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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.

Cfd−d,Cfq−q:“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

08222, Spain (e-mail: fabio.andrade@mcia.upc.edu; romeral@eel.upc.edu;

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 (P−f)and reactive power–voltage (Q−V)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.

See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

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 120◦apart

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

on the direct-axis and quadrature-axis.

The synthesized (by the pulse width modulation-averaged

approximation) voltage reference vector can be represented as

follows:

m =mmax ⎡

⎢

⎣

cos (ωrt+θ)

cos (ωrt−2π/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

ε0→QT=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

dtQd−Qq

dθ

dt −vd

R

iq=d

dtQq+Qd

dθ

dt −vq

R

i0=d

dtQ0−v0

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+Cfd−dvfd

Qq=−Cqvq+Cfq−qvfq

Q0=−C0v0.(4)

Cfd−dand Cfq−qare the equivalent capacitance between the

rotor and the stator in DQ-axes, and the (Cafd−d×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 −Cfd−dvd

Qfq =Cffqvfq −Cfq−qvq.(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.

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

become

id=−Qqωr−1

Rvd

iq=Qdωr−1

Rvq

ifd =1

Rf

vfd

ifq =1

Rf

vfq (7)

and the charges

Qd=−Cdvd+Cfd−dvfd

Qq=−Cqvq+Cfq−qvfd

Qfd =Cffdvfd −Cfd−dvd

Qfq =Cffqvfq −Cfq−qvq.(8)

The aforementioned equations can be used to ﬁnd the ﬁeld

voltages. Replacing the product (–ωrC)−1by the corresponding

reactance Xc,

vfd =1

Xcd

vd−1

Rvq−iqXcfd−d

vfq =1

Rvd+1

Xcd

vq+iqXcfq−q.(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

Xcfq−q

vfq −1

Xcq

vq+1

Rvd

iq=−1

Xcfd−d

vfd +1

Xcd

vd−1

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

Xcfq−q

vfq −j1

Xcfd−d

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,

Xcfd−d=Xcfq−q=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

q−d2

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

+[VqQd−VdQq]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−¯

Te−KDΔ¯ωr

dδ

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

2Jω2

r→H=CV 2

dc

2VA

base

J=CVdc

ωr2→J=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

dω

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

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

id=−Qq−1

0.3vd

iq=Qd−1

0.3vq

ifd =1

10vfd

ifq =1

10vfq (20)

where

Qd=−5.6×10−4vd+2.54 ×10−4vfd

Qq=−5.6×10−4vq+2.54 ×10−4vfd

Qfd =5.6×10−4vfd −2.54 ×10−4vd

Qfq =5.6×10−4vfq −2.54 ×10−4vq(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:

ωbase =2πfbase (elec.radians/seg)

Zsbase =vsbase

isbase

(ohm)

Csbase =1

ωbaseZsbase

(Farads)

Qsbase =Csbase vsbase (Coulombs)

3−ph VAsbase =3

2vsbaseisbase (volt-amperes)

Torquebase =3−ph 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

=3−ph VAsbase

vfbase =Cad

Cafd

vsbase (V)

ifbase =3−ph VA sbase

vfbase

(A)

Zfbase =vfbase

ifbase

(ohm)

Cfbase =1

ωbaseZfbase

(Farads)

Qfbase =Cfbasevfbase (Coulombs).

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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

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´

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´

eC

´

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.