Three-Dimensional Feedforward Space VectorModulation Applied to MultilevelDiode-Clamped Converters

Jose I. Leon, Member, IEEE, Sergio Vazquez, Student Member, IEEE, Ramon Portillo, Student Member, IEEE,Leopoldo G. Franquelo, Fellow, IEEE, Juan M. Carrasco,Member, IEEE,

Patrick W. Wheeler,Member, IEEE, and Alan J. Watson, Student Member, IEEE

Abstract—Simplified space vector modulation (SVM) tech-niques for multilevel converters are being developed to improvefactors such as the computational cost, number of commutations,and voltage distortion. The feedforward SVM presented in thispaper takes into account the actual dc capacitor voltage unbalanceof the multilevel power converter. The resulting technique is alow-cost generalized feedforward 3-D SVM method and is particu-larized for three-phase multilevel diode-clamped converters. Thisnew modulation technique can be applied to topologies where thegamma component may not be zero. The computational cost of theproposed method is similar to those of comparable methods, andit is independent of the number of levels of the power converter.Experimental results using a three-level diode-clamped converterare presented to validate the proposed modulation technique.

Index Terms—Modulation, multilevel systems, voltage control.

I. INTRODUCTION

NOWADAYS, pulsewidth modulation (PWM) and space

vector modulation (SVM) methods coexist and are used

in power converters applied to energy conversion systems [1].

PWM presents the advantage of its extreme simplicity and its

easy and direct hardware implementation [2]–[5]. Level- and

phase-shifted PWMs are the most common PWM techniques

to be applied to multilevel converters using several triangular

carriers with voltage or phase shift [6], [7]. On the other hand,

SVM is a modulation technique where the discrete output

voltages of a power converter are geometrically represented.

The generation of one specific output voltage is made using

a linear combination of the nearest discrete output voltages

located in space. As a result, the averaged modulated voltage

coincides with the desired one. In this way, the modulation

concern is transformed to a mathematical problem, and all

works can be reduced to simple geometrical calculations. The

Manuscript received November 6, 2007; revised June 17, 2008. First pub-lished July 9, 2008; current version published December 30, 2008. This workwas supported by the Spanish Science and Education Ministry under ProjectTEC2006-03863.J. I. Leon, S. Vazquez, R. Portillo, L. G. Franquelo, and J. M. Carrasco are

with the Department of Electronic Engineering, University of Seville, 41092Seville, Spain (e-mail: jileon@gte.esi.us.es).P. W. Wheeler and A. J. Watson are with the School of Electrical and

Electronic Engineering, University of Nottingham, Nottingham NG7 2RD,U.K. (e-mail: pat.wheeler@nottingham.ac.uk.).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/TIE.2008.928110

SVM technique is a software modulation method, and its

computational cost has been one important barrier to using

it in medium/high switching frequency applications. However,

in recent years, several SVM algorithms have been published,

greatly reducing the necessary calculations to the minimum.

Nowadays, a simple and efficient SVM can be executed in a few

microseconds, achieving similar results compared with PWM

methods [8], [9]. Comparing both modulation methods, the

great advantage of using SVM techniques is that some freedom

degrees appear in the modulation process. As the modulation

is done using a switching sequence, the order of the state

vectors into the sequence can be chosen, improving factors as

the number of commutations [10]. In addition, the existence

of redundancies in the control region of the multilevel power

converters can be used in the SVM algorithm to control the dc

voltages of the power converter [11]. Therefore, at the expense

of spending a few microseconds, several advantages can be

achieved using SVM. However, PWM and SVM techniques

for three-phase systems have problems when multilevel power

converters present unbalances in the dc voltages. A perfect

balance of the dc voltages of a multilevel converter cannot

be achieved in all loading conditions. Load imbalances and

nonlinear or transient loads have a significant impact on the

multilevel converter dc voltage ripple (oscillations or actual

values) [12]. In this case, both modulation techniques are not

prepared for this unbalance because they do not take it into

account to carry out the modulation process. In this way, errors

appear in the output modulated voltages because they do not

match to the desired ones when they are averaged over a

switching period. This fact leads to an increase of the harmonic

distortion of the output voltages and currents of the multilevel

converters [12]. This problem has previously been addressed

by other authors, avoiding the influence of the dc-link voltage

ripple on the output signals for two-level converters [13], [14].

Other authors presented works focused on multilevel converters

using multicarrier PWM [15] and SVM [16] for balanced

systems where the neutral is not connected. In this paper, a

3-D feedforward SVM (3D-FFSVM) technique is presented,

taking into account the actual unbalance of the power converter

to carry out the necessary calculations, avoiding errors in the

modulation process. Using the proposed 3D-FFSVM, balanced

and unbalanced systems can be modulated with balanced or

unbalanced dc voltages.

0278-0046/$25.00 © 2008 IEEE

II. SVM TECHNIQUES FOR MULTILEVEL CONVERTERS.

STATE OF THE ART

A. Balanced Systems

The SVM for balanced systems (without the connection of

the neutral point of the load) normally uses a representation

using the alpha–beta frame because the gamma component of

the voltages is zero. In this way, all the discrete positions of

the state vectors of the multilevel power converter are located

on the αβ plane, and 2-D SVM strategies can be used. The

computational cost of this type of SVM methods has been

decreasing, as researchers have introduced new algorithms from

complicated ones to the simplest ones, where the necessary

calculations are very simple [17]–[20].

B. Unbalanced Systems

The unbalanced operation of a multilevel power converter

can be obtained with the neutral of the load connected to the

middle point of the dc link (three-leg four-wire topologies)

or to a new phase of the power converter (four-leg four-wire

topologies). The three-leg four-wire converters can be used as

active power filters and power quality compensators [21]–[24].

On the other hand, the typical applications of four-leg four-

wire converters are distributed power generation, active power

filtering, PWM rectifiers, and common mode noise reduction

[25]. Using these converter topologies, the gamma component

of the voltages is not zero, and the three dimensions have to

be used in the modulation to generate the reference vectors

[25], [26]. First 3-D SVM algorithms were complicated because

they used αβγ components [25], but recent ones achieve high

quality results with a minimum number of calculations using

natural coordinates abc [26], [27]. These simple 3-D SVM tech-

niques carry out a fast geometrical search of the four nearest

state vectors to determine the switching sequence. Using abc

coordinates, the control region is a cube or a prism for three-

leg four-wire or four-leg four-wire topologies, respectively. For

instance, as was introduced in [26], the 3-D control region of a

three-level converter is shown in Fig. 1. The state vectors of the

power converter are denoted as xyz, which means that phases

a, b, and c, respectively, take values between zero and N − 1for an N -level converter.

State zero means that the phase is connected to the lowest

dc voltage level and N − 1 is the highest possible dc voltage.In this way, output voltages Va0, Vb0, and Vc0 are used as

the modulation components, where zero is the point of lowest

voltage of the multilevel power converter. The first task of the

3-D SVM technique presented in [26] (named in this paper as

the conventional 3D-SVM) is to know the subcube where the

reference vector is located. Once this subcube is determined,

it is divided into six tetrahedrons, and the 3D-SVM has to

calculate the tetrahedron where the reference vector is pointing.

These tetrahedrons are shown in Fig. 2.

Finally, depending on the tetrahedron case, a table summa-

rizes all the necessary information to determine the switching

sequence and the corresponding duty cycles [26]. It is important

to notice that these 3D-SVM techniques can be applied to

balanced and unbalanced systems, as the 2-D control region

Fig. 1. Control region of a three-level converter formed by several subcubes.

Fig. 2. Division of each subcube of the control region of a multilevel converterwith equal dc sources.

is indeed included in the 3-D control region. This idea is

graphically shown in [28]. Therefore, the 3D-SVM strategies

can be used for any multilevel power converter with or without

neutral connection, making these techniques very generalized

and useful.

III. THREE-DIMENSIONAL FEEDFORWARD SVM

A. Feedforward Basic Concept for Balanced Systems

The feedforward SVM was first introduced for balanced

systems in [29], working in the 2-D control region in the

αβ plane for three-level neutral-point-clamped converters. This

Fig. 3. Power circuit of the three-level diode-clamped converter.

technique takes into account the actual voltage unbalance of the

three-level diode-clamped converter to represent correctly the

actual 2-D control region. When a voltage unbalance is present

in the power converter, the state vector position changes, and

therefore, the use of the classic 2-D control region introduces

errors in the generation of the modulated output voltage. The

feedforward SVM technique calculates online the actual 2-D

control region, and in this way, the errors in the modulation

process are avoided. In this way, voltage unbalances in the dc

link do not create undesired distortion in the ac output voltages

or output currents. This feedforward 2D-SVM method was

later optimized in [16], where the computational cost of the

technique was reduced.

B. Proposed Feedforward Extension for Balanced and

Unbalanced Systems

In this paper, a simple and generalized 3D-FFSVM is pre-

sented for multilevel converters. In this way, previous works

are improved because the proposed technique can be used for

any number of levels of the power converter, and applications

with the neutral connected can be implemented. The proposed

3D-FFSVM technique is similar to the conventional 3D-SVM

method presented in [26], following a similar notation. The

steps to carry out the proposed 3D-FFSVM are the following.

1) Three-Dimensional Control Region Determination: If

the dc voltages of a multilevel converter are not balanced, the

conventional 3D-SVM technique cannot be used because the

3-D control region changes and it is not formed by regular

cubes. For instance, in the case of a three-level converter, the

control region shown in Fig. 1 changes because the discrete

locations of the state vectors move due to the voltage unbalance.

In the three-level case, two different dc voltages VC1 and VC2

have to be considered for a three-level diode-clamped converter.

The power circuit of the three-level diode-clamped converter is

shown in Fig. 3. The possible Vphase−0 voltages of the converter

are zero, VC1, and VC1 + VC2 (point zero is the lowest voltage

point of the power converter). As in Fig. 1, the phase states can

be represented but using generalized dc voltages VC1 and VC2.

In general, the 3-D control region is a cube with size VC1 +VC2 formed by several rectangular subprisms with different

sizes, depending on the voltage of each dc-link capacitor. This

idea is represented in Fig. 4, where the 3-D control region of

a three-level power converter fulfilling VC1 < VC2 is shown.

Fig. 4. Three-dimensional control region of a three-level converter withvoltage unbalance in the dc link (VC1 < VC2).

The δi values are the sizes of the subprisms that form the 3-D

control region, as shown in Fig. 4, for the three-level case

VDCtotal =N−1∑

i=1

Vci (1)

δi =Vci

VDCtotal

. (2)

A vector Vo containing the possible actual output voltages of the

power converter can be determined. The elements of this vector

are in increasing order from zero to the most positive value. In

the N -level case, this vector is as follows:

Vo = 0, VC1, VC1 + VC2, . . . , VC1 + · · · + VCN−1. (3)

At the same time, vector Vs can be written with the associated

phase states for each output voltage

Vs = 0, 1, 2, . . . , N − 1. (4)

The vector Vo is normalized with respect to the total voltage

of the dc-link generating vector Von

Von =Vo

VDCtotal

=

0,VC1

VDCtotal

,VC1 + VC2

VDCtotal

, . . . ,VC1 + · · · + VCN−1

VDCtotal

= 0, δ1, δ1 + δ2, . . . , δ1 + · · · + δN−1

= 0, δ1, δ1 + δ2, . . . , 1. (5)

2) Normalization of the Reference Vector: The reference

vector calculated by the controller is defined as Vref =Va, Vb, Vc, where Vj is the voltage of phase j with respect to

point zero. This vector is normalized using the total dc voltage

of the power converter. In this way, the normalized positive

reference voltages ua, ub, and uc take values between zero

and one

Vrefn =ua, ub, uc=

Va

VDCtotal

,Vb

VDCtotal

,Vc

VDCtotal

. (6)

3) Determination of the Subprism: An iterative geometrical

search over each component is carried out to find out the vertex

closer to the origin of the subprism, where vector Vrefn is

pointing. For instance, in phase a, it is iteratively asked where

ua is located inside Von vector, comparing with each element.

Finally, the lower and upper closer elements (named Oa and

OSa, respectively) in vector Von of the range where ua is

located can be determined. For instance, for the three-level case,

it follows that

Von =

0,VC1

VC1 + VC2

,VC1 + VC2

VC1 + VC2

= 0, δ1, δ1 + δ2 = 0, δ1, 1. (7)

If δ1 < ua < 1, the factor Oa is δ1 and the factor OSa is

one. This process is repeated for each phase of the reference

vector, calculating the vector Oabc = Oa, Ob, Oc that is thevertex closer to the origin of the rectangular subprism where the

reference vector ua, ub, uc is located. Moreover, the vectorOSabc = OSa,OSb,OSc is also determined. Vectors ∆ =δa, δb, δc and ∆V = ra, rb, rc can be calculated. Doingthis, the coordinate frame can be changed by moving the origin

from (0, 0, 0) to point Oabc, and in this way, vector ∆V is the

positive normalized reference vector Vrefn using this new frame.

∆ vector defines the size of the rectangular subprism where the

reference vector is located

∆ = δa, δb, δc=OSa−Oa,OSb−Ob,OSc−Oc (8)

∆V = ra, rb, rc=ua−Oa, ub−Ob, uc−Oc. (9)

4) Switching Sequence and Duty Cycle Calculation: The

next step of the proposed 3D-FFSVM is to find out the four

nearest state vectors to form the switching sequence to generate

the reference voltage. These four nearest state vectors are

the vertices of a volume which, in a similar way with the

conventional 3D-SVM method, has to be found. In this point

of the modulation process, the subprism where Vrefn is located

is known. The subprism has its origin in point Oabc, and after

the coordinates change, this point is the new origin (0, 0, 0).

The size of the subprism is δa, δb, and δc in components a, b,

and c, respectively. Six tetrahedrons can be used to divide the

subprism, as it is shown in Fig. 5.

The flow diagram to find out the tetrahedron where ∆V

vector is located is shown in Fig. 6. In the worst case, after three

simple comparisons, the tetrahedron is determined. Once the

tetrahedron is determined, the switching sequence to be used

and the corresponding duty cycles can be directly calculated

using a similar process to [26]. The results are summarized

in Table I, where parameters µa, µb, and µc are respectively

defined as

µa =ra

δa

µb =rb

δb

µc =rc

δc

. (10)

Fig. 5. Division of each rectangular subprism of the 3-D control region of amultilevel converter with generalized dc voltages.

Fig. 6. Flow diagram for the proposed 3D-FFSVM technique to find out thetetrahedron where the reference vector is pointing.

The switching sequence is formed by four vectors Spi (i =

1, 2, 3, 4) containing elements Op and OSp factors for phase

p. The last step of the proposed 3D-FFSVM algorithm is to

determine the proper switching corresponding to these factors

Op and OSp. The switching sequence is finally formed by the

phase states included in vector Vs with the same positions

of Op and OSp included in vector Von, respectively. For in-

stance, following the previous example for the three-level case,

working with phase a, the positions of Oa and OSa inside

Von were well known. If Oa = δ1 and OSa = 1, the positionsare two and three, respectively, in Von vector. Therefore, the

elements in positions two and three of Vs are the phase states

to be used in phase a of the power converter, and finally, the

switching sequence for this phase is 1, 2. This process has tobe repeated for the three phases of the power converter.

TABLE ISWITCHING SEQUENCES AND DUTY CYCLESDETERMINED BY THE 3D-FFSVM TECHNIQUE

IV. EXPERIMENTAL RESULTS

The proposed 3D-FFSVM technique has been experimen-

tally tested using the three-leg three-wire back-to-back three-

level diode-clamped converter. The rectifier side is controlling

the dc voltages of the dc-link (VC1 and VC2), and all the

experimental results have been taken from the inverter side.

The control system is based on a TMS320VC33 DSP board,

and the switching frequency is 1.4 kHz. Comparisons with

the conventional 3D-SVM from [26] have been made in order

to show the improvements achieved using the proposed 3-D

feedforward idea.

A. Steady-State Balanced DC Voltages Response

First, both modulation strategies have been tested in steady-

state conditions without any dc voltage unbalance in the power

converter. The total dc-link voltage is 800 V, and therefore,

400 V is the voltage of each half of the dc link. A 50-Hz sinu-

soidal waveform is applied as the reference voltage to be gen-

erated by the inverter side with a modulation index that is equal

to 0.9. The inverter is connected to an RL load (R = 120 Ωand L = 15 mH per phase). The corresponding output phasevoltages and currents of the inverter using 3D-FFSVM are

shown in Fig. 7, and they are equal to the ones obtained

using the conventional 3D-SVM technique. The experimen-

tal results have been taken using a Yokogawa WT1600. In

Fig. 8(a) and (b), the obtained numerical data of the harmonic

spectrum of the phase voltages and currents using the con-

ventional 3-D and feedforward 3-D techniques are presented,

respectively. No relevant changes are noticed between both

modulation techniques obtaining similar values. The obtained

results are similar to conventional SVM methods for multilevel

Fig. 7. Output phase voltages and currents of the three-level power inverterwithout dc voltage unbalance, using the proposed 3D-FFSVM.

Fig. 8. Numerical data of the harmonic spectrum of the output voltages andcurrents of the three-level power inverter without dc voltage unbalance, using(a) conventional 3D-SVM and (b) 3D-FFSVM.

converters [30]. In this way, it is demonstrated that the use of

the proposed 3D-FFSVM does not introduce any additional un-

desired distortion in voltage-balanced steady state conditions.

In addition, it is almost identical to conventional SVM methods

in terms of generality and computational cost.

B. Steady-State Unbalanced DC Voltage Response

Second, the controller of the rectifier in charge of the voltage

balancing is changed on purpose, forcing 100 V of unbal-

ance on the dc-link voltages of the power converter. Under

these conditions, and also imposing the loading conditions

of the experiment shown in Figs. 7 and 8, the conventional

3D-SVM and the 3D-FFSVM techniques are applied. The ob-

tained harmonic spectra of the output phase-to-phase voltages

are shown in Figs. 9 and 10, respectively. It can be seen that,

Fig. 9. Harmonic spectrum of the output phase-to-phase voltages of the three-level power inverter with a 100-V unbalance, using the conventional 3D-SVMtechnique.

Fig. 10. Harmonic spectrum of the output phase-to-phase voltages of thethree-level power inverter with a 100-V unbalance, using the proposed3D-FFSVM technique.

using the conventional 3D-SVM, the second order harmonic

content of the output phase-to-phase voltage achieves 4.6% of

the fundamental.

In Fig. 10, using the proposed 3D-FFSVM, the second

order harmonic content of the output phase-to-phase voltage

is approximately 0.6% of the fundamental. In addition, the

total harmonic distortion (THD) factor of the phase-to-phase

voltage is reduced from 15.3%, using the 3D-SVM method,

to 14.3%, using the 3D-FFSVM technique. The obtained har-

monic spectrum using the proposed 3D-FFSVM technique

shown in Fig. 10 is very similar to the obtained spectrum when

the dc voltages are balanced, as can be seen in comparing with

the values obtained in Fig. 8. As a conclusion, it is shown

that a dc voltage unbalance creates distortion in the output

waveforms if the conventional 3-D modulation is used, whereas

the proposed feedforward modification reduces this distortion

up to the minimum (similar values in Fig. 8).

In order to emphasize the achieved improvements with a

100-V unbalance in the dc-link voltage, a modulation technique

change from conventional 3D-SVM to 3D-FFSVM is applied to

the power converter. The fundamental component, the second

order harmonic value, and the THD of the output phase-to-

phase voltages are shown in Fig. 11(a)–(c), respectively. Using

the conventional 3D-SVM method, the voltage unbalance leads

to a second-order harmonic distortion of around 5%. It is clear

Fig. 11. Forcing a 100-V voltage unbalance; experimental results usingfirst the conventional 3D-SVM and finally the 3D-FFSVM. (a) Fundamentalcomponent of the phase voltage. (b) Second-order harmonic value of the phasevoltage. (c) THD of the phase voltage.

that, when the modulation method is changed using the new

3D-FFSVM, the second order harmonic distortion and the THD

are quickly and drastically reduced to lower values (around

0.5% and 14.5%, respectively) even with the 100-V voltage

unbalance.

Three-dimensional SVM techniques can generate voltages

with γ component different to zero, making these type of SVM

methods very convenient for converters where there is some

zero-sequence current [26], [27]. To emphasize this advantage,

an experiment connecting the neutral point of the load to the

middle point of the dc link has been carried out. A reference

signal composed of a 50-Hz sinusoidal signal with modulation

index that is equal to 0.75, 25% of third harmonic, and 20% of

50-Hz sinusoidal zero-sequence voltage is carried out. Fig. 12

shows the representation of this reference voltage and the αβ

plane. It is clear that the used reference voltage is not restricted

to that plane. The 3D-SVM and 3D-FFSVM techniques are

used, imposing a 100-V voltage unbalance in the dc link. The

harmonic spectra of the obtained phase-to-phase voltages using

the 3D-SVM and the 3D-FFSVM are shown in Figs. 13 and 14,

respectively. Comparing the obtained data, it is clear that, using

the 3D-SVM method, distortion in low order harmonics due to

the dc-link voltage unbalance becomes significant. When the

3D-FFSVM technique is used, this distortion is avoided. The

obtained currents using the 3D-FFSVM technique are shown

in Fig. 15, and it can be observed that they follow the applied

reference voltage with third harmonic content and sinusoidal

zero sequence.

C. Transient DC Voltage Response

The dynamic response of the 3D-FFSVM has been tested

by forcing a variable voltage unbalance in the dc-link voltage

of the power converter. In this way, the voltage unbalance

is changed from 0 to 100V, following a triangular up–down

Fig. 12. Representation of the control region of the three-level converter inbalanced dc voltage condition, the αβ plane, and the used phase-to-neutralreference voltage composed of a 50-Hz sinusoidal with third harmonic content(25%) and 50-Hz sinusoidal zero-sequence voltage (20%). Modulation indexis 0.75.

Fig. 13. Harmonic spectrum of the output phase-to-phase voltage using the3D-SVM technique under a 100-V voltage unbalance generating a 50-Hzsinusoidal voltage with third harmonic content (25%) and 50-Hz sinusoidalzero-sequence voltage (20%). Modulation index is 0.75.

Fig. 14. Harmonic spectrum of the output phase-to-phase voltage using the3D-FFSVM technique under a 100-V voltage unbalance generating a 50-Hzsinusoidal voltage with third harmonic content (25%) and 50-Hz sinusoidalzero-sequence voltage (20%). Modulation index is 0.75.

Fig. 15. Output currents using the proposed 3D-FFSVM technique with100-V voltage unbalance generating a 50-Hz sinusoidal voltage with thirdharmonic content (25%) and zero voltage (20%). Modulation index is 0.75.(From top to bottom) (a) Phase currents. (b) Zero-sequence current.

Fig. 16. DC voltage unbalance changed from 0 to 100 V, following a triangu-lar up–down waveform.

waveform as shown in Fig. 16. The experiment is done by forc-

ing this voltage unbalance and by generating a 50-Hz sinusoidal

waveform with a modulation index that is equal to 0.9. Under

these conditions, as in Fig. 11, a modulation technique change

from conventional 3D-SVM to 3D-FFSVM is carried out, and

the results are shown in Fig. 17. It can be clearly seen that the

dynamic voltage unbalance has a direct relation with the second

order harmonic value and the THD of the output phase-to-phase

voltages. When the 3D-FFSVM is applied, both values are

again attenuated, obtaining the same values in Fig. 11, where

the dc voltage unbalance was fixed to 100 V. Therefore, it is

demonstrated that the 3D-FFSVM technique dynamic response

achieves the same good operation compared with the steady

state response. In this way, a direct consequence is that the

capacitance of the dc-link can be hugely reduced because,

Fig. 17. Applying the dynamic dc voltage unbalance shown in Fig. 16;experimental results using first the 3D-SVM and finally the 3D-FFSVM.(a) Fundamental component of the phase voltage. (b) Second order harmonicvalue of the phase voltage. (c) THD of the phase voltage.

by using the proposed 3D-FFSVM, the possible oscillations

and imbalances of the dc voltage values will not affect the

output voltages and currents of the multilevel power converter.

This advantage was shown in [16] and [29], where the 2-D

feedforward SVM technique was introduced. This fact makes

the use of the proposed feedforward technique really interesting

and reduces the cost and the volume of the power converter.

V. CONCLUSION

DC-link capacitor voltage unbalance in multilevel power

converters can create errors in the modulated voltages, which

can lead to distorted output current waveforms. The basic idea

of feedforward modulation is used to develop a 3D-FFSVM

technique that can be applied to power system applications

with balanced and unbalanced supplies. The proposed method

is a feedforward modulation technique that can be used in

all the multilevel converter topologies even when the gamma

component is not zero.

In this paper, the actual values of the dc voltages are taken

into account in the modulation process, and therefore, the

undesirable output waveform distortion is avoided even in

the worst unbalanced conditions. Experimental results using

a back-to-back three-level diode-clamped converter prototype

are presented. These results validate the proposed modulation

technique. The steady state under balanced or unbalanced volt-

age conditions has been addressed, generating voltage signals

even with low order harmonic content and zero-sequence com-

ponent. In addition, the dynamic response of the system has

been tested, achieving good results. When using the proposed

3D-FFSVM strategy, the dc capacitor voltage unbalance does

not affect the THD of the output waveform, and therefore, the

dc-link capacitance can be minimized, leading to an economical

cost and volume reduction of the power converter. Finally, it is

important to notice that the computational cost of the proposed

modulation is very low and the technique can be applied to

multilevel converters with any number of levels.

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Jose I. Leon (S’04–M’07) was born in Cádiz, Spain,in 1976. He received the B.S., M.S., and Ph.D.degrees in telecommunications engineering from theUniversity of Seville (US), Seville, Spain, in 1999,2001, and 2006 respectively.In 2002, he was with the Power Electronics

Group, US, working on R&D projects. Currently, heis an Associate Professor with the Department ofElectronic Engineering, US. His research interestsinclude electronic power systems, modeling, andmodulation and control of power-electronics con-

verters and industrial drives.

Sergio Vazquez (S’04) was born in Seville, Spain,in 1974. He received the B.S. and M.S. degrees inindustrial engineering from the University of Seville(US), Seville, in 2003 and 2006, respectively.In 2002, he was with the Power Electronics Group,

US, working on R&D projects. He is currently anAssistant Professor with the Department of Elec-tronic Engineering, US. His research interestsinclude electronic power systems, modeling, mod-ulation and control of power electronic convertersand industrial drives, and power quality in renewable

generation plants.

Ramon Portillo (S’06) was born in Seville, Spain,in 1974. He received the Industrial Engineer degreefrom the University of Seville (US), Seville, in 2002.He is currently working toward the Ph.D. degreein electrical engineering in the Power ElectronicsGroup, US.In 2001, he was with the Power Electronics Group,

US, working on R&D projects. Since 2002, he hasbeen an Associate Professor with the Department ofElectronic Engineering, US. His research interestsinclude electronic power systems applied to energy

conditioning and generation, power quality in renewable generation plants,applications of fuzzy systems in industry and wind farms, and modeling andcontrol of power-electronic converters and industrial drives.

Leopoldo G. Franquelo (M’84–SM’96–F’05) wasborn in Málaga, Spain. He received the M.Sc. andPh.D. degrees in electrical engineering from the Uni-versity of Seville (US), Seville, Spain, in 1977 and1980, respectively.In 1978, he joined the US as a Research Assistant,

where he became an Associate Professor in 1982and the Director of the Department of ElectronicEngineering from 1998 to 2005 and has been aProfessor since 1986. He is leading a large researchand teaching team in Spain. In the last five years, his

group activity can be summarized as follows: 40 publications in internationaljournals, 165 in international conferences, ten patents, advisor for ten Ph.D.dissertations, and 96 R&D projects. His current research interests are modula-tion techniques for multilevel inverters and their application to power electronicsystems for renewable energy systems.Dr. Franquelo was the Vice President of the IEEE Industrial Electronics

Society (IES) Spanish Chapter (in 2002–2003) and a Member at Large of theIES AdCom (in 2002–2003). He was the Vice President for Conferences ofthe IES (in 2004–2007), in which he has also been a Distinguished Lecturersince 2006. He has been an Associate Editor for the IEEE TRANSACTIONS ONINDUSTRIAL ELECTRONICS since 2007. Since January 2008, he has been thePresident Elect of the IEEE IES.

Juan M. Carrasco (M’97) was born in San Roque,Spain. He received the M.Eng. and Dr.Eng. de-grees in industrial engineering from the Universityof Seville (US), Seville, Spain, in 1989 and 1992,respectively.From 1990 to 1995, he was an Assistant Professor

with the Department of Electronic Engineering, US,where he is currently an Associate Professor. Hehas been working for several years in the powerelectronics field, where he has been involved in theindustrial application of the design and development

of power converters applied to renewable energy technologies. His currentresearch interests are distributed power generation and the integration ofrenewable energy sources.

Patrick W. Wheeler (M’00) received the B.Eng. de-gree (with honors) and the Ph.D. degree in electricalengineering for his work on matrix converters fromthe University of Bristol, Bristol, U.K., in 1990 and1994, respectively.In 1993, he was a Research Assistant with the

Department of Electrical and Electronic Engineer-ing, University of Nottingham, Nottingham, U.K. In1996, he became a Lecturer in the Power Electronics,Machines, and Control Group, School of Electricaland Electronic Engineering, University of Notting-

ham. Since January 2008, he has been a Full Professor in the same researchgroup. He has published over 200 papers in leading international conferenceproceedings and journals. His research interests are power converter topologiesand their applications.

Alan J. Watson (S’03) received the Masters de-gree in electrical and electronic engineering fromthe University of Nottingham, Nottingham, U.K., in2004, where he is currently working toward the Ph.D.degree in the School of Electrical and ElectronicEngineering.His research interests include multilevel convert-

ers, advanced modulation schemes, and power con-verter control.