In this paper, a dualconverter threephase pulse width modulation (PWM) rectifier based on unbalanced onecycle control (OCC) strategy is proposed. The proposed rectifier is used to eliminate the second harmonic waves of DC voltage and distortion of line currents under unbalanced input grid voltage conditions. The dualconverter PWM rectifier employs two converters, which are called positivesequence converter and negativesequence converter. The unbalanced OCC system compensates feedback currents of positivesequence converter via grid negativesequence voltages, as well as compensates feedback currents of negativesequence converter via grid positivesequence voltages. The AC currents of positive and negativesequence converter are controlled to be symmetrical. Thus, the workload of every switching device of converter is balanced. Only one conventional PI controller is adopted to achieve invariant power control. Then, the parameter tuning is simplified, and the extraction for positive and negativesequence currents is not needed anymore. The effectiveness and the viability of the control strategy are demonstrated through detailed experimental verification.
I. INTRODUCTION
Public power systems are frequently under unbalanced voltage conditions, and singlephase grid voltage sag happens occasionally. Moreover, even harmonic waves appear on DC voltage, as well as distortion on line currents of traditional threephase PWM rectifiers. Therefore, public power systems could be polluted by harmonics, and power loss of PWM rectifier could be increased dramatically. New control strategy and topology for threephase PWM rectifier under unbalanced input voltage conditions must be developed. The control goal is to keep DC voltage constant and sinusoidal currents symmetrical together with unitypower factor. Several control methods have been reported for threephase PWM rectifier under unbalanced input voltage conditions. Relevant experts and scholars have conducted numerous analyses and research.
Proportional resonant (PR) controller was adopted to realize control of current under twophase static coordinate that is free from static error
[1]

[6]
. However, the analysis and design of PR controller are difficult, and the control performance is affected by grid frequency. Furthermore, a DC voltage outer loop PI controller and two PR current inner loop controllers are required. Given that a singleconverter structure is adopted, the threephase grid currents of converter are unsymmetrical, and the workload of the converter switching device is unbalanced. Positive and negativesequence dualcurrent inner loop PI control structures were presented
[7]

[11]
, and static errorfree control for the grid positive and negativesequence currents could effectively prevent DC voltage fluctuation and grid current distortion. However, the control structure and algorithm are complicated. The structure requires four current inner loop controllers and one voltage outer loop controller, namely, five PI controllers. Thus, a large amount of calculation is required. Parameter tuning is also difficult, and the design uses the singleconverter structure as well. Therefore, the converter could not achieve symmetrical threephase currents, thereby leading to unbalanced workload of converter switching devices.
In view of the preceding analysis, the goal of unbalanced control strategy for threephase PWM rectifier under unbalanced grid voltage conditions lies in the following factors: the stability of DC voltage, the sinusoidal AC current, and the threephase symmetric currents, which can balance the workload of the switching devices. The traditional control strategy and the topology could not achieve the three control targets simultaneously.
On the basis of the comprehensive analysis of the preceding papers and research achievements, a novel unbalanced onecycle control (OCC) strategy for dualconverter PWM rectifier under unbalanced voltage conditions is proposed in this paper according to the perspective of the preceding three control targets. The dualconverter PWM rectifier employs two converters, which are called positivesequence converter and negativesequence converter. This unbalanced OCC strategy compensates feedback currents of positivesequence converter via grid negativesequence voltages and compensates feedback currents of negativesequence converter via grid positivesequence voltages. Therefore, the positivesequence currents flow through the positivesequence converter, and the negativesequence currents flow through the negativesequence converter. The currents of every converter are symmetrical, and the workload of every switching device is balanced. The harmonic of DC voltage and the distortion of line currents of threephase PWM rectifier have been eliminated effectively under unbalanced voltage conditions. Unlike the above unbalanced control strategies, the proposed unbalanced OCC strategy uses only one conventional outer loop PI controller, which regulates power to be constant and requires no parameter tuning for current inner loops. The calculations for the AC positive and negativesequence currents are also not required. Thus, the proposed unbalanced OCC strategy achieves high performance with reduced calculation.
II. OCC FOR DUALCONVERTER THREEPHASE PWM RECTIFIER
The topology of OCC for threephase PWM rectifier is shown in
Fig. 1
[12]
. In this figure,
e
_{a}
,
e
_{b}
,
e
_{c}
represent the threephase AC voltages of grid, while
i
_{a}
,
i
_{b}
,
i
_{c}
stand for the grid threephase AC current;
U
_{dc}
is the DC voltage,
R
is the equivalent DC load, and
S_{xy}
is (
x
= a, b, c and
y
= p, n) the switching devices of PWM rectifier.
Topology of OCC for threephase PWM rectifier.
Assuming that grid voltages are under symmetric conditions, the above topology and the traditional OCC system could perform well, and the operating performance would deteriorate sharply under unbalanced grid voltage conditions.
According to the symmetrical component decomposition method, the unbalanced grid voltages could be decomposed into positive and negativesequence as well as zerosequence components. Given that the threephase PWM rectifier adopts threewire connection, no zerosequence components flow through the converter. Therefore, given that the zerosequence voltages are removed, the unbalanced grid voltages are expressed as follows:
where
e
_{x+}
and
e
_{x−}
(
x
= a, b, c) denote the positive and negativesequence voltage component, respectively.
When the grid voltages are under unbalanced condition, the active power of the rectifier must be a constant value by regulating this active power based on power balance principle to restrain the DC voltage second harmonics of the rectifier and the grid currents distortion. In addition, the relationship between positive and negativesequence voltages and currents was derived in
[13]
.
where
φ
_{x+}
(
t
) and
φ
_{x−}
(
t
) denote the instantaneous phase of the grid positive and negativesequence voltages, respectively,
θ
_{x+}
(
t
) and
θ
_{x−}
(
t
) represent the instantaneous phase of the grid positive and negativesequence current, and
i
_{x+}
and
i
_{x−}
are the threephase positive and negativesequence currents (
x
= a, b, c).
The unbalanced OCC strategy for dualconverter threephase PWM rectifier, which is shown in
Fig. 2
, employs the positivesequence converter to regulate the grid positivesequence currents, as well as the negativesequence converter to regulate grid negativesequence currents.
Topology of unbalanced OCC for dualconverter threephase PWM rectifier.
According to the relationship shown in (3), the unitypower factor of the rectifier can be achieved by controlling the phase of the grid positivesequence voltages and currents:
From the first item of (2) and (3), the phase relationship between the negativesequence currents and voltages can be derived as follows:
The grid positive and negativesequence voltages and currents are expressed as follows:
where
R
_{e+}
and
R
_{e−}
denote the equivalent positive and negativesequence power resistors of threephase PWM rectifier, respectively. From the second item of (2), (3), and (4), the following can be obtained:
If
R
_{e+}
=
R
_{e}
, then the following can be obtained:
Threephase currents of the rectifier can be depicted as follows:
Fig. 3
shows the equivalent circuit of the positive or negative converter, which is shown in
Fig. 1
[12
,
16]
. In
Fig. 3
,
d
_{an}
,
d
_{bn}
,
d
_{cn}
are the duty ratios for the converter lower bridge arm switches S
_{an}
, S
_{bn}
, S
_{cn}
, respectively.
Equivalent average model of OCC for threephase PWM rectifier.
The average voltages of rectifier bridge nodes A, B, and C, which are relative to node N, are expressed as follows:
where
X
= A, B, C,
x
= a, b, c. Compared with the neutral point O, grid voltage phasor
u
◌
_{xO}
and converter AC side voltage phasor
u
◌
_{XO}
follow that
According to the analysis in paper
[12]
, the value of the grid side rectifier inductor is actually small (usually approximately 1 mH), and the fundamental wave voltage drops across the inductance can be ignored. Therefore, the converter AC side voltages are approximately equal to the grid voltages:
In
Fig. 3
, the voltages of nodes A, B, C, node N, and node O must be imposed such that
According to (1), the threephase converter does not include zerosequence voltages, and the sum of voltages
u
_{AO}
,
u
_{BO}
, and
u
_{CO}
is equal to zero, that is,
u
_{AO}
+
u
_{BO}
+
u
_{CO}
= 0. From (12), we could obtain the voltages between nodes N and O
When (13), (11), and (9) are substituted into (12), the relationship between the grid voltages and ratios for rectifier lower bridge arm switches can be derived as follows:
Given that the matrix of (14) is singular, no unique solution exists. Therefore, the following possible solution exists:
where
k
_{1}
could be a random constant, because the duty ratio
d
_{xn}
must satisfy the inequality, 0 <
d
_{xn}
< 1 (
x
= a, b, c). When (15) is substituted into this inequality, the following can be obtained:
From (16), to ensure
k
_{1}
is a constant, which would not change along with the grid transient voltages, the following can be acquired:
From (16) and (17), the range of
k
_{1}
can be derived as follows:
where
E
_{mx}
denotes the grid voltage amplitude.
From (18), the relationship between the DC voltage of threephase rectifier based on OCC and the grid voltage amplitude can be expressed as follows:
From (18), the parameter
k
_{1}
could be any numerical value in the range of [
E
_{mx}
/
U
_{dc}
, 1
E
_{mx}
/
U
_{dc}
], but generally for the sake of increasing application, scope for
k
_{1}
,
k
_{1}
should be assigned to the middle point value of the range in (18):
Equations (18) and (19) indicate that the essential condition for the OCC system for threephase PWM rectifier to operate correctly is that the DC voltage is more than double the grid phase voltage amplitude, and then the parameter
k
_{1}
could have a solution.
III. OCC STRATEGY FOR POSITIVESEQUENCE CONVERTER OF PWM RECTIFIER
Only positivesequence current
i
_{x+}
should flow through the positivesequence converter to keep threephase currents of the positivesequence converter symmetrical and sinusoidal. Thus, the relationship between voltages and currents of the positivesequence converter is expressed as follows:
According to (21), the control system compensates feedback currents of the positivesequence converter by grid negativesequence voltages under unbalanced grid voltage conditions
According to (22), the currents of the positivesequence converter are straightforwardly derived as follows:
If
R
_{s}
is the equivalent current sampling resistance
[14]
,
[15]
, then Equation (23) could be expressed as follows:
For the positivesequence converter, Equation (15) could be rewritten as follows:
where
d
_{xn+}
is the duty ratio for the positivesequence converter lower bridge arm switches. When (25) is substituted into (24), the following is produced:
When
d
_{xn+}
= 1 −
t
_{xp+}
/
T
_{s}
is substituted into (26), the following can be obtained:
where
T
_{s}
denotes the switching period of switches, and
t
_{xp+}
is the turnon time of each positivesequence converter upper bridge arm switch in one switching period. From (27), OCC key equations for the positivesequence converter can be obtained:
where
where
u
_{m+}
denotes the positivesequence DC error control voltage, which is used to control output power for rectifier;
τ
_{+}
stands for the integral time constant of integrator in OCC system, and its value is half the switching period; and −
u
_{m+}
e
_{x−}
/(
U
_{dc}
k
_{1}
) is the grid negativesequence voltage compensation item. According to (28), the OCC system structure for the positivesequence converter can be shown as in
Fig. 4
.
Unbalanced OCC system for positivesequence converter.
IV. OCC STRATEGY FOR NEGATIVESEQUENCE CONVERTER OF PWM RECTIFIER
Only negativesequence current
i
_{x−}
should flow through the negativesequence converter to keep threephase currents of the positivesequence converter symmetrical and similar. Thus, the relationship between voltages and currents for the negativesequence converter is expressed as follows:
According to (30), the control system compensates feedback currents of negativesequence converter via grid positivesequence voltages under unbalanced grid voltage condition:
According to (31), the currents of the negativesequence converter are straightforwardly derived as follows:
If
R
_{s}
is the equivalent current sampling resistance, then Equation (32) could be expressed as follows:
where
k
_{2}
and
k
_{3}
are positive constants that are used to control power for negativesequence converter and maximum grid currents, respectively. For the negativesequence converter, Equation (15) could be rewritten as follows:
where
d
_{xp−}
is the duty ratio for the negativesequence converter upper bridge arm switches. When (34) is substituted into (33), the following can be obtained:
where
k
is similar to
k
_{3}
and used to restrain maximum grid currents. Then, substituting the duty ratio
d
_{xp−}
=
t
_{x−}
/
T
_{s}
into (35) can obtain the following control key equation:
where
T
_{s}
denotes the switching period of switches, and txp− is the turnon time of each negativesequence converter upper bridge arm switch in one switching period. From (36), the OCC key equation for the negativesequence converter can be obtained:
where
where
u
_{m−}
denotes the negativesequence DC error control voltage, which is used to control output power for rectifier.
τ
_ is the integral time constant of integrator of OCC system, and its value is half of switching period, and −
u
_{m−}
e
_{x+}
/(
U
_{dc}
(1 −
k
_{1}
)) is the grid positivesequence voltage compensation item. According to (37), the OCC system structure for the negativesequence converter can be shown in
Fig. 5
.
Unbalanced OCC system for negativesequence converter.
According to (28) and (37), the positive and negativesequence converters adopt the same bipolar OCC strategy, and the switching operations of positive and negativesequence converter switches are nearly at the same time based on
Figs. 4
and
5
. Therefore, the highfrequency circulation currents between the positivesequence converter and the negativesequence converter can be eliminated
[16]
.
V. UNBALANCED OCC SYSTEM FOR DUALCONVERTER PWM RECTIFIER
Grid positive and negativesequence voltages could be decomposed under twophase stationary coordinate system from the retained voltage, which is described by (39).
where
e
_{α}
and
e
_{β}
are
α
and
β
axis coordinate components of the grid voltages, respectively, and
e
_{α}
^{⊥}
and
e
_{β}
^{⊥}
denote the
α
and
β
axis coordinate components, which delay onefourth power period. The transient values of the grid positive and negativesequence voltages under threephase stationary coordinate are derived as follows:
According to the characteristics of the dualconverter PWM rectifier, the positive and negativesequence converters must share the same PI outer loop voltage controller. Therefore, the error control voltage of the positive sequence
u
_{m+}
is equal to the error control voltage of the negative sequence
u
_{m−}
. According to (29) and (38),
u
_{m+}
can be made equal to
u
_{m−}
by adjusting the parameters
k
,
R
_{s}
, and
R
_{e}
. If
u
_{m+}
=
u
_{m−}
=
u
_{m}
and given that
k
_{1}
= 0.5, namely,
k
_{1}
= 1 −
k
_{1}
, then parameter
k
is expressed as follows:
The mathematical relationship between error control voltage
u
_{m}
and DC voltage
U
_{dc}
can be derived as follows:
From (43), PI controller can be employed to control the DC voltage for the unbalanced OCC system, and the control equation for DC voltage is expressed as follows:
where
U
^{*}
_{dc}
denotes the set value of DC voltage for rectifier,
U
_{dc}
is the rectifier feedback DC voltage, and
K
_{P}
and
K
_{I}
are the proportionality and integral coefficient for PI controller, respectively. From
Fig. 2
and (43), the following can be obtained:
According to (28) and
Fig. 4
, the feedback currents of the positivesequence converter, which are the positivesequence currents, must be transmitted to the negative inputs of the comparator of the positivesequence controller to ensure that the threephase currents, which flow through the positivesequence converter, are symmetric. According to OCC theory, the positivesequence currents of the positivesequence converter could be used to cancel out the grid positivesequence voltages. The positive and negativesequence voltages exist under unbalanced grid voltage conditions simultaneously, and the grid negativesequence voltage compensation component is transmitted to the positive inputs of comparator of positivesequence controller to cancel out the negativesequence voltages. Therefore, the feedback currents of the positivesequence converter should be compensated by negativesequence components of grid voltages. Similarly, from (37) and
Fig. 5
, the feedback currents of the negativesequence converter, which are the negativesequence currents, must be transmitted to the positive inputs of comparator of positivesequence controller to ensure that the threephase currents, which flow through the negativesequence converter, are symmetric. According to (6), the positivesequence converter serves as a rectifier, whereas the negativesequence converter serves as a gridconnected inverter that inverts the second harmonic power of the DC side back to grid in form of sinusoidal currents. Therefore, the second harmonic waves of DC voltage and distortion of line currents can be eliminated.
According to (28) and (37), the positive and negativesequence currents flow through the positive and negativesequence converters, respectively, and the unbalanced factor for the grid voltages is defined by
[17]
:
where
U
_{+}
and
U
_{−}
denote the grid positive and negativesequence RMS voltages, respectively, whereas
U
_{ab}
,
U
_{bc}
, and
U
_{ca}
are the grid RMS line voltages. The relationship between positive and negativesequence currents can be derived as follows:
According to (47), the negativesequence converter is
VUF
times the capacity of design and the rated currents of switching devices and inductance of positivesequence converter in engineering applications. To achieve consistency, the inductance of negativesequence converter should be designed to be 1/
VUF
times the value of the inductance of the positivesequence converter. The reduction in the rated currents of inductance of negativesequence converter could reduce the size of inductor magnetic core and then greatly reduce the cost. For example, if the unbalanced factor
VUF
is 10% for the design, then the negativesequence converter is 10% the capacity of the positivesequence converter. In addition, the cost of negativesequence converter is greatly reduced. According to (47), because
i
_{x}
is less than
i
_{x+}
, the loss of negativesequence converter is also less than positivesequence converter. Therefore, the extra loss caused by the negativesequence converter increases slightly. In particular, when the grid voltage is under slightly unbalanced condition and
VUF
is nearly zero, the current of the negativesequence converter remains nearly zero, and the loss increases slightly.
VI. EXPERIMENTAL RESULTS
A threephase dualconverter PWM rectifier based on unbalanced OCC strategy is designed in this study to verify the validity of the theoretical analysis and research. The control system for the rectifier adopts the digital signal processor (DSP) TMS320F28335, and the capacity of the rectifier is designed as 5 kVA. The experimental system structure is shown in
Fig. 6
.
Experiment system of unbalanced OCC for dualconverter threephase PWM rectifier.
The control software was completed by TMS320F28335 DSP. Moreover, the digital integrator, data comparator, RS flipflop, and clock generator were completed by using a field programmable gate array. On the basis of the laboratory prototype, we conducted research on the conventional rectifier based on balanced OCC strategy and dualconverter rectifier based on unbalanced OCC strategy.
The grid unbalanced fault was simulated by voltage sag generator, which is shown in
Fig. 7
. The voltage sag generator employed a threephase transformer with 9000 VA capacity. Each phase of the transformer output windings had three taps, and the output voltage amplitudes of every tap were 100, 80, and 60 V.
Voltage sag generator of Phase C.
The grid was initially under balanced voltage conditions, and threephase voltage amplitudes were all 100 V. After the rectifier was running stably, we simulated an unbalanced fault by using voltage sag generator. Moreover, the voltage amplitude of Phase A remained 100 V, the voltage amplitude of Phase B dropped to 80 V, and the voltage amplitude of Phase C dropped to 60 V.
Table I
lists the experimental parameters for the threephase PWM rectifier, and the experimental results are shown in
Fig. 8
.
EXPERIMENTAL PARAMETERS FOR THREEPHASE PWM RECTIFIER
EXPERIMENTAL PARAMETERS FOR THREEPHASE PWM RECTIFIER
Experimental waveforms of threephase PWM rectifier under unbalanced grid voltage conditions.
The experimental waveforms of DC voltage and AC currents of the conventional rectifier based on balanced OCC strategy under unbalanced voltage conditions are shown in
Figs. 8
(a) and
8
(b), respectively, where the second harmonic waves of DC voltage and distortion of line currents appeared. The total harmonic distortion (THD) of the AC currents is nearly 13%, and the oscillation of DC voltage is approximately 19 Vpp. The experimental waveforms of DC voltage and AC currents of the singleconverter rectifier based on PR controller under unbalanced voltage conditions are shown in
Figs. 8
(c) and
8
(d), respectively, where the DC voltage oscillation is approximately 4 Vpp in steady state, which is less than that of the conventional rectifier. The line current is nearly sinusoidal, and the distortion is restrained. However, the line current is unsymmetrical. Thus, the workload of every switching device is unbalanced, and the power rating of every switching device cannot be used to the fullest. The experimental waveform of the DC voltage of the dualconverter rectifier based on unbalanced OCC strategy under unbalanced voltage conditions is shown in
Fig. 8
(e), where the DC voltage enters the steady state rapidly with a slight oscillation, which is only approximately 2 Vpp.
Fig. 8
(f) shows the AC line currents of the dualconverter rectifier, which are the sum of the positive and negativesequence currents under unbalanced voltage conditions. The AC line currents of the dualconverter rectifier are not symmetric but sinusoidal. The threephase AC current waves of the positiveand negativesequence converter are shown in
Figs. 8
(g) and
8
(h), where the threephase AC currents are all symmetric sinusoidal waveforms. The distortion of line currents is eliminated, and the THD is only approximately 2.5%. The power factor is approximately 0.97, and the dualconverter PWM rectifier runs nearly in unitypower factor state.
VII. CONCLUSIONS
An unbalanced OCC strategy for dualconverter threephase PWM rectifier based on grid positive and negativesequence voltage feedforward compensation was proposed in this paper to eliminate the second harmonic waves of DC voltage and distortion of line currents for a threephase PWM rectifier under unbalanced input grid voltage conditions. The unbalanced OCC mathematical model for the positivesequence converter based on the grid negativesequence voltage feedforward compensation and the model for the negativesequence converter based on the grid positivesequence voltage feedforward compensation were presented. Moreover, the current control and parameter tuning method was investigated. The experimental results indicate that unlike other unbalanced control strategies, the proposed unbalanced OCC strategy for dualconverter threephase PWM rectifier needs no calculation for the positive and negativesequence currents and no designation for current inner loop controller. Therefore, the control system for the dualconverter threephase rectifier only has one PI controller, and the control structure is simplified. The second harmonic waves of DC voltage and distortion of AC line currents of rectifier are eliminated, and the harmonic pollution to grid is reduced greatly under unbalanced grid voltage conditions. Moreover, the positive and negativesequence converters both achieved symmetrical threephase sinusoidal currents, which balance the workload of all the switching devices of the converters, and highpower factor can be achieved. With the proposed dualconverter topology and the corresponding unbalanced control strategy, the performance of the threephase PWM rectifier is ultimately optimized under unbalanced grid voltage conditions.
BIO
You Xu was born in Jiangsu, China. He received his B.S. degree in Electrical Engineering from China University of Mining and Technology, Xuzhou, China, in 2002, and his M.S. and Ph.D. degrees in Electrical Engineering from Southeast University, Nanjing, China, in 2007 and 2013, respectively. He is now a lecturer in the School of Automation, Nanjing Institute of Technology. His current research interests include electric power converters for distributed generation, fault ride through for gridconnected inverters, and the application of embedded system for power electronics in power systems.
Qingjie Zhang was born in Jiangsu, China. He received his B.S. in Electrical Engineering from Yancheng Institute of Technology, China, in 2004, and his M.S. in Electrical engineering from Southeast University, China, in 2007. He is now pursuing his Ph.D. in Electrical Engineering at Southeast University, Nanjing, China. He is a lecturer in the College of Engineering, Nanjing Agricultural University. His current interests include power electronics for renewable energy generation, power system stability control, and microgrid technology.
Kai Deng was born in Jiangsu, China. He received his B.S. from Nanjing Agricultural University, Nanjing, China, in 2009, and his M.S. from Beijing Institute of Technology, Beijing, China, in 2012, both in Electrical Engineering. He is now pursuing his Ph.D. in Electrical Engineering in Southeast University, Nanjing, China. His current research interests include Zsource inverters, digital control of power converters, and system integration of modular power converters.
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