophie, Denis Guillaume, Prevost Thibault

R´

eseau de Transport d’Electricit´

e (Research and Development Department)

La D´

efense

marie-sophie.debry / guillaume.denis / thibault.prevost @rte-france.com

Abstract—In upcoming converter dominated power systems,

some devices will be required to function as grid forming to

ensure stability. Several control schemes have been proposed in

the literature and future systems might include a mixture of

them, attending that they fulﬁll grid codes to be deﬁned. Here is

presented a method that derives a simple model based on basic

properties expected from grid forming devices from a system

point of view. This model serves to deﬁne necessary conditions

on local frequency smoothing capability to claim grid-forming

function in a power system. One application case is proposed

to discriminate between the so-called ”grid forming controls”

and ”grid following controls”, on the basis of their external

behavior rather than from their control structure. Eventually, a

cost implication in terms of storage size requirement is discussed.

This novel point of view on grid-forming control will certainly

help TSOs to specify their needs and anticipate the costs for the

upcoming transition.

Index Terms—converter controls, frequency dynamic, power

system dynamic, grid forming, droop control.

I. INTRODUCTION

The share of inverter-interfaced generation from renewable

sources is now greater than 100% of the electrical demand

in some countries [1]. In such systems, TSOs are mainly con-

cerned about the remaining level of inertia and about frequency

stability [2]. Indeed, lower inertia implies higher Rate of

Change of Frequency (RoCoF) and faster frequency transients.

With faster frequency dynamics, frequency-based protections

(e.g. load shedding) may be triggered, endangering system

stability. To allow stable operation of electrical systems with a

high share of converter interfaced generation, some of present

or future inverters will need to upgrade their classical grid

following controls into grid forming controls [3]. Amongst

others, the grid forming function characterizes the ability of

an electrical source to contribute to frequency stability by

making its variation throughout the system smoother. Several

control designs claim to provide a grid forming function to

This work is part of the MIGRATE project (Massive InteGRATion of

power Electronic devices). This project has received funding from the Eu-

ropean Unions Horizon 2020 research and innovation programme under grant

agreement No 691800. This article reﬂects only the authors views and the

European Commission is not responsible for any use that may be made of

the information it contains.

systems (VSM [4], Power Synch. [5], Virtual oscillator control

[7], [8], Matching control [9], Droop ﬁltered [10], SEBIR

[6], Direct voltage [11]) but their effective contribution to

frequency stability has not been highlighted from the system

operator perspective. The present paper proposes to list general

properties expected from grid forming sources to derive a

generic model that embed all the necessary functions for

proper operation. The model can then serve for quantitative

comparison of the grid forming function of different types of

inverter controls when they are connected to a grid. The main

goal of the suggested method is to be not intrusive, so that

only measurements at the point-of-common coupling will be

necessary on an experimental setup to draw conclusions on the

grid forming nature of a source. The method has been only

tested yet by samples from small signal simulations.

The remainder of the paper is organized as follows. The

simpliﬁed model of an ideal grid forming source is derived

in section II. The condition for proper parallel operation are

veriﬁed in section III. The method proposed to assess the

grid forming function provided by an inverter is detailed

in section IV. Section V focuses on simulation results for

different controls existing in the literature. A discussion on

the cost of grid forming function associated with the frequency

smoothing capability is added in VI. Section VII concludes the

paper.

II. MO DE L OF A N ID EA L GRID FORMING SOURCE

A. Necessary properties of source for islanded operation

When a power source is feeding a passive network au-

tonomously, it must comply with electrical quality standards

in terms of voltage and frequency range, so that the connected

load can operate under almost nominal conditions. These

quantities must be contained but also be controllable so that the

system operator can adapt the voltage and the frequency to the

operational situation. For slow dynamics referred traditionally

to quasi-static behavior, a grid forming source must embed

the following basic requirements, simpliﬁed and adapted from

french grid-codes [18] :

(i) The output RMS voltage at the source terminal must

track a reference V∗within a time response of 10

seconds.

(ii) The frequency must track an adjustable reference ω∗,

within a time response of 15 seconds.

For fastest dynamics, the grid forming source must behave

as closed as possible to a voltage source with limited voltage

magnitude and frequency variations. The voltage source be-

havior has never been speciﬁed in grid-codes, because it was

guaranteed by the physics and the design of synchronous gen-

erators, with no need for extra control loop. Observing that the

voltage magnitude response to rated load increase is limited by

the sub-transient reactance of synchronous generators, and that

the frequency dynamics are limited by the mechanical inertia

of rotors, it is proposed here to deﬁne additional requirements

for generic grid forming sources [3]:

(iii) The output voltage magnitude drop following rated load

connection must be contained within ±0.2 pu before the

aforementioned quasi-static control reacts.

(iv) The frequency dynamics at the source output must be

limited. By calling ωm(t)the instantaneous frequency

of the ideally produced voltage waveform, this time limit

will be represented by the time constant Tω.

dωm

dt ≤ωn

Tω

(1)

It corresponds to an initial RoCoF limitation.

The value of the frequency dynamics limit should be ﬁxed

according to criteria detailed later in this paper. One can

already note that the RoCoF level needn’t be as high as

the level ensured by the mechanical inertia of synchronous

generators.

The reason to make the frequency a variable quantity

with limited dynamics, even in the absence of synchronous

generator is three fold. First, it anticipates the fact that grid

forming sources will eventually operate in parallel, and that

they cannot rely on external signals for synchronization. Thus

the frequency cannot be ﬁxed to a single value, but must vary

so that all the frequency references of units stably converge to

a common value. The link between frequency dynamics and

stable synchronization process is discussed in the next section.

Second, the deﬁnition of an observable frequency is pre-

cious to give information on the state of the power system

everywhere on the grid. Taking the present system as an

example, the frequency deviation provides information on

system unbalance, which triggers the support of dispatchable

sources. Therefore, the frequency must be locally deﬁned as a

continuous quantity, i.e. with limited derivative. Furthermore,

if the frequency is expected to be independent from voltage

amplitude, then frequency dynamics are expected to be 5 to

10 times lower than the electrical period [3]. For example, a

limitation of frequency dynamics to 1% of nominal frequency

over one cycle yields (Tac=20 ms):

dωm

ωn

<dt

100Tac

(2)

Considering eq. 1, this limit is equivalent to Tω=100 Tac=2 s.

Third, the limitation of dynamics has regard to measurement

conditions. To ensure fast and common frequency estimation

by sensors and protections, the frequency signal variations

must be limited over the time. The accuracy of present pro-

tection devices is guaranteed for frequency variations slower

than 2 Hz.s−1, which is equivalent to Tω=25 s.

B. Simpliﬁed model of an inverter-based grid-forming source

Fig. 1 represents the assumed minimal model that can be

derived from the necessary conditions (i) to (iv) above. The

model will be later used to assess the grid-forming capability

in grid-connected conditions. The considered assumption are

the following :

Vm

VSC based converter model

Load

e

g

v

src

L

vsrc

vm

Ls

w

m

V*

w

*E

Islanded

Fig. 1. Minimal grid-forming source model under islanded condition

1) The voltage regulation integration time is set to Tv= 10 s.

The voltage Vmwill be considered constant for dynamics

under 1 s.

2) The converter is operating far from its current limit which

implies that the mandatory current limiting strategy is

disregarded.

3) The inductance Ls= 0.15 p.u. represents a accepted

value for the VSC ﬁlter inductance. This low value

will limit the voltage drop above 0.8 p.u. following the

connection of a rated load.

4) The frequency dynamics limitation is set to the most

constraining limit of the previous paragraph, namely

Tω=25 s.

Note that the second assumption is only valid under small-

signal conditions. Thus, the large-signal or transient studies

are out of the scope of the paper.

In ﬁg. 1, vm=Vmcos (θm(t)) is the sinusoidal voltage

equivalent to the modulated voltage of a generic Voltage

Source Converter, edenotes the terminal voltage, vg, the grid

voltage at the point of common coupling, vsrc is the equivalent

voltage of the Thevenin equivalent of the rest of the system.

In simulation, the transformer can also be represented by a

self inductance of Lc= 0.15 p.u.

The following section will investigate the necessary con-

ditions for operation with other grid-forming sources, i.e. in

grid-connected situation.

III. CONDITION FOR SYNCHRONIZATION OF MULTIPLE

GRID FORMING SOURCES

A. Static convergence to a common frequency

To obtain synchronization among grid-forming units on an

inductive transmission grid, linking the delivered active power

Pwith the frequency reference ω∗with a proportional gain mp

can be sufﬁcient, provided there is enough transfer capability

between sources. This additional condition has been added

to our model on ﬁg. 2. Such a simpliﬁed droop controlled

inverter presents the sufﬁcient and necessary conditions for a

stable synchronization with others [17].

P*P

VSC based converter model

Load

e

g

v

src

L

vsrc

Ls

Grid-connected

Vm

vm

w

m

V*

w

*

E

Fig. 2. Full minimal grid-forming source model

Analytically, for small signals and under quasi-static ap-

proximation of the grid, the synchronization mechanism can be

easily explained by the following equation is Laplace domain:

∆ω∗(s) = −mp∆P(s)(3)

∆ωm(s) = −mp

1 + Tωs∆P(s)(4)

Where mpis the droop gain and ∆Pis the power vari-

ation. In quasi-static approach, line dynamics are neglected

compared to frequency variation dynamics. Assuming the two

sources initially don’t exchange power, the power variation

under small grid frequency variations can be simpliﬁed into:

∆P(s) = UmUsrc

X

∆ωm(s)−∆ωsrc(s)

s(5)

Where Umand Usrc are the phase-to-phase RMS voltage,

and Xthe total reactance between the controlled inverter and

the voltage source. Combining (4) and (5) gives the inverter

frequency response to a grid frequency variation:

∆ωm(s) = 1

XTωs2

UmUsrcmp+X s

UmUsrcmp+ 1 ∆ωsrc(s)(6)

The unity static gain of the eq. (6) is the proof of the

synchronization mechanism of the inverter with the grid.

B. Stability condition for synchronization

To validate the simpliﬁed quasi-static approximation above,

we must ensure that frequency variations of the controlled

inverter do not interact with lines dynamics that have been

neglected, and do not interfere with voltage variations.

The ﬁrst condition is ensured by setting Tωto a sufﬁciently

high value, the latter condition is ensured by the assumption

1) of the model where the voltage is considered ﬁxed for fast

dynamics.

The sufﬁcient value of Tωis quantiﬁed in [8], as the

minimum value that let the time to the electrical transients

to vanish. The value chosen of Tω=25 s, largely fulﬁlls this

last condition. Hence, our grid-forming model is considered to

reach steady-state in an asymptotic way validating the previous

equations.

Interestingly, from a rough analysis, the eq. (6) gives that

one of the basic function of a grid forming control is to

locally ﬁlter the frequency variation with, at least, a second

order low pass ﬁlter. The ﬁltering cut off frequency given

by νc=fn.qUmUsrcmp

XTω≈5H z, which quantiﬁes the

grid forming capability regarding frequency stability, or the

frequency smoothing capability. Thus, for islanded and parallel

operation , the frequency dynamics limitation, understood as a

local frequency smoothing capability, is a necessary condition

for providing a grid forming capability to the system.

IV. EXTERNAL CHARACTERIZATION ON FREQUENCY

SMOOTHING CAPABILITY

A. Need for external characterization

The terms grid forming and grid following or grid support-

ing have been employed in many papers to specify inverter

control structures or functions [12]–[14]. However, their deﬁ-

nition is not clearly accepted. A very basic deﬁnition of a grid

forming inverter is an inverter that can operate in islanded

mode, while a grid following inverter must be connected to

the main grid with synchronous machines and/or grid forming

inverters to behave properly. Grid supporting inverters have

further been deﬁned to contribute to grid stability by providing

some ancillary services like voltage or frequency regulation.

Nevertheless, this deﬁnition does not take into account many

other speciﬁcations needed for a 100% power electronics

transmission system [15]. In addition, most of the authors

claim that the function of an inverter is deﬁned by its control

structure, such as in [12], [13]. In practice, a grid-forming-

grid-supporting control and a grid-following-grid-supporting

control can exist and have the same objectives in steady-

state but completely different dynamics, mainly based on the

chosen control time constant. In particular, their contribution

to fast frequency stability differs drastically. The functional

deﬁnitions of the literature are all the more unsatisfying as

some ”hybrid” controls [11] are neither grid forming nor grid

following.

The rest of the paper aims at describing an assessment

method of the ability of different inverter controls to locally

smooth frequency and thus to contribute to its stability. The

method is meant to contribute to clarify the frontier between

grid forming and grid following converters.

B. Principle of the method

From a test bench of ﬁg. 3, the ratio of a frequency variation

of eand vgare plotted on a Bode diagram and compared

VSC based

converter

Load

dc

u

dc

C

e

source

i

g

v

src

L

vsrc

Freq Freq

w

g

w

e

Fig. 3. Simulated system

with the response of the simulated ideal grid-forming model

of section II. In particular, the grid-forming impact on the

frequency is expected to be quantiﬁed by comparison with the

second order low-pass ﬁlter behavior of eq. (6).

The test is meant to be experimental, thus frequency mea-

surement devices are placed to accurately measure the voltage

waveforms eand vgand to derive frequency estimation. At

the present time, the method has been only tested based on

simulated data.

C. Simulation set-up

The operating protocol for the simulation is the following:

1) Building a noisy frequency source around the nominal

frequency:

∆ωg(t) = ωn(Aωcos(νωt)) (7)

2) Obtaining the frequency response a the terminal of the

controlled inverter.

For simplicity, it has been decided as a ﬁrst approach to evalu-

ate rather the frequency of vm, considering that the frequency

is directly given by the control output before modulation of

the VSC.

In our simpliﬁed model of ﬁg 2, the frequency is directly

given by ωm, whereas the tested control were modeled in dq-

frame. We had to take into account the speed of the rotating

frame ωdq and the faster angle variation. Thus :

θm(t) = Zωdq (t) + atan vmq(t)

vmd(t)(8)

ωm(t) = dθm

dt =ωdq (t) + 1

1 + v2

mq(t)

v2

md(t)

(9)

V. GRID FORMING ASSESSMENT RESULTS

A. Example of existing converter controls to be tested:

Three different inverter controls will be assessed in the

paper. All of them are said to be ”grid-supporting” controls:

•a classical grid following based control: synchronization

is ensured using a typical SRF-PLL [16]. The inverter is

controlled as a current source to regulate its active and

reactive power using a PI controller. Outer loops have

been added to provide both reactive power control and

primary frequency control, in both cases using propor-

tional (droop) characteristics.

•a grid forming based control: synchronization is done

using a ﬁltered droop control [10], it therefore provides

primary frequency control. The amplitude of output the

voltage of the inverter is regulated as constant.

•a hybrid approach called direct voltage control [11]: this

control has been speciﬁcally developed for offshore wind

farm to avoid harmonic resonances when the HVDC

inverter of the wind farm switches off for a short period.

It uses a PLL to synchronize to the grid but the voltage,

and not the current, is directly controlled . It is therefore

difﬁcult to classify according to the above deﬁnitions.

For comparison with the present dynamics of the system,

the impact of a synchronous machine regulating both voltage

and frequency will also be tested.

B. Expected results

The comparison between frequency variations Bode plot and

the presented model should be read as follows :

•If for frequency variations speed greater than νc, the gain

of the transfer function is low: the inverter does not react

to the frequency disturbance of the other source. Instead,

the inverter keeps its local frequency very close to 50 Hz,

and thus, smooths the frequency variations in the system.

•If for slow frequency variations under νc, the gain is

close to one: it means that the inverter closely follows the

variation of frequency introduced by the voltage source.

The lower the cutoff frequency, the more the inverter stabilizes

the grid frequency and the more it provides grid forming

function to the system. The method therefore enables to

quantitatively compare the impact of different controls on

frequency stability. If the cutoff frequency is greater than the

nominal frequency of the system, the inverter does not provide

any grid forming function since it only follows the system

frequency even without perturbation.

C. Simulation based results

The Bode plots of the three considered inverter controls and

the one of the synchronous machines are plotted on Fig. 4. The

phase diagram will not be plotted in this article as it does not

bring any additional value for the analysis done below.

As expected, it can be seen on Fig. 4 that the gain of all

transfer functions go to 0 for slow variations of frequency,

which means that the three controlled inverters synchronize to

the grid in steady state. The measured cutoff frequencies are:

•for the grid-forming by ﬁltered droop control: 6.5 Hz

•for the direct voltage control : 7 Hz

•as a reference, for the synchronous machine : 1.5 Hz

The behavior for high frequency may look surprising for the

grid feeding inverter. For frequencies above 70Hz, the PLL

does not capture the change in angle, but the grid voltage

feedforward does it in a grid-feeding based control [16].

Therefore, the gain of the system is close to 1. This inverter

does not impact the behavior of the system for high frequency

events.

Fig. 4. Gain of the transfer function for different inverter controls

On the contrary, the grid forming inverter provides some

damping to the frequency oscillations which frequency is

greater than 6.5 Hz, therefore it limits the rate of change of

frequency. The dynamics that are damped by the synchronous

machine are at even lower frequency.

D. Discussions around the proposed method

The proposed method compares the frequency ﬁltering

effect of different types of generation. This allows a quan-

titative evaluation of the impact of a source, without explicit

knowledge of the control.

•If most of the inverters on the grid are grid following

with fast PLL, the frequency dynamics will only be driven

by the remaining synchronous machines as the inverters

perfectly follow the frequency variations for most of

the frequency range. There is a ﬁltering effect only for

frequency variation between 10 and 70Hz.

•In the case of grid forming inverters, the frequency

dynamics of the system are limited by these inverters.

Such inverters slow down the frequency deviation that

occurs following a loss of generation. As stated in the

previous section, this would only be true if the inverters

have both some available energy and power, but system

wide, we can consider that not all grid forming inverter

would be close to their saturation.

•Direct voltage controlled inverters will also impact the

dynamic of the system, as their cutoff frequency is not so

far from the one of synchronous machines. Their behavior

is quite similar to the grid forming one. Still the slope

the gain of this inverter is less than the one of the grid

forming inverter, which means that for the same cutoff

frequency, it will less ﬁlter high frequency variations.

The system operators (SO) could therefore use this method

to assess if an inverter will improve the dynamics of their

system or not, or even to deﬁne a required cutoff frequency that

an inverter must at least fulﬁll (νc(inv)< νc(req)). Of course

depending on the system characteristics, the requirement of

cutoff frequency should not be the same. An island system

already has very fast frequency dynamics, while the continen-

tal Europe system with its multiple big synchronous machines

has slower dynamics. The cutoff requirement will also depend

on load shedding, islanding or governor control protections, or

on the frequency measurement accuracy of other devices. As

the active power ﬂow and the variation of frequency are linked

(see eq. (14) and (15) below), it can easily be understood that

an inverter that makes its frequency change slowly will not

be able to track its power reference fast and accurately. This

trade-off should be judged by the SO depending on its system

conﬁguration and the capacity of sources to be dispatchable

or not.

VI. DISCUSSION ON THE COST OF FREQUENCY

SMOOTHING CAPABILITY

To locally ﬁlter the frequency the inverter must have:

•Some additional energy buffer to provide to the grid

•Available current capability to increase its power output

when there is a frequency deviation.

The amount of the energy reserve can easily be estimated

by the following consideration. For the simpliﬁed model

of a voltage controlled inverter, subject to grid frequency

variations slower than the nominal frequency (ie, νω< ωn),

the simpliﬁed equation of the active power ﬂow between the

inverter and the source is:

P(t) = UmUsrcsin(θm(t)−θsr c(t))

X(10)

Let’s assume that the inverter perfectly ﬁlters a frequency

oscillation at a given frequency νω1, and that the inverter

frequency is ﬁxed at ωinv =ωn. Further assuming the worse

case where there is no power transfer initially, the phase angle

difference is only provoked by the frequency variation of

the voltage source. Therefore the power exchange can be re-

written as follow:

P(t) = UmUsrcsin(−Rt

0ωnAωcos(νωt)dt)

X(11)

P(t) = UmUsrcsin(−ωn

νωAωsin(νωt))

X(12)

It is worth noting that as νω>1.5Hz which implies

ωn

νωAω<1

3, validating the following approximation:

P(t)≈ − UmUsrc ωn

νωAωsin(νωt)

X(13)

The amount of additional energy reserve ∆Σ that is needed

from the inverter to be able to ﬁlter can then be calculated

based on integrating the power on half a period of the

frequency oscillation.

∆Σ = Zπ

νω

0

−UmUsrc ωn

νωAωsin(νωt)

Xdt(14)

∆Σ = 2UmUsrc Aω

X

ωn

ν2

ω

(15)

Numerial application in our test case, with Um=Usrc =

225kV ,Aω= 0.01,X= 30Ω and the nominal power Pnof

units equal to 1000MW:

∆Σ = Pn

10.6

ν2

ω

J(16)

The inverter must have enough reserve to operate at Pn

during 10.6

ν2

ωs.

VII. CONCLUSION

This article illustrates a method aiming at evaluating the

impact on power systems’ frequency dynamics of different

types of generation connection, interfaced either by a syn-

chronous generator or by power electronics. In the latter case,

the method determines on whether a converter control can

provide grid forming function to the grid or not. Its main

advantage is that it provides quantitative results that can be

easily interpreted without speciﬁc knowledge of the internal

control of the generating units.

The method is based on simulations, the results of which

enable to estimate the Bode magnitude plot of the ratio

between the frequency variations of the voltage imposed by

the considered generating unit and a frequency perturbation

introduced by an ideal voltage source.

The results exhibits that all considered generating units be-

have as low pass ﬁlters in the frequency domain but that their

cut off frequency greatly varies. Synchronous machines and

grid forming inverters smooth frequencies greater than a few

Hertz, whereas grid following inverters smooth frequencies

greater than a few tens of Hertz. The level of attenuation

of a given frequency can easily be deducted from the slope

of the Bode magnitude plot. The obtained results ﬁts the

prediction given by the theoretical transfer function of a basic

grid forming control by ﬁltered droop.

From our ﬁrst view, having a frequency that slowly varies is

of most importance as frequency measurements on short time

scale is very challenging. Having a common frequency that can

be easily and accurately measured would enable any type of

device to help for frequency regulation and would also allow

to keep some operation rules and protection schemes that are

based on frequency measurements.

The method proposed in this paper only covers one of the

aspects of the impact of grid forming inverters on the grid. If

making smoother frequency variations is a necessary condition

for a converter control to be considered as grid forming, it is

however not enough. As mentioned in the paper grid forming

converters must be able to operate in islanded mode and this

is not capture by the frequency characterization. They must be

protected against over currents during large transients which is

out of the scope of small-signal analysis. These two features,

as well as others presented in [10], cannot be detected by

the method proposed in this paper. It should therefore be

completed with other tools to globally characterize the grid

forming behavior of a control.

VIII. ANN EX

A. Parameters of the grid following based inverter

Current loop time response: Tri = 0.01 s, PLL bandwidth:

ωP LL =ωn= 314 rad.s−1,

B. Parameters of the grid forming based inverter

Voltage loop time response: Tr v = 0.05 s, Static droop gain:

mp= 4%, Power ﬁlter cutoff frequency: ωLP =ωn

10

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