A simplified and effective space vector pulsewidth modulation (SVPWM) algorithm with two and three levels for threephase voltagesource converters is proposed in this study. The proposed SVPWM algorithm only uses several linear calculations on threephase modulated voltages without any complicated trigonometric calculations adopted by conventional SVPWM. This simplified SVPWM also avoids choosing the vector sector required by conventional SVPWM. A twolevel overmodulation scheme is integrated into the proposed twolevel SVPMW to generate the output voltage that increases from a linear region to a sixstep state with a smoothly linear transition characteristic and a simple overmodulation process without a lookup table and complicated nonlinear functions. The threelevel SVPWM with a proportionalintegral controller effectively balances the neutral point potential of the neutral point clamped converter. Results from the simulation in MATLAB/Simulink and the experiment based on a digital signal processor are provided to clearly demonstrate the validity and effectiveness of the proposed strategies.
I. INTRODUCTION
Power electronic converters, which have an important role in the field of modern power systems, have been increasingly used in dispersed generation in recent decades, particularly in photovoltaic generators, wind power systems, fuel cells, highvoltage direct current transmission, microgrids, and highly efficient power drives, among others
[1]

[5]
. The performance of power electronic converters relies on the pulsewidth modulation (PWM) technique, which influences converter harmonics, switching losses, and so on.
The PWM technique generally includes two categories, namely, sinusoidal PWM (SPWM)
[6]
,
[7]
and space vector PWM (SVPWM). Compared with SPWM, SVPWM has several merits, such as high utilization ratio of direct current (DC)link voltage, less harmonic, easily digitized implementation, and optimized switching sequence. Therefore, SVPWM is widely used in modern power electronic converters
[8]

[14]
. However, given that the traditional SVPWM technique uses trigonometric functions to calculate the dwell times of selected vectors and distinguish the vector sector, modulation is more timeconsuming and complex than that in the SPWM technique.
To obtain a highly efficient utilization of DClink voltage and increase the output voltage of an inverter, an overmodulation scheme was adopted in
[15]

[18]
. The modulated region of SVPWM was divided into linear and nonlinear regions according to the amplitude of the modulated voltage vector. If the modulated voltage vector trajectory is located at the linear region, then SVPWM is simple; however, DClink voltage cannot be sufficiently utilized. In
[15]
,
[16]
, the overmodulation region was divided into overmodulation modes I and II. Only the amplitude of the reference vector was adjusted in mode I, whereas both the angle and amplitude of the reference vector were rectified in mode II. The methods for rectifying the amplitude and angle of the reference vector are based on the nonlinear functions of the modulation index. Offline calculation and a lookup table are typically employed to reduce calculation. A single overmodulation mode was adopted in
[17]
. Although this mode is simple, harmonics increase and the fundamental voltage gain of the inverter deteriorates. A twomode overmodulation that uses limited trajectories was introduced to achieve linear and realtime modulation in the nonlinear region in
[18]
. Based on
[18]
, the present study proposes a simplified SVPWMbased overmodulation strategy. This strategy obtains linear gains of the fundamental voltage by using simple calculations to rectify modulated voltages. Overmodulation is convenient, simple, and exhibits no complex function.
Threelevel neutral point clamped (NPC) inverters, which produce less harmonics and low du/dt as well as bear a half voltage of the DC link on each power semiconductor switch compared with twolevel inverters, are widely used in mediumvoltage and largecapacity systems
[9]
,
[19]
,
[20]
. In
[21]
, a traditional threelevel SVPWM was introduced. However, this SVPWM is complex and requires trigonometric operations. A threelevel SVPWM was modulated by transforming a normal reference frame into a 60° reference frame in
[11]
. The physical significance of the frame is unclear and the calculation of the phase voltage is complicated in
[9]
. A reference vector decomposition strategy for a threelevel SVPWM was presented in
[19]
. The calculation of the dwell time of the active vectors is the same as that in conventional twolevel SVPWM. The current study analyzes the characteristics of reference vector decomposition, simplifies the modulation process, and presents an absolute twolevel modulation for threelevel NPC converters.
Neutral point potential (NPP) fluctuation is a serious drawback for threelevel NPC converters because it can result in different voltage stresses on semiconductor switches, which may potentially damage semiconductor switches and DClink capacitors. Consequently, converter lifetime can be reduced and system performance can be influenced. Thus, the voltage of the capacitor must be balanced. Methods for NPP balance control have been proposed. Redundant vectors were chosen by a hysteresis controller according to differences in capacitor voltages to balance NPP in
[22]
. Meanwhile, the dwell times of redundant vectors were rearranged according to the direction of a neutral line or a threephase current and voltage deviation of the DC link in
[23]
. Zerosequence voltage was injected in
[24]
. In the present study, NPP is controlled by regulating the dwell time of redundant vectors.
This study proposes a simplified twolevel SVPWM. This SVPWM integrates overmodulation into a twolevel converter and extends to a threelevel SVPWM with NPP control.
The remaining parts of this paper are organized as follows. Section II introduces the simplified SVPWM and its intrinsic relationship with the conventional SVPWM. Section III presents an overmodulation strategy for twolevel converters. The frequency spectrum characteristics in different modulation modes are analyzed based on the proposed SVPWM. The simulation and experimental results are then provided. Section IV presents a threelevel SVPWM. NPP is controlled by a proportional and integral (PI) controller. The simulation and experimental results are shown to demonstrate the validity of the threelevel SVPWM. Finally, Section V concludes the paper.
II. SIMPLIFIED TWOLEVEL SVPWM
 A. Simplified SVPWM Algorithm
A twolevel voltage vector structure is shown in
Fig. 1
(a). When a reference vector is located at sector
N
1, (1) can be obtained via voltsecond balance theory according to the traditional SVPWM algorithm.
T
_{1}
and
T
_{2}
are the dwell times of active vectors
V
_{1}
and
V
_{2}
, respectively. In (2), calculation includes trigonometric functions, which expends the computational periods of the microprocessor. As the solution of dwell times is based on the vector sector, conventional SVPWM complicates the control program and the distribution of the threephase dwell times is not intuitional.
Twolevel space vector. (a) Twolevel voltage vector structure. (b) Relationship between voltage phase angle and the sector. (c) Duration time of high potential in each phase. (d) Duration allocation.
where
V
_{dc}
is the DClink voltage, and
T
_{0,7}
is the dwell time of the zero vector (ZV; 000, 111).
Therefore, a simplified SVPWM is proposed to simplify calculations. The calculation steps of the simplified SVPWM are as follows.
Step 1:
M
_{j}
= min
U
_{j}
^{*}
; (
U
_{j}
^{*}
=
U
_{j}
/
V
_{dc}
); the smallest phase voltage is chosen.
Step 2:
S
_{j}
=
T
_{s}
(
U
_{j}
^{*}

M
_{j}
);
S
_{j}
represents the dwell time of the driver signal that is equal to 1 in phase
j
,
T
_{s}
is a switching period.
Step 3:
t
_{z}
=
T
_{s}
 max
S
_{j}
;
t
_{z}
is the duration of ZV.
Step 4:
t
_{cmj}
= (
T
_{s}

S
_{j}

t
_{z}
/2)/2;
t
_{cmj}
is a compared value of phase
j
for sevensegment switching ,
where
j
= (
a
,
b
,
c
), which represents phases
a
,
b
, and
c
.
Compared with conventional SVPWM, the proposed SVPWM is simpler and easier to digitize.
 B. Validating the Simplified SVPWM
The consistency of the simplified and conventional SVPWM algorithms and the validity of the proposed SVPWM are verified through theoretical deduction in this section.
Fig. 1
(b) shows the relationship between the phase angle of the modulated voltage and the vector sectors. In sector
N
1, the phase angle changes from 90° to 150°, and the relationship of phases
a
,
b
, and
c
is
U
_{a}
>
U
_{b}
>
U
_{c}
, with
U
_{c}
< 0. Min(
U
_{a}
^{*}
,
U
_{b}
^{*}
,
U
_{c}
^{*}
) =
U
_{c}
^{*}
can be obtained according to step 1.
By substituting
U
_{c}
^{*}
into step 2, it can be obtained:
where
S
_{a}
≥
S
_{b}
.
The corresponding dwell times expressed in (3) are shown in
Fig. 1
(c). The three nearest vectors are chosen directly by (3).
Substituting
S
_{a}
into step 3, the dwell time of ZV (000, 111) can be obtained as follows:
According to step 4,
When the conventional SVPWM algorithm is used, (1) can be expressed as follows:
According to the principle of the three nearest vectors and by simplifying (6), it is obtained the following:
where 
V
_{1}
 = 
V
_{2}
 = 2
V
_{dc}
/ 3.
The dwell times of
T
_{0}
,
T
_{1}
, and
T
_{2}
are shown in
Fig. 1
(d). Equations (6) and (7) indicate that sector number should be determined first. Then, based on the sector number, the three nearest vectors are chosen to calculate
T
_{0}
,
T
_{1}
, and
T
_{2}
. Considering that the three nearest vectors vary in different sectors, the calculations of
T
_{0}
,
T
_{1}
, and
T
_{2}
differ in each sector, which increases the complexity of the conventional SVPWM. The calculation of the simplified SVPWM in (3) depends only on the modulated voltage value. Therefore, the simplified SVPWM is indeed simpler than the conventional SVPWM.
Comparing (8) with (3) is difficult because of the variables in different coordinate frames. By transforming the variables in (8) into a threephase stationary coordinate frame to compare the results clearly, it can be obtained the following:
For the proposed SVPWM,
S
_{a}
,
S
_{b}
, and
S
_{c}
in (3) are the durations of the driver signal that are equal to 1 for phases
a
,
b
, and
c
, respectively. In (9),
T
_{0}
,
T
_{1}
, and
T
_{2}
are the dwell times of the three nearest vectors
V
_{0}
(000, 111),
V
_{1}
(100), and
V
_{2}
(110) shown in
Fig. 1
(a), respectively. Without ZV (111), the duration of the driver signal that is equal to 1 for phase a is
T
_{2}
+
T
_{1}
=
T
_{s}
(
U
_{a}
^{*}

U
_{c}
^{*}
) and that for phase
b
is
T
_{2}
=
T
_{s}
(
U
_{b}
^{*}

U
_{c}
^{*}
), which are the same as in (3).
Using a sevensegment switching sequence for modulation, the compared value can be obtained as follows:
Equations (10) and (5) are the same. Based on the deductions and results in this section, vector modulation in other sectors can be validated and is omitted.
 C. Physical Significance of the Simplified SVPWM
To explain the simplified SVPWM intuitively, the synthetic process of reference vector
V
_{f}
is introduced in this section. Given that reference vector
V
_{f}
is located at sector
N
1, as shown in
Fig. 1
(a), calculation is conducted through (3) to (5). The highlevel duration of each phase may be obtained through (3), as shown in
Fig. 1
(c). Based on (3) and
Fig. 1
(c), ZVs (
V
_{0}
,
V
_{7}
) and two vectors
V
_{1}
and
V
_{2}
are used to synthesize reference vector
V
_{f}
, which satisfies the principle of the three nearest vectors. The compared values are obtained via (5) according to sevensegment switching, as shown in
Fig. 1
(d). The relationship between (3) and (5) is shown in
Fig. 1
(c) and 1(d). With the proposed SVPWM, the highlevel duration of each phase can be easily generated by using simple arithmetic operations on the threephase modulated voltages. The preceding analyses reveal that physical significance is clear, and (3) to (5) are intuitively expressed in
Fig. 1
(c) and 1(d).
According to the sevensegment modulation, the relationships among the dwell times of vectors
T
_{0}
,
T
_{1}
, and
T
_{2}
and the compared values of each phase for the conventional SVPWM are shown in
Fig. 1
(d).
III. OVERMODULATION FOR THE SIMPLIFIED SVPWM
The ratio of the actual and maximum amplitudes of the output phase voltage by the converter is defined as the modulation index
[25]
, that is,
where
U
_{r}
is an amplitude of the converter output phase voltage, and
U
_{b}
= 2
^{*}
V
_{dc}
/π is the maximum amplitude of the phase voltage when the converter works at a sixstep state.
According to the magnitude of the modulation index, the modulation region may be divided into three categories, namely, the linear modulation mode, overmodulation mode I, and overmodulation mode II, as shown in
Fig. 1
(a). In the linear modulation region, index
m
is between 0 and 0.969, and voltage vector trajectory is within the inscribed circle of the vector hexagon. In overmodulation mode I, the value of
m
changes from 0.9069 to 0.9514, and its region is between the inscribed circle and the hexagon. In overmodulation mode II,
m
increases from 0.9514 to 1, and vector trajectory is on the hexagon. These descriptions indicate that the trajectory of the voltage space vector is continuous in linear modulation mode and overmodulation mode I. However, the trajectory is discontinuous in overmodulation mode II. The most extreme condition is that the voltage vector is located only at six endpoints of the hexagon when
m
= 1.
 A. Principle of Limited Trajectory
According to linear theory, if a variable
B
∈[
A, C
], then
B
can be expressed as follows
[18]
:
Therefore,
A
,
C
, and
k
can synthesize
B
by using the linear relationship if
B
is between
A
and
C
. If
A
is a constant, then a constant
k
must exist to satisfy the relationship in (12). Similar to the linear relationship among constants, if voltage space vector
is between
and
, then a constant
k
may be found to synthesize
in
[18]
.
By transforming the modulated voltage vector into a threephase stationary reference frame, we obtained the follows:
where
U
_{j_ref}
is the rectified voltage vector; and
U
_{j_in}
and
U
_{j_out}
are the inner and outer boundary voltages, respectively.
For a reference voltage, the modulation index can be obtained and expressed by
B
. By substituting
B
and boundary modulation indices
A
and
C
into (12), an equivalent modulation index
k
is derived. Given that
U
_{j_in}
and
U
_{j_out}
in (13) are the inner and outer boundary voltages with modulation indices
A
and
C
, respectively, the rectified reference vector is achieved by substituting
k
, as well as the inner and outer boundary voltages, into (13). The rectified reference vector satisfies the modulation index requirement and constrains the voltage space vector trajectory within the hexagon.
Based on the previous analyses, an overmodulation strategy is easily integrated into the simplified SVPWM, which may keep vector transits in different modulation modes smooth. The process is studied in detail in the following sections.
 B. Linear Modulation Mode
In linear modulation mode, the boundary conditions of the modulation indices are
A
= 0 and
C
= 0.9069, and the reference vector modulation index is
B
=
m
, where 0 ≤
B
≤ 0.9069. Substituting them into (12) results in
k
=
m
/0.9069. The rectified vector can be obtained by substituting
k
, as well as the inner and outer unit vectors, into (13). The simplified SVPWM strategy can be used directly because the inner vector boundary is zero, and the outer reference boundary is the inscribed circle of the hexagon.
 C. Overmodulation Mode I
Given that the region of overmodulation mode I is between the inscribed circle and the hexagon, two boundary modulation indices exist. The inner one is the maximum linear modulation index, and the external one is the maximum modulation index of overmodulation mode I. These indices are chosen as boundary conditions
A
and
C
, respectively. Thus,
A
= 0.9069,
C
= 0.9514, and
B
=
m
. By substituting
A
,
B
, and
C
into (12), it can be obtained that
k
= (
m
 0.9069)/(0.9514  0.9069). By substituting
k
into (13), the rectified voltage can be expressed as follows:
where voltage vector
U
_{j}
is a unit vector, and 0.577
U
_{j}
is the inner boundary vector.
k
U
_{j}
/
U
_{j1}
is the external boundary vector. The equivalent modulation index
k
is generated from (12), which may guarantee that the amplitude of rectified vector
U
_{j_rec}
is equal to those of the reference vector and the voltage trajectory within the hexagon.
 D. Overmodulation Mode II
Considering that the overmodulation mode II region is outside the overmodulation mode I region, the two boundary conditions are the maximum modulation indices of overmodulation modes I and II, respectively. Therefore,
A
= 0.9514,
C
= 1, and
B
= m. By substituting
A
,
B
, and
C
into (12), we can obtain
k
= (
m
 0.9514)/(1  0.9514). According to (13) and
Table I
, we have
BOUNDARY VALUES OFUA1,UB1,UC1ANDUA2,UB2,UC2
Sectors N and M are based on the voltage space vector. M lags N by 30°. All variables are in per unit and base value is modulated voltage amplitude.
Rectifying the threephase reference voltage by (14) and (15), the amplitude of the rectified voltage is equal to that of the reference voltage, and the trajectory of the rectified reference voltage vector is within the hexagon. The inner and external voltages in different modulation regions are summarized in
Table I
.
 E. Simulation Results
The reference vector in different modulation regions is simulated in MATLAB/Simulink.
Fig. 2
shows the waveforms of the voltage vector trajectories and output phase voltages in the three modulation modes.
Fig. 2
(a) and 2(b) show the results in the linear modulation region. The vector trajectories are circles and the phase voltage waveforms are sinusoidal. When the modulation index increases from 0.5 to 0.9069, the amplitude of the output phase voltage also increases.
Fig. 2
(c) and 2(d) show the voltage vector trajectories and output phase voltages in overmodulation mode I. The trajectory changes from a circle to a hexagon when the modulation index increases from 0.9069 to 0.9514. The vector trajectories and output voltage shown in
Fig. 2
(e) and 2(f) are the results in overmodulation mode II. When modulation index
m
=0.9514, the vector trajectory is a hexagon. When modulation index = 1, the vector trajectory is located at six endpoints of the hexagon and the converter works at a sixstep state.
Fig. 2
shows that the amplitude of phase voltage increases with increasing modulation index.
Different modulation modes. (a) Voltage trajectory at the linear modulation region with m = 0.5, 0.9069. (b) Phase voltage at the linear modulation region with m = 0.5, 0.9069. (c) Voltage trajectory at overmodulation mode I with m = 0.9069, 0.93, 0.9514. (d) Phase voltage at overmodulation mode I with m = 0.9069, 0.93, 0.9514. (e) Voltage trajectory at overmodulation mode II with m = 0.9514, 0.97, 1. (f) Phase voltage at overmodulation mode II with m = 0.9514, 0.97, 1.
 F. Voltage Harmonics Analysis
Based on previous analyses of the overmodulation strategy, the mathematical expressions of a threephase modulated voltage can be determined by (14) and (15), as well as
Table I
, by choosing
U
_{j1}
and
U
_{j2}
. The modulated threephase voltage is expanded by the following Fourier series:
The main components of the phase voltage frequency spectrum are loworder harmonics such as 5th, 7th, 11th, and 13thorder harmonics.
Fig. 3
shows the frequency spectrum of the output phase voltage with different modulation indices. The percentages of harmonics increase correspondingly with increasing modulation index.
Harmonic analysis with different modulation indices.
The linear characteristic of the fundamental components of the output phase voltage versus the modulation index is shown in
Fig. 4
. The effectiveness of the proposed SVPWM based on an overmodulation scheme in obtaining a continuous output voltage from linear modulation to overmodulation with a linear characteristic is highlighted.
Fundamental components of the output voltage as a function of the modulation index.
 G. Experiment Results
A 10 kW twolevel inverter prototype is set up to verify the proposed method by using a 2.2 kW asynchronous induction squirrelcage motor.
Fig. 5
(a) shows thestructure diagram of the twolevel inverter based on a motor drive system. The induction motor is driven via the twolevel inverter controlled by a TMS320LF2812 digital signal processor (DSP, Texas Instruments Inc., Texas, USA) with a 5 kHz switching frequency. The electrical parameters of the motor are given in Table IV. LA28NP and LV28P are used to measure current and voltage, respectively. A BSM 50 GB 120 DLC insulatedgate bipolar transistor (IGBT) is used as a switch for the twolevel inverter. A constant
V
/
f
control is employed to control the motor.
Experimental structure diagram and waveforms. (a) Structure of the motor controlled by the twolevel inverter. (b) Line voltage u_{ab} and phase current i_{a} in the linear modulation region. (c) Line voltage u_{ab} and phase current i_{a} at overmodulation mode I. (d) Line voltage u_{ab} and phase current i_{a} at overmodulation mode II.
Fig. 5
(b) to
5
(d) show the stator line voltage and phase current waveforms when the inverter works at linear modulation mode, overmodulation mode I, and overmodulation mode II, respectively. With increasing modulation index, total harmonic distortion increases in overmodulation modes. The current is smooth and sinusoidal in linear mode. By contrast, the current waveforms are distorted in overmodulation modes I and II.
The stator current may be influenced by the dead time, switch nonlinear characteristic, sampling error and delay, and so on. Given that the induction motor is controlled by a simple and basic
V
/
f
control, speed and current loops are not employed. Stator current may also be directly influenced by stator voltage and other factors. If stator current is controlled by a PI controller, then stator current waveforms will be smooth and stator current distortions will be obviously reduced.
IV. THREELEVEL SVPWM
 A. Simplified ThreeLevel SVPWM
Based on the decomposition algorithm introduced in
[19]
, the simplified threelevel SVPWM is studied in this section. A threelevel vector hexagon is divided into six small subhexagons(labeled with
S
1,
S
2, ...
S
6.). Each hexagon represents a twolevel space vector hexagon, as shown in
Fig. 6
(b). The simplified twolevel SVPWM is employed directly in each small subhexagon. Two power semiconductor switches are attached to each leg of the twolevel converter. An output state variable
S
, with a value equal to 1 or 0, is defined to represent output voltage. When
S
= 1, output voltage is
V
_{dc}
; when
S
= 0, output voltage is 0, but the threelevel converter has three switching states.
Fig. 6
(a) shows that each phase includes four semiconductors; for example,
S
_{a1}
,
S
_{a2}
,
S
_{a3}
, and
S
_{a4}
are in phase α. The switching states of
S
_{a1}
and
S
_{a2}
, as well as of
S
_{a3}
and
S
_{a4}
, are opposite. The output of phase a is simultaneously determined by
S
_{a1}
and
S
_{a2}
. One of these semiconductors is decided by the central vector, which is at the center of the subhexagons (
S
1 to
S
6). Another semiconductor adjusts the amplitude of the output voltage vector. For example, if the central vector is 001 in subhexagon
S
5,
S
_{a1}
= 0,
S
_{b1}
= 0, and
S
_{c2}
= 1 in a switching period. For a central vector expressed by
abc
, if α = 0, then
S
_{1}
= 0; if
α
= 1, then
S
_{2}
= 1. When
S
_{a1}
= 0, phase voltage is 0 or 
V
_{dc}
/2; when
S
_{a2}
= 1, output phase voltage is
V
_{dc}
/2 or 0. The relationship among the threelevel switching variables that are associated with subhexagon number is listed in
Table II
.
Threelevel converter and vector diagram. (a) Structure of the diodeclamped threelevel converter. (b) Region division of the decomposed threelevel space vector diagram. (c) Decomposition of a voltage reference vector.
RELATIONSHIP AMONG SECTOR, CENTRAL VECTOR, AND SWITCHING STATE
RELATIONSHIP AMONG SECTOR, CENTRAL VECTOR, AND SWITCHING STATE
The subhexagon number and central vector can be directly obtained from the modulated voltage. During a switching period, one of the switching variables is constantly equal to 0 or 1 as determined by the subhexagon number. Therefore, only one switching variable remains to be modulated for each leg. The threelevel SVPWM is then equivalent to a twolevel SVPWM.
 B. Neutral Potential Control
A threelevel space vector consists of 27 vectors, which can be divided into four groups, namely, ZV, small vector (SV), middle vector (MV), and large vector, depending on their magnitudes. Given that SV and MV directly connect to the neutral point, the neutral potential can be fluctuant when different SVs and MVs are chosen. Each SV includes a pair of redundant vectors with an opposite influence on the neutral potential and the same effect on synthesizing the reference vector. SVs can be divided into two categories, namely, positive SV (PSV) and negative SV (NSV), according to their influence on the neutral potential. Neutral potential balance may be achieved by adjusting the duration of PSV and NSV during each switching period.
Voltage drop
V
_{L}
across an inductance connected in series with the converter is generally less than 30% of grid voltage
E
. By neglecting the internal resistance of the inductance and maintaining converter work at a unit power factor state, the impedance angle
r
= arctan(
V
_{L}
/
E
) < arctan0.3 = 17.7° is obtained. The waveforms of threephase voltages and currents when the converter works as a rectifier are shown in
Fig. 7
. If the converter works as an inverter,
r
= 180° to 17.7° = 162.3°. According to
Fig. 7
, the relationships between neutral current and NSV in different regions are shown in
Table III
.
Reference voltage and current of the converter (r = 17. 7°).
RELATIONSHIP AMONG SECTORS, NEUTRAL CURRENT, AND PHASE CURRENT
RELATIONSHIP AMONG SECTORS, NEUTRAL CURRENT, AND PHASE CURRENT
NSV increases the neutral potential, whereas PVS decreases it. In
[26]
, an adjusted coefficient
f
was used to control the NPP. The coefficient is given as follows:
where
t
_{n}
and
t
_{p}
are the duration times of NSV and PSV, respectively.
When the coefficient
f
is controlled by a hysteresis
[22]
, that is,
The
f
value is decided by the voltage difference of the upper and lower capacitor voltages in the DC link, as shown in
Fig. 6
(a). If
f
= 1 in (17), then
t
_{n}
= 0 and
t
_{p}
=
t
_{z}
, which indicate that only PSV is selected as an active vector in the current switching period. Otherwise, NSV is chosen. If
f
is equal to 1 or 1, then the switching sequence changes from seven segments to five segments. For example, according to sevensegment switching, a vector sequence, that is, 011, 001, 101, 100, 101, 001, and 011, is used to synthesize a reference vector in
S
1
N
1, as shown in
Fig. 6
(c). When
V
_{dc1}

V
_{dc2}
> 0, f = 1,
t
_{n}
= 0, and
t
_{p}
=
t
_{z}
, the vector sequence becomes 001, 101, 100, 101, and 001. The switch state changes from 1 to 1, which increases harmonics and switching loss.
To avoid the aforementioned problems, a PI controller is used to replace the hysteresis controller based on a control strategy for the neutral potential balance control. The coefficient is redefined as follows:
where
K
_{p}
is the proportional gain of the PI controller, and τ is an integral time constant.
The PI controllerbased voltage balancing method is similar to the hysteresis controllerbased strategy. The difference between them is obtaining coefficient
f
. For the hysteresis controller, when NPP deviates from zero,
f
= 1 or 1; for the PI controller, the coefficient f is within 1 to 1. If
f
= ±1, then the situation is the same as that in the hysteresis controllerbased strategy. The switching mode is changed from sevensegment to fivesegment. The PI controller can guarantee that
f
continuously changes within 1 to 1, and the modulation mode is sevensegment switching at a steady state. Therefore, the PI controllerbased method can improve performance, maintain sevensegment modulation, and reduce the switching loss at a steady state.
The simulation results based on MATLAB/Simulink show the dynamic response of the phase current and DClink voltage of a diode clamped threelevel voltagesource rectifier by using the two methods shown in
Fig. 8
.
Current waveforms of phase a. (a) Neutral potential controlled by the PI controller. (b) Neutral potential controlled in (16).
Neutral voltage deviation is 100 V, and neutral voltage is controlled at 0.16 s. With the hysteresis controller, ripples occur in the DClink voltage, and the alternating current (AC) is distorted at a steady state. With the PI controller, the DClink voltage and AC current are both smooth. The dynamic responses of the two methods are nearly the same because both methods are based on the change in dwell times of PVS and NVS. Hence, the PI controllerbased method may improve performance, particularly at a steady state.
 C. Experiment Results
A threelevel NPC rectifier prototype is built in the laboratory. A TMS320C2812 DSP is used to sample voltage and current, process data, and generate PWM drive signals. A Eupec BSM 50 GB 120 DLC IGBT driven by CONCEPT 2SD315A is used as the power switch. The parameters of the converter are listed in Table V in the Appendix, and the waveforms are obtained by a TPS2024 oscilloscope.
Fig. 9
(a) shows the PWM drive signals, and
Fig. 9
(b) presents the grid current and voltage of phase α. The converter works at a unit power factor state, the peak voltage value is 311V, and the current is 3.9A.
Fig. 9
(c) shows the upper capacitor voltage
V
_{dc1}
and lower capacitor voltage
V
_{dc2}
, which are both 400V. The voltage difference between
V
_{dc1}
and
V
_{dc2}
is smaller than ±1.5V. Thus, the proposed method effectively suppresses neutral potential deviation. The total DClink voltage that is equal to the sum of
V
_{dc1}
and
V
_{dc2}
is smooth, as shown in
Fig. 9
(d).
Waveforms of the threelevel voltagesource converter. (a) PWM of phase a. (b) Voltage and current of phase a. (c) Voltages of V_{dc1} and V_{dc2} and their difference. (d) Voltages of the DC link, V_{dc1}, and V_{dc2}.
V. CONCLUSIONS
This study proposes a simplified SVPWM strategy. The intrinsic relationship between the simplified and conventional SVPWMs is deduced to verify the theoretical validity of the proposed SVPWM. The proposed SVPWM reduces computational requirements, such as avoiding trigonometric operations and vector sector selection. Moreover, digital implementation is easy. The proposed SVPWM is useful for highperformance converters.
A simplified SVPWMbased overmodulation strategy for a twolevel converter is also proposed to increase output voltage with a linear characteristic. This strategy only requires arithmetic calculations to rectify threephase modulated voltages.
Based on the central vector, the threelevel SVPWM is equivalent to a twolevel SVPWM for the NPC converter. A PIbased NPP strategy suppresses voltage deviation at the upper and lower capacitors with nearly no influence on the DClink voltage. This strategy also reduces current harmonics and improves dynamic and steady state performances.
The proposed strategies are all based on simple and linear calculations, which reduce the requirements for the processor and control program design. The simulation and experiment results clearly demonstrate the validity and feasibility of the proposed strategies. The overmodulation control strategy is easily expanded to a threelevel SVPWM. However, this strategy should be combined with neutral potential control, which will be the subject in a future work.
Acknowledgements
This work was supported by the Fundamental Research Funds for the Central Universities (China University of Mining and Technology) (2014ZDPY17). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that help improve the quality of this manuscript.
BIO
RongWu Zhu was born in Xuzhou, China. He received his B.S. and M.S. in Electrical Engineering from Nanjing Normal University in 2007 and China University of Mining and Technology in 2009, respectively. He is currently working toward his Ph.D. at the Department of Information and Electrical Engineering, China University of Mining and Technology. His current research interests include highpower multilevel converters, nonlinear control, and wind power systems.
XiaoJie Wu was born in Hengyang, Hunan Province, China in 1966. He received his B.S. in Industrial Automation from China University of Mining and Technology, Xuzhou, China in 1988, and his M.S. and Ph.D. in Electrical Engineering from the same university in 1991 and 2000, respectively. He was engaged in a postdoctoral research at Tsinghua University, Beijing, China from 2002 to 2004. Since 1991, he has been with the Department of Information and Electrical Engineering, China University of Mining and Technology, where he is currently a professor. His current research interests include stability of AC machines, advanced control of electrical machines, and power electronics.
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