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In this paper, a new method for modeling converter-based power generators in ac-distributed systems is proposed. It is based on the concept of electrostatic synchronous machines. With this new concept, it is possible to establish a simple relationship between the dc and ac side and to study stability in both the small and large signals of the microgrid by considering a dc-link dynamic and high variation in the power supplied. Also, for the purpose of illustration, a mathematical and electrical simulation is presented, based on MATLAB and PSCAD software. Finally, an experimental test is performed in order to validate the new model.
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New Model of a Converter-Based Generator Using
Electrostatic Synchronous Machine Concept
Fabio Andrade Rengifo, Student Member, IEEE, Luis Romeral, Member, IEEE, Jordi Cusid ´
o, Member, IEEE,
and Juan J. C´
Abstract—In this paper, a new method for modeling converter-
based power generators in ac-distributed systems is proposed. It is
based on the concept of electrostatic synchronous machines. With
this new concept, it is possible to establish a simple relationship
between the dc and ac side and to study stability in both the small
and large signals of the microgrid by considering a dc-link dynamic
and high variation in the power supplied. Also, for the purpose of
illustration, a mathematical and electrical simulation is presented,
based on MATLAB and PSCAD software. Finally, an experimental
test is performed in order to validate the new model.
Index Terms—Microgrid control, microgrid model, stability of
RES renewable energy system.
vfd,vfq: instantaneous electrostatic field voltage in
vd,vq,v0:instantaneous electrostatic stator voltage in
ifd,ifq:electrostatic field current in DQ-axis.
id,iq,i0: instantaneous electrostatic stator currents in
Cffd,Cffd:self-capacitance of rotor in DQ-axis.
CdCqCo:self-capacitance of stator in DQ0-axis.
Cfdd,Cfqq:“mutual” capacitances between stator and rotor
R: armature resistances.
Rf:rotor resistances.
0:demand charge in each DQ0-circuit.
fq:available charge of rotor in DQ0-axis.
Θ: angle by D-axis leads the axis of a-phase.
ωr: rotor angular velocity.
THE “inertialess” and intermittent production of renew-
able generators has increased significantly in recent years.
Manuscript received February 25, 2013; revised October 4, 2013 and De-
cember 20, 2013; accepted January 27, 2014. Date of publication February 14,
2014; date of current version May 15, 2014. Paper no. TEC-00084-2013.
F. A. Rengifo, L. Romeral, and J. J. C´
ardenas are with the MCIA
Innovation Electronics, Technical University of Catalonia, Barcelona
08222, Spain (e-mail:;;
J. Cusid´
o is with the Area of Energy of CTM Centre Technologic, Manresa
08242, Spain and also with the MCIA Innovation Electronics, Technical Univer-
sity of Catalonia, Barcelona 08222, Spain (e-mail:
Color versions of one or more of the figures in this paper are available online
Digital Object Identifier 10.1109/TEC.2014.2303827
Based on electronic converter interfaces, intermittent genera-
tors such as photovoltaic devices or windmills are connected and
managed by a smart control system. When the smart control sys-
tem also manages local consumers and energy storage devices,
the system constitutes a microgrid. These modern power gen-
erators are highly efficient, reliable, modular, environmentally
friendly, noiseless, and controlled with high precision. Because
of this, they will be a significant competitor in future power
Nevertheless, it is possible that an increased predominance
of these kinds of generators may have a negative impact on the
stability of the distribution network. Therefore, stability analysis
and robust control are essential issues which must be considered.
Typically, the generator interface is controlled by a dc/ac con-
verter with a decentralized control system. This control system
manages the active and reactive shared power in the microgrid.
Generally, it uses a hierarchical control which divides the tasks
among three levels that make the microgrids “smarter” [1]. The
primary control has two loops. The inner loop is used for reg-
ulating current and voltage. The outer loop uses the PQ droop
and virtual impedance for sharing active and reactive power with
only local variables [2], [3]. A second level is located at the com-
mon coupling point and handles synchronization or islanding
algorithms [4]–[8]. Finally, the third control deals with energy
management, the electricity market, and importing/exporting
energy to the utility [9], [10].
This strategy allows the inverters to share the active and re-
active power demanded by loads in the microgrid while consid-
ering the maximum ratings.
Currently, several models and control issues have been stud-
ied for converter-interface-based generators. Some of these
include decentralized control techniques [2], active power
frequency (Pf)and reactive power–voltage (QV)droop
controls [11]–[14], as well as linear state-space models [19] In
general, these proposals are based on averaged differential equa-
tions and artificial droop curves. The proposed models emulate
the behavior of electromagnetic synchronous machines and the
natural coupling between frequency and power.
Based on recent publications related with the stability issues
of microgrid systems, different methodologies have been ap-
plied to study the behavior of them. A dynamic analysis by
means of voltage and current phasors has been applied on [15],
while on [16], [17], a droop control analysis on inverters has
been implemented. The load sharing management by means of
droop control is proposed on [18], although there is a nega-
tive impact on global stability. Most of this research is based
on dynamical models. These models require a high amount of
state variables and neglect the dynamic behavior of the dc link.
0885-8969 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
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Moreover, these are complex models that are difficult to work
with. Therefore, it is necessary to make some assumptions, such
as considering small signals [15]–[19] to perform the analysis
while disregarding large signal excursions.
Recently, Vandoorn et al. proposed a control strategy which
modifies the set value of the ac microgrid voltage in function of
the dc-link voltage [20], [21]. They used a P/V-droop control
for power sharing in the microgrid, as well as a Vdc/Vrms -droop
control in order to avoid frequent changes in the power delivered.
However, this paper considers only a low-voltage microgrid with
a resistive network rather than an inductive network [22].
In this paper, the authors present a new generator model that
is equivalent to an electrostatic synchronous machine. This per-
mits an analysis and design of stable control methods for any
microgrid condition. The model establishes a direct and easy
relationship for dynamic analysis of converting energy from the
dc link to ac power via an electronic converter. Furthermore,
this new model makes it possible to study the dynamics, ro-
bustness, and stability of these generators by using the classical
techniques which are traditionally used for synchronous ma-
chines [23], [24]. The proposed model has the advantage of
integrating the motion equation for stability studies with the dy-
namics of the dc link without increasing the number of poles. In
this way, it is possible to analyze several generators connected
in a microgrid and to determine the global stability of the system
while taking into account the energy available from generators
as well as the power consumed by loads on the microgrid. The
outline of the paper is as follows. The next section describes
the mathematical model of an ideal electrostatic machine. In
Section III, the relationships between the electrostatic machine
and the power converter interface are presented, which al-
lows defining the new power interface model former described.
Section IV contains simulation and experimental results that
validate the new model. Finally, Section V concludes by high-
lighting the main contributions of the paper.
A. Mathematical Description
A new model of RES including its power converter interface
is proposed, which is based on an ideal electrostatic machine.
To obtain the model, a rotating reference frame (DQ0) is applied
to dc-side and ac side, which includes the three-phase inverter
of Fig. 1. The stator circuits consist of three-phase armature
capacitors carrying an electric charge Q.
It may be considered that electrostatic machine is equivalent
physically to a magnet synchronous machine with two pair of
field poles.
The use of an electrostatic equivalent machine instead of a
magnetic model allows having a direct relationship between the
dc capacitor and the electrostatic machine rotor. The rotor is
supplied by a direct voltage, which produces an electric field
that induces alternating charges in the armature circuit. The
three-phase surfaces of the armature are distributed 120apart
in space. As the rotor models the energy supplied by the RES,
large excursion of renewable energy can be represented on the
dc-voltage bus.
Fig. 1. Electronic converter interface for renewable power source.
Fig. 2. Representation of the electrostatic machine.
The proposed model does consider the dynamics of the PQ
control and the output filter, but it does not take into ac-
count the VI control’s dynamics because of their high-frequency
The dc/ac converter interface consists in a three-phase inverter
with a space vector modulation (SVM) module responsible of
managing the states of the inverter. It can synthesize the output
voltages from eight discrete voltage levels (U0,U1,U2,U3,
7). In the considered model, the electric field
magnitude and speed of rotation parameters are related with the
dc capacitor voltage and the reference vector (m).Fig.2shows
the functional diagram of this interface, the eight states in the
space representation, and the electrostatic machine equivalent
If it is taken into account that the switching frequency in the
SVM module is a decade higher than the working frequency, it
is possible to use the fundamental harmonic of each signal to
obtain algebraic models and state equations of the plant [25].
This permits to use the averaged values of the state variables to
the proposed model.
Fig. 3. Representation of the electrostatic machine by its equivalent capacitors
on the direct-axis and quadrature-axis.
The synthesized (by the pulse width modulation-averaged
approximation) voltage reference vector can be represented as
m =mmax
cos (ωrt+θ)
cos (ωrt2π/3+θ)
cos (ωrt+2π/3+θ)
In steady-state conditions, the reference vector rotates at a
constant angular speed (ωr), which defines the frequency of the
output voltages.
In the literature, a methodology was used to explain the dc/dc
converter’s behavior by the use of the dc transformer’s concept.
This paper uses the same methodology to define the concept
of the electrostatic machine and to describe the behavior of the
dc/ac converters.
The total charge (QT)on each surface (a, b, c) according to
the Gauss law is
ε0QT=ε0|E||S|cos θ. (2)
Electrostatic machine includes “self” and “mutual” capaci-
tances with equivalent behavior of self and mutualinductance in
the synchronous machine. Therefore, RC circuits that describe
the behavior of the surfaces and the electric field are considered
in order to obtain mathematical relationships between rotor and
stator. The electrostatic machine in the DQ rotating frame is
presented in Fig. 3.
B. DQ0 Frame Representation
In this paragraph, it is used as a per unit transformation to
normalize systems variables.Therefore, the equations are in per
unit. Line to neutral voltage, line current, and grid frequency are
used as base quantities for the stator. Fig. 3 shows the current
equations of the stator circuits in the electrostatic machine.
dt vd
dt vq
Fig. 4. D-axis and q-axis equivalent circuits of a power electronic based
The charge in each circuit at any instant is given by
Cfddand Cfqqare the equivalent capacitance between the
rotor and the stator in DQ-axes, and the (Cafdd×vfd)product
is equivalent to the transfer charge from the rotor to the stator.
The current equations of the rotor in the electrostatic machine
can be expressed as follows:
ifd =d
dtQfd vfd
ifq =d
dtQfq vfq
The equivalent charges Qfd and Qfq are given by
Qfd =Cffdvfd Cfddvd
Qfq =Cffqvfq Cfqqvq.(6)
Equation (6) gives the details of the electrical charge transfers
from the rotor to the stator in the DQ frame. Each rotor circuit
component has an associated electrical charge which depends
on its charge storage capacity and the electrical charge delivered
to the circuit of the stator.
A. General Description
Fig. 1 illustrates a renewable source generator based on elec-
tronic power interface. This kind of generator has an outer con-
trol for allowing it to work like a plug and play system. The
generator is synchronized by means of a phase-locked loop
(PLL) block with the electric utility. Furthermore, droop curves
are often used to share the active and reactive power in the mi-
crogrid. The inner control is used to track the reference signals
and the SVM module for managing the states of the inverter.
A renewable prime source, solar or wind, delivers energy to
the dc link, which is modeled by a current source. The dc voltage
is regulated by means of the corresponding control block. DC
link and control systems could be seen like a rotating electric
field which gives an electric charge to the three-phase system.
The speed of the electric field is equal to the reference frequency
in the inner loop of control that is changed according to the
shared power by means of the droop curves.
B. Equivalent Circuits for Direct and Quadrature Axes
The equivalent circuits and their complete characteristics, in-
cluding the current equations and the output filter are presented
in the Fig. 4. The circuits represent the complete electrical char-
acteristics and provide a visual description of the power elec-
tronics interface.
C. Steady-State Analysis
The behavior of electronic interface under balanced steady-
state conditions may be analyzed by applying per unit (3)–(6).
Since rotor and stator quantities are constant under steady state,
all time derivative terms drop out of the model. Also, zero-
sequence components are not present and ωr=1pu. There-
fore, per unit equations under balanced steady-state conditions
ifd =1
ifq =1
vfq (7)
and the charges
Qfd =Cffdvfd Cfddvd
Qfq =Cffqvfq Cfqqvq.(8)
The aforementioned equations can be used to find the field
voltages. Replacing the product (ωrC)1by the corresponding
reactance Xc,
vfd =1
vfq =1
Also, the stator voltage and current may be written as phasor
representation “vt” and “
it” where vt=vd+vqand
iq. The relationships between equations according to (5) can be
vfq 1
vfd +1
The equivalent circuit of the stator is shown in Fig. 5.
Fig. 5. Steady-state equivalent circuit of an electrostatic machine.
The current and voltage equations are
Requiv +jXcequiv ·vt
Substitution of (10) in (11), R=Requiv, and Xcq=Xcd=
Xcequiv, followed by reduction of the resulting expression,
yields the following expression foriiin phasor form with DQ-
axes as reference:
vfq j1
So far, the power interfaces have been modeled assuming a dc
source from the prime energy side. Under steady-state, the dc-
voltage bus is constant and the energy coming in is equal to the
energy that is getting out. Additionally, it has been considered
the capacitor of the dc-bus like the rotating reactance along the
D-axis. Thus,
Xcfdd=Xcfqq=Xcdc =(ωrCdc)1
vfq =Vdc
2dqvfd =Vdc
where diis the average value for the duty cycle.
The equivalents current source, resistance, and admittance
are expressed as
2Xcdc jVdcdq
Xcequiv =Xcdc d2
dRequiv =Xcdc 2dqdd
D. Electrical Power and Torque
Equation (3) can be derived an expression for the power trans-
ferred between the dc-side and the ac-side, which is
dt +Vq
Rate of Ch ange
Armatu re Electric
Power Transf.
from dc ac
Fig. 6. Outer power control loop of an inverter-based generator.
The electrostatic machine has an “air-gap torque” Tethat is
obtained by dividing the power transferred from dc to ac side
by the rotor speed (or reference frequency of the control).
E. Equation of Motion
The swing equation for magnetic synchronous machine is
also used in this electrostatic machine concept, it is
dt =1
dt =ω0Δ¯ωr.(16)
The aforementioned equation is normalized in term of per
unit inertia constant H, defined as the kinetic energy in watt-
seconds at rated speed divided by the VA base. The Hvalue
is calculated for the electrostatic machine by making equal the
stored energy and the kinetic energy.
2CV 2
dc =1
rH=CV 2
where Jis the virtual moment of inertia of the virtual generator
and it permits study of the dynamics of the system to maintain
the required voltage and frequency of the ac systems.
F. Model Parameters
The parameters that describe the inertial and damping values
of the generator are expressed as Mand D, respectively, on the
following equations.
The Dconstant indicates a frequency’s deviation in func-
tion of the delivered power by each generator. In these type of
generators, that are based on dc/ac interfaces, the frequency’s
deviation occurs because of the droop curves that are used in
the control of the shared power. The block diagram of Fig. 6
presents the relation of the reference’s frequency in function of
the power that is shared to the microgrid. The relations that ex-
press the power and the frequency deviation (Δω) are presented
on (18).
dt 1
KpΔω. (18)
Combining the equation that describes the nachine’s move-
ment (16) and the equation of the power control, is able to
calculate the Miand Diparameters as follows:
,D =1
The power Pmcan be determined with the maximum value
of the available power on the dc bus and the maximum values
of the signals ddand dqof the control.
To compare the proposed model with the conventional para-
metric one, both simulations of mathematical and electrical
models have been carried out by means of PSCAD X4 and
MATLAB/Simulink environment.
The aim of these simulations was to verify that the virtual
machine equations can describe the behavior of a generator
with a converter interface, such as the electrical model does.
It has been considered a generator working in grid connected
mode. The parameters used in the simulation are listed in Table I.
Fig. 1 shows the generator configuration of the simulated
system. The model has used the set of equations in (3)–(6) and
the values in Table I and it was described as
ifd =1
ifq =1
10vfq (20)
Qd=5.6×104vd+2.54 ×104vfd
Qq=5.6×104vq+2.54 ×104vfd
Qfd =5.6×104vfd 2.54 ×104vd
Qfq =5.6×104vfq 2.54 ×104vq(21)
A. Simulation
In order to validate the proposed mathematical model, the
generator was connected to the utility grid, with a resistive load
Fig. 7. Simulations of voltages and currents in time domain with a 20% of input power disturbance.
connected in parallel. Two different simulation tests were made.
In the first one, the behavior of the system was analyzed, ap-
plying a 20% disturbance in the input energy, while the system
was working in stable state. In the second test, the system was
working at nominal power when a 100% disturbance appeared
in the input energy.
The mathematical model includes the transfer functions of
each controller and the PLL for the synchronization to the grid.
On Fig. 7, a comparison of the Vdc,Vd, and VVqsignals is
depicted, obtained by the mathematical model and the electric
With the system running in steady state, a disturbance in the
renewable energy was forced at the tenth second. The dc-voltage
controller applied a corrective action, increasing the DQ current
references in order to maintain constant dc–dc level. This action
provoked a reduction of the power that is delivered to the load
by the renewable generator. After that, the utility grid had to
increase the power level to balance the system.
The proposed mathematical model describes the behavior of
the generator and its electrical formulation.
Fig. 8 presents the voltage and current signals that were mea-
sured during the second test, with a 100% disturbance magnitude
in the primary energy. In this case, the control of the genera-
tor was programmed to deliver only active power to the utility.
The values of the currents and voltages DQ, obtained by the
simulation and the mathematical model can be seen in Fig. 7.
B. Experimental Results
This section provides the experimental results that were used
to validate the model. The simulation cases and the presented
model were tested in laboratory conditions. The test bench was
composed by a three-phase inverter connected to the grid, a dc
power source which was used to emulate the renewable energy,
and a resistive load of 1 kW. The main electrical signals were
acquired by a dSPACE 1006. The energy disturbance was intro-
duced to the input by the use of a dc power source. During the
test, the source generated two disturbances of 20% and 100%
magnitude (of the nominal power), respectively.
The generator was operating at rated power. Under this state,
the Vdand Vqvalues were about 1 and 0.3 p.u., respectively.
Initially, the microgrid was injecting power to the grid (Id=0.7
Iq=0). For the first disturbance test, a 20% reduction of the
renewable power input was applied. After the reduction, a tran-
sient state appeared with a duration of approximately 5 s. After
the transitory period, the values of the current Idand voltage Vq
fell to 0.508 and 0.205 p.u., respectively. Comparing the steady
state values of the system before the disturbance as well as the
ones after the time of transition with the simulations, it was
observed that results are similar as are presented in Fig. 7. On
the other hand, the experimental test of this scenario, Fig. 9(a)
shows the output voltages and currents of the inverter when a
20% reduction of the renewable power input was applied. Sig-
nals Vd,I
q, and Vdc showed a small variation that is difficult
Fig. 8. Simulations of voltages and currents in time domain with a 100% of input power disturbance.
to be identified in the measurements, due to the existence of
electrical noise with similar magnitude.
For the second case study, a 100% disturbance in the re-
newable input current was applied. Fig. 9(b) shows the output
voltages and currents of the inverter when the system stops to
deliver energy to the grid. With this large disturbance, all signals
have significant variation.
A comparison of Figs. 8 and 9(b) indicates that the voltage
and current values are the same in simulation and in the real
experiment for both transient and stable states. The same values
were calculated also by the use of the proposed model.
Finally, a numerical comparison between simulated and real
data is performed, being the simulated data those obtained from
analytical model developed in the paper. Thus, the steady-state
error at rated output power and transient-state error under dis-
turbances are considered. The steady-state error is calculated
into a time window of 2 s before the disturbance appears.
During this time, instantaneous error is calculated by sub-
tracting the two databases of simulated and real results at every
instant. The root mean square error (RMSE) is obtained using
where Ierr =Instantaneous error.
To calculate the transient-state error, a time window equal
to the settling time (according to 5% criteria) has been cho-
sen. The settling time is now 5 s. Then, the RMSE is again
obtained. Table II shows the error values for both steady-state
and transient-state for currents and voltages.
It can be concluded that the proposed model can predict the
behavior of a generator based on an inverter with a very low
error. Thus, analytical expressions of the model can be used to
analyze the stability of the renewable system for a wide range
operation, and to develop stable and accurate control of the RES
The main contribution of this paper is the presentation of
a useful mathematical model that enables the analysis of re-
newable power sources for any operational point and energy
Fig. 9. (a) Experimental measurements of voltages and currents in time domain with a 20% of input power disturbance. (b) Experimental measurements of
voltages and currents in time domain with a 100% of input power disturbance.(a) 20% disturbance in prime input energy. (b) 100% disturbance in prime input
delivered, even with variable or discontinuous energy supply
(nonconstant dc bus). Currently, models consider constant dc-
bus voltage, which implies discarding real dynamics in renew-
able source, i.e., at the input of power converter. On the contrary,
proposed model allows introducing new equations to directly re-
late input and output power converter dynamics. The model has
been obtained by extending the concept of the electromagnetic
machine to the electrostatic machine. By this way, changes on
voltages and charges (i.e., energy) can be introduced and a direct
and fast relationship between the ac and dc side can be defined.
Furthermore, a set of equations have been defined that permit
to model this relationship between the dc-side and ac-side and
the prime energy. This will allow generating new algorithms of
stability, control, and energy management taking into account
large and real variations in both loads and power sources.
Simulation and experimental comparisons have been carried
out to verify the goodness of mathematical model under dif-
ferent conditions. The model can be used in both isolated and
connected to the grid modes. In both cases, it allows studying the
stability in both small and large signal. The proposed model is a
powerful tool to find operational converter limits such as control
signal saturation, to perform large signal stability analysis, and
to study fast transients in power sources.
The following base quantities for the stator:
vsbase peak value of rated line to line neutral voltage (V);
isbase peak value of rated line current (A);
fbase rated frequency (Hz).
The base values of the remaining quantities are automatically
set and depend on the aforementioned base quantities as follows:
ωbase =2πfbase (elec.radians/seg)
Zsbase =vsbase
Csbase =1
Qsbase =Csbase vsbase (Coulombs)
3ph VAsbase =3
2vsbaseisbase (volt-amperes)
Torquebase =3ph VAbase
tbase =1
The per unit time derivative is given by
Rotor base quantities:
vfbase =Peak value of rated dc voltage (V)
vfbase ifbase =3
=3ph VAsbase
vfbase =Cad
vsbase (V)
ifbase =3ph VA sbase
Zfbase =vfbase
Cfbase =1
Qfbase =Cfbasevfbase (Coulombs).
[1] J. M. Guerrero, J. C. Vasquez, J. Matas, L. G. de Vicuna, and M. Castilla,
“Hierarchical control of droop-controlled ac and dc microgrids—a general
approach toward standardization,” IEEE Trans. Ind. Electron., vol. 58,
no. 1, pp. 158–172, Jan. 2011.
[2] K. Jaehong, J. M. Guerrero, P. Rodriguez, R. Teodorescu, and
N. Kwanghee, “Mode adaptive droop control with virtual output
impedances for an inverter-based flexible ac microgrid,” IEEE Trans.
Power Electron., vol. 26, no. 3, pp. 689–701, Mar. 2011.
[3] J. He and Y. W. Li, “Generalized closed-loop control schemes with em-
bedded virtual impedances for voltage source converters with LC or LCL
filters,” Trans. Power Electron., vol. 27, no. 4, pp. 1850–1861, 2012.
[4] B. Bahrani, H. Karimi, and R. Iravani, “Nondetection zone assessment
of an active islanding detection method and its experimental evaluation,”
IEEE Trans. Power Del., vol. 26, no. 2, pp. 517–525, Mar. 2011.
[5] I. J. Balaguer, L. Qin, Y. Shuitao, U. Supatti, and Z. P. Fang, “Control for
grid-connected and intentional islanding operations of distributed power
generation,” IEEE Trans. Ind. Electron., vol. 58, no. 1, pp. 147–157, Jan.
[6] A. Samui and S. R. Samantaray, “Assessment of ROCPAD relay for is-
landing detection in distributed generation,” IEEE Trans. Smart Grid,
vol. 2, no. 2, pp. 391–398, Mar. 2011.
[7] M. A. Zamani, A. Yazdani, and T. S. Sidhu, “A control strategy for en-
hanced operation of inverter-basedmicrogrids under transient disturbances
and network faults,” IEEE Trans. Power Del., vol. 27, no. 4, pp. 1737–
1747, 2012.
[8] M. Savaghebi, A. Jalilian, J. C. Vasquez, and J. M. Guerrero, “Secondary
control scheme for voltage unbalance compensation in an islanded droop-
controlled microgrid,” IEEE Trans. Smart Grid, vol. 3, no. 2, pp. 797–807,
[9] C. Chen, S. Duan, T. Cai, B. Liu, and G. Hu, “Smart energy management
system for optimal microgrid economic operation,” IET Renewable Power
Generation, vol. 5, no. 3, pp. 258–267, May. 2011.
[10] A. Colet-Subirachs, A. Ruiz, G. Bellmunt, O. A. Cuevas, and F. Sudria-
Andreu, “Centralized and distributed active and reactive power control
of a utility connected microgrid using IEC61850,” IEEE Syst. J.,vol.6,
no. 1, pp. 58–67, Mar. 2012.
[11] J. M. Guerrero, J. Matas, L. G. Vicuna, M. Castilla, and J. Miret, “Decen-
tralized control for parallel operation of distributed generation inverters
using resistive output impedance,” IEEE Trans. Ind. Electron., vol. 54,
no. 2, pp. 994–1004, Mar. 2007.
[12] Y. Mohamed and E. F. El-Saadany, “Adaptive decentralized droop con-
troller to preserve power sharing stability of paralleled inverters in dis-
tributed generation microgrids,” IEEE Trans. Power Electron., vol. 23,
no. 6, pp. 2806–2816, Dec. 2008.
[13] D. De and V. Ramanarayanan, “Decentralized parallel operation of invert-
ers sharing unbalanced and nonlinear loads,” IEEE Trans.Power Electron.,
vol. 25, no. 12, pp. 3015–3025, Dec. 2010.
[14] G. Diaz and C. Gonzalez-Moran, “Fischer–Burmeister-based method
for calculating equilibrium points of droop-regulated microgrids,” IEEE
Trans. Power Syst., vol. 27, no. 2, pp. 959–967, May 2012.
[15] S. V. Iyer, M. N. Belur, and M. C. Chandorkar, “A generalized computa-
tional method to determine stability of a multi-inverter microgrid,” IEEE
Trans. Power Electron., vol. 25, no. 9, pp. 2420–2432, Sep. 2010.
[16] Y. Mohamed and E. Saadany, “Adaptive decentralized droop controller
to preserve power sharing stability of paralleled inverters in distributed
generation microgrids,” IEEE Trans. Power Electron., vol. 23, no. 6,
pp. 2806–2816, Nov. 2008.
[17] E. Barklund, N. Pogaku, M. Prodanovic, C. Hernandez-Aramburo,
and T. C. Green, “Energy management in autonomous microgrid using
stability-constrained droop control of inverters,” IEEE Trans. Power Elec-
tron., vol. 23, no. 5, pp. 2346–2352, Sep. 2008.
[18] R. Majumder, B. Chaudhuri, A. Ghosh, R. Majumder, G. Ledwich, and
F. Zare, “Improvement of stability and load sharing in an autonomous
microgrid using supplementary droop control loop,” IEEE Trans. Power
Syst., vol. 25, no. 2, pp. 796–808, May 2010.
[19] N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, analysis and
testing of autonomous operation of an inverter-based microgrid,” IEEE
Trans. Power Electron., vol. 22, no. 2, pp. 613–625, Mar. 2007.
[20] T. L. Vandoorn, B. Meersman, L. Degroote, B. Renders, and
L. Vandevelde, “A control strategy for islanded microgrids with dc-link
voltage control,” IEEE Trans. Power Del., vol. 26, no. 2, pp. 703–713,
Apr. 2011.
[21] T. L. Vandoorn, B. Meersman, J. De Kooning, and L. Vandevelde, “Anal-
ogy between conventional grid control and islanded microgrid control
based on a global dc-link voltage droop,” IEEE Trans. Power Del., vol. 27,
no. 3, pp. 1405–1414, Jul. 2012.
[22] T. L. Vandoorn, J. D. M. De Kooning, B. Meersman, J. M. Guerrero, and
L. Vandevelde, “Automatic power-sharing modification of P/V droop con-
trollers in low-voltage resistive microgrids,” IEEE Trans. Power Del.,
vol. 27, no. 4, pp. 2318–2325, Oct. 2012.
[23] P. Kundur, Power System Stability and Control. New York, NY, USA:
McGraw-Hill, 1994.
[24] S. V. Iyer, M. N. Belur, and M. C.Chandorkar, “Analysis and mitigation of
voltage offsets in multi-inverter microgrids,” IEEE TransEnergy Convers.,
vol. 26, no. 1, pp. 354–363, Mar. 2011.
[25] D. Montesinos-Miracle and O. Gomis-Bellmunt, “Control scheme of
three-level NPC inverter for integration of renewable energy resources
into ac grid,” IEEE Syst. J., vol. 6, no. 2, pp. 242–253, Jun. 2012.
Fabio Andrade Rengifo (S’06) received the B.Sc.
degree in electronic engineering and the Master’s de-
gree in engineering with emphasis on Automatic Con-
trol from the Universidad Del Valle, Cali, Colombia,
in 2004 and 2007, respectively, and the Ph.D. de-
gree from the Universitat Polit`
ecnica de Catalunya,
Barcelona, Spain, in 2013.
In 2009, he joined the Motion Control and In-
dustrial Centre Innovation Electronics, Universitat
ecnica de Catalunya, as a Researcher, where
he is currently working in power electronic appli-
cations to improve the integration of renewable energy systems to the grid.
His main research interests include modeling, analysis, design, and control of
power electronic converters/systems, especially for dc/dc power conversion,
grid-connection of renewable energy sources, and microgrid application.
Luis Romeral (M’98) received the M.S. degree in
electrical engineering and the Ph.D. degree from
the Universitat Polit`
ecnica de Catalunya, Barcelona,
Spain, in 1985 and 1995, respectively.
In 1988, he joined the Department of Electronic
Engineering, UPC, where he is currently an Asso-
ciate Professor and the Director of the Motion and
Industrial Control Group, whose major research ac-
tivities concern induction and permanent magnet mo-
tor drives, enhanced efficiency drives, fault detection
and diagnosis of electrical motor drives, and improve-
ment of educational tools. He has developed and taught postgraduate courses
on programmable logic controllers, electrical drives and motion control, and
sensors and actuators. He is a Member of the European Power Electronics and
Drives Association and the International Federation of Automatic Control.
Jordi Cusid´
o(S’06–M’10) received the degree in
electrical engineering from the Technical University
of Catalonia (UPC), Barcelona, Spain, in 2005.
Since 2005, he has been with the Department of
Electronic Engineering, UPC, where he is currently
an Assistant Professor teaching courses on analogue
electronics for aeronautical applications. He belongs
to the Motion and Industrial Control Group, Depart-
ment of Electronic Engineering, UPC. He is a Mem-
ber of the TechnologicalCentre of Manresa, Manresa,
Spain, where he is responsible for people for techno-
logical assistance to several industries and university departments in fields of
automotive and aeronautical applications. He participates in European-Union-
and Spanish-government-funded projects. He has also participated as an Engi-
neer or has been responsible for research and development projects funded by
local private companies in the areas of electrical-machines design and industrial
control. He is a Member of the IEEE Industrial Electronics Society and the
IEEE Aerospace and Electronic Systems Society.
Juan Jos´
ardenas received the Electronic En-
gineering degree from Universidad del Valle, Cali,
Colombia, in 2006, and the Ph.D. degree in elec-
tronics from the Universitat Polit`
ecnica de Catalunya
(UPC), Barcelona, Spain, in 2013.
In 2008, he joined the UPC–MCIA research group
in the area of energy management and optimization.
His main research interests include intelligent energy
management systems, energy optimization and load
modeling and forecasting on the user side, supported
by computational intelligence technologies, and sig-
nal processing and statistics.
... The stability assessment for VSG-IIDG depends on the nonlinear models. Andrade proposed the idea of modeling the inverter as an electrostatic machine [22]. The concept of electrostatic machine establishes a direct relationship between the DC and AC side of the inverter. ...
... Briefly, the changes in VSG-IIDG are represented by the changes in charges. Further derivation in [22] takes IIDG as a current source → i i with equivalent resistance R eq and admittance X eq relevant to parameters of the inverter: ...
... Energies 2018, 11, x FOR PEER REVIEW 4 of 16 changes in VSG-IIDG are represented by the changes in charges. Further derivation in [22] takes IIDG as a current source  i i with equivalent resistance eq R and admittance eq X relevant to parameters of the inverter: ω ω ...
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Inverter-interfaced distributed generators (IIDGs) have been widely applied due to their control flexibility. The stability problems of IIDGs under large signal disturbances, such as large load variations and feeder faults, will cause serious impacts on the system. The virtual synchronous generator (VSG) control is an effective scheme for IIDGs to increase transient stability. However, the existing linearized stability models of IIDGs are limited to small disturbances. Hence, this paper proposes a Lyapunov approach based on non-linearized models to assess the large signal stability of VSG-IIDG. The electrostatic machine model is introduced to establish the equivalent nonlinear model. On the basis of Popov’s theory, a Lyapunov function is derived to calculate the transient stability domain. The stability mechanism is revealed by depicting the stability domain using the locus of the angle and the angular frequency. Large signal stability of the VSG-IIDG is quantified according to the boundary of the stability domain. Effects and sensitivity analysis of the key parameters including the cable impedance, the load power, and the virtual inertia on the stability of the VSG-IIDG are also presented. The simulations are performed in PSCAD/EMTDC and the results demonstrate the proposed large signal stability assessment method.
... While the overall MG stability has been analyzed in a limited amount of works, which usually involve complex techniques such as the genetic algorithm [13] or Takagi-Sugeno multi-modelling approach [14], the number of studies involving the stability individual sub-sys tems is surely higher. In particular, droop controlled inverters have been investigated in [15][16][17]. More in detail, in [15,16] an equivalence between the inverter DC side capacitance dynamics and the rotating machine rotor one is established, while the authors of [17] have developed a reduced order model for the system that allowed finding the energy function and the stability conditions. ...
... In particular, droop controlled inverters have been investigated in [15][16][17]. More in detail, in [15,16] an equivalence between the inverter DC side capacitance dynamics and the rotating machine rotor one is established, while the authors of [17] have developed a reduced order model for the system that allowed finding the energy function and the stability conditions. Unfortunately, some of the assumptions on which it is based e.g. ...
... Otherwise, if f(y 0 )≠0, the solution is easily obtained as: (13) with (14) whose invertibility is a consequence of the continuity of f and therefore the existence of a neighborhood of y 0 where f has constant sign. Observing from (12) that coefficients B and C cannot be contemporary null, it is possible to rewrite f as: (15) with ϕ such that: (16) it is possible demonstrating that the following. ...
Microgrids (MGs) control is one of the most important and challenging topics in nowadays power system engineering research. The most common and well-known primary control logic is the droop one, since it allows MG sources to share the power request and to participate to the frequency and voltage regulation without a dedicated communication network. Therefore, the stability of a droop-controlled MG working point is a crucial issue that is typically assessed resorting to the small-signal stability theory. However, such approach does not allow to understand whether an equilibrium point can be reached starting from a specified initial condition. In this paper, this problem is faced proposing a stability condition that relies on some simplifying assumptions. The proposed solution does not need any numerical methods or software simulators (often highly CPU demanding). The developed methodology is validated with simulations performed in the PSCAD-EMTDC environment representing the MG components with a high level of detail.
... Small cogeneration heat plants (CHP) powered by biofuels can support grid stability to some extent, but burning biofuels is harmful to the environment and, thus, not ideal [33]. Furthermore, CHPs have a thermal energy component that limits their size and broader use. ...
Full-text available
This article describes a simulation of energy distribution in an average household where electricity is produced with a small wind generator or purchased from the public electricity grid. Numerical experiments conducted within an average of five minutes were performed using annual production and consumption graphs. Virtual storage devices, a water tank and a battery were used to buffer energy inside the household. The energy required for non-shiftable consumption and hot water consumption were taken directly from the utility grid. Surplus energy remaining from wind generator production after providing for consumption and storage needs were redirected there. A cover factor was used as a measure of the efficiency of energy distribution. One of the aims of the article was to determine by simulations the change of the cover factor in a virtually designed situation where the expected energy output of the wind generator was known in advance over one to three hours. The results found that for the configuration of the proposed nanogrid option, the positive results were readily achieved when the expected wind generator production was known an hour ahead. Then, the cover factor increased from 0.593 to 0.645. The side result of using pro-jected/expected production is an increase in asymmetrical energy exchanges bilaterally between nanogrid and utility grid in favour of grid sales. Another finding was that the cover factor depended on the wind generator's production intensity but less on the intensity of consumption within the household.It is hoped/expected that future research will address the prediction of output using mathematical methods.
... The traveling wave induction machine model was further developed into a two phase ( − ) model with slip [12]. Separately excited synchronous machine dq-axis models appear in power systems as equivalencies of power electronic inverters or synchronous condensers [13], but are not developed from the physical capacitances of machines to provide torque production insight. The primary contributions of this paper are divided among the following sections: ...
The use of advanced dielectric liquids and manufacturing techniques for enhanced surface area per unit volume, along with other multiplicative gains, is enabling macro-scale electrostatic machines with competitive torque density. This emergence prompts the need for circuit modeling that guides machine design and drive controls, rooted in the canon of well-established electromagnetic machinery practices. Prior dielectric and manufacturing innovations are combined here with a newly developed unified electrostatic machine dq-axis framework to form a machine that demonstrates industrial utility. Design, equivalent circuits and performance are validated by an experimental prototype intended for low speed direct drive servo actuators. A prototype electrostatic machine was constructed entirely of aluminum and printed circuit boards and possesses active material torque densities $\ge1.4\,\rm N-m/kg$ and $\ge 2.65\,\rm N-m/L$ without the need for forced air or liquid cooling. Additional features include near zero loss at stall and low torque ripple.
... where i ldi , i lqi , v bdi and v bqi are the direct and quadrature terms of the inverter current i li and bus voltage v bi in Fig. 1 [15], [31] respectively. u vi represents the secondary voltage control input. ...
This paper presents a novel distributed secondary control method for both voltage and frequency regulation in islanded microgrids. Firstly, the large-signal dynamic model of inverter-interfaced DG is formulated in the form of multi-input multi-output nonlinear system, which can be converted to a partly linear one using input-output feedback linearization. Then, the linear distributed model predictive controller is designated in each DG to realize the secondary voltage control, by incorporating the forecasted behaviors of the local and neighboring DG units. Through the receding optimization index every update process, the implementation of optimal control action accelerates the convergence rate for voltage magnitudes to the reference value. Following, after transforming the nonlinear DG dynamics into a first-order linear system, a distributed proportional integral algorithm is introduced in the frequency restoration while maintaining the accurate active power sharing. Our approach utilizes the distributed architecture, which indicates superior reliability and flexibility compared to the centralized manner; moreover, it can accommodate diverse uncertainties in communication links, model parameters, and time delays. Simulation results are provided to verify the effectiveness of the proposed control methodology. OAPA
... Then the set points of the converter current controller are modified accordingly. For an observer in the grid side, this real time simulation and subsequent fast control of the inverter phase currents gives an impression of an actual synchronous generator connected at PCC. Although, this is the general idea behind VSM, several different implementations of VSM can be found in (Qing-Chang & George, 2011;Fabio, et al., 2014;Yong, et al., 2012;Hans-Peter & Ralf, 2007). ...
The demand for electrical energy is growing exponentially worldwide, putting a lot of burden on the conventional sources of energy. However, since these sources of energy are not inexhaustible, the need to explore alternate sources of energy is increasingly becoming a necessity. The conventional sources of electrical energy are also big sources of pollution of various types resulting in harmful effects on climate and the health of all living beings. The ever‐increasing demand for electrical power and a heightened concern about the climate is a motivation to look for such sources of energy that are inexhaustible, or can be replenished naturally and quickly as per human time scale, as well as clean. The natural sources of energy such as sun, wind or rain are renewable and have less impact on the climate as compared to the conventional sources of energy. However, new challenges are bound to emerge in an environment powered by the implementation of clean energy resources. The transition to renewables to help combat climate change and fulfil the energy demand of future generations needs to be viable. Integrating renewables with the conventional grids is resulting in sustainable microgrids. Stability of the microgrid is critical as it affects the quality of the power supply. However, owing to its unique characteristics, microgrids pose stability issues distinct from those identified in bulk electrical systems. Different methods of computation, control and analysis are being employed throughout the world to address these challenges posed by microgrids. This chapter envisages presenting the stability concerns and issues associated with microgrids along with a state‐of‐the‐art review of the techniques employed for improving stability of microgrids working in either islanded or grid‐connected mode.
Self-governing microgrids may consist of several con-ventional synchronous generators, converter-based distributed resources, and energy storages. In this paper, an eigenvalue-ori-ented stability assessment method is established to study the small signal stability of such systems that have complex dynamic char-acteristics. A new index referred to as the microgrid’s marginal frequency stability index is established. This index is then used to determine the influence and correlation between sizes, quantities and internal parameters of the resources and loads on the mi-crogrid’s stability. Likewise, using this index, the impact of the isolation and coupling impedance amongst resources and loads are studied on the microgrid’s stability. The findings of this study can be employed when designing microgrids to realize more sta-ble systems.
Full-text available
This paper addresses the low-frequency relative stability problem in paralleled inverter-based distributed generation (DG) units in microgrids. In the sense of the small-signal dynamics of a microgrid, it can be shown that as the demanded power of each inverter changes, the low-frequency modes of the power sharing dynamics drift to new locations and the relative stability is remarkably affected, and eventually, instability can be yielded. To preserve the power sharing stability, an adaptive decentralized droop controller of paralleled inverter-based DG units is presented in this paper. The proposed power sharing strategy is based on the static droop characteristics combined with an adaptive transient droop function. Unlike conventional droop controllers, which yield 1-DOF tunable controller, the proposed droop controller yields 2-DOF tunable controller. Subsequently, the dynamic performance of the power sharing mechanism can be adjusted, without affecting the static droop gain, to damp the oscillatory modes of the power sharing controller. To account for the power modes immigration at different loading conditions, the transient droop gains are adaptively scheduled via small-signal analysis of the power sharing mechanism along the loading trajectory of each DG unit to yield the desired transient and steady-state response. The gain adaptation scheme utilizes the filtered active and reactive powers as indices; therefore, a stable and smooth power injection performance can be obtained at different loading conditions. The adaptive nature of the proposed controller ensures active damping of power oscillations at different operating conditions, and yields a stable and robust performance of the paralleled inverter system.
This paper proposes an enhanced control strategy for electronically coupled distributed energy resources that improves the performance of the host microgrid under network faults and transient disturbances. The proposed control strategy does not require controller mode switchings and enables the electronically coupled distributed energy resources to ride through network faults, irrespective of whether they take place within the host microgrid or impact the upstream grid. Moreover, the proposed control ensures acceptable power quality for the duration of the faults, which is an important feature for protection against certain classes of faults, as well as for sensitive loads. Further, the paper proposes a supplementary control loop that improves the microgrid post-fault recovery. The effectiveness of the proposed control strategy is demonstrated through a comprehensive set of simulation studies, conducted in the PSCAD/EMTDC software environment.
In this paper, a generalized closed-loop control (GCC) scheme is proposed for voltage source converters (VSCs) with LC or LCL output filters. The proposed GCC scheme has a single-loop control of inverter output (voltage or current) and two parallel virtual impedance terms using additional measurements. The virtual impedance can be the equivalent internal impedance or external impedance (or both), depending on their control term and feedback variable selection. The internal impedance term is mainly responsible for providing desired damping to the filter circuit, and the external virtual impedance term can effectively adjust the converter system closed-loop output impedance. As each term in the GCC scheme can be controlled independently, the proposed GCC scheme has great flexibility and can easily realize and explain the performances of the traditional single- and multiloop control schemes and their different variations. Moreover, the GCC scheme provides a distinct physical meaning of each control term, which makes the control parameter tuning more straightforward and robust. Additionally, as shown in this paper, the proposed GCC scheme can tackle some traditionally challenging control objectives by avoiding the harmonics filtering or derivative terms. Experimental results from laboratory VSC prototypes are obtained to validate the proposed GCC scheme.
This paper gives theoretical foundation to a procedure for modeling in a simply way the behavior of a droop-regulated islanded microgrid when, due to reserve scheduling considerations, the power reserves might be exhausted. The main problem observed for computing the operating point of those microgrids is that the droop formulation that is to be entered in the computation depends on the knowledge of the final result, which indeed means that the solution by means of Newton-Raphson-like methods is hindered. The proposed procedure reduces the complexity by formulating the problem as a complementarity problem. A discussion is offered then on the specific problem of droop formulation: two states are possible (with and without power limit reached), with the particularity that the power must be considered constant only when the power limit is reached, whereas the frequency can freely vary in both states, searching for an equilibrium in the load share among all the generation units endowed with droop regulation. Further, the problem is simplified by resorting to the use of Fischer-Burmeister NCP-functions (NCP for nonlinear complementarity problem), which substitutes the piecewise-defined droop function by an only scalar function (Fischer-Burmeister function) that makes the problem tractable to be solved by Newton-Raphson-like methods. The paper concludes with an exposition of numerical simulations in which the consequences of considering the power exhaustion on stability and operating points are demonstrated.
For islanded microgrids, droop-based control methods are often used to achieve a reliable energy supply. However, in case of resistive microgrids, these control strategies can be rather different to what conventional grid control is accustomed to. Therefore, in this paper, the theoretical analogy between conventional grid control by means of synchronous generators (SGs) and the control of converter-interfaced distributed generation (CIDG) units in microgrids is studied. The conventional grid control is based on the frequency as a global parameter showing differences between mechanical power and ac power. The SGs act on changes of frequency through their P/f droop controller, without interunit communication. For CIDG units, a difference between dc-side power and ac-side power is visible in the dc-link voltage of each unit. Opposed to grid frequency, this is not a global parameter. Thus, in order to make a theoretical analogy, a global measure of the dc-link voltages is required. A control strategy based on this global voltage is presented and the analogy with the conventional grid control is studied, with the emphasis on the need for interunit communication to achieve this analogy. A known control strategy in resistive microgrids, called the voltage-based droop control for CIDG units, approximates this analogy closely, but avoids interunit communication. Therefore, this control strategy is straightforward for implementation since it is close to what control engineers are used to. Also, it has some specific advantages for the integration of renewables in the network.
Microgrids are receiving an increasing interest to integrate the growing share of distributed-generation (DG) units in the electrical network. For the islanded operation of the microgrid, several control strategies for the primary control have been developed to ensure stable microgrid operation. In low-voltage (LV) microgrids, active power/voltage (P/V) droop controllers are gaining attention as they take the resistive nature of the network lines and the lack of directly coupled rotating inertia into account. However, a problem often cited with these droop controllers is that the grid voltage is not a global parameter. This can influence the power sharing between different units. In this paper, it is investigated whether this is actually a disadvantage of the control strategy. It is shown that with P/V droop control, the DG units that are located electrically far from the load centers automatically deliver a lower share of the power. This automatic power-sharing modification can lead to decreased line losses; therefore, there is overall better efficiency compared to the methods that focus on perfect power sharing. In this paper, the P/V and P/f droop control strategies are compared with respect to this power-sharing modification and the line losses.
The concept of microgrid hierarchical control is presented recently. In this paper, a hierarchical scheme is proposed which includes primary and secondary control levels. The primary level comprises distributed generators (DGs) local controllers. The local controllers mainly consist of power, voltage and current controllers, and virtual impedance control loop. The central secondary controller is designed to manage the compensation of voltage unbalance at the point of common coupling (PCC) in an islanded microgrid. Unbalance compensation is achieved by sending proper control signals to the DGs local controllers. The design procedure of the control system is discussed in detail and the simulation results are presented. The results show the effectiveness of the proposed control structure in compensating the voltage unbalance.