Third Harmonic

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013

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Voltage Balancing control in 3-Level Neutral-Point
Clamped Inverters Using Triangular Carrier PWM
Modulation for FACTS Applications
Javier Chivite-Zabalza, Pedro Izurza-Moreno, Danel Madariaga, Gorka Calvo,
and Miguel Angel Rodr´ıguez, Member, IEEE

Abstract—In this paper, a novel technique to balance the voltage
of the two split dc capacitors of a 3-Level neutral-point-clamped
inverter using triangular carrier pulse width modulation is presented. This technique, suitable for reactive power compensation
and for inverters operating with a relatively low switching frequency, consists in adding a square wave at six times the output
frequency. Subsequently, this paper presents a comparison with
two already known strategies in which a sinusoidal waveform at
two and six times the output frequency are injected. The current
contribution to the midpoint of the dc bus is then analyzed for
different modulation indexes and operating conditions. Based on
this analysis, a small-signal averaged model, suitable for control
design purposes is presented. Finally, simulation and experimental
results on a 690-Vac, 120-kVA test bench that validate the theory
are shown.
Index Terms—FACTS, insulated gate bipolar transistors
(IGBT), multilevel inverters, SHE, static synchronous series compensator (SSSC), STATCOM, VSC.

I. INTRODUCTION
HE development of FACTS devices, which aim for the
transmission network to operate close to its thermal limit
by providing a fast dynamic control [1]–[3], has been boosted
in recent times by the emergence of new semiconductor devices such as high-voltage insulated-gate bipolar transistors (IGBTs) [4]. The well-known neutral point clamped (NPC) 3-level
(3-L) inverter has been widely used in this type of applications [1], [2], and [4]. These inverters are often based on wellproven solutions, also used in other high-integrity demanding
applications such as steel mill processes, ship propulsion, and
ship dredger pumps [6]. An example of such equipment, are the
Ingedrive MV100 range of Ingeteam, shown in Fig. 1. These
are 3-L NPC IGBT-based inverters, where each has a power in
the range of 3.5 MVA.
One of the main control challenges in 3-L NPC inverters is to keep the voltage across the two dc-split capacitors

T

Manuscript received July 10, 2012; revised November 9, 2012; accepted
December 26, 2012. Date of current version March 15, 2013. Recommended
for publication by Associate Editor F. Wang.
The authors are with Ingeteam Power Technology S.A., Parque tecnol´ogico
de Bizkaia, 48170 Zamudio, Spain (e-mail: [email protected];
[email protected]; [email protected]; gorka.calvo@
ingeteam.com; [email protected]).
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/TPEL.2012.2237415

Fig. 1. Main power stack characteristics for the MV100-INGECON WIND
converter system.

balanced [5], [7]–[29]. The use of space vector pulse width modulation (SVPWM) techniques, widespread these days, presents
limitations to control the midpoint voltage at very low power
factor operation and high modulation indexes, as is often the
case of FACTS devices [7]–[9]. Moreover, there are applications where the use of traditional triangle carrier modulation is
preferred. For instance when the output waveforms are nonsinusoidal, as is the case of active filters, or when the topology
requires additional balancing loops, for instance antisaturation
control of magnetic devices connected at the inverter output,
i.e., coupling transformers.
The addition of a dc-offset to the reference waveform is a
recurred technique in applications with a power factor close to
unity, but that has no effect on pure reactive compensation [26].
The next section describes three voltage-balancing techniques
suitable for this purpose, two of them have been found in the
literature and the third one is a contribution of this paper.
II. VOLTAGE BALANCING TECHNIQUES FOR REACTIVE
COMPENSATION
This section begins by describing the compensation techniques based on sinusoidal second and sixth harmonic injection,
followed by the newly proposed compensation method. The explanation of the voltage balancing technique is based on the
3-L NPC inverter seen in Fig. 2. The explanation assumes that

0885-8993/$31.00 © 2012 IEEE

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013

The effect of this technique in the modulation reference waveforms is seen in Fig. 3(a), where the modulation index for the
reference and injected waveform is 0.6 and 0.1, respectively.
The magnitude of the injected waveform in these figures has
been exaggerated for clarity purposes.
One of the disadvantages of this method is that second harmonic voltage components are heavily penalized in most power
quality legislations. To overcome this problem, the injection of
a sinusoidal waveform at six times the supply frequency can be
used instead, as explained in the next section.
Fig. 2. Three-level NPC inverter configuration with ideal sinusoidal current
sources.

the inverter is supplied by a three phase set of pure sinusoidal
currents (1). However, the experimental results show later on
in this paper that the analysis and proposed techniques are also
valid for an unbalanced supply system.
Also, the output voltage of the inverter is produced by comparing the reference waveform showed in (2), with two levelshifted triangular carrier waveforms to obtain a 3-L output [5].
In addition, the injection of a third harmonic waveform, as seen
in (3), is a well recurred method to increase the magnitude of
the output voltage, about 15% higher where K3 takes a value of
1/6 [5]
iA = Iˆ sin (ωt + φ)


iB = Iˆ sin ωt − 2π/3 + φ


iC = Iˆ sin ωt + 2π/3 + φ

(1)

vA = E/2 m1 sin (ωt)


vB = E/2 m11 sin ωt − 2π/3


vC = E/2 m1 sin ωt + 2π/3

(2)

v3 = E/2 m1 K3 sin (3ωt)

(3)

B. Sinusoidal Sixth Harmonic Injection
It is well known that harmonics multiple of three times the
output frequency are zero sequence, that is, they are in phase
across all three phases and are, therefore, inherently cancelled
out at the output of a three-phase system [5]. Moreover, even
harmonics contribute with a nonzero average value of current
to the midpoint in the same way as explained in the previous
section. This method consists in injecting a sinusoidal waveform
at six times the supply frequency as seen in (5) [29]. The effect
of this method on the output waveform is showed in Fig. 3(b)
for a m1 index of 0.6 and a modulation index of the injected
waveform m6 of 0.1
vA −6 sin = vA + E/2m6−sin sin (6ωt + φ6−sin )
vB −6 sin = vB + E/2m6−sin sin (6ωt + φ6−sin )
vC −6 sin = vC + E/2m6−sin sin (6ωt + φ6−sin )

where m1 is the modulation index and K3 is the amount of third
harmonic injection

(5)

where m6 is the modulation index for the compensation waveform.
One of the potential disadvantages of using this method in
large power inverter systems is the relatively high switching
frequency that is required to materialize the sixth harmonic component. This has a frequency of 300 and 360 Hz, respectively
for supply frequencies of 50 or 60 Hz. Large power inverters
normally operate at relatively low switching frequencies, below
1 kHz. The next section proposes a voltage balancing method
that aims to overcome this problem.

A. Sinusoidal Second Harmonic Injection

C. Squared Sixth Harmonic Injection

This method consists in injecting a negative sequence second
harmonic voltage waveform, superimposed to the fundamental
output waveform defined in (4). Since the second harmonic
components are negative sequence, a negative sequence of those
will be in phase with the positive fundamental sequence [27],
[28]

This technique can be seen as an evolution of that presented in
the previous section and consists in injecting a squared wave at
six times the supply frequency. Since the injection of a squared
wave can be regarded as the addition of a dc offset of a given
magnitude to the reference waveform that changes sign every
30 electrical degrees, its practical implementation is likely to be
simpler, especially in inverters operating with a low switching
frequency. This injected waveform can be expressed as

vA −2 sin = vA + E/2 m2 sin (2ωt + φ2 )

 

vB −2 sin = vB + E/2 m2 sin 2 ωt + 2π/3 + φ2
 


vC −2 sin = vC + E/2 m2 sin 2 ωt − 2π/3 + φ2

vA −6sq = vA + v6−sq
(4)

where m2 is the modulation index for the compensation waveform.

vB −6sq = vB + v6−sq
vC −6sq = vA + v6−sq
v6−sq = E/2m6−sq sign [sin (6ωt + φ6−sq )] .

(6)

CHIVITE-ZABALZA et al.: VOLTAGE BALANCING CONTROL IN 3-LEVEL NEUTRAL-POINT CLAMPED INVERTERS

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Fig. 3. Line-to-midpoint output voltage for: (a) sinusoidal second harmonic injection; (b) sinusoidal sixth harmonic injection; (c) squared sixth harmonic
injection. Continuous line: v A ; dashed line: v B ; dotted line: v C , normalized with respect to E/2.

The effect of this technique on the inverter output waveform is
illustrated in Fig. 3(c) for a m1 index of 0.6 and a modulation
index of the injected waveform m6 of 0.1. The next section evaluates the ability of the three compensation approaches described
so far to control de midpoint voltage.
III. ANALYSIS OF THE AVERAGE CURRENT CONTRIBUTION
TO THE MIDPOINT AND AN AVERAGE MODE
This section begins by analyzing the current contribution to
the midpoint for the three voltage balancing techniques explained in the previous section. Subsequently, a sensitivity analysis for different modulation indexes and phase-shift angles is
performed. Next, a small signal averaged model of the midpoint
balance control loop, suitable for control design purposes, is
proposed.

Fig. 4.

Second harmonic compensation for m 1 = 0.6 and m 2 = 0.3.

Fig. 5.
= 0.2.

Sixth harmonic sinusoidal compensation for m 1 = 0.6 and m 6 − sin

A. Analysis of the Current Contribution to the Midpoint
As already explained in [27], [28], the instantaneous duty
ratio d of the portion of the phase input current that flows to the
midpoint is defined as

⎫
⎧
−E/ ≤ vK < 0 ⎬
⎨ 1 + 2vK/E
2
dK =
(7)


⎩
0 ≤ vK ≤ −E/ ⎭
1 − 2vK/E
2
where k = A, B, or C
Consequently, the instantaneous contribution of any phase to
the midpoint current is obtained by multiplying the duty ratio
dK by the phase current (1)
iM −K = dK iK .

(8)

Since the voltage drift of the midpoint is a comparatively slower
varying process than the instantaneous voltage ripple present in
the dc bus, the effectiveness of the midpoint voltage control can
be evaluated by considering the mean dc midpoint current over
an entire line period T . This is calculated in a straightforward
manner as
T
¯iM −K = 1
iM −K dt
(9)
T 0
where the total average current contribution to the midpoint is
¯iM = ¯iM −A + ¯iM −B + ¯iM −C

1 T
=
(iM −A + iM −B + iM −C ) dt.
T 0

(10)

The effect that the injection of a negative sequence second harmonic, sinusoidal sixth harmonic and squared sixth harmonic,
have on the midpoint current can be visually seen by looking at
the waveforms in Figs. 4–6 respectively, corresponding to phase
A.
These waveforms have been computed using the MATLAB
software using (1)–(10) for a modulation index of the fundamental waveform of m1 = 0.6. The modulation index for the
compensation waveforms where m2 = 0.3, m6−sin = 0.2, and
m6−sq = 0.2 and the phase angle of the injection waveforms
was set to zero, that is, Φ2 = 0, Φ6−sin = 0, and Φ6−sq = 0. The
first two waveforms on top of every figure show the fundamental and the compensating waveforms in continuous and dotted
lines. The second waveform depicts the instantaneous duty ratio
d of the current to the midpoint. The third waveform shows the
current iA flowing into phase A, 90◦ out of phase with respect
to the output voltage. The fourth waveform shows the current

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013

TABLE I
IM /m 2 AND IM /m 6 GAINS IN THE FIRST LINEAR OPERATING REGION

Fig. 6. Sixth harmonic squared injection at six times the supply frequency for
m 1 = 0.6 and m 6 −sq = 0.2.

contribution of phase A to the midpoint, calculated by direct
multiplication of the second and third waveforms, as seen in
(8). Finally, the last set of waveforms show the instantaneous
overall current contribution to the midpoint made by all three
phases, using a continuous trace and its average value over the
entire period, using a dotted trace. The voltages and currents
are normalized with respect to E/2 and the phase peak current,
respectively. It can be seen by visual inspection how the average
current has a negative value, more significant in the case of the
second harmonic injection Fig. 4(a). The usual third harmonic
current ripple, which creates a third harmonic voltage ripple in
the midpoint is also present.
The next section carries out a sensitivity analysis of the key
parameters involved in the midpoint balance mechanism.
B. Sensitivity Analysis of the Current Contribution
to the Midpoint
A phase-shift angle sensitivity analysis using numerical calculation for second harmonic injection was carried out first, by
varying the phase-shift angle Φ2 between the injected and reference waveforms and the phase angle Φ between the voltage
and current. The results for modulation indexes of the reference
and injected waveform of 1 and 0.4, respectively, are showed in
Fig. 7(a) and (b). The continuous and dashed lines in Fig. 7(a)
correspond to a pure capacitive and inductive load, respectively.
Also, the reference and injection waveforms in Fig. 7(b) are in
phase, that is Φ2 = 0. It can be seen that the largest contribution
of current to the midpoint takes place for pure reactive loads
where the reference and compensating waveforms are in phase.
Also, there is no current contribution for active loads. Similar
results are obtained for sixth harmonic injection and for different modulation indexes. For that reason, it is proposed in this
paper to choose an injection waveform that is in-phase with the
reference waveform, that is, with Φ2 = 0, Φ6−sin = 0, and Φ6−sq
= 0, as it has the greater compensating effect on the midpoint.
The results of Fig. 7(b) suggest that the voltage balancing
control can also be achieved by phase shifting the injection
waveform rather than by adjusting its amplitude as proposed
in this paper. However, this method has not been adopted by
the authors as it would entitle using an injection waveform of

a constant, and relatively high magnitude at all times, even
if the midpoint capacitors are balanced. This would cause a
degradation of the steady-state harmonic content in the case of
second harmonic injection and a permanent limitation on the
maximum modulation index in all cases.
An interesting observation that can be drawn by looking at
the top waveforms in Figs. 4–6 is that, since the injection waveforms of the three proposed strategies are even harmonics of the
fundamental, they have zero crossings at π/2 and 3π/2, where
the magnitude of the fundamental has its peaks. It follows that
their contribution to the peak value of the resultant reference
waveform will be the same, whether the injection waveform is
in phase, or 180◦ out of phase with respect to the fundamental,
or whether its amplitude is positive or negative.
The variation of the modulation index m2 , m6−sin , and m6−sq
of the injection waveform, defined in (4)–(6), has also been studied and the results are shown in Fig. 8. The plots on the top and
bottom rows have been obtained without and with third harmonic injection. Also, the first, second, and third columns in the
figure show the results for second harmonic, sinusoidal sixth
harmonic and squared sixth harmonic injection, respectively.
The continuous and dotted lines correspond to capacitive compensation where the modulation indexes for the reference waveform were 0.8 and 0.5, respectively. Also, the dashed–dotted
and dashed lines correspond to inductive compensation where
the modulation indexes for the reference waveform are 0.8 and
0.5, respectively. Moreover, the thicker continuous traces correspond to the range of balancing waveform available without
entering in overmodulation. This limitation will be explained in
greater detail in Section III-C.
Looking at the figure, it can be seen how the second harmonic compensation introduces an average value of current to
the midpoint IM significantly higher than that provided by its
sixth harmonic injection counterparts. It can also be appreciated, how all three compensation techniques present a linear
region for low amplitudes of the injected waveform, where the
current IM is proportional to the modulation index of the injected signal. In the case of the second harmonic compensation,
the slope of the curves takes a lower value at a given m2 value.
However, in the case of sixth harmonic injection, it reaches a
peak and then gradually falls down until it collapses to zero or
even becomes negative. The values of the first linear portion of
the curve, namely KM , defined in (11) for all three modulation techniques, with and without third harmonic injection, are
shown in Table I. This value is used in the next section to obtain
an averaged model of the midpoint voltage balance control loop
KM =

IM
IM
IM
or KM =
or KM =
.
ˆ 2
ˆ 6−sin
ˆ 6−sq
Im
Im
Im

(11)

CHIVITE-ZABALZA et al.: VOLTAGE BALANCING CONTROL IN 3-LEVEL NEUTRAL-POINT CLAMPED INVERTERS

4477

Fig. 7. Current to midpoint im against phase-shift angle Φ 2 for second harmonic sinusoidal injection. (a) Current contribution IM (p.u.) against Φ 2 for a pure
capacitive (continuous trade) and inductive (dashed trace) load. (b) Current contribution IM (p.u.) against Φ.

Fig. 8. Variation of the average current to the midpoint against modulation index of the injected waveform. (a) Second harmonic compensation. (b) Second
harmonic compensation using third harmonic injection. (c) Sinusoidal sixth harmonic compensation. (d) Sinusoidal sixth harmonic compensation using third
harmonic injection. (e) Squared sixth harmonic compensation. (f) Squared sixth harmonic compensation using third harmonic injection.

Fig. 9. Variation of the p.u. average current to the mid-point against the modulation index of the injected waveform, using third harmonic injection, for
an unbalanced set of currents represented by the addition of a 0.5 p.u. inverse sequence. (a) Second harmonic compensation. (b) Sinusoidal sixth harmonic
compensation. (c) Squared sixth harmonic compensation.

By looking at Fig. 8 and Table I, it can be concluded that the
injection of a second harmonic waveform is more suitable for
control purposes since not only it contributes with a comparatively higher current to the midpoint, but also its linear operating
region extends for a wider range of values of the compensating waveform. Moreover, it does not become negative nor it

collapses to zero. Also, the squared harmonic injection technique provides a slightly higher midpoint current IM than the
sinusoidal one.
The reason for this behavior is the linearity property
of the integral appearing in (9), resembling the calculation
of a Fourier series coefficient, which can be expressed as

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013

follows:
¯iM −A =

1
2·π

1
=
2·π


0



0

2·π

As long as this condition is met, the previous assumption will be
valid, and the values of IM will be proportional to m2 or m6−sin .
In our particular case, when the sinusoidal second and/or sixth
harmonic are added to the fundamental, this equation becomes

iM −A (ωt) · dωt

2·π

2 · vA
1−
E



(12)
where the time integral has been multiplied by the angular speed
ω, to get an angle integral. This can be simplified to
ˆ 2·π
¯iM −A = − 1 2 · I
vA · sign(vA ) · cos(ωt) · dωt
2·π E 0
(13)
finally resulting in
ˆ π
¯iM −A = − 2 · I
vA · cos(ωt) · dωt
(14)
E·π 0
where it has been assumed that vA is positive for angles ωt in
the range 0 < ωt ≤ π, and negative for angles π < ωt ≤ 2 π.
This assumption will be justified later, and its range of validity
calculated accordingly.
Note that, due to the waveform symmetry, the upper integration limit has been changed from 2 π to π, and the value of
the resulting integral has been doubled accordingly. Thus, the
sign(vA ) function can be suppressed. Therefore, the expression
in (14) becomes linear in the amplitudes of the different harmonics which may form vA . This is expected as odd voltage
harmonics forming vA do not contribute to the integral of (14),
since only even voltage harmonics are expected to contribute to
the balance of the dc voltage of a 3-L-NPC inverter.
For phase symmetry reasons (phase voltages and currents
are supposed to be direct-sequence or zero-sequence, but not
negative-sequence), the average current entering the midpoint
of the dc-bus provided by phases B and C must be the same
as that provided by phase A. Consequently, the values obtained
previously must by multiplied by three to get the total contributions.
This behavior will persist as long as the voltage harmonics
added to the fundamental signal do not force a sign change in the
total voltage waveform vA at any angle, as previously assumed.
Since all harmonics are set up in sine phase, this is, all cross the
zero level, at the same angle and with positive slope, the limit
for that condition is met when the derivative gets close to zero
at the opposite angle (ωt = π), when the waveform crosses zero
from positive to negative values).
Therefore, an algebraic expression for the validity of the linearity condition is obtained as follows:

mh · sin(h · ωt)
V =
h

dV
=
dωt



h · mh · cos(h · ωt)

h




dV 
=
h
·
m
−
h · mh < 0. (15)
h
dωt ω t=P i
h·even

2 · m2 + 6 · m6 < 1 · m1 .

· sign(vA ) · Iˆ · cos(ωt) · dωt

h·o dd

(16)

So, in this linear region, valid for small enough amplitudes of
the injected harmonics, the gains obtained with this method are
therefore
E
E
· m1 · sin(ωt) + · m2 · sin(2 · ωt)
2
2

6 · Iˆ π
= 3 · ¯iM −A = −
vA · cos(ωt) · dωt
E·π 0

6 · Iˆ π E
=−
· m2 · sin(2 · ωt) · cos(ωt) · dωt
E·π 0 2

3 · Iˆ · m2 π
=−
sin(2 · ωt) · cos(ωt) · dωt
π
0
¯iM
4
4
(17)
= · Iˆ · m2 → KM =
=
ˆ
π
π
I · m2

vA =
¯iM

¯iM

for the second harmonic compensation, and
E
E
· m1 · sin(ωt) + · m6 · sin(6 · ωt)
2
2
π
ˆ
6·I
= 3 · ¯iM −A = −
vA · cos(ωt) · dωt
E·π 0

6 · Iˆ π E
=−
· m6 · sin(6 · ωt) · cos(ωt) · dωt
E·π 0 2

3 · Iˆ · m6 π
=−
·sin(6 · ωt) · cos(ωt) · dωt
π
0
¯iM
36
36
· Iˆ · m6 → KM =
(18)
=
=
35 · π
35 · π
Iˆ · m6

vA =
¯iM

¯iM

for the sinusoidal sixth harmonic compensation. This confirms
the values shown in Table I, directly measured from the slopes
in Fig. 8(a)–(d).
The case of the squared sixth harmonic compensation is a
little more complex to calculate, since this waveform is composed of infinite harmonics with weights inversely proportional
to their respective harmonic order. That property makes that,
even when very small amplitudes of the squared sixth harmonic
are added, the voltage waveform changes sign in the vicinity
of ωt = π. Therefore, the integral of (14) no longer holds, and
that of (13) applies instead, which is nonlinear due to the sign
function appearing into its integral. Therefore, the behavior of
that strategy is not strictly linear from the very beginning. However, a tangent slope can be calculated for small amplitudes of
the squared sixth harmonic compensating waveform, as already
seen in Fig. 8(e) and (f) and in Table I.
A disadvantage of injecting a second harmonic voltage is
that, since it is not a zero sequence harmonic, it appears at
the inverter output. It is well known that second harmonic
currents (created by second harmonic voltages) are restricted

CHIVITE-ZABALZA et al.: VOLTAGE BALANCING CONTROL IN 3-LEVEL NEUTRAL-POINT CLAMPED INVERTERS

in most power quality standards. However, the magnitude of the
compensating waveform is expected to be small, and therefore
negligible at the output. Moreover, it is usually injected during
the short transient periods required to balance the midpoint of
the dc-link, out of the steady state operating regime evaluated
by most power quality standards.
The operation of the three techniques under discussion has
also been tested in the presence of unbalance currents by adding
a negative sequence component to the currents seen in (1)
iA = Iˆ sin (ωt + φ) + IˆINV sin (ωt + φINV )




iB = Iˆ sin ωt − 2π/3 + φ + IˆINV sin ωt + 2π/3 + φINV




iC = Iˆ sin ωt + 2π/3 + φ + IˆINV sin ωt − 2π/3 + φINV .
(19)
Using those, the variation of the modulation index m2 , m6−sin ,
and m6−sq of the injection waveform, defined in (4)–(6) and
represented in Fig. 9 has been repeated. The plots on the first,
second, and third columns in the figure show the results for
second harmonic, sinusoidal sixth harmonic and squared sixth
harmonic injection, respectively, where third harmonic injection
has been used.
These have been obtained for inductive compensation and
using modulation indexes of 1 and 0.5 for the reference and
injection waveforms, respectively. The dashed line corresponds
to a balanced set of line currents, that is, where IˆINV = 0. Also,
the dotted and dashed–dotted lines correspond to a ratio of inverse to direct sequence current ratio IˆINV /Iˆ of 0.5. In the
former, the negative sequence currents are in phase with the
direct sequence currents whereas in the latter the negative sequence currents are 90◦ lagging. These results are mirrored for
capacitive reactive compensation. It is clear by looking in the
figures how the presence of a negative sequence has a negligible
effect on the balancing current that flows to the midpoint of the
dc link, particularly, in the linear region of operation. This is
also justified by looking at the symmetry features of (13), which
is zero for the product of odd voltage and current harmonics.
The next Section looks at the maximum waveform that can
be injected for the proposed techniques
C. Limit of the Injection Waveforms
The sum of the reference and compensating waveform should
not exceed the dc-bus voltage E/2. Consequently the compensating waveform must comply with the following expression at
all times
vCOM P ≤ E/2 − vA ,B ,C

(20)

where vCOM P can be either vA ,B ,C −2sin , vA ,B ,C −6sin , or
vA ,B ,C −6sq as defined in (4)–(6), respectively. Also, vA ,B ,C
is expressed in (2) and (3) if third harmonic injection is used.
The maximum modulation index for the injection waveforms
has been calculated by solving (20) for the angle (ωt)M AX
that achieves the peak value in expressions (4)–(6). This angle has been calculated by solving the equation that equates the

4479

angle derivative expression to zero, following basic mathematic
principles.
Using those, the maximum modulation index for the compensating waveform has been calculated for several values of the
modulation index
√ m1 of the reference waveform, close to the
maximum 2/ 3, where third harmonic injection has been used
and the results are shown in Fig. 10. The second harmonic, sinusoidal sixth harmonic, squared sixth harmonic and SVPWM are
represented using continuous, dashed, dashed–dotted and dotted traces, respectively. Also, in that figure, the resultant average
current that flows to the midpoint IM , normalized with respect to
the peak value of the line current is also shown. These results are
compared with the current that is obtained using the well-known
SVPWM technique for a 3-L-NPC inverter [7]–[10].
In that figure, it can be seen how the maximum compensating
waveform that can be injected is gradually reduced as the modulation index for the reference waveform
√ is increased, until it
reaches its maximum value of m1 = 2/ 3, where no compensating waveform can be introduced. On the contrary, SVPWM
can produce a mid-point balancing current even at the maximum
modulation index. It is also seen in Fig. 10 how second harmonic
injection has a higher balancing effect, whereas sinusoidal and
squared harmonic injection can produce compensating currents
with similar values. This is explained, despite squared sixth
harmonic injection having a higher balancing action than sinusoidal sixth harmonic injection, because the latter allows having
an injection waveform of a magnitude higher than that of the
former.
It is important to mention that the contribution of the proposed techniques to the midpoint balancing current can be easily increased at certain times at the expenses of sacrificing the
magnitude of the reference waveform. That is, the value of
m2 , m6−sin , or m6−sq , depending on whichever technique is
used, can be temporarily increased by subsequently reducing
the value of m1 , to bring the midpoint back into balance, as it
will be seen in the experimental results seen in Section IV. This
feature is not straightforward to implement in SVPWM.
An example of the maximum current contribution to the midpoint for a modulation index of m1 = 0.9 and second harmonic,
sinusoidal sixth harmonic, squared sixth harmonic compensation and SVPWM is shown in Fig. 11. In that figure, the instantaneous current is depicted using a continuous trace and its
average value over a supply line cycle is seen with a dotted trace.
The next Section obtains a small-signal averaged model that
can be used for control design purposes.
D. Small Signal Averaged Model of the Split Capacitors
Midpoint Balance
The averaged model of the split capacitors midpoint balance
loop is obtained by averaging the output capacitor currents and
voltages over a supply cycle, that is, by neglecting the instantaneous ripple. The analysis also assumes that the dc bus voltage
vo is stable and has a constant average value of E. This assumption is justified, even at the low switching frequencies typical
of large converter systems, by the criteria chosen to design
the dc-link capacitors. This, with the purpose of avoiding any

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Fig. 10. (a) Maximum compensating modulation index and (b) produced compensating IM current (normalized against the peak value of the line current),
for the modulation index of the reference waveform. Continuous line: second harmonic compensation, dashed line: sinusoidal sixth harmonic compensation,
dashed-dotted line: squared sixth harmonic compensation, dotted line: SVPWM.

Fig. 11. Maximum instantaneous (continuous trace) and average (dotted trace) current contribution to the mid-point of the dc-capacitors for m 1 = 0.9.
(a) Second harmonic compensation using third harmonic injection for m 2 = 0.237. (b) Sinusoidal sixth harmonic compensation using third harmonic injection
for m 6 −sin = 0.236. (c) Squared sixth harmonic compensation using third harmonic injection for m 6 −sq = 0.221. (d) SVPWM.

overvoltage across the power devices, aims to reduce the voltage switching ripple to a very low value by including enough
capacitance. For that reason, the voltage unbalance within one
switching cycle for the different modulation methods has not
been considered.
By looking at the diagram of Fig. 1, it can be easily deducted
that voltage balance of the midpoint capacitors is
v¯C 2 − v¯C 1 =

1
C


0

T

1
(¯iC 2 − ¯iC 1 ) dt =
C



T

¯iM dt

(21)

0

Expressing this equation in the S domain, and relating the average midpoint current IM to the peak phase current and the
modulation index of the compensating waveform by means of
the KM ratio (11) seen in Table I, the following transfer function
is obtained:
Iˆ sin (φ) KM
VC 2 − VC 1
=
m2 (or m6 )
CS

Fig. 12. Control loop of the split capacitors mid-point balance for reactive
power compensation.

(22)

Based on this transfer function, a control loop structure valid
for reactive power compensation is proposed in Fig. 12, where a
usual PI controller is employed in combination with a low-pass,
first order filter, to suppress the voltage ripple components (23).
The control output is normalized by dividing it by the peak
reactive component of the phase current, a straightforward operation in usual digital control systems. An optional offset has
been added to the setpoint to allow introducing step changes to
evaluate the performance of the control loop. The cross-over frequency of the first order filter is selected to mitigate the third harmonic ripple components present across the capacitors (150 Hz

CHIVITE-ZABALZA et al.: VOLTAGE BALANCING CONTROL IN 3-LEVEL NEUTRAL-POINT CLAMPED INVERTERS

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Fig. 13. Step response of the split capacitors voltage balance loop. (a) positive step response and (b) positive and negative step response. The thick continuous
line: averaged response; the thin continuous line: instantaneous response; the tick dotted line: setpoint.

for a 50 Hz system), for example at 15 Hz or 94.25 rad/s. Subsequently, the PI loop can be tuned using conventional techniques,
for instance, aiming to achieve a reasonable Phase Margin [30].
Although, as explained in the next Section IV, the dynamic response achieved using this control strategy is sufficient for most
large power electronics applications, other suitable strategies
compatible with the proposed compensating techniques could
also be devised
Controller T F = 

1

 KP
S/ + 1
ωP

(S + iTn )
S

(23)

The next section shows simulation and experimental results that
proof the operation of this modeling and control approach.
IV. SIMULATION AND EXPERIMENTAL EVALUATION
The operation of the control strategy proposed in Fig. 12 has
been validated, first in simulation and then experimentally, on
a 690 Vac, 100 A, 120 kVA, 3-L NPC inverter, based on IGBT
devices switching at 600 Hz. Each of the split capacitors had a
capacitance value of 6.6 mF and the positive and negative dclink terminals of the inverter were supplied by a 950 V external
rectifier unit, with the midpoint left floating. The phase-shift
angle of the injection waveforms with respect to the reference
waveform has been chose to be Φ2 = 0, Φ6−sin = 0, Φ6−sq = 0
for a greater compensation effect, as explained in Section III-B.
The simulation results for a step response when second harmonic compensation is selected, are shown in Fig. 13. The
inverter was supplied by a set of ideal current sources of 90-A
RMS and the RMS phase-to-neutral output voltage was 310 V.
To validate the averaged model described in the previous section, the control loop as shown in Fig. 12 has been made run
in parallel with the complete inverter simulation. In that figure, the left and right-hand plots show the response to a 0 to
50 V and 0 to 50 to 0 V offset step respectively. The continuous
thin and thick lines represent the instantaneous and averaged
vC 2 − vC 1 waveforms, which show close correspondence. The
step setpoint is depicted in the figure using a dotted thin trace.
The control parameters where chosen as ω P = 94.24 rads/s;
KP = 0.0863 and iTN = 2.93. The same results were obtained
for sixth harmonic squared and sinusoidal compensation, but
are not shown in this paper for the shake of briefness. The
control loop achieves an open-loop bandwidth of 2.64 Hz, a

settling time of 0.784 s (for a 1% error criteria) and an overshoot of 12%. Therefore, with the proposed adjustment, it will
take about 40 line cycles to complete a step response. Although
this may seem a long time, in practice, due to the amount of
capacitance present in the dc link, as explained in this Section III-D, the dc-link midpoint voltage balance changes relatively slowly. This is confirmed in the experimental section and
in several industrial applications where this strategy has been
implemented.
In the experimental validation, the inverter output terminals
were connected to an 18 mH inductive load and it was operated in current control mode. To verify the ability to control
the midpoint, an offset step of 50 V was introduced when the
output current was controlled to be 70-A RMS and the results
are shown in Fig. 14(a)–(c), for second harmonic, sinusoidal
sixth harmonic, and squared sixth harmonic compensation, respectively.
The first two waveforms show the capacitor voltages vC 1 and
vC 2 , the third waveform the inverter output voltage with respect
to the midpoint of the dc-link capacitors vAG and the bottom
waveform shows the output current iA . The proportional gain
KP for second and sixth harmonic was chosen to be 0.3 and
0.85, respectively, whereas the integral term iTn and the filter
crossover frequency were 2.93 and 94.24 rad/s, respectively, in
all three cases (23). These waveforms show how the midpoint
voltage balance can be successfully controlled for all the proposed strategies. A similar response was obtained for a falling
50 V to 0 V step response. By looking at the results, it is very hard
to appreciate substantial differences between response obtained
using sinusoidal and squared sixth harmonic injection. Therefore, the main difference between these two techniques lies in the
easier implementation offered by the squared harmonic injection at low switching frequencies. In the waveforms of Fig. 14,
a dip in the output current during the midpoint transient can be
observed, which highlights a limitation of the proposed compensation techniques. This is due to the output voltage required
to control the output current, already operating close to its limit
in this experiment, diminishing to accommodate the midpoint
compensating waveforms. This effect is significantly greater in
the case of the sixth harmonic injection strategies as their compensation power is about three or four times smaller than that of
the second harmonic strategy. This limitation is only likely to
appear when operating at very high modulation indexes and is

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013

Fig. 14. Instantaneous response for a mid-point imbalance step of 0 to 50 V, 70 A RMS. (a) sinusoidal second harmonic injection; (b) sinusoidal sixth harmonic
injection; (c) squared sixth harmonic injection. First waveform: v c 1 (100 V/div), second waveform: v c 2 (100 V/div), third waveform v A G (500 V/div), fourth
waveform: iA (100 A/div). Time scaling: 40 ms/div.

Fig. 15. Instantaneous response for a mid-point imbalance step of 0– 50 V, 35 A RMS, unbalanced load: (a) sinusoidal second harmonic injection; (b) sinusoidal
sixth harmonic injection; (c) squared sixth harmonic injection. First waveform: v c 1 (100 V/div), second waveform: v c 2 (100 V/div, third waveform iA (50 V/div),
fourth waveform: iB (50 A/div). Time scaling: 40 ms/div.

Fig. 16. Instantaneous response for a step change on the phase current, second harmonic compensation K P = 0.3, iTn = 2.93, ωP = 94.24 rad/s. (a) 10 to
75 A RMS, K P = 0 (controller disabled); (b) 10 to 75 A RMS (c) 75 to 10 A RMS. First waveform: v c 1 (100 V/div), second waveform: v c 2 (100 V/div), third
waveform: v A G (500 V/div), fourth waveform: iA (100 A/div); Time scaling: (a): 40 ms/div. (b) and (c): 100 ms/div.

related to the nature of the transient, the control loop adjustment
and the dc-bus capacitance.
In addition, to verify the level of second harmonic introduced
by the second harmonic injection strategy, the harmonic content of the voltage and current waveforms shown in Fig. 14(a)
has been measured using a high-bandwidth Yokogawa PM3000
power quality analyzer. These measurements, carried out at
steady-state operating conditions, show that the second harmonic voltage and current content, expressed as a percentage of
the fundamental is 1% and 0.4% respectively. These are residual values, not necessarily related to the midpoint balancing
strategy but to the asymmetries introduced by the low switching
frequency that is used [1].

The next test consisted in evaluating the performance midpoint balance control loop for a sudden step change of the output
current. The waveforms seen in Fig. 16(a) show that, for a rising step from 10 to 75 A, under no midpoint control (KP =
0), there was a drift of the midpoint voltage balance, eventually causing a inverter trip. The order of the waveforms is the
same as that explained for Fig. 14. However, when the midpoint voltage balance control is activated, in this case for second harmonic compensation, the midpoint remained balanced
for both a rising current step (10 to 75 A), Fig. 16(b), and a
falling step (75 to 10 A), Fig. 16(c). Similar results were obtained when sinusoidal or squared sixth harmonic injection was
used.

CHIVITE-ZABALZA et al.: VOLTAGE BALANCING CONTROL IN 3-LEVEL NEUTRAL-POINT CLAMPED INVERTERS

To evaluate the performance when supplying an unbalanced
load, the midpoint response was tested by connecting an unbalanced inductive load at the secondary of a star-delta transformer.
This particular test setup was chosen due to availability reasons.
The RMS value of the A, B, and C phases was 30, 34, and 34 A,
respectively, which created the circulation of 32.7 A of positive
sequence, 2.6 a of negative sequence and 0.3 A of zero sequence,
as measured by a Yokogawa WT3000 precision power analyzer.
The results are showed on Fig. 15 where the top two waveforms
represent the dc-link capacitor voltages vC 1 and vC 2 and the
top two waveforms represent the two output currents iA and iB .
It can be seen in that figure how the midpoint voltage can be
controlled for second harmonic (a) sinusoidal sixth harmonic,
(b) squared sixth harmonic injection and (c) under an unbalanced load situation, validating the analysis shown in Section
III-D. Again, similar results were obtained for sixth harmonic
injection.
V. CONCLUSION
This paper has presented a novel technique to balance the voltage of the two split dc capacitors of a 3-L neutral-point clamped
inverter, suitable for reactive power compensation, when triangular carrier PWM modulation is employed. It consists in
injecting a squared waveform at six times the supply frequency.
Subsequently, it has been compared with two already known
strategies based on the injection of a negative-sequence second
harmonic and a sinusoidal sixth harmonic waveform. The contribution of current to the inverter midpoint of these techniques has
been analyzed as a performance measure. Subsequently a smallsignal averaged model, suitable for control design purposes has
been presented and a control strategy has been proposed. Finally, these techniques have been evaluated both in simulation
and in a 690-V ac, 120 kVA experimental setup when supplying
both a balanced and an unbalanced inductive load. The results
conclude that, although the three techniques are valid, second
harmonic injection has a major effect in balancing the inverter
midpoint and the even harmonics that it introduces are negligible and only present during transients. However, out of the two
sixth harmonic injection methods that do not produce even harmonics at the inverter output, the proposed squared waveform
technique is preferred, as it has a slightly greater compensation
effect on the dc midpoint and is easier to implement, particularly,
in systems employing a low modulation index.
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Javier Chivite-Zabalza received the B.Sc. Eng. degree in electrical and electronic engineering from
Mondragon University, Mondragon, Spain, in 1993,
the M.Sc. degree in power electronics and drives
from the Universities of Birmingham and Nottingham (joint degree), Birmingham, U.K., in 2003, and
the Ph.D. degree from the University of Manchester,
Manchester, U.K., in 2006.
He was a Project Engineer in the field of industrial
automation and drives from 1994 to 1999 in Spain,
and from 1999 to 2003 in the U.K. From 2003 to
2006, he was with Goodrich ESTC, Birmingham, U.K. where he was involved
in the development of high power-factor rectifiers for aerospace applications.
From 2006 to 2008, he was with the Rolls-Royce University Technology Centre,
University of Manchester, where he was involved in research on more electric
concepts for autonomous aerospace power systems. In 2008, he joined Ingeteam
Technology S.A., in Spain, developing power electronic converters for FACTS
devices and motor drives applications, where he is currently In-Charge of the
voltage source converter development for the 47 MVAR SSSC demonstrator.
Dr. Chivite-Zabalza is a Registered Chartered Engineer in the U.K. and a
member of the Institution of Engineering and Technology.

Pedro Izurza-Moreno was born in Bilbao, Spain, in
1982. He received the M.Sc. degree in automation
and industrial electronics from Mondragon University, Mondragon, Spain, in 2007.
In 2007, he joined the Industrial and Marine
Drives Department, Ingeteam Technology, S.A., Zamudio, Spain, where he was involved in research
on high-power inverters, especially on vector control, firmware design and programming, pulse width
modulation (PWM), space-vector PWM, selective
harmonic elimination modulation, and three-level
neutral-point-clamped inverters.

Danel Madariaga was born in Bilbao, Spain, in
1973. He received the M.Sc. degree in industrial engineering from the University of the Basque Country
(EHU-UPV), Bilbao, Spain, in 1998, and the M.Sc.
degree in physics from the Universidad Nacional de
Educaci´on a Distancia, Madrid, Spain, in 2008.
In 1998, he joined R&D Department, Ingeteam
Technology, S.A., Zamudio, Spain, where he was
mainly involved in research on high-power inverters, especially on vector control, firmware design
and programming, mathematical modeling, spacevector pulsewidth modulation (PWM), and three-level neutral-point-clamped
(3-L NPC) inverters. He was a part-time Science Teacher for four years. His
current research interests include solving the polynomial equation systems appearing in selective harmonic elimination PWM techniques, and modulation
techniques for balancing the dc bus of 3-L NPC inverters.

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 10, OCTOBER 2013

Gorka Calvo received the B.Sc. degree in electronics
engineering from the Escuela de Ingenier´ıa T´ecnica
Industrial de Bilbao, Bilbao, Spain, in 2005, and the
M.Sc. degree in automatics and industrial electronics
from the Escuela T´ecnica Superior de Ingenier´ıa de
Bilbao, Bilbao, in 2009.
Since 2009, he has been with the Ingeteam Power
Technology S.A., Zamudio, Spain, where he is engaged in high-voltage and high-power converters,
for industrial and FACTs applications. From 2007
to 2009, he was with Supsonik S.L, Sondika, Spain,
as RDi Engineer of power electronics, designing and testing hardware and control of rectifiers, UPS, voltage-frecuency converters, and PV inverters of power
converters. His research interests include power electronics, control of power
converters, and modeling.

Miguel Angel Rodr´ıguez (M’06) was born in San
Sebastian, Spain, in August, 1966. He received the
B.Sc. (Eng.) degree in electronic engineering from
Mondragon University, Mondragon, Spain, in 1989,
the M. Sc degree in electrical engineering from the
Swiss Federal Institute of Technology Lausanne,
Lausanne, Switzerland, in 1992, and the Ph.D. degree in industrial engineering from the University of
Zaragoza, Zaragoza, Spain, in 2000.
From 1992 to 2008, he was an Associate Professor in the Department of Electronics, University of
Mondragon and participated in different research projects in the field of wind
energy systems, lift drives, and railway traction. In September 2008, he joined
Ingeteam Power Technology S.A., Zamudio, Spain. He is currently the Power
Electronics Systems Manager at the Power Grid Automation Business Unit of
Ingeteam Power Technology, responsible for developing new power electronics solutions for transmission and distribution grid applications. His research
interest includes modeling and control of voltage source converters for FACTS
applications such as SSSC, STATCOM and energy storage systems, and the grid
integration studies for those systems.