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Transactions on Power Systems

1

Aggregated Model of Virtual Power Plants for

Transient Frequency and Voltage Stability Analysis

Junru Chen, Member, IEEE, Muyang Liu, Member, IEEE, Federico Milano, Fellow, IEEE

Abstract— The Virtual Power Plant (VPP) has been proposed

to aggregate Distributed Generations (DGs) to act like a single

power plant, thus, also has functions on the frequency and

voltage support. The previous models of the VPP are static

and focus on the energy trading and management. For the

system transient response analysis, a dynamic VPP model must

be needed. The paper proposes a reduced-order yet accurate

aggregated model to represent VPP transients for the stability

analysis of power systems. The goal is to provide a model that is

adequate for system studies and can serve to the Transmission

System Operator (TSO) to evaluate the impact of VPPs on

the overall grid. The proposed model can accommodate the

transient response of the most relevant controllers included

in the distributed generators that compose the VPP. Using a

comparison with a real-time detailed Electro-Magnetic Transients

(EMT) models of the VPP conﬁrms the validity of the proposed

aggregated model. The case studies based on the IEEE 39-bus

system veriﬁes the accuracy of the proposed aggregated model

on the system stability analysis.

Index Terms— Virtual Power Plant (VPP), Fast Frequency

Response (FFR), voltage stability, frequency stability.

I. INTRO DUC TIO N

A. Motivation

In the context of the power system migrating into higher

Distributed Generations (DGs), the concept of the Virtual

Power Plant (VPP) has been proposed to aggregate these

DGs units and/or load, and to coordinate to act like a single

power plant [1]. In order to maintain the system frequency

and voltage stability, VPPs like any other power plants, are

expected to have frequency and voltage support capabilities.

However, the VPP consists of a number of DGs, each of which

has its own transient response. A model able to represent the

transient response of a VPP as a single dynamic device is

sought by TSOs but still missing. This paper addresses this

issue and proposes an aggregated VPP model for transient

stability analysis.

B. Literature Review

The vast majority of existing aggregated VPP models are

aimed to solve the economic dispatch and energy management

problems and are thus steady-state models [1], [2]. Instead,

to date, the transient analysis of the high renewable system

is based on the separated DG models. Based on their control,

M. Liu and J. Chen are currently with Xinjiang University, China.

At the time of preparing this work, the authors were with the School

of Electrical and Electronic Engineering, University College Dublin, Ire-

land. (E-mails: junru.chen.1@ucdconnect.ie, muyang.liu@ucd.ie and fed-

erico.milano@ucd.ie).

This work is supported by the European Commission, by funding Junru

Chen and Federico Milano under the Project EdgeFLEX, Grant No. 883710;

and by the Science Foundation Ireland, by funding Muyang Liu and Federico

Milano, under Investigator Program Grant No. SFI/15/IA/3074.

i.e. current sources or voltage sources, these DGs are classiﬁed

into Grid-Following (GFL) and Grid-Forming (GFM) respec-

tively [3]. References [4] and [5] propose a detailed full-order

model for the GFL-DG and GFM-DG accordingly.

Since converter dynamics are fast with respect to the elec-

tromechanical modes and regulators of synchronous machines,

the dominant dynamics of a DG comes from their controllers.

Based on this observation, references [6] and [7] propose a

2nd-order model of the GFL-DG and GFM-DG. However,

even with such second-order DG models, the computational

burden may still be considerable if the number of units that

compose the VPP is high. Moreover, from the viewpoint of a

TSO, it is not viable to model the transient behavior of each

small unit included in a VPP. TSOs, in fact, only need to know

the transient response of the VPP as a whole.

Aggregating several small units into a simple(er) model is

common practice. TSOs often employ aggregated grid models

for dynamic security assessment. For example, the 179-bus

Western Electricity Coordinating Council (WECC) system is

aggregated from the original 10,000+ bus transmission system.

Fast frequency response analysis is based on the aggregated

models, for example, reference [8] proposes a transfer function

to aggregate all the synchronous generators and reference [9]

proposes a generic transfer function to represent all of gen-

erations in the power system. For a more accurate frequency

analysis in time domain simulation, reference [10] proposes

a model to aggregate the multiple wind machine system into

a single wind generator, reference [11] proposes method to

aggregate the multiple grid-feeding converter system into a

second order model and reference [12] proves that the virtual

inertia response of the wind turbine can be aggregated into a

similar form of the swing equation.

Most of the work interests on the system frequency response

thus above models are based on the assumption that the voltage

of the DG keeps constant. However, in reality, the occurrence

of any contingency disturbs the grid voltage and enforces the

DG reaction on the voltage. The transient voltage regulation

in the DG will affect its active power output and further

affects the grid frequency. This interaction differs with the

DG controls.

the DG works on the GFL mode and its voltage keeps

constant. With regard to the VPP, it actually mixes GFL and

GFM units with various frequency and voltage controls, so

that its response is more involved. In order to model the

entire VPP accurately, the system identiﬁcation, such as inertia

estimation [13] and grid impedance estimation [14], is required

to determine the parameters of the aggregated model. To the

best of the authors’ knowledge, such a model has not been

proposed so far.

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Transactions on Power Systems

2

C. Contributions

This work proposes an aggregated VPP model able to

accurately reproduce the transient response of a VPP at the

fundamental frequency for the transient frequency and voltage

stability analysis. The secondary control of the VPP can ensure

a solid output from all the internal DGs in the forthcoming

commission period. The use case is to analyses the system

response in the situation of the contingency occurrence, i.e.

generator outage, load change and line outage. Thus, the ag-

gregated model is based on the assumption that all the internal

DG units working in unsaturated situation and working in

the symmetrical situation. Since the units in a VPP can be

either GFL or GFM, the proposed VPP model consists of

a voltage source and a current source in parallel to emulate

synchronization transients separately. The load in the VPP is

modeled according to their location to minimize the effect of

the voltage-dependent load on frequency and voltage response

[15]. The proposed VPP model is veriﬁed via a fully-ﬂedged

EMT model and via a RMS transient stability model based on

the IEEE 39-bus system.

D. Organization

The remainder of the paper is organized as follows. Section

II reviews the basic concept of VPP and analyzes the transients

of its internal units. Section III proposes the aggregated

VPP model. Section IV validates the proposed aggregated

model by comparing it with a detailed VPP EMT model

in Matlab/Simulink. In Section V, the IEEE 39-bus system

serves to show the accuracy of the proposed model for the

system dynamic security assessment. Conclusions are drawn

in Section VI.

II. VIRT UAL POWER PLANT A ND I T S CON TROL

A VPP is a cluster of DGs with several different technolo-

gies, e.g. wind generator (WT), PV panels, electric vehicle

(EV) chargers, electrical storage system (ESS) and loads as

shown in Fig. 1. VPP can coordinate their internal units via

the dual-directional communication system. The control tasks

of the VPP are separated in its time scales, using threefold hi-

erarchical layout including primary control, secondary control

and tertiary control [16]. The latter two controls are centralized

and implemented by Distribution System Operators (DSOs)

and are not further considered in this work.

The primary control, on the other hand, is implemented

into the individual units to achieve fast frequency response,

and primary frequency and voltage response. For the system

dynamic security assessment by TSOs, the knowledge of the

frequency and voltage supporting capabilities, the transient re-

sponse and the grid power injection at the Point of Connection

(POC) is required. The remainder of this section provides a

brief review on the general DG control strategies, i.e. Grid-

Following (GFL) and Grid-Forming (GFM).

A. Grid following DG

The GFL-DGs is widely used in wind farms, PV plants and

EV charger stations. It behaves like a current source, delivering

PMU

VPP

Operator

WG

PV

EV

ESS

Load

VPP

Transmission

grid

Power line

Communication line

POC

PCC

PCC

PCC

PCC

PCC

Fig. 1: VPP Structure.

the assigned current id,iqor power p,qinto the grid. Its grid

synchronization is based on the voltage, using a Phase-Locked

Loop (PLL) to track the phase of the voltage at the Point of

Common Coupling (PCC). Note, PCC is a point of a single

DG connecting to the rest of the VPP, while POC is a point of

VPP connecting to the utility grid. Assuming that the phase

angle of the PCC voltage is the reference, the phase of the

VPP at the POC is −δ. Then the q-axis PCC voltage in the

synchronous dq-frame can be written as follows:

vd=vpoc cos(−δ)−(ωg+ ∆ω)lgiq,(1)

vq=vpoc sin(−δ)+(ωg+ ∆ω)lgid,(2)

where ωgis the grid frequency, ∆ωis the PLL frequency

deviation to the grid in transients, i.e. ∆ω=ω−ωg,lgis the

grid inductance from the DG to the POC. The synchronization

of the GFL-DG enforces vqto be null as indicated in Fig. 2

and its time constant depends on the PI parameters, Kp,Ki

normally in the range [50,100] ms [17]. Of course, there are

advanced PLLs [18] for the purpose of lower harmonics, but

essentially, they all contain a proportional part for a quick

stabilization and an integral part for a zero steady-state error

on the grid frequency deviations. Thus, here we use the generic

PI-based PLL for simplicity.

When the converter is perfectly synchronized with the grid,

the q-axis PCC voltage vqis null while the d-axis PCC voltage

vdequals to the voltage magnitude at the POC. Moreover, the

active power and reactive power are fully decoupled in the

steady state or vq= 0.

The reference currents of the converter control are given by:

iref

d=pref

vd

, iref

q=−qref

vd

,(3)

where the reference active and reactive powers are given by:

pref =p∗+Kd(ω∗−ω)−M˙ω , (4)

qref =q∗+Kq(v∗−vd),(5)

where ω∗and v∗are the nominal frequency and voltage,

respectively. The active power reference (4) contains the feed-

forward power p∗from the DC source of the DG and the power

for the fast frequency response, i.e. df/dt response (inertia

emulation) M, and the primary frequency control, i.e. f-P

droop control Kd, where the frequency signal ωis is the

grid frequency detected by the PLL or Phasor Measurement

Unit (PM U); and the reactive power reference (5) contains

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Transactions on Power Systems

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the reactive power set point q∗and the compensated reactive

power for the voltage support, i.e. the V-Q droop control Kq.

Note, the virtual inertia in GFL could be achieved besides

using ESS, the kinetic energy stored in turbine [19] and the

de-loading control [20], but these all can be represented in the

form of (4) [12].

The power delivered by the GFL-DG to the POC is:

p=idvpoc cos(−δ) + iqvpoc sin(−δ),(6)

q=−iqvpoc cos(−δ) + idvpoc sin(−δ).(7)

The resulting model of the GFL-DG is shown in Fig. 2,

where the dynamics of the converter current controller have

been neglected as their time scale is of the order of 1ms

and thus much faster than the synchronization dynamics and,

hence, it has been assumed that id=iref

dand iq=iref

q[6].

The computation of this GFL-DG reduced model includes

3 differential equations (two for PLL and one for RoCoF

computation) and 7 algebraic equations.

Fig. 2: GFL-DG dynamic model.

B. Grid forming DG

The GFM-DG is widely used in microgrids and is aimed

at substituting the Synchronous Generator (SG) to impose the

voltage and frequency to the grid. Its grid synchronization is

based on the same principle as the SG, i.e. based on the power

balance. A particular synchronization method for GFM-DGs

is the Virtual Synchronous Generator (VSG), which consists

in emulating the inertia through a swing equation. Again

assuming that the PCC is the reference, the phase of the VPP

at the POC is −δ. Then the synchronization of the VSG is

given by:

M˙ω=p−p∗+Kd(ω∗−ωg) + D(ω−ωg),(8)

where Dis the damping coefﬁcient. Note, the virtual inertia

in GFM could be achieved besides using ESS, the power

synchronization control [21] and DC voltage-based inertia

emulation [22], but these all can be represented in the form of

(8) [23]. The voltage support in the GFM-DG is a Automatic

Voltage Regulation (AVR) with gain Kvas in SGs:

v=v∗+Kv(v∗−vpoc ).(9)

Since the GFM-DG controls the voltage directly, its reactive

power couples to the active power and the power at the POC

is the consequence of the voltage difference between the PCC

and POC. Assuming the system impedance is solely reactive,

namely lg, the power at the POC is:

p=v vpoc

ω lg

sin(δ),(10)

q=v vpoc

ω lg

cos(δ)−v2

poc

ω lg

.(11)

The dynamics of the converter voltage controller are of the

order of 10 ms and are thus negligible. For the same reason,

also the dynamics of the current controller are not considered,

as in the model of the GFL-DG [7]. The resulting GFM-DG

model is shown in Fig. 3. The computation of this GFM-DG

reduced model includes 2 differential equations and 3 algebraic

equations.

Fig. 3: GFM-DG dynamic model.

III. AGGR EGATE D VPP MODE L

As discussed above, the GFL-DG and GFM-DG have dif-

ferent dynamic responses. To properly capture their transients,

thus, the proposed aggregated VPP model includes one current

and one voltage source. Then, the distributed loads in a VPP

can be represented with three aggregated loads, according to

their locations as indicated in Fig. 4, where the impedance

connecting to the POC is used to represent the effect of the

VPP system impedance on the DG dynamics as shown in

Fig. 2 and Fig. 3.

Fig. 4: VPP equivalent model.

Note that, in steady-state, since the GFL-DGs controls active

and reactive power directly, the aggregated current source is

modelled as PQ bus with negative powers; whereas, since

Transactions on Power Systems

4

GFM-DGs controls the active power and the voltage directly,

the aggregated voltage source model is modelled as a PV bus.

The remainder of this section discusses the deﬁnition of the

parameters of the aggregated VPP model shown in Fig. 4.

A. Aggregated Current Source Model

Let us assume that the VPP includes nGFL-DGs. Taking

the POC as the observation point, these GFL-DGs can be

represented by a Norton equivalent circuit where the ncurrent

sources are connected in parallel. The resulting aggregated

model can be easily obtained as the sum of their currents and

multiple of their impedances, as follows:

ia

d=Pn

i=1 p∗

i+Pn

i=1 ∆pi

vd

,(12)

ia

q=−Pn

i=1 q∗

i+Pn

i=1 ∆qi

vd

,(13)

and

1

la

gI

=

n

X

i=1

1

lg,i

,(14)

where, ∆pi(∆qi) is the active (reactive) power compensation

of the GFL-DG ifor the frequency (voltage) support. Consid-

ering the power compensation from each GFL is (4) and (5),

the total power compensation can be computed as:

n

X

i=1

∆pi=

n

X

i=1

Kd,i(ω∗−ωi)−

n

X

i=1

Mi˙ωi,(15)

n

X

i=1

∆qi=

n

X

i=1

Kq,i(v∗−vp oc ),(16)

where we assume that the set points ω∗and v∗are the same

for all GFL-DGs. In turn, the f-P gain, Rate of Change of

Frequency (RoCoF) gain and V-Q gain of the aggregated

current source are the sum of the values of all GFL-DG

included in the VPP.

The synchronism difference of the GFL-DGs mainly de-

pends on the term lgidin Fig. 2. The larger lgid, the longer the

synchronizing dynamics, in fact, the feed-forward loop of the

GFL-DG synchronizing dynamics depends on lgid(Kp+Ki

s).

Based on this observation, the aggregated PLL parameters can

be computed as the weighted sum of the PI parameters of all

GFL-DGs in the VPP as indicated in (17). Since id,i in reality

is variable, for simplicity, in the computation of the synthetic

PLL parameter, assuming the active and reactive power is

decoupled, the active current then can be represented by the

active power reference as follows:

Ka

p+Ka

i

s=Pn

i=1 lg,iid,i (Kp,i +Ki,i

s)

Pn

i=1 lg,iid,i

≈Pn

i=1 lg,ip∗

i(Kp,i +Ki,i

s)

Pn

i=1 lg,ip∗

i

.

(17)

Substituting the synthetic parameters ia

d,ia

q,la

g,Ka

pand Ka

i

into Fig. 2, we obtain the aggregated current sources and (1),

(2) and (12)-(7) constitute the resulting Differential-Algebraic

Equation (DAE) model of the aggregated current source.

B. Aggregated Voltage Source Model

Let us assume that the VPP includes mGFM-DGs. Taking

the POC as the observation point, the GFM-DGs can be repre-

sented by a Thevenin equivalent circuit where the mvoltage

sources are connected in parallel. The resulting aggregated

model is a voltage source connected to the POC through a

line. The active power of the aggregated voltage source is the

sum of the active powers of all GFM-DGs:

pa=

m

X

i=1

pi≈vivpoc sin δi

m

X

i=1

1

ωolg,i

,(18)

where, for simplicity the ωi’s of the GFM-DGs have been

approximated as the reference angular speed, namely ωi≈ωo.

Similarly, the reactive power at the POC is given by:

qa=

m

X

i=1

qi≈vivpoc cos δi

m

X

i=1

1

ωolg,i

−v2

poc

m

X

i=1

1

ωolg,i

.

(19)

From (18) and (19), the aggregated line impedance la

gVof

the aggregated voltage source is computed similarly by (14)

using all the Thevenin equivalent impedance of the GFM-DGs.

Then, substituting (9) into (11) and (19), we obtain the reactive

power due to the voltage change (∆v=v∗−vpoc) at the POC

as follows:

∆qa= ∆v

m

X

i=1

Kv,i cos δi

ωolg,i

,(20)

where, again ωi≈ωo,∀i= 1, . . . , m. Finally, the aggregated

AVR gain Ka

vis computed as the weighted sum of the AVR

gain Kv,i of all GFM-DGs:

Ka

v=Pm

i=1(Kv,i cos(δi)/ωolg ,i)

Pm

i=1 cos(δi)/ωolg,i

≈Pm

i=1 Kv,i qi

Pm

i=1 qi

.(21)

As shown in Fig. 3, the dynamics of the frequency response

and synchronization are in the same loop, whose dynamic

behavior is determined by the inertia, damping and droop

coefﬁcients. The inertia Maand droop Ka

dof the aggregated

voltage source is simply the sum of inertias and droops of all

GFM-DGs:

Ma=

m

X

i=1

Mi, Ka

d=

m

X

i=1

Kd,i .(22)

The equivalent angular speed (frequency) of the aggregated

voltage source is obtained as weighted sum of the frequencies

of all GFM-DGs, similarly to the concept of the Center of

Inertia (CoI), as follows:

ω=Pm

i=1 Miωi

Pm

i=1 Mi

.(23)

The synthetic damping within VPP consists of the damping

in the DGs and the losses from the VPP grid resistances,

thus, cannot be simply computed as the sum of the damping

coefﬁcients of the GFM-DGs. Essentially, damping is the

friction power to prevent the frequency change. Based on

this, we propose to compute the damping through the swing

equation (8), as follows:

Da=Pm

i=1 pi−p∗−Ka

d(ω∗−ω) + ˙ωM a

ω−ωg

.(24)

Transactions on Power Systems

5

Substituting the synthetic parameters Ka

v,Ma,Ka

dand Da

into the scheme represented in Fig. 3, we obtain the aggregated

voltage source. Equations (8), (9) and (18)-(24) constitute the

resulting DAE model of the aggregated voltage source.

C. Load

The GFL- or GFM-interfaced loads can be modelled and

aggregated into previous current or voltage source model

respectively with negative generation. While other lloads are

generally modelled as voltage-dependent load:

pl,i =pl0,i vi

v∗αi, ql,i =ql0,i vi

v∗βi.(25)

Where αi/βiis the voltage coefﬁcient of the active/reactive

power. When αi=βi= 0, it is constant power loads,

αi=βi= 1, it is constant current loads, αi=βi= 2,

it is constant impedance loads. Loads are distributed in the

region of the VPP and we assume that their voltage depends

on the nearby DG. GFL-DGs control the power directly, while

its terminal voltage is passively controlled as the consequence

of the the assigned power set point. On the other hand, GFM-

DGs control the voltage directly. Based on these observations,

loads can be classiﬁed into three categories, according to the

location:

i. viis set to the voltage at the POC if the load is closed

to the POC;

ii. viis set to the voltage of the aggregated current source

output if the load is closed to the GFL-DG;

iii. viis set to the voltage of the aggregated voltage source

output if the load is closed to the GFM-DG;

Then, the distributed loads are aggregated into three clusters,

namely at the buses of the POC, current source and voltage

source, respectively, as shown in 4. Note that, in reality, viis

not exactly equal to the aggregated bus voltage. Hence a small

mismatch on the loading is inevitable.

D. System Identiﬁcation

A VPP generally measures (vi, id,i, iq,i )or (pi, qi, ωi)at

the terminal of each DG, and commands the set-point p∗

i,

q∗

i,v∗,ω∗and the primary control gains Kd,i,Kq,i ,Kv,i to

each DG according to the unit commitment of its secondary

control. However, the internal dynamic parameters, inertia

Mi, input inductance lg,i and damping Diare not known

by the VPP. For example, some of DGs use the adaptive

and alternating inertia, and some feed forward the inertia of

the turbine for the grid inertia provision. In the aggregated

model, the damping can be identiﬁed with (24) but the inertia

Miand input impedance lg,i still need to be identiﬁed. The

power system identiﬁcation is a broad topic and still under

researching. There are many methods in the literature. Here

we only introduce one method based on [24]:

˙

Mi=Tm,i(∆Pi−Kd,i (ω∗

i−ωi)−Mi˙ωi)sign( ˙ωi).(26)

where Tm,i is the time constant of the inertia estimation.

The system reactance from each DG to POC can be com-

puted via a derivation of the power ﬂow equation (10) [25],

[26]:

lg,i =vivgsin(δi)

∆pi

≈vivgR(ωi−ωg)

∆pi

.(27)

The aggregated VPP model is completed. In comparison

with a VPP full model, the construction of the model from

3n+ 2mdifferential equations plus 7n+ 3m+ 2lalgebraic

equations is reduced to 5 differential equations plus 16 alge-

braic equations. The larger number of the units in the VPP,

the larger computational relief using the aggregated model.

IV. MOD E L VALIDATION

A real-time simulation solved in Matlab/Simulink is utilized

to validate the proposed aggregation model against fully-

ﬂedged EMT models that represent each element of the VPP.

The tested VPP is shown in Fig. 5. It consists of 2GFL-

DGs, 2GFM-DGs and 4loads connected to an inﬁnite bus.

This inﬁnite bus is modelled as an ideal voltage source, with

controlled frequency and voltage. The nominal frequency is

50 Hz. The base voltage is 10 kV and base power is 1MW.

The DG parameters as well as the aggregated voltage/current

source model settings are given in Table I. For simplicity, the

converter parameters are the same for all DGs and are given

in Table II. The system parameters are shown in Fig. 5. The

perturbances are a voltage change and frequency change of

the ideal source.

TABLE I: DGs and aggregated model parameters.

Unit M KdD Kv/Kqlg

[MWs/rad] [MW/rad] [MW/rad] [H]

GFL1 1.0 110 -600 0.157

GFL2 1.5 80 -367 0.193

GFM1 0.6 50 50 0.9 0.135

GFM2 0.3 80 80 0.9 0.160

I-model 2.5 190 -967 0.087

V-model 0.9 130 163 0.9 0.066

TABLE II: Converter parameters of DGs.

Parameter Value Parameter Value

LCL ﬁlter (µH/L) 0.5/3/0.1Voltage controller P/I 0.01/10.4

Current controller P/I 5000/10400 GFL1 PLL P /I 144/2560

GFL2 PLL P/I 126/2240 I-model PLL P /I 138/2450

A. Transient Response following Frequency Variations

This section veriﬁes the accuracy of the aggregated model in

response to a frequency change at the POC. The initial steady-

state is characterized by 50 Hz frequency and 1pu voltage at

the POC. At t= 2 s, the frequency of the ideal source starts

decreasing from 50 Hz s to 49.85 Hz with a 0.3Hz/s slope.

The frequency variation stops at 2.5s. For simplicity and to

better illustrate the dynamics, the results below only show the

fast frequency response.

Figure 6 shows the comparison of the active power tran-

sients between the total power of the GFL-DGs (GFM-DGs)

of the detailed model and the aggregated current (voltage)

Transactions on Power Systems

6

j0.0377 pu j0.0346 pu

j0.0817 pu

j0.0377 pu

j0.0911 pu j0.0534 pu

j0.0408 pu j0.0157 pu

P=0.7 pu

Q=0.2 pu

P=0.4 pu

Q=0.1 pu

P=0.9 pu

Q=0.6 pu

P=0.5 pu

Q=0.11 pu

GFM1

p*=1 pu

v*=1 pu

GFM2

p*=0.5 pu

v*=0.5 pu

GFL1

p*=0.7 pu

q*=0 pu

In nite bus

v=1+j0 pu

POC

VPP

j0.011 pu

GFL2

P*=0.3 pu

q*=0 pu

j0.01pu

j0.001pu

Fig. 5: VPP topology for EMT simulation.

1.5 2 2.5

0

0.02

0.04

Active power (pu)

Time (s)

Total GFL

Current source model

(a) GFL-DGs vs. aggregated current source model.

1.5 2 2.5

0

1

2

3x 10−3

Aactive power (pu)

Time (s)

Total GFM

Voltage source model

(b) GFM-DGs vs. aggregated voltage source model.

Fig. 6: Active power transient response after a frequency variation.

source. The aggregated models appears to capture accurately

the dynamics of the overall VPP.

Figure 7 shows the results of the reactive power transients

between the detailed VPP and the proposed aggregated model.

Both the detailed VPP and the aggregated model present a

reduction of the reactive power. This reduction is due to the

power coupling of the GFMs and the voltage dependent loads.

Although the voltage level of the aggregated voltage source is

set to match the initial reactive power of the detailed model,

local GFM-DG voltages may be different in the detailed VPP

model. Consequently, the reactive power response can show

some small difference in the two models, as illustrated in (19).

Similarly, as mentioned above, the aggregated load voltage

may be different from the local voltages of the actual loads.

These are the reasons why Fig. 7 shows a reactive power

mismatch between the aggregated model and the detailed VPP.

However, since the voltage differences in the GFM-DGs and/or

in the load are generally small, such a mismatch is negligible.

1.5 2 2.5

−2.2

−2.18

−2.16

−2.14

−2.12

−2.1

Reactive power (pu)

Time (s)

Detailed VPP

Aggregated model

Fig. 7: Reactive power transient response after a frequency change.

B. Transient Response following Voltage Variations

This section veriﬁes the accuracy of the aggregated model

in response to voltage step variations. The initial steady-state

is characterized by 50 Hz frequency and 1pu voltage at the

POC. The voltage of the ideal source jumps from 1pu to 0.9

pu at t= 2 s and recovers to 1pu at t= 2.5s.

Figure 8 shows the active power transients following the

voltage variations. The dynamic response of the GFL-DGs

show two aspects: (i) the negative feedback of the PLL

synchronization; and (ii) the reactive power compensation.

Because of these effects, the active and reactive powers of the

GFL-DG are coupled during the transient. This leads to the

spike in the active power at the instant of the voltage change.

The aggregated model shows a lower peak on the active power

than the detailed VPP, because it neglects the transients of the

current controller.

With regard to the GFM-DGs, the steps in the voltage

activates the AVR that, as a consequence, increases the output

voltage of the DGs. This voltage regulation leads to the

active power change as indicated in (18). Then the power

synchronization of the GFM-DG moves the phase to re-track

its power reference. This transient behavior is well reﬂected

in the aggregated voltage source model. The sub-transient

oscillation is due to the dynamic coupling between the swing

equation and the system impedance [27]. This behavior is also

accurately captured by the aggregated model.

1.5 2 2.5

0.9

1

1.1

Active power (pu)

Time (s)

Total GFL

Current source model

(a) GFL-DGs vs. aggregated current source model.

1.5 2 2.5

1.2

1.4

1.6

1.8

Time (s)

Reactive power (pu)

Total GFM

Voltage source model

(b) GFM-DGs vs. aggregated voltage source model.

Fig. 8: Active power transient response after voltage variations.

Figure 9 shows the reactive power transient of the detailed

and aggregated VPPs. The voltage support of the VPP com-

Transactions on Power Systems

7

pensates the reactive power following the POC voltage change.

The aggregated model accurately captures such a reactive

power compensation. Since the grid voltage before t= 2 s

and after t= 2.5s is the same, the reactive power trajectories

at post fault are perfectly matched. However, during the POC

voltage sag, there is a small mismatch due to the different

voltage levels in the VPP grid. Again, this mismatch is small

and does not affect the overall accuracy of the aggregated

model.

1.5 2 2.5

−2.5

−2

−1.5

−1

−0.5

0

Reactive power (pu)

Time (s)

Detailed VPP

Aggregated model

Fig. 9: Reactive power transient response after voltage variations.

C. Transient Response following Grid Impedance Variation

This section veriﬁes the accuracy of the aggregated model

in response to the grid impedance variation. The inﬁnite bus

now connects to the VPP via a parallel transmission line, of

which value is 0.01 pu and 0.001 pu. The line of 0.001 pu is

cut off at 2 s.

Figure 10 shows the active power transients following the

line outage. The aggregated model can basically capture the

response of the VPP, but shows a higher mismatches for the

aggregation of the GFL-DGs. This is because the phase at this

time is jumped at the POC and the grid impedance increases.

This lead to the transient current in the GFL-DGs lower its

reference, thus, resulting in a less peak than the aggregated

model with reference current output.

1.5 1.6 1.7 1.8 1.9 2

0.9

0.95

1

Active power (pu)

Time (s)

Total GFL

Current source model

(a) GFL-DGs vs. aggregated current source model.

1.5 1.6 1.7 1.8 1.9 2

−0.04

−0.02

0

0.02

Active power (pu)

Time (s)

Total GFM

Voltage source model

(b) GFM-DGs vs. aggregated voltage source model.

Fig. 10: Active power transient response after line outage.

Figure 11 shows the reactive power transients following

the line outage. Since GFL-DGs applies the constant power

control, the reactive power variation is insigniﬁcant, only in

the order of 10−3pu. The aggregated current source model

captures the synchronous resonance occurred in the GFL-

DGs. On the other hand, the grid impedance change leads to

the reactive power change in the GFM-DGs and this is well

represented by the voltage source model as indicated in (20).

1.5 1.6 1.7 1.8 1.9 2

0

10

20

x 10−4

Reactive power (pu)

Time (s)

Total GFL

Current source model

(a) GFL-DGs vs. aggregated current source model.

1.5 1.6 1.7 1.8 1.9 2

0.2

0.4

0.6

0.8

Reactive power (pu)

Time (s)

Total GFM

Voltage source model

(b) GFM-DGs vs. aggregated voltage source model.

Fig. 11: Reactive power transient response after line outage.

V. CA SE STU DY

This case study validates the ﬁdelity of the aggregated

model for system-wide applications, e.g. the dynamic security

assessment solved by the TSOs. The grid is a modiﬁed New

England 10-machine system with inclusions of 3VPPs. Three

SGs with nearby loads are replaced with the VPPs. The

topology of the VPPs and the overall grid is shown in Fig. 12.

For simplicity, the topology of each VPP is identical, but the

capacities and parameters of the DGs are different (see Table

III) as well as the distributed loads in the VPP.

For the detailed grid formulation, the GFL- and GFM-

DGs are represented with full-order models [28], including

the dynamics of voltage and current controllers and converter

ﬁlters. The model of the SGs and their primary controls,

i.e. AVRs and Turbine Governors (TGs), are given in [29].

This case study aims to verify the accuracy of the proposed

aggregated model in different scenarios, namely, contingencies

occurring at supply and demand buses as well as in the

transmission system. The grid frequency is estimated with CoI.

Finally, simulations are solved with Dome, a Python-based

power system software tool [30].

The computation is carried out by the Dell Inspiron 15-3567

with 4 Intel Core i5-7200U 2.5 GHz. The computational time

of the full model is 35.267 s while that of the aggregated model

is 26.582 s. There is a 24.6 % reduction on the computational

time. This reduction can be further increased for the modern

power system with increasingly high penetration of VPP.

A. Eigenvalue Analysis

Before doing the time domain simulation, this section com-

pares the eigenvalue of the system using aggregated models

with that using detailed VPP models.

Transactions on Power Systems

8

TABLE III: VPPs and its aggregation model parameters.

Unit M KdD Kv/Kqlgω

VPP1-GFL1 0 35 -100 1.12 ·10−3

VPP1-GFL2 20 52 -88 6.99 ·10−4

VPP1-GFL3 45 44 -65 6.89 ·10−4

VPP1-GFM1 20 35 35 0.9 2.17 ·10−4

VPP1-GFM2 42 50 50 0.9 9.86 ·10−4

VPP1-I-model 65 131 -253 2.65 ·10−4

VPP1-V-model 62 85 85 0.9 6.38 ·10−3

VPP2-GFL1 21 46 -73 4.97 ·10−4

VPP2-GFL2 10 82 -81 7.25 ·10−4

VPP2-GFL3 0 44 -37 2.75 ·10−3

VPP2-GFM1 75 55 55 0.9 8.78 ·10−4

VPP2-GFM2 52 34 34 0.9 6.39 ·10−3

VPP2-I-model 31 172 -191 2.66 ·10−4

VPP2-V-model 127 89 89 0.9 6.69 ·10−3

VPP3-GFL1 15 36 -23 1.09 ·10−3

VPP3-GFL2 18 22 -31 1.32 ·10−3

VPP3-GFL3 8 19 -17 2.48 ·10−3

VPP3-GFM1 31 42 42 0.9 2.02 ·10−3

VPP3-GFM2 27 18 18 0.9 1.56 ·10−3

VPP3-I-model 41 77 -71 4.81 ·10−4

VPP3-V-model 58 60 60 0.9 9.45 ·10−3

Gen 1

Gen 2

Gen 3

Gen 4

Gen 5

Gen 6

Gen 7

Gen 8

Gen 10

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25 26

27

28

30

31

32

33

34

35

36

37

39

VPP 1

VPP 2

VPP 3

DG 1

DG 2

DG 3DG 4

DG 5

Fig. 12: VPPs in the modiﬁed New England system.

Figure 13 shows the comparison results of the critical

eigenvalues. Due to the simplicity, the aggregated model

has less number of the eigenvalues, but the dominated or

right most eigenvalues are well represented. However, the

detailed VPP, due to the control interactions, presents extra

a pairs of eigenvalues in the poor damped area, for example

at -0.17207±j7.097100. This will lead to oscillations at the

corresponding frequency, i.e. 2.42 Hz, as proved latter in

Fig. 14b, Fig. 15b and Fig. 16b.

B. Scenario 1: Generator Outage

This scenario considers the outage of the machine Gen 8at

t= 1 s. Figure 14 shows the trajectories of the grid frequency,

the active power and reactive power of the VPP 1at its POC,

and the voltages at the nearby grid buses.

The active power response of the VPP can be accurately

captured by the aggregated model, as shown in Fig. 14b, so

−4.0−3.5−3.0−2.5−2.0−1.5−1.0−0.50.0 0.5

Re

−20

−15

−10

−5

0

5

10

15

20

Im

Detailed VPP

Aggregated Model

Fig. 13: Eigenvalue comparison between the detailed VPP and

Aggregated Model.

that the grid frequency response (see 14a) is identical to the

one obtained with the detailed model. On the other hand, as

expected, the reactive power, e.g. see Fig. 14d, obtained from

the aggregated model shows a small mismatch with respect to

that of the detailed VPP, thus, resulting in a small mismatch

on the voltage response, as shown in Fig. 14c.

0 5 10 15 20

Time [s]

0.992

0.994

0.996

0.998

1

ωCoI [pu (Hz)]

Detailed VPP

Aggregated Model

(a) System frequency.

0 5 10 15 20

Time [s]

4.5

5

5.5

6

6.5

7

7.5

8

8.5

pVPP 1 [pu (MW)]

Detailed VPP

Aggregated Model

(b) VPP1 active power at POC.

0 5 10 15 20

Time [s]

0.95

1

1.05

1.1

vBus [pu (kV)]

Bus 28

Bus 19

Bus 22

(c) Bus voltages.

0 5 10 15 20

Time [s]

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

qVPP 1 [pu (MVar)]

Detailed VPP

Aggregated Model

(d) VPP1 reactive power at POC.

Fig. 14: Modiﬁed New England system, Scenario 1: generator outage.

Transactions on Power Systems

9

C. Scenario 2: Load Outage

This scenario considers the outage of the load at bus 8at t=

1s. To avoid repetitions, we only show the system frequency

and bus voltages. After the contingency, since the generation is

greater than the loading, the system frequency increases. The

aggregated model accurately captures such a dynamic response

of the frequency as well as the voltage response as shown in

Fig. 15.

0 5 10 15 20

Time [s]

1.000

1.001

1.002

1.003

1.004

1.005

1.006

1.007

1.008

ωCoI [pu (Hz)]

Detailed VPP

Aggregated Model

(a) System frequency.

0 5 10 15 20

Time [s]

0.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

vBus [pu (kV)]

Bus 28

Bus 19

Bus 22

(b) Bus voltages.

Fig. 15: Modiﬁed New England system, Scenario 2: load outage.

D. Scenario 3: Line Outage

This scenario considers the outage of the line connecting

buses 2 and 25 at t=1 s. Fig. 16 shows the system response

obtained with the detailed and the proposed aggregated VPP

models. In this case, the frequency response of the aggregated

model presents a mismatch in the ﬁrst 7 s after the contingency.

This mismatch is due to the converter controller dynamics,

whose dynamic effect is also shown in the EMT results

discussed in Section IV. This mismatch, however, is very small

in percentage and absolute values. It is visible only because

of the small scale of the y-axis in Fig. 16a.

VI. CO N CL USI ON

This paper proposes an aggregated low-order model of VPP

that is able to accurately capture the transient response of

VPP with respect to the system contingencies. The proposed

aggregated model consists of a current source to represent

GFL-DG dynamics and a voltage source to represent GFM-

DG dynamics in the VPP. Loads are also properly represented

in the proposed aggregated model. Simulation results indicates

that, with this model, TSOs can study the dynamic response

of the grid without loss of accuracy and with no need to

model in detail the network and the various units in the VPP.

In the future work, the effect of the VPP system resistance

on the damping coefﬁcient of the aggregated model could

be further investigated. Besides, based on this model, the

secondary control of the VPP could also be aggregated, and

0 5 10 15 20

Time [s]

0.999970

0.999975

0.999980

0.999985

0.999990

0.999995

1.000000

1.000005

1.000010

ωCoI [pu (Hz)]

Detailed VPP

Aggregated Model

(a) System frequency.

0 5 10 15 20

Time [s]

1.01

1.02

1.03

1.04

1.05

1.06

vBus [pu (kV)]

Bus 28

Bus 19

Bus 22

(b) Bus voltages.

Fig. 16: Modiﬁed New England system, Scenario 3: line outage.

then frequency and voltage stability in a national grid could

be analyzed.

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Muyang Liu (S17-M20) received the ME and

Ph.D. in Electrical Energy Engineering from Uni-

versity College Dublin, Ireland in 2016 and 2019.

Since December 2019, she is a senior researcher with

University College Dublin. Her scholarship is funded

through the SFI Investigator Award with title “Ad-

vanced Modeling for Power System Analysis and

Simulation.” Her current research interests include

power system modeling and stability analysis.

Junru Chen (S17-M20) received the ME and Ph.D.

degree in Electrical Energy Engineering from Uni-

versity College Dublin in 2016 and 2019. He was

exchanging student at Kiel University (Germany) in

2018 and at Tallinn University of Technology (Esto-

nia). He is currently a senior researcher at University

College Dublin and a visiting scholar at Aalborg

University, Denmark. His current research interests

in Power electronics control, modeling, stability and

application.

Federico Milano (S’02, M’04, SM’09, F’16) re-

ceived from the University of Genoa, Italy, the ME

and Ph.D. in Electrical Engineering in 1999 and

2003, respectively. From 2001 to 2002, he was with

the Univ. of Waterloo, Canada. From 2003 to 2013,

he was with the Univ. of Castilla-La Mancha, Spain.

In 2013, he joined the Univ. College Dublin, Ireland,

where he is currently Professor of Power Systems

Control and Protections and Head of Electrical

Engineering. His research interests include power

systems modeling, control and stability analysis.