Stability Analysis of Two Types of

Grid-Forming Converters for Weak Grids

Sulaiman Almutairi, Zhixin Miao, Lingling Fan†

Department of Electrical Engineering, University of South Florida

Tampa FL USA 33620

Correspondence Email: †linglingfan@usf.edu

Abstract

Grid-forming converters are designed to be capable of operating without a main grid. It is known that

grid-following converters operating in weak grid conditions may experience dynamic voltage stability

issues. Low-frequency oscillations have been observed in the real world. The objective of this paper

is to examine grid-forming converters’ weak grid operating characteristics. To this end, two types of

grid-forming converters are examined for weak grid operation. Firstly, the steady-state operation limits

are identified relying on optimization problem formulation and solving. It is found that both grid-forming

converters may reduce the steady-state operation limit, compared to a grid-following converter. Secondly,

dynamic stability limits are identified through electromagnetic transient (EMT) simulations. Furthermore,

each converter’s frequency-domain admittance and impedance characteristics are characterized using a

data-driven system identification method. s-domain admittance-base eigenvalue analysis confirms the

dynamic stability limit for each converter. It is found that low-frequency oscillations appear in one type

of converters while do not appear in another type of converters. The two grid-forming converters can

enhance dynamic stability comparing to the specific grid-following converter.

Index Terms

Voltage-source converter, , low-frequency oscillations, admittance measurement, dq domain, grid-

forming, Grid-following, droop control, stability analysis.

1

I. INTRODUCTION

VOLTAGE-source converters (VSCs) have been rapidly penetrating into the electric power grids.

High penetration of VSCs introduced new dynamics and stability problems. Common stability

issues include voltage stability and dynamic stability due to the interconnections of VSCs into weak

grids [1]. Appendix B of [1] lists several real-world stability issues. Classic voltage stability issues were

observed by ERCOT and BPA. In 2010 in BPA, with wind power ramped up, voltage at the interconnected

230 kV system dropped. In addition, low-frequency oscillation related to weak grid conditions have also

been observed in Texas [2].

The low-frequency dynamic stability phenomenon has been thoroughly investigated in the literature

(e.g. [3]–[6]). The mechanism of instability was pinpointed as the point of common coupling (PCC)

voltage reduction upon power-order increase. This power and voltage coupling is due to grid-following

control structure and weak grid effect. The coupling becomes stronger when grid is weaker. In another

word, the sensitivity of dV/dP becomes high for weak grids. The grid strength is generally indicated by

the short circuit ratio (SCR), which can be given as 1/Xg, where Xg is the aggregated transmission line

reactance. According to the IEEE Standard 1204-1997 [7], the AC grid can be classified as weak grid if

SCR < 3.

Currently, most of the IBRs adopt grid-following control. In this control mode, a converter has no

capability to provide frequency and hence it cannot work as a sole source to serve loads. In grid-

following control, phase-locked-loop (PLL) is required to synchronize the VSC with the grid. The

grid provides frequency support. With constantly increasing penetration of IBRs, the portion of the

traditional synchronous generator-based resources keeps reducing, which poses challenges for operation.

One such challenge is on providing frequency and voltage support relying on traditional resources only

[8]. To deal with these concerns, grid-forming controllers have been introduced to provide frequency

and voltage regulation. According to National Renewable Energy Lab’s recent roadmap report on grid-

forming converter, there are a variety of grid-forming controls that have been proposed in the literature,

e.g., droop-based controllers, virtual synchronous machines (also known as synchronverters), and virtual

oscillator controllers [9]. Compared with the last two types, the droop-based converter control is the most

popular control and they have been used in microgrids, as indicated by [10]. Thus, our paper focuses

on grid-forming converters that are based on P-f and V-Q droop control. This mode can participate in

frequency regulation and voltage support [11] [12].

There are two types of grid-forming converter control designs: the first type orients from grid-following

2

control with additional droop characteristics; the second orients from grid-forming control with additional

droop controls.

The objective of this paper is to investigate grid-forming converters’ operation in weak grids to see if

they can provide stability enhancement to the system. To this end, first, steady-state analysis is carried out

to examine the system steady-state limits for the grid-following converter and two types of grid-forming

converters. Second, several EMT testbeds are set up to examine the operation of grid-following VSC

and two types of grid-forming VSCs in weak grid interconnection mode. The dynamic system limits are

identified. Finally, the admittance models of the three converters are obtained via a data-driven system

identification method. The frequency-domain admittance and impedance characteristics are compared for

three converters. With admittance model available, s-domain admittance based eigenvalue analysis [13],

[14] further confirms the dynamic stability limits.

Since one type of grid-forming converter is based on grid-following converter control with additional

droop control, open-loop system analysis is also conducted in this research to demonstrate the effect of

droop control on stability.

The contribution of this paper is three-fold.

• First, this paper designed an optimization problem formulation to identify steady-state operation

limits for grid-following and grid-forming converters. The analysis results indicate that contrary to

intuitive perception, grid-following converter has a larger steady-state stability margin compared to

grid-forming converters equipped with droop controls.

• Second, this paper finds out that both type of grid-forming converters can enhance dynamic stability

comparing to grid-following converter.

• Third, this paper demonstrates why droop controls can enhance dynamic stability through a data-

driven open-loop system analysis.

The remainder of the paper is organized as follows. The VSC structure and the overall control configura-

tion of the VSC grid-following and grid-forming controllers are presented in Section II. The steady-state

calculation is giving in Section III. Section IV presents EMT study of VSC operation in weak grids.

Section V presents the comparison of three admittance/impedance models obtained via data-driven system

identification method. In addition, eigenvalues-based stability analysis using the measured admittance

confirms EMT study results on system dynamic stability limit. An additional open-loop system analysis

is also carried out in Section V to demonstrate the influence of droop control on system stability. The

paper is concluded in Section VI.

3

II. SYSTEM STRUCTURE AND CONTROL

In Fig. 1, a VSC with RL filter is connected to an ac grid. The ac grid is modeled as a voltage

source (“infinite bus”) with a voltage magnitude and frequency fixed at 1 p.u and 60 Hz. Lf and Rf

are the filter inductor and resistor, respectively. Cf is the shunt capacitor that provides reactive power.

Rg and Lg are the equivalent resistance and inductance of the grid transmission line. i, vPCC and ig

are the converter-side current, the PCC voltage and the grid-side current. The EMT testbed is developed

using MATLAB/SimPowerSystems 100-kW grid-connected inverter demo. The converter control and

transmission line topology have been modified for the purpose of this study. The inverter is supplied by

an ideal 500 V DC source. The 500 DC voltage is converted to a three-phase 260 ac voltage. The system

parameters are presented in Table I.

TABLE I: System Parameters

Description Parameters ValueRated Power Sb 100 kW

Reference Power P ∗ 0.9 puFrequency ω 377 rad/s

Converter filterRf 0.15/50 puLf 0.15 pu

Shunt capacitor Cf 0.25 puDC voltage VDC 500 V

Line impedance Rg 0.1Xg

Power loop Kpp, Kip 1, 100Inner loop Kpi, Kii 0.3, 5

Converter 1 Voltage loop Kpv, Kiv 1, 100Converter 2 Voltage loop Kpv, Kiv 0.5, 50

Converter 3 dq-axis voltageKpd, Kid 1, 100Kpq, Kiq 1, 100

P-f droop m 0.2V-Q droop n 0.1

In this paper, three types of VSC control are examined: grid-following and two grid-forming controls.

All controls adopt vector control. The grid-following control (notated as converter 1 in Fig. 1(a)) is based

on PCC voltage oriented control [11] [15]. Phase-locked-loop (PLL) synchronizes the VSC to the grid.

The converter control consists of two cascade control loops. The inner control loop is for fast current

control and the outer control regulates real power and PCC voltage magnitude.

The first grid-forming converter design (notated as converter 2 in Fig. 1(a)) is based on the grid-

following control with additional droops to adjust power order and voltage order. The second grid-forming

converter design (notated as converter 3 in Fig. 1(b)) is based on grid-forming converter with additional

droops. A significant difference that differentiates Converter 3 from Converter 2 is that the PLL is not

4

𝑅𝑓 𝐿𝑓

𝐶𝑓

𝑉𝑃𝐶𝐶

𝑅𝑔 𝐿𝑔

𝑖 𝑖𝑔 𝑉𝑔

𝑇1 𝑇2

𝑉𝐷𝐶

𝑉𝑆𝐶

𝑃𝑊𝑀 𝑃𝑊𝑀

𝑉𝑃𝐶𝐶,𝑎𝑏𝑐

𝑖𝑎𝑏𝑐

𝑉𝑑 𝑖𝑑 𝑖𝑞

∫

𝜔∗

𝑖𝑑

𝑖𝑞

PI

PI

PI

PI

𝜔𝐿𝑓

𝜔𝐿𝑓

𝑉𝑑

𝑉𝑞

𝑢𝑑∗

𝑢𝑞∗

𝑖𝑑𝑟𝑒𝑓

𝑖𝑞𝑟𝑒𝑓

+ -

+

+

- +

+

𝜃𝑃𝐿𝐿

𝑃∗

𝑃

-

𝑉𝑝𝑐𝑐

𝑎𝑏𝑐 𝑉𝑞 ∆𝜔

𝜃𝑃𝐿𝐿

𝑉∗

PI

+

-

+

𝑑𝑞

𝑎𝑏𝑐

𝑑𝑞 𝜔

Power/Voltage controlCurrent control

𝑎𝑏𝑐

𝑑𝑞

+

- 𝜔

𝜔∗

+

- 𝜔

𝜔∗

1/𝑚

- 𝑄∗

𝑄

V-Q droop control

𝑄∗

𝑄

V-Q droop control

𝑛

P-f droop control

Converter 1

Converter 2

𝑉𝑃𝐶𝐶,𝑎𝑏𝑐

𝑖𝑎𝑏𝑐

𝑉𝑑 𝑖𝑑 𝑖𝑞

∫

𝜔∗

𝑖𝑑

𝑖𝑞

PI

PI

PI

PI

𝜔𝐿𝑓

𝜔𝐿𝑓

𝑉𝑑

𝑉𝑞

𝑢𝑑∗

𝑢𝑞∗

𝑖𝑑𝑟𝑒𝑓

𝑖𝑞𝑟𝑒𝑓

+ -

+

+

- +

+

𝜃𝑃𝐿𝐿

𝑃∗

𝑃

-

𝑉𝑝𝑐𝑐

𝑎𝑏𝑐 𝑉𝑞 ∆𝜔

𝜃𝑃𝐿𝐿

𝑉∗

PI

+

-

+

𝑑𝑞

𝑎𝑏𝑐

𝑑𝑞 𝜔

Power/Voltage controlCurrent control

𝑎𝑏𝑐

𝑑𝑞

+

- 𝜔

𝜔∗

1/𝑚

- 𝑄∗

𝑄

V-Q droop control

𝑛

P-f droop control

Converter 1

Converter 2

+ -

LoadLoad

(a)

𝑉𝑃𝐶𝐶 ,𝑎𝑏𝑐 𝑖𝑎𝑏𝑐

𝑉𝑞 𝑉𝑑 𝑖𝑑 𝑖𝑞

𝑃∗

𝑃

𝑚

𝑛 𝑄∗

𝑄

𝜔𝐵𝑎𝑠𝑒 ∫ 𝑎𝑏𝑐/𝑑𝑞

𝜔∗

𝑉∗

𝑉𝑑

𝑉𝑞

𝑉𝑞 ,𝑟𝑒𝑓

𝜔𝐶𝑓

𝜔𝐶𝑓

𝑖𝑑

𝑖𝑞

PI

PI

PI

PI

𝜔𝐿𝑓

𝜔𝐿𝑓

𝑉𝑑

𝑉𝑞

𝑢𝑑∗

𝑢𝑞∗

𝑖𝑑𝑟𝑒𝑓

𝑖𝑞𝑟𝑒𝑓

+

+ -

+

+ +

-

-

+

+

-

+

+

+

+

+

+

+ -

-

-

- + +

𝜃

Current control Power/Voltage control

V-Q droop control

P-f droop control

𝐸

𝑎𝑏𝑐

𝑑𝑞

𝑃𝑊𝑀 𝑃𝑊𝑀

𝑅𝑓 𝐿𝑓

𝐶𝑓

𝑉𝑃𝐶𝐶

𝑅𝑔 𝐿𝑔

𝑖 𝑖𝑔 𝑉𝑔

𝑇1 𝑇2

𝑉𝐷𝐶

𝑉𝑆𝐶

LoadLoad

Converter 3

(b)

Fig. 1: Control structure of the VSC controls: (a) Converter 1 & 2, (b) Converter 3.

used for synchronization. Rather, a given frequency is used in grid-forming control. For Converter 3, this

frequency order is modulated based on the error between the power measurement and the power order.

The inner current control structure is same as that in the grid-following control while the outer control

regulates PCC voltage magnitude. With the q-axis voltage order set as 0, d-axis voltage order determines

the steady-state PCC voltage magnitude.

The full details of all controls are shown in Fig.1.

5

III. STEADY-STATE ANALYSIS

Assume that the three VSCs are connected to a grid with the same real power order, voltage order,

and reactive power order setting. For grid-connected operation, the PCC bus has no loads connected.

The loads will be connected to the PCC bus only when the converters are in autonomous mode. The

steady-state operation limit of the three VSC controllers is denoted by the maximum grid impedance Xg.

This limit can be obtained using an optimization problem shown in 2.

The complex power from the PCC bus to the grid can be written as follows: S = V PCCIg∗, where

V PCC = vd + jvq and Ig = igd + jigq are the voltage and current phasors, and

Ig =V PCC − V g

Rg + jXg

The real and reactive power from the converter to the PCC bus can be expressed as:

P = real(S) = vdigd + vqigq,

Q = imag(S) +BfV2PCC = −vdigq + vqigd +BfV

2PCC.

For grid-forming converters, at steady-state, the relationship of the PCC voltage magnitude and reactive

power is determined by the droop control characteristics.

VPCC = V ∗ + n(Q∗ −Q). (1)

For grid-following converters, the above relationship becomes VPCC = V ∗ or it is equivalent that

n = 0.

Due to grid-connected operation mode, the system frequency is always at its nominal. Hence, real

power always follows the power order (P = P ∗).

The formulation is as follows:

maximize Xg

subject to P ∗ − P = 0

P = vdigd + vqigq

Q = −vdigq + vqigd +BfV2PCC

VPCC − V ∗ − n(Q∗ −Q) = 0

Ig =V PCC − V g

Rg + jXg

(2)

6

The decision variables are the PCC voltage VPCC, its angle θPCC, line impedance Xg and reactive

power Q. A set of power and voltage orders are given as follows. The real and reactive power are set

as: P ∗ = 0.9 pu, Q∗ = 0.4 pu, whereas the reference voltage is set as: V ∗ = 1.0 pu. Table II presents

the steady-state limit for each converter at the respective power and voltage order setting.

TABLE II: Steady-state limits

Control mode Xg Q VPCC θPCC nGrid-Following 1.22 0.56 1.0 1.67 0Grid-forming 1.20 0.49 0.99 1.58 0.1

It can be seen that the steady-state limit for the grid-following converter is about Xg = 1.22 pu while

for the grid-forming converter is about Xg = 1.20 pu.

Remarks: The above analysis shows that grid-forming converters do not enhance steady-state operation

margin.

IV. EMT STUDY ON WEAK GRID OPERATION

In this section, we examine weak grid operation for three converters. The EMT testbed results are

shown in Fig. 2. The dynamics of the PCC voltages, exported real and reactive power from the converter

and the system frequency are presented. A small perturbation is applied to the system by a step change

in the reference power P ∗ from 0.9 p.u to 0.91 p.u at t = 2 sec. The line impedance is Xg = 1.1. It

is observed that when the step change is applied, converter 1 is experiencing oscillations around 6 Hz

and becomes unstable, while converter 2 and 3 are kept stable. The dynamics response results show that

converter 1 limit is reduced to Xg = 1.1 pu.

Fig. 3 presents the simulation results with different grid strength and compares between the other two

converters: converter 2 and 3. The line impedance is Xg = 1.17 pu. A step change is applied to the

reference power P ∗ from 0.9 p.u to 0.91 p.u . From the dynamic responses, around 2 Hz oscillation can

be observed when the step change is applied for converter 2, whereas a stable operation is observed for

converter 3.

Finally, Fig. 4 depicts the dynamic responses of converter 3 with two different grid strengths: one is

Xg = 1.17 pu and another is Xg = 1.19 pu. After a step change in the power reference P ∗ at t = 2 sec,

the system stays stable for Xg = 1.17 pu, while it loses stability when Xg = 1.19 pu.

Remarks: Based on the simulation results, converter 3 can provide better power transfer capabilities

comparing to the other two types. The dynamic stability limits for the three converters are Xg = 1.1 pu

7

Fig. 2: Dynamic responses of the EMT testbed with all control modes, Xg = 1.1 pu: PCC voltage, real and reactive power toPCC bus and system frequency.

Fig. 3: Dynamic responses of the EMT testbed of converter 2 and 3, Xg = 1.17 pu: PCC voltage, real and reactive power toPCC bus and system frequency.

for converter 1, Xg = 1.17 pu for converter 2, and 1.19 > Xg > 1.17 for converter 3. It can be seen

low-frequency oscillations appear for converter 2. On the other hand, converter 3 can operate close to

the steady-state limit and low-frequency oscillations do not appear.

V. STABILITY ANALYSIS

In this section, admittance models of the three converters will be compared. The admittance models

are obtained via a measurement testbed. With the admittance model available, eigenvalue analysis is

8

Fig. 4: Dynamic responses of the EMT testbed of converter 3, Xg = 1.17 & 1.19 pu: PCC voltage, real and reactive powerto PCC bus and system frequency.

𝑃𝐶𝐶 𝑉𝑔

𝑉𝐷𝐶

𝑖𝑑

𝑖𝑞 𝑑𝑞

𝑎𝑏𝑐

𝑉𝑆𝐶

𝑑𝑞

𝑎𝑏𝑐 𝑣𝑑

𝑣𝑞

𝑣𝑎𝑏𝑐

𝑖𝑎𝑏𝑐 𝜔𝑡 𝜔𝑡 Controlled

Voltage Source

𝑌𝑣𝑠𝑐 𝑌𝑔

Fig. 5: An admittance measurement testbed of VSC.

conducted to demonstrate grid strength’s impact on stability. The analysis results corroborate with the

EMT simulation results in Section III. Finally, droop control’s effect on stability is also examined using

open-loop system analysis.

A. Measuring admittance

Impedance or admittance-based model analysis has been introduced in the literature [16]. Frequency-

domain measurement is the most commonly used technology today. To obtain impedance or admittance

measurements, two types of techniques are used. The first one is the harmonic injection technique, also

known as frequency scanning [17], [18]. In this technique, a measurement testbed for an IBR is required.

First, a device is connected to an ideal voltage source with a nominal frequency 60 Hz to operate at a

specific operation condition. Another voltage source at a specific frequency is connected in series with the

main voltage source to create voltage perturbation. Fast Fourier transform (FFT) is conducted to measure

the current component at that frequency. The ratio between voltage and current at the injected frequency

9

is then obtained. To obtain an input/output model defined in either state-space or transfer function, an

extra step of frequency-domain data fitting, vector fitting [19], must be performed.

The second technique uses an external hardware device called impedance measurement unit (IMU).

The impedance is extracted while the system is online. Thus, a separate measurements testbed is not

required. The device is installed in the system and injects voltage perturbation. The frequency domain

information from both voltage and current measurements is gathered using FFT and the impedance in

dq-frame is then computed [20]–[22].

Although these methods are widely adopted, they have a time-consuming process because they are based

on repeated injections. Moreover, an analytical form of the impedance is required to conduct stability

analysis. To address these challenges, an efficient tool to measure the dq-frame admittance matrix using

step responses proposed in [23], [24] is applied in this paper.

A measurement testbed of VSC is set up and its topology is shown in Fig. 5. The VSC is connected

to an ideal voltage source and operates at desired condition. Between the PCC bus and the ideal voltage

source, there is a small impedance. Two experiments are required to generate time-domain step responses

for admittance identification. The first experiment is to apply a step change in the d-axis voltage of the

ideal voltage source, and the second one is to apply a step change in the q-axis voltage. Each experiment

produces two time-domain responses: one is the d-axis line current i(k)d (t) and the other is the q-axis line

current i(k)q (t), where k = 1, 2 indicates the index of each event number.

For each experiment, the Laplace domain expression can be obtained: id(s), iq(s). Then, the admittance

matrix can be found as:

Yvsc = −

i(1)d (s)

v(1)d (s)

i(2)d (s)

v(2)q (s)

i(1)q (s)

v(1)d (s)

i(2)q (s)

v(2)q (s)

= −s

p

i(1)d (s) i(2)d (s)

i(1)q (s) i

(2)q (s)

(3)

where p is the size of the perturbation. The negative sign is owing to the current direction since it is

going out of the converter.

In this research, a 2% step change is applied to vgd. Line current id and iq are measured. Then, a 2%

step change is applied to vgq and the line current id and iq are measured. Fig. 6 presents the data related

to three converters. The data are fed to the system identification algorithm, e.g., eigensystem realization

algorithm (ERA), and the signals’ s-domain expressions can be found. Furthermore, the admittance

models can be found.

10

1.9 2 2.1 2.2 2.3 2.4 2.5

0.8

0.82

0.84

i d1

measurements

reconstructed data

1.9 2 2.1 2.2 2.3 2.4 2.50.45

0.5

0.55

i q1

1.9 2 2.1 2.2 2.3 2.4 2.5

0.8

0.82

0.84

i d2

1.9 2 2.1 2.2 2.3 2.4 2.5

Time (s)

0.45

0.5

0.55i q2

(a)

1.9 2 2.1 2.2 2.3 2.4 2.5

0.8

0.82

0.84

i d1

measurements

reconstructed data

1.9 2 2.1 2.2 2.3 2.4 2.50.45

0.5

0.55

i q1

1.9 2 2.1 2.2 2.3 2.4 2.5

0.8

0.82

0.84

i d2

1.9 2 2.1 2.2 2.3 2.4 2.5

Time (s)

0.45

0.5

0.55

i q2

(b)

1.9 2 2.1 2.2 2.3 2.4 2.5

0.8

0.82

0.84

i d1

1.9 2 2.1 2.2 2.3 2.4 2.50.45

0.5

0.55

i q1

1.9 2 2.1 2.2 2.3 2.4 2.5

0.8

0.82

0.84

i d2

1.9 2 2.1 2.2 2.3 2.4 2.5

Time (s)

0.45

0.5

0.55

i q2

measurements

reconstructed data

(c)

Fig. 6: Dynamic responses of id and iq for two events: i(1)d and i(1)q for event 1, and i

(2)d and i

(2)q for event 2. The step change

of 2% is applied at t = 2 sec. a) converter 1, b) converter 2, c) converter 3.

11

In Fig. 6, the blue curves are the measured data and the red curves are the reconstructed data using the

identification algorithm. Data from 2 to 2.5 s are used for analysis. With 18th system order assumption and

2500 Hz sampling frequency, the identified admittance matrix is obtained. The Bode plots are presented

in Fig. 7. The inverse of the admittance matrix is the impedance matrix. Fig. 8 presents the Bode plots

of the impedance models for three converters.

-40

-20

0

20

To: O

ut(

1)

From: In(1)

converter 1

converter 2

converter 3

-180

-90

0

90

180

To: O

ut(

1)

-40

-20

0

20

To: O

ut(

2)

100

102

-180

-90

0

90

180

To: O

ut(

2)

From: In(2)

100

102

Bode Diagram

Frequency (Hz)

Ma

gn

itu

de

(d

B)

; P

ha

se

(d

eg

)

Fig. 7: Bode plots of the admittance models.

-20

0

20

To: O

ut(

1)

From: In(1)

converter 1

converter 2

converter 3

-180

-90

0

90

180

To: O

ut(

1)

-20

0

20

To: O

ut(

2)

100

102

-180

-90

0

90

180

To: O

ut(

2)

From: In(2)

100

102

Bode Diagram

Frequency (Hz)

Ma

gn

itu

de

(d

B)

; P

ha

se

(d

eg

)

Fig. 8: Bode plots of the impedance models.

Remarks: It is found that the absolute value of the negative resistance in Zqq at a low-frequency

12

range is effectively reduced for grid-forming control. It is well known from the existing literature that the

negative resistance in Zqq causes low-frequency oscillational instability [25]. Thus, from the impedance

models, it can be projected that grid-forming converters do increase small-signal stability.

B. Eigenvalue-based stability analysis using identified admittance

From the measuring point, the system can be represented by two shunt admittances: Yvsc and Yg, where

Yg is the transmission line admittance:

Yg,dq =

Rg + sLg −ωoLg

ωoLg Rg + sLg

−1

(4)

The relationship between the small current injection and the voltage at the measurement point can be

expressed as follows:

iinj,dq = (Yvsc,dq + Yg,dq)︸ ︷︷ ︸Ytotal

Vpcc,dq (5)

Treating the current injection as an input and the voltage as an output, the transfer function of the

whole system is Y −1total. Therefore, the poles of the transfer function Y −1

total are the eigenvalues or the

zeros of Ytotal. This eigenvalue computing strategy has been proposed in [13] back in 1999. It was

recently compared with Bode plots and Nyquist diagrams in [14] and demonstrated as an accurate and

straightforward stability analysis tool.

1) Converter 1: Fig. 9a presents the trajectory of the eigenvalues of the system in grid-following mode

when the grid-side impedance Xg is varying from 0.45 pu to 1.16 pu. The figure shows that as Xg is

increased, a pair of eigenvalues in about 6 Hz mode moves toward the right half-plane. The marginal

condition is found when Xg = 1.1 pu. The unstable mode observed in the eigenvalue analysis matches

the EMT dynamic responses shown in Fig. 2, where 6 Hz oscillations have been observed.

2) Converter 2: Fig. 9b shows the trajectory of the eigenvalues of Converter 2. The grid strength Xg

is varying from 0.45 pu to 1.24 pu. The plot indicates that a pair of eigenvalues around 2 ∼ 3 Hz moves

toward the right half-plane. The marginal condition of the system is observed when Xg = 1.17 pu. The

observation almost corroborates with EMT results shown in Fig. 3, where 2 Hz is observed.

3) Converter 3: The trajectory of the eigenvalues is plotted in Fig. 9c for Converter 3 when the system

is in grid-forming mode. The grid strength Xg is varying from 0.45 pu to 1.24 pu. The plot indicates that

a pair of eigenvalues move toward the right half-plane. The marginal condition of the system is observed

when Xg = 1.19 pu. The eigenvalue analysis aligns with EMT results in Fig. 4,

13

-300 -200 -100 0

Real Axis

-200

-100

0

100

200Im

ag

ina

ry A

xis

(H

z)

(a)

-200 -150 -100 -50 0

Real Axis

-150

-100

-50

0

50

100

150

Ima

gin

ary

Axis

(H

z)

(b)

-300 -200 -100 0

Real Axis

-100

-50

0

50

100

Ima

gin

ary

Axis

(H

z)

(c)

-40 -30 -20 -10 0 10

Real Axis

-30

-20

-10

0

10

20

30

Ima

gin

ary

Axis

(H

z)

Xg=1.1 pu

(d)

-30 -20 -10 0 10

Real Axis

-20

-10

0

10

20Im

ag

ina

ry A

xis

(H

z) Xg=1.17 pu

(e)

-10 -8 -6 -4 -2 0 2

Real Axis 10-3

-1.5

-1

-0.5

0

0.5

1

1.5

Ima

gin

ary

Axis

(H

z)

10-3

Xg= 1.19 pu

(f)

Fig. 9: Trajectory of eigenvalues when Xg increases. Column 1: converter 1 (a)(d). Column 2: converter 2 (b)(e). Column 3:converter 3 (c)(f). Lower row figures (d)(e)(f) are the zoom-in of the upper row figures (a)(b)(c).

C. Droop control impact on stability

In this subsection, the impact of adding P-f and V-Q droop control on stability will be evaluated. To

apply the analysis, the loop gain of the system should be identified. The open-loop system (Converter 1

grid integrated system) is shown in Fig. 10 as G1. By treating the reference real power P ∗ and reference

voltage V ∗ as the inputs and ∆ω and the reactive power Q as the outputs, the closed-loop system can

be obtained as in Fig. 10. Based on the diagram, the loop gain of the system is G1G2, where G2 is:

G2 =

1m 0

0 n

(6)

Hence, the eigenvalues of the system are the zeros of (I +G2G1), where I is the identity matrix.

The ERA method can be utilized to obtain the plant model G1 by injecting a step change to the inputs

P ∗ and V ∗ and the outputs’ data ∆ω and Q are recorded.

The EMT testbed in Fig. 1 for Converter 1 is used to do the experiments. The line impedance Xg = 1.08

14

∆𝜔

−

∆𝑄

𝑛

1/𝑚

𝑃∗

𝑉∗ 𝐺1 −

Fig. 10: Closed-loop system.

2.4 2.5 2.6 2.7 2.8 2.9 3-1

0

1

2.4 2.5 2.6 2.7 2.8 2.9 3

0.22

0.24

Q

2.4 2.5 2.6 2.7 2.8 2.9 3-1

0

1

2.4 2.5 2.6 2.7 2.8 2.9 3

Time (s)

0.22

0.24

Q

measurements

reconstructed data

Fig. 11: Dynamic responses of ∆ω and Q for two events. The step change of 1% is applied at t = 2 sec.

pu. A 0.01 p.u step change is applied to the reference real power P ∗, and the data are recorded. Then,

another 0.01 p.u step change is applied to the reference voltage V ∗ and the data are recorded. The

recorded data are fed to the ERA toolbox. The original and the estimated data from the ERA are plotted

in Fig. 11. The results show an excellent match.

Fig. 12a presents the Root loci of the open-loop system from the input P ∗ to the output ∆ω. As it can

be seen, increasing the P-f droop coefficient can move the dominant mode (6 Hz) to the left half-plane.

Therefore, adding the P-f droop control can enhance the small-signal stability of the system.

The Root loci of the open-loop system from the input V ∗ and the output Q is depicted in Fig. 12b.

Unlike the impact of the P-f droop control on the small-signal stability, when the V-Q droop coefficient

increases, the dominant mode moves to the right half-plane which may deteriorate small-signal stability.

Fig. 13 gives the trajectory of the eigenvalues of the closed-loop system when the P-f droop coefficient

1/m varies and keeping n constant. The plot indicates that adding the droop control with proper selection

15

-40 -30 -20 -10 0

-40

-20

0

20

40

Root Locus

Real Axis (seconds-1

)

Imagin

ary

Axis

(seconds

-1)

(a)

-50 -40 -30 -20 -10 0

-40

-20

0

20

40

Root Locus

Real Axis (seconds-1

)

Imagin

ary

Axis

(seconds

-1)

(b)

Fig. 12: Root loci of the open-loop system, (a) from P ∗ to ∆ω, (b) from V ∗ to Q.

-60 -40 -20 0

Real Axis

-15

-10

-5

0

5

10

15

Ima

gin

ary

Axis

(H

z)

Fig. 13: Eigenvalues of (I + G2G1) when increasing P-f droop coefficient 1/m and keeping n constant.

of droop coefficient can improve the small-signal stability. The dominant mode moves to the range of

2 ∼ 4 Hz. The results align with the observation in the paper in Fig 2-3.

VI. CONCLUSION

In this paper, grid-forming converters operation in weak grids are investigated. Firstly, steady-state

analysis is presented to show that grid-forming converters do not increase steady-state operating margin.

Then, EMT studies are presented to show that grid-forming converters do increase small-signal stabil-

ity margin. To further compare the converters with different controls, admittance/impedance frequency

responses of the three types are compared. It is found that the absolute value of the negative resistance

in Zqq at low-frequency range is reduced for grid-forming control. s-domain admittance model-based

eigenvalue analysis results corroborate with the EMT simulation results. Furthermore, the impact of

16

adding P-f and V-Q droop control on stability is evaluated using open-loop system analysis. The results

indicate that P-f droop control can improve the small-signal stability.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are openly available in Github at

https://github.com/fllseu/grid forming.

REFERENCES

[1] NERC, “Integrating Inverter-Based Resources into Low Short Circuit Strength Systems,” no. December,

p. 47, 2017. [Online]. Available: https://www.nerc.com/comm/PC Reliability Guidelines DL/Item 4a. Integrating

Inverter-Based Resources into Low Short Circuit Strength Systems - 2017-11-08-FINAL.pdf

[2] S.-H. Huang, J. Schmall, J. Conto, J. Adams, Y. Zhang, and C. Carter, “Voltage control challenges on weak grids with

high penetration of wind generation: Ercot experience,” in 2012 IEEE Power and Energy Society General Meeting. IEEE,

2012, pp. 1–7.

[3] L. Fan and Z. Miao, “An explanation of oscillations due to wind power plants weak grid interconnection,” IEEE Transactions

on Sustainable Energy, vol. 9, no. 1, pp. 488–490, 2018.

[4] L. Fan, “Modeling type-4 wind in weak grids,” IEEE Transactions on Sustainable Energy, vol. 10, no. 2, pp. 853–864,

2019.

[5] L. Fan and Z. Miao, “Wind in weak grids: 4 hz or 30 hz oscillations?” IEEE Transactions on Power Systems, vol. 33,

no. 5, pp. 5803–5804, 2018.

[6] Y. Li, L. Fan, and Z. Miao, “Wind in weak grids: Low-frequency oscillations, subsynchronous oscillations, and torsional

interactions,” IEEE Transactions on Power Systems, vol. 35, no. 1, pp. 109–118, 2020.

[7] “Ieee guide for planning dc links terminating at ac locations having low short-circuit capacities,” IEEE Std 1204-1997, pp.

1–216, 1997.

[8] B. Kroposki, B. Johnson, Y. Zhang, V. Gevorgian, P. Denholm, B. M. Hodge, and B. Hannegan, “Achieving a 100%

Renewable Grid: Operating Electric Power Systems with Extremely High Levels of Variable Renewable Energy,” IEEE

Power and Energy Magazine, vol. 15, no. 2, pp. 61–73, 2017.

[9] Y. Lin, J. H. Eto, B. B. Johnson, J. D. Flicker, R. H. Lasseter, H. N. Villegas Pico, G.-S. Seo, B. J. Pierre, and A. Ellis,

“Research roadmap on grid-forming inverters,” National Renewable Energy Lab.(NREL), Golden, CO (United States),

Tech. Rep., 2020.

[10] J. Rocabert, A. Luna, F. Blaabjerg, and P. Rodrguez, “Control of power converters in ac microgrids,” IEEE Transactions

on Power Electronics, vol. 27, no. 11, pp. 4734–4749, 2012.

[11] R. Lasseter, Z. Chen, and D. Pattabiraman, “Grid-forming inverters: A critical asset for the power grid,” IEEE Journal of

Emerging and Selected Topics in Power Electronics, pp. 1–1, 2019.

[12] J. Rocabert, A. Luna, F. Blaabjerg, and P. Rodriguez, “Control of power converters in ac microgrids,” IEEE transactions

on power electronics, vol. 27, no. 11, pp. 4734–4749, 2012.

[13] A. Semlyen, “S-domain methodology for assessing the small signal stability of complex systems in nonsinusoidal steady

state,” IEEE Transactions on Power Systems, vol. 14, no. 1, pp. 132–137, 1999.

17

[14] L. Fan and Z. Miao, “Admittance-based stability analysis: Bode plots, nyquist diagrams or eigenvalue analysis?” IEEE

Transactions on Power Systems, pp. 1–1, 2020.

[15] L. Fan, Control and dynamics in power systems and microgrids. CRC Press, 2017.

[16] J. Sun, “Small-signal methods for ac distributed power systems–a review,” IEEE Transactions on Power Electronics, vol. 24,

no. 11, pp. 2545–2554, 2009.

[17] J. Huang, K. A. Corzine, and M. Belkhayat, “Small-signal impedance measurement of power-electronics-based ac power

systems using line-to-line current injection,” IEEE Transactions on Power Electronics, vol. 24, no. 2, pp. 445–455, 2009.

[18] B. Badrzadeh, M. Sahni, Y. Zhou, D. Muthumuni, and A. Gole, “General methodology for analysis of sub-synchronous

interaction in wind power plants,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 1858–1869, 2012.

[19] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Transactions

on power delivery, vol. 14, no. 3, pp. 1052–1061, 1999.

[20] Z. Shen, M. Jaksic, B. Zhou, P. Mattavelli, D. Boroyevich, J. Verhulst, and M. Belkhayat, “Analysis of phase locked loop

(pll) influence on dq impedance measurement in three-phase ac systems,” in 2013 Twenty-Eighth Annual IEEE Applied

Power Electronics Conference and Exposition (APEC). IEEE, 2013, pp. 939–945.

[21] M. Jaksic, Z. Shen, I. Cvetkovic, D. Boroyevich, R. Burgos, C. DiMarino, and F. Chen, “Medium-voltage impedance

measurement unit for assessing the system stability of electric ships,” IEEE Transactions on Energy Conversion, vol. 32,

no. 2, pp. 829–841, 2017.

[22] Z. Shen, M. Jaksic, P. Mattavelli, D. Boroyevich, J. Verhulst, and M. Belkhayat, “Three-phase ac system impedance

measurement unit (imu) using chirp signal injection,” in 2013 Twenty-Eighth Annual IEEE Applied Power Electronics

Conference and Exposition (APEC). IEEE, 2013, pp. 2666–2673.

[23] L. Fan and Z. Miao, “Time-domain measurements-based dq-frame admittance model identification of inverter-based

resources,” IEEE Transactions on Power Systems, pp. 1–1, 2020.

[24] L. Fan and Z. Miao, “A modular small-signal analysis framework for inverter penetrated power grids: Measurement,

assembling, aggregation, and stability assessment,” arXiv preprint arXiv:2003.03860, 2020.

[25] B. Wen, D. Boroyevich, R. Burgos, P. Mattavelli, and Z. Shen, “Analysis of dq small-signal impedance of grid-tied

inverters,” IEEE Transactions on Power Electronics, vol. 31, no. 1, pp. 675–687, 2015.