Full text of DOI:10.1051/epjap:2001172

Eur. Phys. J. AP 15, 105–115 (2001)
THE EUROPEAN
PHYSICAL JOURNAL
APPLIED PHYSICS
c
ꢀ EDP Sciences 2001
Modelling and control of a seven level NPC voltage source
inverter. Application to high power induction machine drive
H. Gheraia1,a, E.M. Berkouk1, and G. Manesse2
1
Laboratoire de Commande des Processus, ENP, 10 avenue Hassen Badi, El Harrach, BP 182, Alger, Algeria
´
Laboratoire d’Electricit´e Industrielle, CNAM-Paris, 292 rue Saint Martin, 75141 Paris Cedex 03, France
2
Received: 1 December 1999 / Revised: 2 November 2000 / Accepted: 26 January 2001
Abstract. In this paper, we study a new kind of continuous-alternating converters: a seven-level neutral
point clamping (NPC) voltage source inverter (VSI). We propose this inverter for applications in high
voltage and high power fields. In the first part, we develop the knowledge and the control models of
this inverter using the connections functions of the semi-conductors. After that, we present two pulse
width modulation (PWM) algorithms to control this converter using its control model. We propose these
algorithms for digital implementation. This multilevel inverter is associated to the induction machine. The
performances obtained are full of promise to use it in the high voltage and high power fields of electrical
traction.
PACS. 84.30.Bv Circuit theory (including computer-aided circuit design and analysis) – 84.30.Jc Power
electronics; power supply circuits
1 Introduction
its control one. As an application of the control model, we
propose two algebraic algorithms to control the seven-level
NPC VSI, and we study the behaviour of the induction
machine fed by this inverter.
The variable speed control of electrical machines has great
advantages in the industrial processes. Mainly it improves
their static and dynamic performances. The improvement
in power electronic components has quickly led to find bet-
ter solutions to the speed and position regulation of the di-
rect current (DC) machine. However, the use of this motor
is limited to the low power and speed applications. In fact,
these limitations and the presence of the collector have
led to the development of speed regulators for alternating
current (AC) machines (induction and synchronous ma-
chines). The apparition of new power components such as
the gate turn-off thyristors (GTO) and the insulated gate
bipolar transistors (IGBT) have led to the conception of
new and fast converters for high power applications. Thus,
the variator (static converter-AC machine) has seen its
cost decreasing considerably. The progress accomplished
in the micro-computer tools (DSP, micro-controllers, ...)
has led to the synthesis more performant and robust con-
trol algorithms for sets of converter-machine. In this pa-
per, we study the performances of the induction machine
drive fed by a new seven-level NPC VSI. In the first part,
we introduce the induction machine model. The second
part constitutes the originality of this work which is the
modelling and control of the three-phase seven-level NPC
VSI. Thus, in this part we firstly develop the function-
ing model of this multilevel inverter in control mode using
the Petri nets. Then, we develop a knowledge model and
2 Model of the induction machine
2.1 Simplified hypothesis
We formulate the following hypothesis:
1- The magnetic circuit saturation and Foucault currents
are neglected.
2- The windings resistance do not vary with the temper-
ature.
3- We consider only the first space harmonic of the mag-
netomotrice power distribution of the stator and the
rotor.
4- Air gap is constant and the mutual inductances are
sinusoidal functions of angles between the stator and
the rotor axes frame.
2.2 Electrical equations of the induction machine
Figure 1 represents the six windings of the three-phase
induction machine. The general voltage equations of the
induction machine are obtained by writing that the volt-
age applied to each six windings is the sum of the ohmic
a
e-mail: agheraia@yahoo.com

106
The European Physical Journal Applied Physics
i
s
iar
r
i
q
G
s
G
G
r
r
s
i
r
G
i
r
i
c r
s
Fig. 1. Schematic representation of an induction machine.
and induction drops. Thus, the induction machine model induction machine model is given as follow:
is defined by the following system [1–3]:

˙
X = A · X + B · U







d
dt



[V ] = [Rs] · [Is] + [Φs]
dω
dt
Y = X
1
s
(3)
=
· (p · Cem − p · Cr − f · ω)

(1)
J




d
[Vr] = [Rr] · [Ir] + [Φr].
dt
with:
The fluxes are linked to the currents by the inductance
matrix [L(θ)].
X = [Ids Iqs Φds Φqs] .
ꢀ
ꢁ
ꢀ
ꢁ
[Φs]
[Φr]
[Is]
[Ir]
= [L(θ)] ·
·
(2)
The electromagnetic torque is given by
Cem = p · (Φqr · Idr − Φdr · Iqr)
The inductance matrix [L(θ)] can be decomposed to four
blocs as follow:
"
#
[Ls] [Msr]
[Mrs] [Lr]
A =
ꢂ
ꢃ
[L(θ)] =



−1
1
Ts
1
Tr
1
ω
·
+
ωgl

σ
σ · Ts · Tr σ · Ls










ꢂ
ꢃ
with:
−1
1
Ts
1
Tr
ω
1





−ωgl
−Rs
0
·
+
−
Ls Ms Ms
Lr Mr Mr
σ
σ · Ls σ · Tr · Ts
0












M L M
s
M L M
r r r
[L ] =
;
[L ] =
r

s
s
s
0
ωs




Ms Ms Ls
Mr Mr Lr
−Rs
U =
−ωs
0
and


[Msr] = [Mrs]t =
1

0
ꢂ
ꢃ
ꢃ
ꢂ
ꢃ
ꢃ












ωgl = ωs − ω
ωs = dθs/dt
ω = dθ/dt.
σ · Ls
2π
2π
3



ꢀ
ꢁ
cos θ
cos θ +
cos θ −
1
Vds
Vqs








3
0
0
0

B =
;
;
σ · Ls 




ꢂ
ꢃ
ꢂ



0
2π
3
2π
3

M · cos θ −
cos θ
cos θ +

0






ꢂ
ꢃ
ꢂ
2π
3
2π
cos θ +
cos θ −
cos θ
3
3 Seven-level NPC voltage source inverter
model
θ: is the angle between the stator and the rotor frames.
The seven-level NPC inverter (VSI) is a new structure of
power conversion used to feed great power AC loads. Fig-
ure 2 presents the structure of the three phases seven-level
2.3 State equations
By introducing the Park transformation in the system (1) NPC inverter. Several complementary laws are possible for
and using the frame linked to the rotating field, the the seven-level NPC VSI. The optimal complementary law

H. Gheraia et al.: Modelling and control of a seven level NPC voltage source inverter
107
3
T
T
T 3 4
4
4
4
4
3 4
3
T 3
T
T
3
3
3 3
T
T
T
T
T
T
T 3 3
3
3
3
T
3
T 3
T 3
T
3
T 3
3
A
3
T
T
T
T 3
3
3
3
3
T
T
T
3
3
T 3
T 3
T
T
T 3
3
T 3
T
T 3
T
T
T
3
Fig. 2. A seven-level NPC voltage source inverter associated to the induction machine.
Table 1. The excitation table of an arm “1” of the seven-level NPC VSI.
VAM
−3UC
−2UC
−UC
0
F11 F12 F13 F14 F110 F19 F15 F16 F17 F18 F111 F112
0
0
0
1
1
1
1
0
0
0
0
1
1
1
x
x
x
x
1
1
0
x
x
x
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
1
1
1
0
0
0
0
1
1
0
x
0
0
x
1
0
x
x
x
0
x
0
0
1
0
0
0
0
0
1
0
0
0
0
0
2UC
3UC
UC
used for this converter is presented below.
3.1 A working model of the seven-level NPC VSI using
Petri nets
Bi5 = Bi2
Bi6 = Bi1
Bi7 = Bi4
Bi8 = Bi3
To modelize the seven-level NPC voltage source inverter,
we use the DESIGN method [4,5]. This one is very simple
to manage and to integrate on the computerized tools to
simulate this converter. The principle of this method con-
sists to search the different configurations resulting from
the turn on or off of the different semi-conductors, and
Bis: control signal of the semiconductor TDis.
With these complementary laws, every arm the conditions of transition from a configuration to an
of the inverter (which is multi-tripole cell) can other one. The topology analysis of one arm of the inverter
be considered as equivalent to six simple tripole in controllable mode shows eight configurations. Table 2
cells [4,5]. These six cells are constituted re- shows the electrical quantities which characterise every
spectively by the following pairs of switches configuration (ViM is the voltage of the phase i relatively
{(TDi1, TDi6), (TDi2, TDi5), (TDi3, TDi8), (TDi4, TDi7)}. to the middle point M). The transition receptivities of the
Table 1 defines the excitation table of an arm (arm 1) of Petri nets associated to the working model of an arm of the
the inverter.
seven-level inverter are defined by using the commutation

108
The European Physical Journal Applied Physics
E
E
E
0
0
0
0
0
0
E0
0
0
0
0
0
0
E
E
E
Fig. 3. A Petri network of an arm (1) of an NPC seven-level inverter.
condition of tripole commutation cells as follow (Fig. 3):
Table 2. A known grandeur for each of configuration of an
arm “i” of the seven-level NPC VSI.
R01 = B12, R10 = B12,
E0
E1
E2
E3
E4
E5
E6
Vout = 0
Vout = Uc1 = UC
R02 = (B12 & B13), R20 = (B12 & B13)
R03 = (B12 & B13 & B14), R30 = (B12 & B13 & B14)
R04 = (B11 & B13 & B14), R40 = (B11 & B13 & B14)
R05 = (B11 & B13), R50 = (B11 & B13),
R06 = B11, R60 = B11
Vout = Uc1 + Uc2 = 2UC
Vout = Uc1 + Uc2 + Uc3 = 3UC
0
Vout = −Uc1 = −UC
0
0
Vout = −(Uc1 + Uc2 ) = −2UC
0
0
0
Vout = −(Uc1 + Uc2 + Uc3 ) = −3UC
R12 = B13, R21 = B13,
R13 = (B13 & B14), R31 = (B13 & B14)
R14 = (B11 & B12 & B13 & B14),
3.2 Knowledge model of a seven-level NPC VSI
In a controllable mode using the proposed complementary
laws, we define for each semi-conductor TDis a connection
function Fis as follow [6,7]:
R41 = (B11 & B12 & B13 & B14)
R15 = (B11 & B12 & B13), R51 = (B11 & B12 & B13)
R16 = (B11 & B12), R61 = (B11 & B12)
R23 = B14, R32 = B14, R24 = (B11 & B12 & B14),
R42 = (B11 & B12 & B14)
(
1
0
if the semi-conductor TDis is turned on
if it is turned off.
Fis
=
(i: number of the arm and s: number of the semiconduc-
R25 = (B11 & B12), R52 = (B11 & B12),
R26 = (B11 & B12 & B13), R62 = (B11 & B12 & B13)
R34 = (B11 & B12), R43 = (B11 & B12),
R35 = (B12 & B13 & B14), R53 = (B12 & B13 & B14)
tor).
The voltages of the three phases A, B, and C, rela-
tively to the middle point M, are given by the following
system [8]:

R36 = (B11 & B12 & B13 & B14),
R63 = (B11 & B12 & B13 & B14)
V
= F11 · F12 · F13 · F14 · Uc1

AM




+F11 · F12 · F13 · F14 · (Uc1 + Uc2)




R45 = B14, R54 = B14, R46 = (B13 & B14),
R64 = (B13 & B14)
+F11 · F12 · F13 · F14 · (Uc1 + Uc2 + Uc3)

0
−F15 · F16 · F17 · F18 · Uc1





0
0
0
R56 = (B13), R65 = B13.
−F15 · F16 · F17 · F18 · (Uc1 + Uc2
)



0
0
−F15 · F16 · F17 · F18 · (Uc1 + Uc2 + Uc3
)

H. Gheraia et al.: Modelling and control of a seven level NPC voltage source inverter
109



0
V
= F21 · F22 · F23 · F24 · Uc1
 VBM = (F29 · Uc1 − F211 · Uc1
)
BM





0
0
+(F210 · (Uc1 + Uc2) − F212 · (Uc1 + Uc2 ))
+F21 · F22 · F23 · F24 · (Uc1 + Uc2)






+(F21 · (Uc1 + Uc2 + Uc3) − F2b0 · (Uc1 + Uc2 + Uc3 ))
b
0
0
0
+F21 · F22 · F23 · F24 · (Uc1 + Uc2 + Uc3)
(4)
(8)

0
−F25 · F26 · F27 · F28 · Uc1





0
0
0
−F25 · F26 · F27 · F28 · (Uc1 + Uc2
)




0
0
0
VCM = (F39 · Uc1 − F311 · Uc1
)
−F25 · F26 · F27 · F28 · (Uc1 + Uc2 + Uc3
)
0
0
+(F310 · (Uc1 + Uc2) − F312 · (Uc1 + Uc2 ))

+(F31 · (Uc1 + Uc2 + Uc3) − F3b0 · (Uc1 + Uc2 + Uc3 )).
b
0
0
0


V
CM
= F31 · F32 · F33 · F34 · Uc1




+F31 · F32 · F33 · F34 · (Uc1 + Uc2)




The system (8) shows that a seven-level NPC VSI is
equivalent to six two-level or three three-level NPC VSI
in series. For the last part of the paper, we suppose:
Uc1 = Uc2 = Uc3 = Uc1 = Uc2 = Uc3 = UC. Thus,
the system (8) becomes:
+F31 · F32 · F33 · F34 · (Uc1 + Uc2 + Uc3)

0
−F35 · F36 · F37 · F38 · Uc1





0
0
0
0
0
0
−F35 · F36 · F37 · F38 · (Uc1 + Uc2
)



0
0
−F35 · F36 · F37 · F38 · (Uc1 + Uc2 + Uc3 ).
ꢄꢅ
ꢆ

VAM
VBM
VCM
=
=
=
F19 + 2F110 + 3F1b1
− F111 + 2F112 + 3F1b0 · UC
F29 + 2F210 + 3F2b1
− F211 + 2F212 + 3F2b0 · UC
F39 + 2F310 + 3F3b1
− F311 + 2F312 + 3F3b0 · UC.

In order to reduce the above equation system, we define
the half arm connection functions Fib1 and Fib0 associated
respectively to the upper and lower half arms.



ꢅ
ꢆ







ꢄꢅ
ꢆ


Fib1 = Fi1 · Fi2 · Fi3 · Fi4
(9)
ꢅ
ꢆ


Fib0 = Fi5 · Fi6 · Fi7 · Fi8.
(5)
(6)




ꢄꢅ
ꢆ



We note that:




ꢅ
ꢆ
Fi9 = Fi1 · Fi2 · Fi3 · Fi4
Fi10 = Fi1 · Fi2 · Fi3 · Fi4
Fi11 = Fi5 · Fi6 · Fi7 · Fi7
Fi12 = Fi5 · Fi6 · Fi7 · Fi8.
For the equation (9), we define a global half arm connec-
tion function:
bT
F
F
= Fi9 + 2Fi10 + 3Fib1
= Fi11 + 2Fi12 + 3Fib0.
i1
bT
i0
(10)
Thus the system (4) becomes:

bT
i1
When F equal to “0”, the upper half arm is opened, and
VAM = (F19 · Uc1 + F110 · (Uc1 + Uc2)




bT
i0
if F equal to “0”, the lower half arm is opened too. The
simple voltages of the three phases seven-level NPC VSI
are given by the following matricial system:
+F1b1 · (Uc1 + Uc2 + Uc3))


0
0
0
−(F111 · Uc1 + F112 · (Uc1 + Uc2
)


b
0
0
0
+F10 · (Uc1 + Uc2 + Uc3 ))




bT
11
bT
10


F
− F
2 −1 −1
−1 2 −1
−1 −1 2
VA
VB
VC









bT
21
bT
20



= 1/3 ·
· F − F
· UC. (11)
VBM = (F29 · Uc1 + F210 · (Uc1 + Uc2)
+F2b1 · (Uc1 + Uc2 + Uc3))





bT
31
bT
30
F
− F
(7)
0
0
0

−(F211 · Uc1 + F212 · (Uc1 + Uc2
)



b
0
0
0
+F20 · (Uc1 + Uc2 + Uc3 ))
3.3 Control model of the seven-level NPC VSI

The model established above is discontinued. It is used
to simulate PWM strategies. In order to develop a con-
trol model of this inverter, we define the average model of
the knowledge one. Thus, we define the generating func-
tion “xg” of a discontinued one “x”, as the mean value of
“x” on a modulation period “Th” supposed very small as
follow [7,8]:
VCM = (F39 · Uc1 + F310 · (Uc1 + Uc2)
+F3b1 · (Uc1 + Uc2 + Uc3))





0
0
0
−(F311 · Uc1 + F312 · (Uc1 + Uc2
)



b
0
0
0
+F30 · (Uc1 + Uc2 + Uc3 )).
The system (7) can be written as follows:

Z
Th
0
VAM = (F19 · Uc1 + F111 · Uc1
)

xg = (1/Th)
x dt.
(12)
0
0
+(F110 · (Uc1 + Uc2) − F112 · (Uc1 + Uc2 ))
0


+(F11 · (Uc1 + Uc2 + Uc3) − F1b0 · (Uc1 + Uc2 + Uc3 ))
b
0
0
0
Thus, a generating connection function (Fis)g of a switch
TDis (resp. (Fibm
of an arm) is a continuous function
)
g

110
The European Physical Journal Applied Physics
U
I
é
C
é
V
As
I
V
V
b T
im
s
s
b T
i m
F
F
ꢀ
ꢀ
F
N
I
[
is
g
Fig. 4. A causal informational graph associated to the control model of the seven-level NPC VSI.
representing the mean value of the discontinued function 4.1 Algorithm 1
Fis (resp. Fibm) on a modulation period “Th”. The control
model of the three-phase seven-level NPC VSI, deduced
from the system (11), is given by the following equation
4.1.1 Principle
where VA, VB and VC represent the mean value of the
This algorithm is based on the triangulo-sinusoidal strat-
egy with one carrier [5]. The different blocks of Figure 5
are expressed for this algorithm as follow:
instantaneous ones.
ꢇ
ꢇ
ꢇ
ꢈ
ꢈ
ꢈ
ꢇ
− F
ꢇ
ꢈ
ꢈ
ꢈ






bT
bT
F
F
F
11
10
2 −1 −1
−1 2 −1
−1 −1 2
VA
g
g
g
g
g
g














bT
21
bT
20
− F
ꢇ
= 1/3 ·
·
· UC
(13)
V
B
Step 1: Compute the simple generating conversion func-
bT
31
bT
30
tions ng :
VC
− F
i
ng = vref /UC
with: i = 1, 2, 3
i
i
with:
ꢇ
ꢇ
ꢈ
ꢈ
ꢇ
ꢈ
bT
i1
= (Fi9)g + 2 (Fi10)g + 3 Fib1
= (Fi11)g + 2 (Fi12)g + 3 Fib0
Step 2: Compute the half arm and switch generating con-
nection functions:
F
g
g
g
ꢇ
ꢈ
bT
i0
F
.
(14)
g
•0 ≤ ng ≤ 1 ⇒ (Fi9) = ng
i
i
g
We define the simple generating conversion matrix [N]g
as follow:
(
)
ꢇ
ꢇ
ꢇ
ꢈ
ꢈ
ꢈ
ꢇ
− F
ꢇ
ꢈ
ꢈ
ꢈ


(Fi9) + 2 (Fi10) = ng


i
g
g
bT
bT
10
F
F
F
•1 < ng ≤ 2 ⇒
11
2 −1 −1
−1 2 −1
−1 −1 2
g
g
g
g
g
g
i
(Fi9)g + (Fi10)g = 1










bT
21
bT
20
− F
ꢇ
[N] = 1/3 ·
·
·
(15)
g
ꢇ
ꢈ
(
)
bT
31
bT
30
2 (Fi10) + 3 Fb = ng
− F
i
i1
g
g
ꢇ
ꢈ
•2 < ng ≤ 3 ⇒
i
(Fi10)g + Fib1 = 1
Figure 4 represents the causal informational graph associ-
ated to the seven-level NPC inverter. The relation (R1),
(R2), (R3) and (R4) correspond respectively to the re-
lations (10), (12), (15) and (13). The relation (R5) rep-
resents the state model of the inverter’s load (induction
machine). We note that the seven-level inverter presents
a unity feedback as with the two-level and the three-level
inverters [5]. It proves that the seven-level NPC VSI is
strongly non linear.
g
• −1 < ng ≤ 0 ⇒ (Fi11) = ng
i
i
g
(
)
(Fi11) + 2 (Fi12) = ng
i
g
g
• −2 < ng ≤ −1 ⇒
i
(Fi11)g + (Fi12)g = 1
ꢇ
ꢈ
(
)
2 (Fi12) + 3 Fb = ng
i
i0
g
g
ꢇ
ꢈ
• −3 ≤ ng ≤ −2 ⇒
i
(Fi12)g + Fib0 = 1
4 PWM Strategies of the seven-level
NPC VSI: algebraic modulation
g
Step 3: Compute of the half arm and switch instanta-
The different triangulo-sinusoidal strategies can be
neous connection functions [7]:
achieved with digital mean by sampling reference voltages
√
vref such as: vref
= Veff 2 sin(ωt − 2Π(i − 1)/3). In Figure 6 describes the passage from the generating func-
i
i=1,2,3
this paper, we develop two digital PWM algorithms which tion (Fis)g to the instantaneous connection function Fis.
use the control model developed previously. The general This latter is a typical pulse width modulation signal
flow chart of an algebraic PWM using this control model (PWM). The time t is initialised at the end of each pe-
is presented in Figure 5. The modulation index “m” and riod “Th”. If the sampling have been made between the
the modulation rate “r” are defined as for the triangulo- time “(q − 1) · Th” and “(q · Th)”, (Fis
)
represents
(q−1)
g
sinusoidal strategies [5,9,10].
the amplitude of the sampling generating function. The

H. Gheraia et al.: Modelling and control of a seven level NPC voltage source inverter
111
re
V
i
Compute of t e
Compute of t e
o
e
o
e e
t
fu
ngi
t o
tep
tep
o
e t o
e e
t
fu
t o
F
, F
F,
F
F
F,
ꢀ
i9
i
i
i
i
i
Compute of t e
t
t
ou
o
e t o fu
t o
tep
tep
F
, F
,F
F
F
F,
i
ꢀ
i9
i
i
i
i
Compute t e omm
of tffee e t
t
e
Bis
Fig. 5. The flow chart of an algebraic modulation using the control model of the seven-level NPC VSI.
mean value of the connection function should be equal to
(F
is
(
(Fis
)
(q−1). It is given as follow:
g
Z
q·Th
ꢇ
ꢈ
Fis
= (1/Th) ·
Fis · dt = δ1(q−1) (16)
g
(q−1)
(F
is (q -
(q−1)·Th
where δ1(q−1) represents the cyclic ratio. There are many
studies dealing with the position of the pulse and all these
studies found that the best result for the harmonics spec-
trum are obtained with a pulse position in the center [7].
Thus, the period “Th” is odd multiple of the fundamental
period of the output wave of the inverter.
.
F
is
4.1.2 Simulation results
Figures 7 and 8 represent the simple output voltage of the
seven-level NPC VSI controlled by the proposed algebraic
modulation for m = 39, 42 and r = 0.8. We notice that:
– If “m” is even, the simple output voltage has a symme-
try at π/2 and π. So we have only the odd harmonics;
– If “m” is odd, we have no symmetry. So we have an
even or odd harmonics.
@
1(q
1
Fig. 6. Elaboration of the instantaneous connections functions
from the generating one.
4.2 Algorithm 2
The voltage harmonics gather by families around fre-
quencies multiple of mf (Figs. 7 and 8). The first family
has the biggest amplitude. For m = 39, the rank of har-
monics which have the most important amplitude are 38
and 40 (Fig. 9). The adjusting characteristic is linear from
r = 0 to r = 1, and the harmonics rate decreases when
“r” increases (Fig. 10). Figure 11 shows the electromag-
netic torque in steady state. The frequency of the torque
can be six or three times the frequency of the output volt-
age of the seven-level NPC VSI. Figure 12 represents the
possibility of inversion of the induction machine speed.
4.2.1 Principle
For the implantation of the vectorial modulation type 2
using one carrier [4,5], we propose this algorithm using
the control logic model of the seven-level NPC VSI.
Step 1: Compute the simple generating conversion func-
∗
∗
∗
tions ng : ng = vref /UC
with: i = 1, 2, 3
i
i
i
∗
vref = vref + V0
i
i
ꢄ
ꢇ
ꢈ
ꢇ
ꢈ
V0 = − max vref
+ min vref
/2 (17)
i=1,2,3
i=1,2,3

112
The European Physical Journal Applied Physics
VA(v)
0
0
0
0
0
(
)
Fig. 7. The output voltage and it’s harmonics spectrum (for: m = 39, r = 0.8).
VA(v)
0
0
0
0
0
(
)
Fig. 8. The output voltage and it’s harmonics spectrum (for: m = 42, r = 0.8).
0
0
0
0
0
0
0
0
0
0
0
0
40
4
0
0
2
r
r
Fig. 9. Variation of the output voltage harmonics amplitude (for: m = 39).
Step 2: Compute the half arm and switch generating con-
nection functions:
∗
∗
i
• −1 ≤ ng ≤ 0 ⇒ (Fi11)g = ng
i
(
)
∗
(Fi11)g + 2 (Fi12)g = ng
(Fi11)g + (Fi12)g = 1
i
∗
• −2 ≤ n < −1 ⇒
∗
∗
i
•0 ≤ ng ≤ 1 ⇒ (Fi9)g = ng
gi
i
ꢇ
ꢈ
(
)
2 (Fi12)g + 3 Fib0 = ng
∗
(
)
i
g
∗
(Fi9) + 2 (Fi10) = n
gi
∗
ꢇ
ꢈ
• −3 ≤ n < −2 ⇒
g
g
gi
(Fi12)g + Fib0 = 1
∗
•1 < ng ≤ 2 ⇒
i
g
(Fi9)g + (Fi10)g = 1
Step 3: Compute the half arm and switch instantaneous
connection functions:
The change from generating function (Fis)g to instanta-
neous connection function Fis is made by the same proce-
dure of the precedent algorithm.
ꢇ
ꢈ
(
)
2 (Fi10)g + 3 Fib1 = ng
∗
i
g
∗
ꢇ
ꢈ
•2 < ng ≤ 3 ⇒
i
(Fi10)g + Fib1 = 1
g

H. Gheraia et al.: Modelling and control of a seven level NPC voltage source inverter
113
(
s)
Mr
Fundamental amplitude
0
0
0
0
a m ni
ate
r
0
0
0
0
0
0
t (s)
Fig. 10. The reglage characteristic of the output voltage (for:
m = 39).
Fig. 12. The induction machine speed (for: m = 39, r = 0.8).
Cem (Nm)
5 Conclusion
In this paper, we have studied a new DC-AC power con-
verter intended for high power applications: a seven-level
NPC VSI. Thus, we have defined its operation model in
controllable mode using Petri nets, and we have devel-
oped a knowledge model of this inverter using the con-
nection function of the switches and half arm. We have
shown that the seven-level NPC VSI is equivalent to a
six two-level inverters or three three-level NPC VSI in se-
ries. Also, we have elaborated a control model using the
generating functions. As application of this control model,
we have proposed two algebraic modulation algorithms for
numerical applications. The first algorithm gives a linear
characteristic of the output voltage of the inverter from
r = 0 to r = 1. However the second algorithm allows to
widen the linear zone of the output voltage characteristic
about 20%. The performances obtained for the behaviour
of the induction machine fed by the seven-level NPC VSI
which is controlled by one of these algorithms are full of
promise to use this inverter in high power fields as electri-
cal traction.
( )
Fig. 11. The induction machine torque (for: m = 39, r = 0.8).
4.2.2 Simulation results
Figures 13 and 14 represent the simple output voltage
of the seven-level NPC VSI controlled by the proposed
algebraic modulation for m = 39, 42 and r = 0.8. We
note that:
– If “m” is even, the simple output voltage has a
symmetry at π/2 and π. So we have only the odd
harmonics;
Appendix: List of principal symbols
Rs, Rr: stator, rotor resistance
Ls, Lr: stator, rotor inductance
Ts, Tr: stator, rotor time constant
M: magnetizing inductance
– If “m” is odd, we have no symmetry. So we have an
even or odd harmonics.
The voltage harmonics gather by families around fre-
quencies multiple of mf (Figs. 13 and 14). The first fam-
ily has the biggest amplitude. For m = 39, the rank of
harmonics which have the most important amplitude are
38 and 40 (Fig. 15). The adjusting characteristic is linear
from r = 0 to r = 1, 2, and the harmonics rate decreases
when “r” increases (Fig. 16). Figure 17 shows the elec-
tromagnetic torque in steady state. The frequency of the
torque can be six or three times the frequency of the out-
put voltage of the seven-level NPC VSI. Figure 18 repre-
sents the possibility of inversion of the induction machine
speed.
Vds, Vqs: stator voltage d−q axis component
Ids, Iqs: stator current d−q axis component
Idr, Iqr: rotor current d−q axis component
Φds, Φqs: stator flux d−q axis component
Φdr, Φqr: rotor flux d−q axis component
ωs: stator angular frequency
ω: rotor speed
σ: total leakage coefficient (= 1 − M2/Ls · Lr)
J: total inertia
Cr: load torque
f: coefficient of friction
p: number of pole pairs.

114
The European Physical Journal Applied Physics
VA(v)
0
0
0
0
0
Fig. 13. The output voltage and it’s harmonics spectrum (for: m = 39, r = 0.8).
(v)
VA
0
0
0
0
0
Fig. 14. The output voltage and it’s harmonics spectrum (for: m = 42, r = 0.8).
0
0
0
0
0
0
0
0
0
0
0
0
0
3
5
3
0
0
3
r
31
r
Fig. 15. Variation of the output voltage harmonics amplitude (for: m = 39).
Cem (Nm)
Fundamental amplitude (P.U)
0
0
0
0
a m ni
ate
r
0
( )
Fig. 16. The reglage characteristic of the output voltage (for:
m = 39).
Fig. 17. The induction machine torque (for: m = 39, r = 0.8).

H. Gheraia et al.: Modelling and control of a seven level NPC voltage source inverter
115
Mr
3. M. Boussak, Doctorat Thesis, University of Paris VI, 1989.
4. E.M. Berkouk, Doctorat Thesis, University of CNAM-
Paris, 1995.
5. H. Gheraia, Magister Thesis, Ecole Nationale Polytech-
nique, Alger, 1999.
´
6. E.M. Berkouk, Y.B. Romdhane, G. Manesse, SAMS J. 18-
19, 511 (1995).
7. J.P. Caron, J.P. Hautier, Mod´elisation et Commande de la
´
Machine Asynchrone (Editions Technip-Paris, 1995).
8. H. Gheraia, E.M. Berkouk, G. Manesse, in Proceedings of
the 6th International Conference on Modelling and Simu-
lation of Electric Machines, Converters and Systems, Lis-
bon, Portugal, 1999.
Fig. 18. The induction machine speed (for: m = 39, r = 0.8).
9. E.M. Berkouk, Y.B. Romdhane, G. Manesse, in Proceed-
ings of the 6th European Conference on Power Electronics
and Applications, Sevilla, Spain, 1995, p. 706.
10. H. Gheraia, E.M. Berkouk, G. Manesse, in Proceedings of
the 1st International Conference on Electrotechnics, Oran,
Algeria, 1999, p. 190.
References
1. P.C. Krause, Analysis of Electric Machinery (Editions
McGraw-Hill, 1987).
2. J. Chatelain, Machine Electrique (Editions Dunod, 1983),
tome 1.
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