A novel carrierbased PWM (CBPWM) strategy of a threelevel NPC converter is proposed in this paper. The novel strategy can eliminate the lowfrequency neutral point (NP) voltage oscillation under the entire modulation index and full power factor. The basic principle of the novel strategy is introduced. The internal modulation wave relationship between the novel CBPWM strategy and traditional SPWM strategy is also studied. All 64 modulation wave solutions of the CBPWM strategy are derived. Furthermore, the proposed CBPWM strategy is compared with traditional SPWM strategy regarding the output phase voltage THD characteristics, DC voltage utilization ratio, and device switching losses. Comparison results show that the proposed strategy does not cause NP voltage oscillation. As a result, no lowfrequency harmonics occur on output linetoline voltage and phase current. The novel strategy also has higher DC voltage utilization ratio (15.47% higher than that of SPWM strategy), whereas it causes larger device switching losses (4/3 times of SPWM strategy). The effectiveness of the proposed modulation strategy is verified by simulation and experiment results.
I. INTRODUCTION
Threelevel NPC converter is one of the most commonly used multilevel converters, of which the main circuit is shown in
Fig. 1
. Compared with traditional twolevel converter, threelevel NPC converter has larger output power, smaller output phase voltage THD, less device voltage stress, and lower system EMI. Therefore, the latter is widely applied to highpower system, active power filter, and power system’s reactive power compensation
[1]

[5]
.
Neutral point (NP) voltage generally has to be kept at
U_{dc}
/2 for normal operation of a threelevel NPC converter. However, obvious NP voltage oscillations occur in real systems because the NP current
i_{o}
is not zero in most cases. If the NP voltage is voltage will be unbalanced
[6]
,
[7]
. The influence will be not controlled, oscillation will occur and the DC capacitor transferred to ac side; thus, lowfrequency harmonics will appear in the output voltage.
Three methods can be used to control NP voltage oscillations. The first method is achieved by adding an additional hardware NP voltage control circuit in the system
[8]
. With energy absorbing and feedback from energy storage elements to DC capacitors, the NP voltage is clamped to
U_{dc}
/2. However, this method increases the hardware cost and control complexity of the system. Therefore, this method is not commonly used in NPC converters.
The second method is to add a software NP voltage control scheme
[9]

[11]
. The control scheme normally modifies the reference signals or output switching states of the modulation strategy to eliminate NP voltage oscillation. In traditional threelevel SPWM strategy, zerosequence component injection scheme is typically used as an NP voltage control scheme. However, considering the fact that the amplitude of the final reference signals should not be greater than that of the carriers, the scheme cannot be used in highmodulation index and lowpower factor situations
[12]

[14]
.
The final scheme is to design a modified modulation strategy with a capability of DC voltage balance
[15]
. In
[16]
and
[17]
, the lowfrequency oscillation in NP voltage was eliminated using virtual vectors in spacevector modulation strategy. Nevertheless, the algorithm was finally implemented using a carrierbased PWM. This algorithm also has to deal with angles and trigonometric functions, which complicates its application. In
[18]
and
[19]
, a threelevel carrierbased PWM (CBPWM) strategy is proposed by dividing the single modulation signal of traditional threelevel SPWM strategy into two decoupled signals. However, this strategy only gives a specific solution rather than all solutions to the two modulation waves. Moreover, they did not study the output THD performance, which is an important evaluation index.
The current study investigates a novel CBPWM strategy, which can eliminate lowfrequency NP voltage oscillation with any modulation index and power factor. The remaining of this paper is divided into the following sections. Section II studies the modulation wave relationship between the novel strategy and traditional SPWM strategy based on voltage equivalent and NP voltage balance principle. Section III deduces 64 modulation wave solutions of the novel PWM strategy. Section IV analyzes the output phase voltage THD characteristics, DC voltage utilization ratio, and system device switching losses of the novel strategy. Section V verifies the correctness of the theoretical analysis by simulation and experiment results. Finally, Section VI concludes.
II. MODULATION WAVE RELATIONSHIP BETWEEN THE NOVEL STRATEGY AND THE SPWM STRATEGY OF A THREELEVEL NPC CONVERTER
The novel strategy is based on the use of two modulation signals for each phase of the converter, as shown in
Fig. 2
. The two modified modulation signals of each phase satisfy the following expression:
where
U_{x}
stands for the modulation wave of traditional SPWM strategy;
U_{xp}
and
U_{xn}
stand for the two modulation signals of the phase
x
(
x
=
a
,
b
,
c
) in the novel strategy. The signals with subscript
p
will only cross the upper carrier to control the switching devices
S
_{x1}
and
S
_{x3}
, whereas the signals with subscript
n
will only cross the lower one to control the switching devices
S
_{x2}
and
S
_{x4}
.
U
_{xpmax}
and
U
_{xnmin}
are the maximum and minimum values of the two modulation signals, respectively.
During modulation, the variable
s_{xo}
defines whether the output state of the phase
x
is “O,” as shown in
Fig. 1
,
s_{xo}
∈ {0, 1}. When
s_{xo}
has a unity value, the subsequent phase current
i_{x}
will flow through NP; otherwise, the phase current
i_{x}
has no effect on NP current. Therefore, the NP current
i_{o}
can be expressed as follows:
Scheme of the threelevel NPC voltage source. Converter.
To preserve NP voltage balance, the locally averaged NP current must be zero. Therefore, the averaged NP current, instead of the instantaneous current, must be utilized. The averaged NP current is obtained by using the moving average operator, which is expressed as follows:
Applying this operator to Equation (2), we have
where
d_{xo}
=
for
x
= {a, b, c}.
and
define the averaged NP and phase currents, respectively. Assuming that the frequency of the carriers in the novel strategy is higher than that of the modulation signals, the duty cycle
d_{xo}
can be expressed as follows:
The following equation is obtained from Equations (2) to (5):
To remove the lowfrequency NP voltage oscillation under any modulation index and power factor, the average NP current
should always be zero. Given that the load is starconnected or triangleconnected (
Fig. 1
), no zerosequence component exists in the current
i_{x}
. Thus, the sum of the average phase currents (
) is always zero, that is,
If
then Equation (6) would be
Considering that different circuits have varied load conditions and power factors, Equation (9) is a simple and useful solution of
=0. Substituting Equation (9) into Equation (1), the two modulation signals of each phase in the novel CBPWM strategy are obtained as follows:
In this section, the relationship between the modulation wave
U_{x}
of traditional SPWM strategy and the modulation waves
U_{xp}
and
U_{xn}
of the CBPWM strategy is deduced, as shown in Equation (10). Only one variable (
k
) is presented in Equation (10). The solution of
k
will be studied in Section III.
III. MODULATION WAVE SOLUTIONS OF THE NOVEL PWM STRATEGY
In Section II, the expressions of modulation wave of the novel strategy have been deduced. The final problem is to find available solutions to
k
shown in Equation (10). An infinite number of solutions can generally be found. In the present section, solutions of
k
will be studied in detail.
In traditional threelevel SPWM strategy, each phase has only one modulation signal. If
U_{dc}
/2 is selected as the base value, the threephase positivesequence reference voltages
U
_{a0}
,
U
_{b0}
, and
U
_{c0}
in traditional SPWM strategy can be normalized as
where
M
is the modulation index, and
ω_{s}
is the angular frequency of fundamental wave. To utilize DC voltage and improve the system DC voltage utilization ratio, a common method is to inject a zerosequence voltage (
U_{z}
) into the threephase reference voltages. Hence, the actual reference voltages can be given by
In this situation, the modulation waves
U_{xp}
and
U_{xn}
of the novel strategy can be expressed as Equation (13). Two variables
k
and
U_{z}
are presented in the equation. Their appropriate values will make
U_{xp}
and
U_{xn}
equal to their extreme values (0 or 1 for
U_{xp}
, 0 or −1 for
U_{xn}
), which are only the extreme values of the upper and lower carriers, as shown in
Fig. 2
. Consequently, the output switching state will remain unchanged, which will decrease the system switching losses. In such cases,
k_{xy(j)}
, which defines the solution of
k
when the value of modulation waves
U_{xy}
(
x
=
a
,
b
,
c
;
y
=
p
,
n
) is
j
(
j
= 1, 0, −1), is expressed as Equation (14).
Diagram of the novel strategy of a threelevel NPC.
In Equation (14), two values of
k_{xy(j)}
can form simultaneous equations to solve the zerosequence voltage (
U_{z}
). As a result, the variable
U_{z}
yields two solutions. The first solution (
U
_{z1}
) is obtained by solving two simultaneous equations of
k_{xy(j)}
of the same phase, as shown in Equation (15). The second solution [
U
_{z2}
in Equation (15)] is yielded by solving two simultaneous equations of
k_{xy(j)}
of different phases. Further research shows that
U
_{z1}
will cause the duty cycle
d_{xo}
equals to zero, which will make the threelevel PWM strategy degenerate into twolevel PWM strategy. Therefore,
U
_{z1}
should be eliminated.
The variables
k
and
U_{z}
in Equation (13) should realize the modulation waves
U_{xp}
and
U_{xn}
with values between their maximum and minimum. Thus, the appropriate
k
and
U_{z}
can be obtained with different ranges of
ω_{s}t
.
The solutions to
k
and
U_{z}
are piecewise functions, as shown in Equation (16). In each π/3 radian of
ω_{s}t
,
k
has two solutions, whereas
U_{z}
has only one solution. Totally,
k
has 64 solutions in each fundamental cycle. Most of these solutions will cause discontinuity of
U_{xp}
and
U_{xn}
. If
U_{xp}
and
U_{xn}
are continuous, they should satisfy Equation (17), where the subscripts “+” and “” stand for the right and left limits of the signal, respectively. Only two solutions make
U_{xp}
and
U_{xn}
satisfy Equation (17), which are shown in Equation (18). Equation (19) and
Fig. 3
show the continuous situation of
U_{xp}
and
U_{xn}
(
x
=
a
,
b
,
c
,) when the modulation index
M
equals 1. In other cases,
U_{xp}
and
U_{xn}
are discontinuous. Equation (20) and
Fig. 4
show the discontinuous situations of
U_{xp}
and
U_{xn}
in the novel strategy (
M
= 1) when k satisfies Equation (21). As shown in
Fis. 3
and
4
, the modulation waves of the novel strategy remain the same (maximum or minimum) in 1/3 fundamental cycle.
Continuous situations of U_{xp} and U_{xn} of the CBPWM strategy (M = 1).
Discontinuous situations of U_{xp} and U_{xn} of the CBPWM strategy (M = 1).
In this part, the modulation wave solutions of the novel CBPWM strategy are discussed. In general, 64 modulation wave solutions are obtained, among which 2 are continuous and the other 62 are discontinuous. The system linetoline voltage applying each of the 64 modulation wave solutions is the same as that of the traditional SPWM strategy because
k
and
U_{z}
are eliminated in the linetoline voltage expression.
IV. CHARACTERISTIC ANALYSIS OF THE NOVEL STRATEGY
In Section III, the modulation wave solutions of the novel PWM strategy have been deduced. To give a comprehensive evaluation of the novel strategy, the characteristics of DC voltage utilization ratio, device switching losses, and output phase voltage THD characteristics are analyzed in the current section.
 A. DC Voltage Utilization Ratio and Device Switching Losses of the Novel PWM Strategy
DC voltage utilization ratio is an important evaluation index of PWM strategy. This ratio equals to the ratio of fundamental amplitude of output linetoline voltage and total DC voltage, that is,
where
U
_{ab1}
is the fundamental amplitude of the output linetoline voltage.
According to Equation (16), the expression of the zerosequence voltage injected into the phase voltage is the same as the modulation wave difference between traditional threelevel SVPWM strategy and SPWM strategy. As a result, the DC utilization ratio of the novel strategy (all of the 64 modulation wave solutions) is the same as that of the SVPWM strategy (15.47% higher than that of the threelevel SPWM strategy).
System device losses are another important evaluation standard for modulation strategy. The preceding analysis indicates that the threephase modulation waves of the proposed CBPWM strategy (all of the 64 modulation wave solutions) remain the extreme values in 1/3 fundamental cycle. As a result, the output switching states of each phase remain unchanged in 1/3 fundamental cycle. By contrast, when the threelevel SPWM strategy is used, the output switching states of each phase remain the same in 1/2 fundamental cycle. Therefore, the total switching losses of the novel CBPWM strategy are 4/3 times of the traditional threelevel SPWM strategy assuming that the system switching losses are proportional to the switching frequency.
 B. Phase Voltage THD Characteristics of the Novel PWM Strategy
Output phase voltage (the earth point is the NP “O” in
Fig. 1
) THD characteristic is another important evaluation standard of modulation strategy. In this study, doubleFourier analysis is used to analyze the output phase voltage THD characteristic of the novel PWM strategy. The expression and coefficients of doubleFourier analysis are shown in Equations (23) and (24), respectively. In these equations,
ω_{s}
stands for the modulation wave angular frequency, and
ω_{c}
stands for the carrierwave angular frequency.
A
_{00}
is the coefficient of the zerosequence components;
A
_{0n}
and
B
_{0n}
,
A
_{m0}
and
B
_{m0}
, and
C_{mn}
are the coefficients of baseband, carrier, and sideband harmonics, respectively.
During coefficient solving, the trigonometricfunction integral that contains triangle variables [such as cos(
ξ
cos
θ
) and sin(
ξ
cos
θ
)] must frequently be solved. In these situations, Jacobi–Anger expansions, which are shown in Equation (25), can be used to expand the triangle variables into Bessel series forms
[20]
.
Two modulation waves of each phase exist in the novel strategy. As a result, the doubleFourier analysis results of the two modulation waves should be added to obtain the phase voltage THD. Based on Equations (13) and (19), all of the 64 modulation wave solutions of the novel strategy have the same THD performance. The coefficients of harmonic components are shown in Equations (26) and (27).
Fig. 5
shows the output phase voltage THD performance of the novel PWM strategy when the modulation index (
M
) equals to 0.8 and 0.9. According to the output phase voltage waveform expressions (26) and (27) and
Fig. 7
, for a given value of M, the following findings are obtained:
Diagram of phase voltage THD characteristics of the novel strategy.
1) Compared with the output phase voltage harmonic components of the traditional threelevel SPWM strategy, the output phase voltage of the novel strategy also contains odd carrier harmonics, even sideband harmonics (
n
is even) around the odd carrier multiples (
m
is odd), and odd sideband harmonics (
n
is odd) around the even carrier multiples (
m
is even). The angular frequencies of the main harmonics are
ω_{c}
,
ω_{c}
±2
ω
_{s}
, and 2
ω
_{c}
±
ω_{s}
. These components will be greatly reduced by output filter because their frequencies are higher than the fundamental frequency. The only difference is that 6
i
− 3 (
i
= 1, 2, 3…) multiples of baseband harmonics are remain in the output phase voltage because of the existence of zerosequence component in the threephase reference voltage. These lowfrequency harmonics will not appear in the linetoline voltage and phase current because the load is starconnected or triangleconnected.
2) With the increase of
M
, the output phase voltage THD of the novel strategy gradually decreases. This characteristic is consistent with traditional SPWM strategy.
3) The THD performances of the novel CBPWM strategy and traditional SPWM strategy are difficult to compare because the NP voltage oscillation caused by the SPWM strategy will influence its THD characteristic.
In this part, the DC voltage utilization ratio, device switching losses, and output phase voltage THD characteristics of the novel strategy are studied. Compared with traditional SPWM strategy, the novel PWM strategy has higher DC voltage utilization ratio but larger device switching losses. In terms of output phase voltage THD, the novel strategy does not cause NP voltage oscillations. The output phase voltage THD of the novel strategy is only influenced by the system modulation index (
M
). On the contrary, the output phase voltage THD of the SPWM strategy is influenced by the system modulation index (
M
) and NP voltage oscillations. Moreover, the output linetoline voltage and phase current will not contain lowfrequency harmonics if the novel PWM strategy is used.
V. SIMULATION AND EXPERIMENT VERIFICATIONS
The effect of the proposed novel PWM strategy is verified by simulation and experiment. The diagram of the simulation and experimental power circuits is shown in
Fig. 1
. The values of the dclink voltage and capacitors are
U_{dc}
= 400 V and
C
_{1}
=
C
_{2}
= 1000 μF, respectively. The load is connected to the threelevel NPC VSC through an LCL filter (
L
_{1}
=
L
_{2}
= 3 mH,
C
= 17 uF) with star connection. The proposed novel CBPWM strategy is implemented in a fully digital system using a TMS320F2812 DSP. The switching frequency is 2 kHz.
Figs. 6
and
7
show the simulation results of output phase voltage (
U_{ao}
), line voltage (
U_{bc}
), DC capacitor voltages (
U
_{c1,2}
), and load current (
i_{a,b,c}
) of the standard SPWM strategy and the novel PWM strategy proposed in this paper. The modulation index of the system is 0.8. Threephase resistive load (
R
= 15 Ω, starconnected) and inductive load (
L
= 20 mH, starconnected) are used in
Figs. 6
and
7
, respectively. The modulation strategy changes from the SPWM strategy to the novel PWM strategy (
k
=
k
_{1}
) when
t
= 0.2 s. The simulation results indicate that the novel CBPWM strategy can completely eliminate the lowfrequency NP voltage oscillation compared with traditional SPMW strategy.
Simulation results of the standard SPWM and novel PWM strategies (resistor load, M = 0.8, k = k_{1}).
Simulation results of the standard SPWM and novel PWM strategies (inductive load, M = 0.8, k = k_{1}).
Fig. 8
shows the contrastive experiment results of the standard SPWM and novel PWM strategies in different modulation indices (
M
= 0.8 and 0.9) with threephase resistive load (
R
= 15 Ω, starconnected).
Figs. 8
(a) and(b) show the results obtained by applying the standard SPWM strategy. The variables shown are a linetoline voltage
U_{cb}
, the output phase voltage
U_{ao}
, the voltage on the DC capacitor
U
_{c1}
, and the output load current
i_{c}
.
Figs. 8
(c) to(j) show the same results when the proposed novel PWM strategy is applied.
Table I
shows the main harmonics of the experimental DC capacitor voltage results when traditional SPWM strategy and the novel PWM strategy are used.
Fig. 8
and
Table I
indicate that significant lowfundamental frequency (the fundamental frequency is 50 Hz in this study) voltage oscillations occur on the DC capacitor when traditional SPWM strategy is used. Moreover, the DC components of the two DC capacitor voltages are not equal. When the novel PWM strategy is used, the voltage on the DC capacitor does not contain any lowfrequency oscillation but only highfrequency ripples, which are related to the switching frequency. The unbalance of DC capacitor voltage is also greatly reduced. However, the switching frequencies of switching devices increase when the novel PWM strategy is applied compared with those when traditional SPWM strategy is used. The reason is that the two modulation waves in the novel PWM strategies are not clamped to their extreme values in half of the fundamental cycle.
Experimental results of the standard SPWM and novel PWM strategies.
MAIN HARMONIC COMPONENTS OF THE EXPERIMENTAL DC CAPACITOR VOLTAGES USING THE NOVEL PWM STRATEGY AND SPWM STRATEGY
MAIN HARMONIC COMPONENTS OF THE EXPERIMENTAL DC CAPACITOR VOLTAGES USING THE NOVEL PWM STRATEGY AND SPWM STRATEGY
Table II
shows the THD characteristic comparison of the results of the experiment applying the novel PWM strategy.
Tables I
and
II
indicate that the four modulation wave solutions with the same modulation index (
M
) have the same THD characteristics, which is consistent with the preceding analysis.
THD CHARACTERISTIC COMPARISON OF THE RESULTS OF THE EXPERIMENT APPLYING THE NOVEL PWM STRATEGY
THD CHARACTERISTIC COMPARISON OF THE RESULTS OF THE EXPERIMENT APPLYING THE NOVEL PWM STRATEGY
Threephase inductive load (20 mH, starconnected) is used to confirm the DC voltage control effect of the novel PWM strategy under highmodulation index and lowpower factor, in which the lowfrequency NP voltage oscillations cannot be eliminated by traditional SPWM strategy with a zerosequence component injection scheme
[13]

[15]
.
Fig. 9
(a) shows the experiment results of the traditional SPWM strategy, and those of the SPWM strategy with zerosequence component injection are shown in
Fig. 9
(b).
Figs. 9
(c) and 9(d) show the experimental results when the novel PWM strategy is used. All of the four experimental results are obtained under the same system condition (
U_{dc}
= 200 V,
M
= 0.8, power factor angle
φ
= 90°).
Comparison of SPWM and the novel PWM strategy under highmodulation index and lowpower factor (M = 0.8, cosφ = 0).
The experimental results indicate that when the system modulation index (
M
) is high and the output power factor is low, traditional SPWM strategy with zerosequence component injection cannot eliminate the lowfrequency oscillation existing in the DC capacitor voltage. The reason is that in such a case (
M
= 0.8, cos
φ
= 0), the maximum of the phase reference signal
U_{x}
(x = a, b, c), which is the sum of positivesequence reference signal
U
_{x0}
and the zerosequence component
U_{z}
, is larger than that of the carrier
[13]

[15]
. By contrast, the novel PWM strategy proposed in this paper can solve the problem very well.
VI. CONCLUSIONS
This paper presents a novel PWM strategy that can eliminate lowfrequency NP voltage oscillations of a threelevel NPC converter under any system modulation index and output power factor. The modulation wave relationship between the novel PWM strategy and traditional SPWM strategy is first investigated. The operation principle and modulation wave solving process of the proposed PWM strategy is then analyzed in details. The proposed PWM strategy are also compared with traditional SPWM strategy in terms of output phase voltage harmonic characteristic, device switching losses, and DC voltage utilization rate. Finally, numerical simulation and experiment results are given to verify the theoretical analysis.
With the proposed PWM strategy solutions, which can be directly implemented in a simple microprocessor and does not cause lowfrequency voltage oscillation in the NP, threelevel NPC VSC is a promising and competitive topology for wide range power conversion in renewable energy applications and many other lowpower applications. The proposed PWM strategies can also be extended to the application of threelevel NPC SVG, threelevel NPC APF, and threelevel ANPC VSI
[21]
–
[23]
.
Acknowledgements
This work was supported in part by the State Key Laboratory of Electrical Insulation and Power Equipment (China) and by the National “863” Program of China under Project 2012AA050206.
BIO
Ning Li (S’09) received his B.S. and M.S. degrees in Electrical Engineering from Xi’an Jiaotong University, Xi’an, China in 2006 and 2009, respectively. He is currently working toward his Ph.D. degree at the same university. His current research interests include multilevel converter, wind power generation, and power electronics.
Yue Wang (M’05) received his B.S. degree from Xi’an Jiaotong University, Xi’an, China in 1994; M.S. degree from Beijing Jiaotong University, Beijing, China in 2000; Ph.D. degree from Xi’an Jiaotong University in 2004. He is currently an Associate Professor at Xi’an Jiaotong University. His current research interests include active power filters, wind power generation, motor drives, multilevel converters, and flexible ac power transmission.
Wanjun Lei received his B.Sc., M.Sc., and Ph.D. degrees in Electrical Engineering from Xi’an Jiaotong University, China. He is an assistant professor in the Department of Industry Automation, Power Electronics, and Renewable Energy Research Center at this university. He is a member of China Power Supply Society and IEEE. His current research focuses on power electronics inverter and power quality control technique.
Ruigen Niu received his B.S. degree from Xi’an Jiaotong University, Xi’an, China in 2008, where he is currently a Researcher at the School of Electrical Engineering. His current research interests include power quality of wind power generation.
Zhao’an Wang (SM’98) received his B.S. and M.S. degrees from Xi’an Jiaotong University, Xi’an, China in 1970 and 1982, respectively. He also received his Ph.D. degree from Osaka University, Osaka, Japan in 1989. From 1970 to 1979, he was an Engineer at Xi’an Rectier Factory. Since 1982, he has been a Lecturer at Xi’an Jiaotong University, where he is also currently a Professor. He is the author and coauthor of more than 150 technical papers and has led numerous government and industrysponsored projects in the areas of power and industrial electronics. His research interests include power conversion systems, harmonic suppression, reactive power compensation, motor drives, power electronic integration, and active power filters.
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