chapter 4 a svpwm algorithm for npc-mli in over
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140
CHAPTER 4
A SVPWM ALGORITHM FOR NPC-MLI IN OVER
MODULATION REGIONS
4.1 INTRODUCTION
Further avid and investigation on any superior performance PWM
method, is towards widening output controllable range of MLI. Any attempt in
driving the output range to higher range requires thorough investigation and
operation in over modulation regions. The SVPWM is the better choice due to its
merits namely high inverter energy efficiency, waveform quality and voltage
linearity. However, these performance merits may not be guaranteed when it is
extended for OVM regions. The PWM strategy, which performs well at Linear
Modulation (LM) may not perform well in OVM. In this context two issues are to be
taken earnestly. The first one is the SVPWM switching sequences and vector
choices need to be carefully done without affecting the performance. Secondly, the
performance characteristics of the modulator in OVM must be understood well. It
can be interpreted from this discussion that performances of SVPWM in 2-level VSI
is well understood while the same at MLI needs further investigations.
The main issues involved with OVM and its implementation with the
associated literature survey are described in section 2.2.2. This chapter proposes a
simple algorithm for over modulation regions, which is just an intelligence extension
of the linear mode SVPWM (detailed in chapter 3). The proposed algorithm easily
determines the location of reference vector and calculates on-time. It employs a
simple mapping algorithm to generate gating signals for the inverter. The proposed
algorithm can be easily extended to an n-level inverter. It is applicable to 3-level
NPC-MLI topology. Simulation and experimental results are provided for 3-level
141
NPC inverter. The analytical results prove the development of simple yet highly
efficient modulation strategy.
4.2 OVER MODULATION STATUS AND REVIEW
In SVPWM, the normalized output voltage of the inverter is described
with Ma ranging from zero to unity and sub-divided as LM and over OVM [2].
Computation of output voltage more than 0.866, is named as OVM and
corresponding Ma equals 0.907. For a targeted output voltage, a suitable Vref is
fixed. The tip of this Vref traces a circle. Depending upon the Ma values this circle
may be an incumbent or lying out of the hexagon. During the effort of fixing the Vref
in OVM, at any sector (span of 600), the output synthesizes lesser voltage than
desired in mid span of angle and higher voltage on the end spans (LV sides) as
detailed in Figure 4.1.
Figure 4.1. SVD in entire modulation
142
If the SVD hexagon overlapped an OVM boundary circle, then the two
regions (areas) may be noticed in any of the sector. The first area (volt-sec) is
enclosed by the circular boundary and excess to hexagonal boundary (in the middle
of the sector). The second area is the addition of two areas at both ends of the sector
(closer to LVs), which is caused due to the difference between the oversized
hexagonal boundary and the circle. Operations in these two areas/regions are termed
as OVM-I and OVM-II respectively [97].
The OVM scheme is essential, if the drive is required to operate at
extended speed including the field-weakening region and with higher torque and
power characteristics [40], [187]. Hence the researchers tread in this field to give the
significant performance of the MLI in OVM [133], [187], [188].The over
modulation range refers to the operating regions of the SVPWM beyond the linear
range which indicates the inscribed circle of the hexagon. In SVPWM, the
normalized output voltage of the inverter can be described by Ma which ranges from
0 to 1. According to the modulation index, inverter output range is divided into
three regions as shown in Figure 4.1 and described in detail in [2].
The range 0 to 0.907 is LM and 0.907 to 1.0 is called OVM region. The
OVM is classified as OVM-I (Ma: 0.907 to 0.953) and OVM-II (Ma: 0.954 to 1).
Hence the Ma describes the voltage utilization level of the modulator. In this study
Ma is defined as the ratio of the magnitude of the fundamental component of the
inverter output voltage (V1) to (maximum possible) fundamental voltage under six-
step operation (Vsix-step=(2/π)Vdc) for given DC bus voltage [2]. For a given DC-link
voltage Vdc, Ma is given as
(4.1)
If Ma is greater than 1, the desired circular trajectory of Vref crosses the
SVD hexagonal boundary. The available virtual vectors can only synthesize Vref
when the tip of this vector lies inside the hexagonal boundary. Even at this situation
Vref can synthesize using the clever usages of available vector. Even though the Vref
refa
dc
VM =
(2/π)V
143
over a cycle could be synthesized by exiting vector, instantaneous volt-sec balance
cannot be met out (unlike in LM) when Vref exceeds the hexagonal boundary.
Therefore, there would be portions of the line cycle where the desired Vref could not
be synthesized. The solution to this problem consists of modifying the desired
trajectory of Vref in a way that it lies within the SVD large hexagon and it produces
the same fundamental voltage. This can be done at the expense of introducing low
frequency harmonics in the AC voltages, which are absent in the under modulation
range [125].
OVM-I: A modified voltage vector reference is derived from the
reference voltage vector by changing its magnitude, whereas the phase angle is kept
at its original value. [132].
OVM -II: An actual voltage reference vector is kept at a vertex of the
hexagon for a particular time and a change in the phase angle of the modified
reference vector is required. [132].
4.3 CHALLENGES AND PROBLEM FORMULATION
Many types of SVPWM schemes for a 3-level NPC-MLI with OVM
operation has been developed [105], [127]. The OVM has been first attempted by
Seo et al. for a 3-level MLI [105]. The scheme obtains the Vref magnitude directly
from LM operation which offered difficult trigonometric duty cycle division,
nevertheless it cannot be applied directly to an n-level inverter. And another host of
OVM algorithms are presented in [125]-[130], [187] which allows to extent the
linearity of the inverter output voltage. However, these schemes use complex
trigonometric functions in calculating the modified voltage vector and the duty
cycle.
Mondal et al. [132] have proposed a SVPWM based OVM for a 3-level-
NPC-MLI, where on-time calculation of equations differ for every triangular section
at respective Ma values.
144
a b
Figure 4.2. SVD for sector-1; a. OVM-I, b. OVM-II
Here, the voltage reference V* can be expressed as
(4.2)
Where V1 is the magnitude of original reference voltage and ʽαc’ is the cross over
angle. Since V1 is larger than Vdc/√3, during the rotation of the reference vector, the
vector goes outside the hexagon during the period 1 to 2 as depicted in Figure 4.2.a.
During the period 1 to 2, the reference magnitude need to be limited at hexagonal
boundary (Vdc/√3 cos (π
6− θ)) and the averaged reference magnitude in this period
becomes less than V1 (loss in voltage). They have used V* for Vref, V1 for magnitude
of Vref.
jθ
2 c
* dcc c
jθ
jθ
2 c
V e ; 0 θ α
V πV ; α θ α
π 33 cos θ e
6
π π V e ; α θ
3 3
(4.3)
Where αc is the crossover angle. Concept of Mondal et al. scheme is gaining the loss
voltage in periods P-1 and 2-Q through the circular boundary of higher radius (still
* jθ
1 c V = Ve 0; θ<α
145
within the hexagonal boundary). As the trajectory in the period 1 to 2 is resituated at
hexagon, it is immaterial whether the circular boundary is increased or not.
Therefore a modified voltage is obtained by Equation (4.3) and Figure 4.2.b where
V2 represents the boosted voltage magnitude to compensate the loss of magnitude in
Vref during period 1-2. Hence during the periods A-B and C-D the reference vector
having a V2 magnitude is located inside the hexagon, and during the period B-C the
reference voltage moves along (C=60-αc). As the modified voltage reference (V*) is
computed by Equation (4.3), it is necessary to calculate trigonometric functions such
as cos (π/6−θ) with the information of magnitude and the angle of Vref [188], [189].
Hence several steps are involved to determine the reference voltage vectors.
Furthermore, this technique utilizes the minimum phase error projection scheme for
generating the duty ratios.
To remove this complexity, Dong-MyungLeehe et al. have proposed
static OVM to generate a new boosted voltage vector with a simple procedure [126].
The scheme obtains the magnitude of a new reference voltage directly from the
modulation index, which is predetermined by the result of the Fourier Series
Expansion (FSE) method and eliminates the decision procedure for the angle and
magnitude of the reference vector. From Figure 4.3 value of V2 is directly obtained
from Ma without finding αc and the angle of the reference vector. During
implementation, these values are obtained from approximate equations of the
magnitude and phase angle which are functions of Ma values.
Figure 4.3. SVD for sector-1
146
In Dong-Myung Leehe et al. method, the required reference value, i.e. V2
which makes the fundamental voltage magnitude after boosting equivalent to V1 is
calculated by employing the FSE Equation (4.4),
(4.4)
Here, the constant coefficients of the FSE on both sides are omitted. This
makes the poor reference voltage estimation. In addition the scheme uses several
steps to determine the reference voltage vectors. Due to the increased computational
complexity, it is cumbersome to extend this scheme to an n-level inverter. To avoid
this computational complexity, some of the researchers have attempted to estimate
the boosted voltage region through magnitude correction factor and angle correction
factor [125]-[130]. The duty ratio calculations are different for each triangles and
hence as the level increases it needs more on-time equations, which increases the
complexity [125]. S B Monge et al. have proposed a SVM for an n-level inverter
including OVM with hexagonal-boundary compression factor calculations and
deliberated the effect of OVM on neutral-point voltage fluctuation in a 3-level
converter [127]. Amit Kumar Gupta et al. proposed an OVM scheme with a neutral
voltage balancing, which does not use the medium voltage vectors that affect the
inverter output voltage [128]. Conversely, the proposed magnitude correction
methods are indirect in nature. Despite the fact, it is obvious that any
simulation/hardware implementation requires correction in terms of duty cycle and
angle results in the poor THD values [131]. .
From the detailed study on OVM the following observations are noted,
i. The magnitude of the Vref samples which are outside the hexagon are
modified by using a Magnitude Correction Factor (MCF) and Angle
Correction Factor (ACF). The duty ratio calculations are different
for different triangles and hence as the level increases the technique
3 3 3
1 2 2
0 0)
3
3 cos(6
cc
cc
j j j j j j j jdc
c
VV e e d V e e d e e d V e e d
147
needs more number of timing equations which results in increased
complexity.
ii. The presented magnitude correction methods are indirect in nature.
It is an obvious expectation that any simulation/hardware
implementation requires correction in terms of duty cycle. The other
options like angle correction results in poor valuation of THD.
iii. In general the OVM range is nonlinear in nature. To handle this
nonlinearity, the magnitude and angle of the Vref are modified. Due
to the composite geometry it needs a pre-processor which makes the
implementation complex and expensive.
4.4 DIRECT MODULATION DEPTH CORRECTION PROCEDURE
FOR OVER MODULATION
The main idea behind the proposed OVM technique is incorporating the
magnitude correction in the timings (on-time of the switches) of constituting vectors
using MCF equations, LUTs and pre-defined calculations. It uses a simple mapping
to generate gating signals for the inverter in LM and OVM regions. This scheme
accurately determines the location of the Vref and calculates on-time. Firstly, the
boundary restriction inherited in the linear modulation hexagon is redefined as
Modulation Depth Loss Factor (MDLF) and quantified for various OVM indices.
Next, the appropriate time correction is carried out in the switching vector timing.
The proposed timing corrected SVPWM strategy hosts a simple assessment and duty
cycle pre-calculation procedure for determining on-times. The Timing Correction
Algorithm (TCA) ingeniously corrects these magnitude losses and gains in terms of
addition and subtraction of a corrective term in the on-time respectively.
The proposed OVM schemes operate the zones of OVM-I and OVM-II
separately. The hexagonal SVD of a three-phase MLI reprises similar tactic of
working in all the six sectors. Hence the operation is explained for the first sector in
the proposed schemes and the same is applicable for other sectors too.
148
4.4.1 Over Modulation-I Zone (0.907 ≤ Ma ≤ 0.953)
When Ma value lies between 0.907 to 0.953, it is called as OVM-I region.
The dotted circle in Figure 4.4 shows the original trajectory of Vref in OVM zone. In
this region, the Vref traverse outside the hexagon (span-2) for some time and for the
remaining time it lies within the hexagon (span-1 and span-3). Multiplication time t
with this span represent area (volt-sec). It is known that OVM comprises of circular
and hexagonal trajectories and the transition from circular trajectory to the
hexagonal trajectory happens at αc which is constant-unique angle for any given Ma
which is given by Equation (4.5). αc is the angle where the Vref first crosses the
hexagon track as shown in the Figure 4.4. In the conventional OVM techniques, the
samples (p, q, r, s) which are outside the hexagon are limited to lie on the hexagonal
boundary (p|, q|, r|, s|). In this proposed work the magnitude of the voltage reference
samples which are outside the hexagon are modified by using a MCF. However
when the original Vref traverses outside the hexagon, the time average equation gives
an unrealistic on-time.
In Figure 4.4, for a particular value of Ma in the OVM, the path K-B-p-q-
r-s-C-L shows the original trajectory of the Vref and the trajectory is modified to
follow the path K|-B|-C|-L| only by modifying the on-time equations. In OVM-I,
only the magnitude of the Vref is changed whereas the angle is chosen without any
changes. If the reference samples lie in area-1 or area-3, then these samples are
synthesized by available space vectors using NTV scheme. For a chosen angle in the
trajectory, pieces lie between (0 to αc) and (π/3-αc to π/3), the vector moves on
circular track and the modulation strategy is same as that of LM and for remaining
part (αc to π/3-αc)of the sector on hexagonal track.
1
c
a
π πα cos
6 (2 3M
(4.5)
The Si identification and ∆i,j determination are same for both the
trajectories and they differ only in on-time calculation. Null vectors will never
participate when the trajectory is along the hexagon.
149
Figure 4.4. First sector 3-level SVD at OVM-I, (0.907≤ Ma ≤0.953)
4.4.1.1 Circular trajectory
If the reference angle 𝛾 does not satisfy the condition αc ≤𝛾< π/3-αc then
the Vref follows the circular trajectory. Identification of Si, determination of ∆i,j and
on-time calculations of the Vref are explained in the following sections.
4.4.1.2 Sector judgment
For applying the SVPWM technique, first it is required to determine the
sector in which the voltage vector lies. For any given Vref, the angle γ and its Si can
be determined by using the same Equation (4.2) and Equation (4.3).
4.4.1.3 Triangle determination
After the sector identification, the sub-triangle determination is the most
significant one which is discussed in section 3.3.2 with the help of X1 and X2
integers.
150
4.4.1.4 On-time calculations for circular trajectory
On-time calculations are based on the location of the Vref. As mentioned
before, there is a loss of fundamental voltage in the region where the Vref exceeds
the hexagon boundary. To compensate this loss, the Vref amplitude is increased to
K|-B| from K-B by using MDLF as shown in Figure 4.4. So the magnitude of the Vref
is changed from the original Vref. A modified Vref trajectory proceeds partly on the
hexagon and partly on the circle.
a
f
M 0.907MDLF D
0.045
(4.6)
When the trajectory remains on the circular part, the switching time of
Ta, Tb, and To in Equation (4.15)-(4.17) are modified to compensate for the loss in
volt-seconds by introducing a MDLF. For a given Ma, the value of the MDLF is
constant and it is given by Equation (4.6). It is also called as Df, where 0.045 is the
value between maximum value of LM zone to minimum value of OVM-II zone.
Figure 4.5 displays the variation between MDLF and Ma. The MDLF increases
when Ma is increased. The modified on-time equations for down triangle are:
s
βos 2
am s αo f o
VT T V 0.5D T
3
(4.7)
s
βo 2
bm s f o
V T T 0.5D T
h
(4.8)
om s am bmT T T T (4.9)
The modified on-time equations for upper triangle are:
s
βos 2
am s αo f o
VT T V 0.5D T
3
(4.10)
151
s
βo 2
bm s f o
V T T 0.5D T
h
(4.11)
om s am bmT T T T (4.12)
Where, Tam, Tbm and Tom are modified value of Ta, Tb and To
Figure 4.5. MDLF versus Ma
4.4.1.5 Hexagonal trajectory
If the angle 𝛾 satisfies the condition αc ≤𝛾<π/3-αc then the Vref follows the
hexagonal trajectory. The Si judgment and ∆i,j determination can be done in the
similar manner as in the circular trajectory. During hexagonal trajectory, the
coordinates of the Vref are given in terms of angle 𝛾 and level of inverter output n
which are expressed as follows:
α
3 n 1 V
( 3 tan γ)
(4.13)
β
3 n 1 tan γ V
3 tan γ
(4.14)
152
4.4.1.6 On-time calculations for Hexagonal trajectory
The on-time durations for the region where the modified reference
trajectory is moved along hexagonal track can be derived as
s
βos
a s αo
V T T V
3
(4.15)
b s a T T T (4.16)
oT 0 (4.17)
When Vref lies outside the hexagon, the summation of Ta+Tb exceeds Ts and this
begins to appear when the length of the Vref becomes greater than the radius of the
hexagon-inscribed circle (i.e., OVM)
4.4.2 Over Modulation-II Zone (0.953˂Ma≤ 0.999)
In some applications such as field weakening vector control, the drive is
required to operate at extended speeds, and hence the OVM with Ma extending near
to unity is essential [3]. In the OVM-II, the controller affects both the magnitude and
the angle of the Vref [2]. After OVM-I has reached the maximum limit as Ma=0.953,
OVM-II becomes active. Under OVM-II, if the Vref lies in the area-4, then the
essential feature is that the particular active large vector (LV1) closer to the Vref is
used with increasing duty cycle and angle. In OVM-II, the circular part of the
trajectory vanishes and the switching in this region is characterized by a holding
angle (αh).The αh is expressed as a function of Ma, i.e., αh=f (Ma). Its value is limited
between [0 to π/6] as shown in Figure 4.6. Figure 4.7 shows the first sector of 3-
level SVD at OVM II.
153
4.4.2.1 Holding angle calculation
Normally αh is a nonlinear function of modulation index and obtained by
a lookup table. In this chapter, the holding angle αh is obtained using a strategy
similar to one discussed in [133]. αh is the angular velocity of the Vref . The angular
velocity is proportional to modulation index Ma [130].
Figure 4.6. αh versus Ma
(4.18)
hh h
a
h h
a
h h
a
h
a
h
a
h
π π - 2α
α α3 3 = + +M 1 1 0.9535
2 α 2 α π π- = -
1 0.9535 3 M 3 (0.9535)
1.907 α - 2 α π 1 1 = -
0.9535 3 M (0.9535)
π π- 0.2920 α =
0.9535 M
1 2.99α = 0.314-
0.2789 M
10.72α = 11.26-
M
a
h
a
h a
10.72α = 11.26-
M
α = 10.72(1.05-1 /M
)
154
Hence, in Equation (4.18), for a given Ma only two arithmetic operations
i.e. one division and one subtraction are required to obtain the holding angle αh. It
shows the simplicity of implementing SVPWM in overmodulation mode II.
Figure 4.7. First sector of 3-level SVD at OVM-II, (0.953˂ Ma ≤1)
When the Vref is in the span (area-6), it contains an angle from (π/3-αh) to
π/3 such that the vector is made to hold at its respective nearest large vector (LV2).
If the chosen angle is between αh and π/3-αh (area-5) then the Vref follows the
hexagonal path and the Ta and Tb are calculated by using the same on-time
calculation as that during the hexagonal trajectory in OVM-I Equation (4.15)-(4.18).
For a given switching frequency, the current distortion increases with the Ma value.
The distortion strongly increases when the reference waveform become
discontinuous in the OVM-II. During 0≤ 𝛾<αh and (π/3-αh)≤ 𝛾<π/3, the reference
vector is held at one of the six vertices of the hexagon. At Ma=1.0, hexagonal track
vanishes and vector Vref is only held at the six large vectors sequentially. This is six-
step operation similar to 2-level inverter. Therefore, a multilevel inverter when
operated at Ma=1.0, mislays its MLI characteristics [131].
155
4.4.2.2 On-time calculations for hexagonal trajectory
During the on-time for the region where the modified reference trajectory
obeys conditions: 0≤ 𝛾<αh and (π/3-αh)≤ 𝛾<π/3, the reference vector is held at one of
the six vertices of the hexagon and the on-time values are given by,
h a s b o 0 γ α ; T T , T T 0 (4.19)
h b s a o
π π α γ ; T T ,T T 0
3 3 (4.20)
Figure 4.8. Flowchart of calculating on-time in the entire Ma
Go to Δi,j calculation
for linear modulation
Go to Cross over
angle (αc) calculation
Go to Holding
angle(αh)
calculation
NO
LM OVM-I OVM-II
Va,Vb,Vc,
Ma
abc to dq
Transformation
Complex to
polar
Sector identification
Gamma calculation
Uref = 3Ma (n-1)/ π
Ma>0.90 Ma>0.95
Ta = Ts(Vαos − Vβo
s √3 )
Tb = Ts(Vβos h )
To = Ts − Ta − Tb
0 ≤ 𝛾 < 𝛼ℎ; 𝑇𝑎 = 𝑇𝑠, 𝑇𝑏 = 𝑇𝑜 = 0
𝛱 3 − 𝛼ℎ ≤ 𝛾 < 𝛱 3 ; 𝑇𝑏 = 𝑇𝑠, 𝑇𝑎 = 𝑇𝑜 = 0
Ta = Ts(Vαos − Vβo
s √3 )
Tb = Ts − Ta To = 0
Vector mapping
YES NO YES
156
This is summarized as a flowchart and shown in Figure 4.8 and 4.9. It
determines parameters such as sectors and triangles, and calculates on-time. These
details are subsequently used by mapping unit to generate gating signals. Table 4.1
shows the switching sequences of all the sectors for a 3-level SVM in the entire
modulation region.
Figure 4.9. Flowchart of calculating ∆i,j and on-times in OVM-I and OVM-II
αc ≤ γ< π 3 − αc
Compensation
factor
(Cf) calculation
Cross over
angle
Calculation
Holding angle
Calculation
αh ≤ γ< π 3 − αh
Calculation of
modified
on-times
MDLF
𝑉𝛼 = √3 (𝑛 − 1) (√3 + tan 𝛾)
𝑉𝛽 = √3(𝑛 − 1) tan 𝛾 (√3 + tan 𝛾)
𝛼𝑐 = (𝛱 6) − 𝑐𝑜𝑠−1(𝛱 2√3𝑀𝑎 ) 𝛼ℎ = 10.5(1.05 − 1 Ma )
X1 = (n − 2) X2 = int(Vβ h )
Vαos = Vα − X1 + 0.5X2 Vβ0s = Vβ − X2h
Δj = X12 + 2X2
Return
Vector mapping
YES YES NO NO
157
Table 4.1. ∆i,j and on-time calculation for OVM-I and OVM-II
Modulation Timing calculations Si,j
Vector mapping
Linear
modulation(LM)
Ta = Ts(Vαos − Vβo
s √3 )
Tb = Ts(Vβos h )
To = Ts − Ta − Tb
[0-1-1] [0-10] [1-10] [100] [101]
[111] [101] [100] [000] [0-10] [0-1-1] [111]
[101] [1-11] [1-10] [0-10]
[100] [1-10] [1-1-1] [0-1-1]
OVM
-I
Circular
trajectory
Tam = Ta + 0.5Cf2To
Tbm = Tb + 0.5Cf2To
Tom = Ts − Tam − Tbm
[-101] [1-11] [1-10] [0-10]
[0-1-1] [1-10] [1-1-1] [100]
Hexagon
al
trajectory
Ta = Ts(Vαos − Vβo
s √3 )
Tb = Ts − Ta
To = 0
[1-11] [1-10] [1-10] [1-11]
[1-10] [1-1-1] [1-1-1] [1-10]
OVM
-II
Hexagon
al
trajectory
Ta = Ts(Vαos − Vβo
s 3)
Tb = Ts − Ta
To = 0
[1-10] [1-1-1] [1-1-1] [1-10]
[1-11] [1-10] [1-10] [1-11]
Holding
trajectory
Ta = Ts , Tb = To = 0
Tb = Ts , Ta = To = 0
[1-11]
[1-1-1]
S1,1
VZ
VS1 VM6 VS1 VS6 VS6
VS6 VL6 VM6 VS6
VS1 VM6 VL1 VS1
VL6 VL6 VM6 VM6
VM6
VM6 VL1 VL1
VL6 VL6 VM6 VM6
VM6
VM6 VL1 VL1
VL1
VL6
VZ
S1,2
S1,3
S1,4
S1,3
S1,4
S1,3
S1,4
S1,3
S1,4
S1,3
S1,4
To/3 To/3 To/3 Ta/2 Ta/2 Tb/2 Tb/2
Tom/2 Tom/2 Tam Tbm
Tom/2 Tom/2 Tam Tbm
Tb2
Ta/2 Tb/2 Tb/2 Ta/2
Ta/2 Tb/2 Tb/2 Ta/2
Ta
VS6 VS1 VS1 VZ VS6
VL1
VS6 VM6
VM6 VS1 VS6
VS6 VL6
To/2 To/2 Ta/2 Ta/2 Tb
To/2 To/2 Tb Ta
To/2 Ta Tb To/2
Ta/2 Tb/2 Ta/2 Tb/2
Tb/2 Ta/2 Ta/2
Tb
158
4.5 SIMULATION STUDY
The performance of the proposed MLI SVPWM algorithm is investigated
and simulated by MATLAB 11.b/Simulink for 3-level 12 switch NPC-MLI drive
with 300V DC-link, two 100µF capacitors, and 3kHz switching frequency.
4.5.1 Simulink Schematization of Proposed OVM
Figure 4.10. Flow chart for OVM- MATLAB implementation
3-2 Coordinate
Transformation
𝑉∗, 𝛼 Calculator
V∗=
√Vα2 + Vβ
2
γ=
tan−1 V ∝Vβ
Sector
Detector
Module
𝛾 Identifier
K1,K2
Calculator
𝑉𝛼𝑖,𝑉𝛽𝑖
Calculator
Sub
triangle
Identifier
𝑉𝛼𝑜𝑠 ,𝑉𝛽𝑜
𝑠
Calculator
Switching Pattern
for S1
Switching Pattern
for S2
Switching Pattern
for S3
Switching Pattern
for S4
Switching Pattern
for S5
Switching Pattern
for S6
Sub triangle Switching State
Timing Calculations
𝑉𝛼 𝑉𝛽
𝑉∗
𝑉𝛼𝑜𝑠
𝑉𝛽𝑜𝑠
Ts
Ta Tb To
Sector & 𝜸 Identifier Module
Local Vector Generator Module Sector Selector Module
Switching Pulse Generator
Si
Si
𝛾𝑖
SA1
SA2
SA3
SA4
SB1
SB2
SB3
SB4
SC1
SC2
SC3
SC4
Switching
Sequence Park’s Transformation
OVM-I
OVM-II
Trajectory Identifier
Ma
𝛾𝑖
𝛾𝑖
159
Figure 4.10 shows the MATLAB/Simulink simulation model of OVM.
This implementation is similar to LM except the 4th block (sector selector module).
The difference lies in the inclusion of OVM region trajectory identification and its
on-time calculations. The block includes modified trajectory identifier, its angles (αc
and αh ), MCF and the modified switching time calculator which are used to
calculate the timing of the OVM-I and OVM-II regions. The inputs of this block are
types of the sub-triangles, Vαos , Vβo
s and the sampling time period (Ts= 1/fs). In the
same block, the sample and hold blocks are used after Ta and Tb calculator block.
The purpose of these blocks is to hold the values of Ta and Tb fixed during each
PWM period.
The times To, Ta, and Tb are obtained from the calculations and Ts is the
sampling time of inverter. By using these timing values the switching sequence is
determined. Then this timing values are given to the different sub-triangles based on
the sub-triangle block.
4.5.2 Simulation Results
The performance of the proposed scheme is studied and understood by an
extensive simulation study by using MATLAB/Simulink 11.b in entire modulation
ranges, including OVM region with 3kHz switching frequency. Initially, the
simulation study conceded for maximum LM range (0.907), Vline is resulted as
268.4V with THD value of 15.66%. When the inverter is operated in proposed OVM
scheme at Ma values 0.95, 0.98 and 0.99, the Vline(peak voltage) values are observed
as 285V, 294V and 297V respectively. The aforementioned simulation results
confirm the theoretical values.
Figure 4.11 shows the Vline and Iline waveforms for OVM-I at Ma=0.953.
As a result the Vline is observed as 284.7V with VTHD as 19.70% as shown in Figure
4.12. It is demonstrated that the proposed OVM-I scheme utilizes the maximum
modulation boundary till 0.953.
160
Figure 4.11. Simulation results for OVM-I with Ma=0.953; Three-phase Vline
[200V/div] [1ms/div], Iline [2A/div] [1ms/div]
161
Figure 4.12. Simulation results for OVM-I; Harmonic spectrum of Vline at
Ma=0.953
Figure 4.13. Simulation results for OVM-I with Ma=0.99; Three-phase Vline
[200V/div] [1ms/div], Iline [2A/div] [1ms/div]
162
Figure 4.14. Simulation results for OVM-II; Harmonic spectrum of Vline at
Ma=0.99
When Ma value is increased above 0.953, then the proposed scheme
works under OVM-II zone with usages of LVs and this mode extends up to 1.
Superfluously, the proposed SVM-OVM-II scheme works in extreme modulation
range with peak amplitude of 0.99. Figure 4.13 shows the Vline and Iline waveforms
for OVM-I at Ma=0.99. Here, Vline is measured as 297.1V. The harmonic spectrum
of OVM-II is revealed as 27.70% in Figure 4.14 which is less than previously
reported OVM schemes [125], [188], [189]. The line voltage and THD performance
analysis of suggested OVM schemes are carried out and plotted for different Ma.
Figure 4.15 shows the chart of fundamental voltages versus Ma. Based on the study,
it could be understood that the fundamental voltage increases on increasing the Ma.
Figure 4.16 shows the chart of voltage THD for the different Ma values. It is
understood that, as the THD values are increasing, the inverter operates in higher
modulation indices. Particularly in OVM-II zone, the THD is higher because of
pulse dropping.
163
Figure 4.15. Fundamental Vline versus Ma
Figure 4.16. Line voltage THD versus Ma
4.6 IMPLEMENTATION OF PROPOSED OVER MODULATION
FOR NPC-MLI
This section discusses about the experimental setup of SVPWM fed
NPC-MLI drive and FPGA implementation of the suggested OVM scheme is
provided. The test is performed on the same laboratory prototype 2kW NPC-MLI
drive system. Corroboration is done using a SPARTAN III-3A XC3SD1800A-
FG676 DSP-FPGA processor. FPGA implementation of this OVM scheme is done
using Xilinx ISE 10.1 system generator navigator tool.
164
4.6.1 FPGA Implementation Results
4.6.1.1 RTL view
Figure 4.17 shows the overall RTL view of proposed OVM architecture.
This implementation is similar to LM with a small alteration. The difference lies in
the inclusion of OVM region’s trajectory identification and its calculations. The
block includes modified trajectory identifier (circular and hexagonal), its angles (αc
and αh), MCF and the modified switching time calculator, which is used to calculate
the timing of the OVM-I and OVM-II regions arguments. Figure 4.18 shows
ModelSim 5.8e hardware pulse pattern (S1A - S4C) for OVM-II scheme at 0.99 Ma.
Figure 4.17. Overall RTL view of proposed OVM implementation
Figure 4.18. Hardware pulse pattern (S1A-S4C) for OVM-II scheme at 0.99 Ma
using ModelSim 5.8e
165
4.6.1.2 Device utilization report
The recommended work has also considered some key design measures
for improving computation accuracy and simplifying hardware design. Fixed-point
arithmetic unit is adopted for implementing the calculations on FPGA. The observed
(V*) and αc calculations are influenced only by the Ma value. As an outcome, the
modified Vref and associated duty cycle calculation of vectors require very simple
equations as compared to the existing schemes. Hence the proposed OVM scheme
uses less hardware resources, which is 8% of the total resources avilable on the
SPARTAN–III-3A XC3SD1800A-FG676 DSP-FPGA processor board. This is 2%
higher than LM scheme resource utilization on this same FPGA processor. Figure
4.19 depicts the resource utilisation of the proposed algorithm. In addition, due to
simple calculations, the proposed OVM-I and OVM-II algorithms exhibit the
processing speed of 15.4µsec and 16.03µsec respectively which is less compared to
reported scheme [126]. Hence it is very suitable for motor control applications.
Figure 4.19. Resource utilization of the proposed OVM algorithm
166
4.7 EXPERIMENTAL RESULTS AND THEIR ANALYSIS
The corroborating experimental results are captured using six channel
YOKOGAWA make power analyzer. Initially the investigation is carried out for
maximum LM range Ma of 0.907. The output voltage is obtained with help of
maximum SV and MV duty cycle and minimum LV duty cycle as 265.48V peak
Vline with 16.57% THD spectrum. The next experiment is carried out for the
proposed schemes with OVM-I and OVM-II zones. When the drive is accelerated to
OVM-I by increasing Ma to 0.953, the proposed simplified OVM scheme helps to
increase the voltage profile by utilizing maximum LV duty cycle from 265.48V with
minimum computational process. As a result the Vline is observed as 284.24V and
fortified VTHD is 19.94%. Figure 4.20 shows the Vline and Iline waveforms for OVM-I
at Ma=0.953. It is demonstrated that the proposed OVM-I scheme utilizes the
maximum modulation boundary till Ma=0.953. When Ma value is increased above
0.953, then the proposed scheme works under OVM-II zone with the help of 100%
LV duty cycle and this mode extends up to 1. Superfluously, the proposed SVM-
OVM-II schemes works in extreme modulation range with Ma of 0.999.
a
Figure 4.20. (Continued)
167
b
c
Figure 4.20. Experimental results of OVM-I with Ma = 0.953; a. Vline
[300V/div] [1ms/div], Iline [3A/div] [1ms/div], b. Vline [300V/div]
[1ms/div] and its harmonic spectrum, c. Three-phase Vline
[300V/div] [1ms/div], Iline [3A/div] [1ms/div]
168
a
b
Figure 4.21. (Continued)
169
c
Figure 4.21. Experimental results of OVM-II with Ma = 0.998; a. Vline
[300V/div] [1ms/div], Iline [3A/div] [1ms/div], b. Vline [250V/div]
[1ms/div] and its harmonic spectrum, c. Three-phase Vline
[300V/div] [1ms/div], Iline [3A/div] [1 ms/div]
Figure 4.21 shows the Vline, Iline and harmonic spectrum of OVM-II at Ma
=0.99. Here Vline and its harmonic spectrum is measured as 296.35V and 28.64%.
These both OVM-I and OVM-II experimental results are in good agreement with the
simulation results.
Table 4.2 lists the values of peak Vline magnitudes and THD at different
Ma for the proposed SVPWM technique. From the table, it is observed that the
output voltage is equal to 272.87V at Ma=0.917 and with Ma=0.937, it is found that
the Vline=280.42V. This voltage is further increased to a value of 284.25V and
293.23V when the inverter is operated at OVM-I(0.953) and OVM-II(0.985)
respectively. Hence it is evident that the output voltage of the inverter increases as
the operating value of Ma moves towards the OVM region up to 0.99. Based on
these analysis, the Ma value moves towards the OVM region and output voltage
magnitudes obtained are high when compared to LM by properly utilizing the DC-
170
link voltage.The Table 4.2 shows the Vline and its VTHD for the inverter, for different
Ma values. It can be seen that VTHD increases when Ma is increased. At higher Ma
(OVM region), due to result in pulse dropping the THD value is high in OVM.
Table 4.2. Experimental results for the proposed OVM
Zone Ma Vline(V) VTHD (%)
OVM-I
0.917 272.87 17.94
0.927 276.82 18.51
0.937 280.42 18.99
0.947 282.40 19.45
0.953 284.25 19.94
OVM-II
0.968 287.27 22.46
0.975 290.09 24.34
0.985 293.23 26.97
0.995 296.35 28.64
4.8 SUMMARY
In this chapter, the theoretical analysis, design habitat, FPGA
implementation and experimental verification of the proposed OVM, based on
standard 2-level SVPWM, has been successfully exhibited. This scheme uses a
timing correction algorithm (TCA). The on-time calculation equations do not change
with triangle. A simple method of calculating on-times in the over modulation range
is used. Hence a solution to complex equations and LUTs are not required. This
leads to ease of implementation in FPGA. The proposed implementation is general
in nature and can be applied to a variety of modulation schemes. The simulation and
experimental results are presented in order to confirm the performance merits of the
offered OVM at different modulation ranges of the inverter.
171
The salient features of the proposed OVM strategies are as follows:
Duty ratio calculations are similar to 2-level inverter and can be
easily validated for any number of levels n ≥3. There are no
significant changes in computation with the increase in number of
levels.
The considered OVM scheme delivered maximum Vline compared to
the existing schemes [125],[126],[189]
The proposed OVM scheme has low THD compared to the present
SVPWM schemes [125],[189], the existing zero-sequence voltage
injection technique [190] and the optimal carrier based PWM strategy
[191].
Computational Complexity: The proposed algorithm reduces the
memory space in the real time implementation as 8%. In addition, the
algorithm minimizes the processing time of OVM-I and OVM-II as
15.4μsec and 16.03μsec respectively which is less than reported
seheme [126]
The proposed algorithm is tested with open loop three-phase squirrel
cage induction motor speed control drive, accelerated from the LM to
OVM and then to SM, unlike in general OVM reported schemes.
This elucidates that the suggested SVPWM can be directly
functioning to open-loop drives.
The proposed OVM approach maintains the volt-sec balance equation
throughout the modulation range.
The main benefits of the proposed strategy is its simplicity and that it
can be implemented for field weakening operations.
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