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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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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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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
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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)
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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)
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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-
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/
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Transactions on Power Systems
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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.
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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.
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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.
