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# New Model of a Converter-Based Generator Using Electrostatic Synchronous Machine Concept

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## Abstract and Figures

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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344 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 2, JUNE 2014
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´
ardenas
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
microgrids.
NOMENCLATURE
RES renewable energy system.
vfd,vfq: instantaneous electrostatic ﬁeld voltage in
DQ0-axis.
vd,vq,v0:instantaneous electrostatic stator voltage in
DQ0-axis.
ifd,ifq:electrostatic ﬁeld current in DQ-axis.
id,iq,i0: instantaneous electrostatic stator currents in
DQ0-axis.
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
surfaces.
R: armature resistances.
Rf:rotor resistances.
Qd,Q
q,Q
0:demand charge in each DQ0-circuit.
Qfd,Q
fq:available charge of rotor in DQ0-axis.
Θ: angle by D-axis leads the axis of a-phase.
ωr: rotor angular velocity.
I. INTRODUCTION
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
juan.jose.cardenas@mcia.upc.edu).
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: jordi.cusido@ctm.com.es).
Color versions of one or more of the ﬁgures in this paper are available online
at http://ieeexplore.ieee.org.
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
markets.
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.
RENGIFO et al.: NEW MODEL OF A CONVERTER-BASED GENERATOR USING ELECTROSTATIC SYNCHRONOUS MACHINE CONCEPT 345
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.
II. CONCEPT OF ELECTROSTATIC MACHINE
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
components.
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,
U4,U5,U6,U
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
system.
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.
346 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 2, JUNE 2014
Fig. 3. Representation of the electrostatic machine by its equivalent capacitors
The synthesized (by the pulse width modulation-averaged
approximation) voltage reference vector can be represented as
follows:
m =mmax
cos (ωrt+θ)
cos (ωrt2π/3+θ)
cos (ωrt+2π/3+θ)
.(1)
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
ΦE=
E·
S=QT
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.
id=d
dtQdQq
dt vd
R
iq=d
dtQq+Qd
dt vq
R
i0=d
dtQ0v0
R.(3)
Fig. 4. D-axis and q-axis equivalent circuits of a power electronic based
generator.
The charge in each circuit at any instant is given by
Qd=Cdvd+Cfddvfd
Qq=Cqvq+Cfqqvfq
Q0=C0v0.(4)
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
Rf
ifq =d
dtQfq vfq
Rf
.(5)
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.
III. EQUIVALENT OF DG BASED IN ELECTRONIC INTERFACE
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
RENGIFO et al.: NEW MODEL OF A CONVERTER-BASED GENERATOR USING ELECTROSTATIC SYNCHRONOUS MACHINE CONCEPT 347
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.
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
become
id=Qqωr1
Rvd
iq=Qdωr1
Rvq
ifd =1
Rf
vfd
ifq =1
Rf
vfq (7)
and the charges
Qd=Cdvd+Cfddvfd
Qq=Cqvq+Cfqqvfd
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
Xcd
vd1
RvqiqXcfdd
vfq =1
Rvd+1
Xcd
vq+iqXcfqq.(9)
Also, the stator voltage and current may be written as phasor
representation “vt” and “
it” where vt=vd+vqand
it=id+
iq. The relationships between equations according to (5) can be
expressed:
id=1
Xcfqq
vfq 1
Xcq
vq+1
Rvd
iq=1
Xcfdd
vfd +1
Xcd
vd1
Rvq.(10)
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
ii=
it+1
Requiv +jXcequiv ·vt
vt=vd+jvq
it=id+jiq.(11)
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:
it=1
Xcfqq
vfq j1
Xcfdd
vfd.(12)
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
2dd(13)
where diis the average value for the duty cycle.
The equivalents current source, resistance, and admittance
are expressed as
it=Vdcdd
2Xcdc jVdcdq
2Xcdc
Xcequiv =Xcdc d2
qd2
d
d2
q+d2
dRequiv =Xcdc 2dqdd
d2
q+d2
d.(14)
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
Pt=Vd
dQd
dt +Vq
dQq
dt

Rate of Ch ange
Armatu re Electric
Energy
+[VqQdVdQq]ωr

Power Transf.
from dc ac
V2
d
R+V2
q
R

R.Loss
.
(15)
348 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 2, JUNE 2014
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
dΔ¯ωr
dt =1
2H¯
Tm¯
TeKDΔ¯ωr
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.
1
2CV 2
dc =1
22
rH=CV 2
dc
2VA
base
J=CVdc
ωr2J=2H
ω2
rVA
base
(17)
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).
Δω=ωf
s+ωf
(KpΔP)
ΔP=1
ωfKp
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
TAB LE I
SIMULATION PARAMETERS
calculate the Miand Diparameters as follows:
M=2HKpωf+1
Kpωf
,D =1
Kp
.(19)
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.
IV. SIMULATION STUDY AND EXPERIMENTAL EVALU ATIO N
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
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
id=Qq1
0.3vd
iq=Qd1
0.3vq
ifd =1
10vfd
ifq =1
10vfq (20)
where
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
RENGIFO et al.: NEW MODEL OF A CONVERTER-BASED GENERATOR USING ELECTROSTATIC SYNCHRONOUS MACHINE CONCEPT 349
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
simulation.
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
350 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 2, JUNE 2014
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
(22)
RMSE =
1
N
N
i
Ierr2
i(22)
where Ierr =Instantaneous error.
TAB LE I I
ROOT MEANS SQUARE ERROR (RMSE)
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
system.
V. CONCLUSION
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
RENGIFO et al.: NEW MODEL OF A CONVERTER-BASED GENERATOR USING ELECTROSTATIC SYNCHRONOUS MACHINE CONCEPT 351
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
energy.
352 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 2, JUNE 2014
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.
APPENDIX
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:
Zsbase =vsbase
isbase
(ohm)
Csbase =1
ωbaseZsbase
Qsbase =Csbase vsbase (Coulombs)
3ph VAsbase =3
2vsbaseisbase (volt-amperes)
Torquebase =3ph VAbase
ωbase
(Newton-meters)
tbase =1
ωbase
seg.
The per unit time derivative is given by
¯p=1
ωbase
p.
Rotor base quantities:
vfbase =Peak value of rated dc voltage (V)
vfbase ifbase =3
2vsbaseisbase
=3ph VAsbase
Cafd
vsbase (V)
ifbase =3ph VA sbase
vfbase
(A)
Zfbase =vfbase
ifbase
(ohm)
Cfbase =1
ωbaseZfbase
Qfbase =Cfbasevfbase (Coulombs).
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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
Polit
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´
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´
eC
´
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.
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