Threelevel PWM rectifiers applied in medium voltage applications usually operate at low switching frequency to keep the dynamic losses under permitted level. However, low switching frequency brings a heavy crosscoupling between the current components
i
_{d}
and
i
_{q}
with a poor dynamic system performance and a harmonic distortion in the gridconnecting current. To overcome these problems, a mathematical model based on complex variables of the threelevel voltage source PWM rectifier is firstly established, and the reasons of above issues resulted from low switching frequency have been analyzed using modern control theory. Then, a novel control strategy suitable for the current decoupling control based on the complex variables for
i
_{d}
and
i
_{q}
is designed here. The comparisons between this kind of control strategy and the normal PI method have been carried out. MATLAB and experimental results are given in detail.
1. Introduction
Nowadays, PWM rectifiers have been widely applied in series of industrial fields, such as Static Reactive Power Generator (SVG), Active Power Filter (APF), Unified Power Flow Controller (UPFC), High Voltage DC Transmission (HVDC) and some renewable energy generation system
[1

6]
. The key issues concerning the control of a threelevel voltage source PWM rectifier are about neutral potential balancing, switch losses and the crosscoupling between
i
_{d}
and
i
_{q}
. In order to improve the output power of the converters, the switching frequency of the semiconductors (IGBT, IGCT or GTO) are usually kept low to restrain switch losses at permitted level
[7

9]
. However, low switching frequency not only brings about a heavy crosscoupling between the current components
i
_{d}
and
i
_{q}
as well as a poor dynamic system performance
[10
,
11]
, also a severe harmonic distortion, especially the low order harmonics, in the gridconnecting current
[12]
.
Considering the situation that rectifiers usually being connected to grid, a heavy harmonic distortion can’t meet the grid standards, and will be harmful to other grid loads. A novel dead beat control scheme combining with repetitive control was presented in
[13]
, while it didn’t take consideration of the influence of low switching frequency. Literature
[14]
put forward a model predictive control strategy in static coordinates, which had a nice performance with a low switching frequency, however, its characteristic of frequency fluctuation brought about some other difficulties in filter design and issues of electromagnetic compatibility. Two kinds of approaches have been proposed in
[15]
to improve the harmonic distortion resulted from low switching frequency, in which a LCL filter was considered, whose reasonable parameters and volume could be a problem for its development.
Studies on a novel control scheme for a highpower threelevel voltage source PWM rectifier with a low switching frequency have been carried out in this paper, which are focusing on current decoupling controller. Section 2 establishes a complex model with low switching frequency for a threelevel PWM rectifier case considered in this paper, and Section 3 analyze the problems resulted from low switching frequency in detail. Section 4 describes a novel complex current controller to realize the decoupling between id and iq, along with some simulation verification. The whole control scheme is shown in Section 5 and the performance are evaluated based on a experimental platform whose microprocessor being DSP and FPGA, at switching frequency
f
_{s}
=500Hz.
2. Complex Model for Threelevel PWM Rectifier
The topology of a threelevel PWM voltage source rectifier studied in this paper is shown in
Fig. 1
.
Topology of the threelevel PWM voltage source rectifier
Where,
e
_{a}
,
e
_{b}
,
e
_{c}
represent the grid voltages;
i
_{a}
,
i
_{b}
and
i
_{c}
are gridside currents;
ν
_{aO}
,
ν
_{bO}
,
ν
_{cO}
are ac voltages;
L
and
R
are the filter inductance and resistance respectively;
R
_{L}
stands for load while
i
_{L}
is the load current; idc is the current of DClink and
V
_{dc}
represents the DClink voltage;
C
_{1}
and
C
_{2}
are DClink capacitors whose voltages are
u
_{c1}
and
u
_{c2}
; O stands for clamped point while O’ is the midpoint of the grid.
The traditional models for PWM rectifier are described in dq coordinates as following
Where,
e
_{d}
and
e
_{q}
are the dq components of the grid voltage;
ν
_{d}
,
ν
_{q}
represent the dq components of the ac voltage respectively and
i
_{d}
,
i
_{q}
are the gridside currents in dq coordinates;
p
is differential operator;
ω
_{s}
stands for grid angular frequency and S
_{dP}
, S
_{qP}
, S
_{dN}
, S
_{qN}
represent the switch status also in dq coordinates.
Defining complex variables as
With the definitions in Eq. (3), the complex models can be obtained without the consideration of the influence of switching frequency
Where, τ
_{s}
= L/R is the time constant of the ac side;
e
_{s}
,
i
_{s}
,
ν
_{s}
are the complex variables of grid voltage, gridside current and ac voltage respectively;
S
_{P}
and
S
_{N}
are complex variables about the switch statuses.
From Eq. (4), the complex transform of the current loop can be obtained as
A low switching frequency influence the control system in terms of an effect on the PWM link and we use Eq. (6) to make an approximation, which containing a crosscoupling factor j
ω
_{s}
τ
_{d}
ν
_{s}
.
While,
ν
*
_{s}
is the reference voltage vector resulted from the current controller;
τ
_{d}
stands for the delay time with the low switching frequency and the sampling delay, normally,
τ
_{d}
=0.75/
f
_{s}
(
f
_{s}
is switching frequency)
[11
,
16]
. And the complex transform of Eq. (6) is
The complex transform of the whole current loop can written as
3. Crosscoupling Problem with a Low Switching Frequency
The corresponding open loop zeropole of Eq. (8) is obtained as shown in
Fig. 2
, in which, complex roots p
_{1}
= −1/
τ
_{d}
− j
ω
_{s}
, p
_{2}
= −1/
τ
_{s}
− j
ω
_{s}
, whose positions have relationships with the time constants
τ
_{d}
and
τ
_{s}
. When
f
_{s}
is high,
τ
_{s}
>>
τ
_{d}
, the pole p
_{2}
is the dominant one while p
_{1}
has a little effect on the system performance. Reducing
f
_{s}
makes p
_{1}
be near to the zero shaft, and has a gradually enhanced influence on the system performance. Actually, the complex factor j in Eq. (8) is the essential reason why does the cross  coupling exists.
ZeroPole map of the unregulated open loop
In order to analyze the influence on crosscoupling in further, a coupling function in frequency domain is defined to describe the coupling degree.
The coupling degree between
i
_{d}
and
i
_{q}
for this threelevel PWM rectifier with a low switching frequency can be obtained by substituting Eq. (8) into Eq. (9), which is shown in
Fig. 3
. From
Fig. 3
, it can be seen that low switching frequency brings a more serious crosscoupling.
Crosscoupling degree of the current loop at different switching frequency
4. Complex Current Controller Design
 4.1 Normal PI current controller
A normal PI with a feedforward compensation can realize the decoupling control for the current components
i
_{d}
and
i
_{q}
. For the complex transform of the current loop described as Eq. (5), a feedforward compensation j
ω
_{s}
τ
_{s}
i
_{s}
is introduced firstly, and then, a PI controller is given as
By designing
k
_{0}
and
τ
_{i}
properly, the current loop can realize a nice control performance. However, with the consideration of a low switching frequency, extra crosscoupling, j
ω
_{s}
τ
_{d}
i
_{s}
exists as shown in Eq. (8). There are two parameters need to be adjusted along with double feedforward compensations if using normal PI control.
The Bode diagram and Step response of normal PI current with different switching frequency
f
_{s}
are shown in
Fig. 4
.
From
Fig. 4 (a)
, it can be known that the control bandwidth will reduce along with the decrease of the switching frequency, while the rising time, peak time, adjusting time and overshoot increase shown in
Fig. 4 (b)
, which mean that normal current controller doesn’t suitable anymore with a low switching frequency.
Influence on normal current controller with different switching frequency: (a) Bode analysis; (b) Step response
By tuning the gain value
K
_{iP}
of the PI controller can get good performance without considering the switching frequency, as shown in
Figs. 5(a)
and
5(b)
. However, when the switching frequency is low, the adjustment of gain tuning can’t improve the system performance to a great degree compared with the complex current controller. The comparisons between the normal PI controller and the complex one are shown in
Fig.6
when
f
_{s}
= 500 Hz (both of which have achieved the best effect).
Influence on normal current controller with different gain value (a) Bode analysis; (b) step response
Comparisons between the normal PI controller and the complex one
 4.2 Complex current controller design
From above theoretical analysis, we know that crosscoupling has closely relationship with imaginary parts existing in the complex transform. Given that a novel controller
F
_{r}
(
s
) designed to emit two complex poles at the same time, the coupling phenomenon would be eliminated fundamentally.
In this paper,
F
_{r}
(
s
) is redesigned as Eq. (11), which is also called a complex current controller.
Where, there is only one parameter
k
_{0}
need to be designed and the complex signal graph of the current control system is shown in
Fig. 7
.
Complex signal flow graph of the current control system with a complex current controller
Substituting (11) into (8), the open loop current transform with a complex controller can be obtained as
Both complex poles now are eliminated and the coupling issue is solved.
In order to design the parameter
k
_{0}
, a closed loop transform with current controller is given as
Where, 2
ζ
ω
_{n}
= 1/τd ,
ω
_{n}
^{2}
=
k
_{0}
/(
τ
_{s}
t
_{d}
)
Eq. (13) shows that this kind of closed loop is a typical second order system, whose damping coefficient
ζ
should be designed as 0.707 to reach an optimal performance
[17]
, and then, the theoretical
k
_{0}
should be designed as
Considering the parameters of the threelevel PWM rectifier are: amplitude of
e
_{s}
is 690V,
L
= 5mH,
R
= 0.1 Ω,
C
_{1}
=
C
_{2}
= 6800μF,
V
_{dc}
= 1800V,
R
_{L}
= 0.1 Ω,
f
_{s}
=500Hz, so the calculated
τ
_{s}
= 0.05 and
τ
_{d}
= 0.0015, the theoretical
k
_{0}
is 16.67, and the influence for different
k
_{0}
is shown in
Fig. 8
.
Influence on the control system with different k_{0}
From
Fig. 8
, it can be seen that, larger
k
_{0}
can brings a more broad control bandwidth and a faster response, however, the overshoot increases.
Fig. 8
gives a recommendation that
k
_{0}
in this paper is 20, which also verify the robustness of the designed complex current controller.
 4.3 Simulation comparisons
The whole control scheme is designed as shown in
Fig. 9
, in which, the PI controller is used to control the DClink voltage
V
_{dc}
. The power factor is set as Pf =1 with taking the output of the PI voltage controller as the given value of
i
_{d}
and the given value of
i
_{q}
setting as zero.
Whole control scheme of the proposed strategy
Simulation comparisons have been carried out in MATLAB/simulink according to the control scheme as shown in
Fig. 9
, between the normal PI controller and the complex one while
f
_{s}
= 500Hz, and a conventional PI controller is used for voltage outer loop. The simulation parameters are as same as the ones in the part 5; at
t
= 0s,
R
_{L}
= 500Ω, at
t
=1.2s,
R
_{L}
=100Ω.
The current components
i
_{d}
and
i
_{q}
are shown in
Fig. 10
, the grid voltage ea and the enlarged gridside current
i
_{a}
are shown in
Fig. 11
,
Fig. 12
is the current trajectory of the gridside current with the complex current controller.
Current components id and i_{q} with different current controllers
Grid voltage e_{a} and the enlarged gridside current i_{a} with different current controllers
Current trajectory of the gridside current with the complex current controller
From
Fig. 10
, it can be known that, this kind of complex current controller can realize the decoupling with a better characteristic compared with the normal PI controller. The harmonic distortion of gridside current
i
_{a}
also has been improved as shown in
Fig. 11
. The dynamic current trajectory shown in
Fig. 12
verifies that this kind of complex controller has a nice dynamic performance.
5. Experimental Verification
Experimental platform has been established as shown in
Fig. 13
, where, a TI TMS320F28335 DSP is used to realize the control algorithm and a Xilinx Spartan3EFPGA is adopted to implement A/D,D/A, pulses generation. Type of the Power devices is Infineon 450A, 1700V IGBT, whose drivers are the type of Concept (3W, 20A). The detailed scale of the experimental platform is shown in
Table 1
.
Experimental platform
Detail experimental parameters
Detail experimental parameters
The implementation of the complex current controller is the most challenging part during the digital implementation. Firstly, Eq. (11) can be decomposed into the real and imaginary components. According to the definition in Eq. (3), that a real component corresponds the directaxis one (d coordinate) while an imaginary part corresponds the quadrature axis one (q coordinate). Then, the complex current controller could be designed in dq coordinates.
The steady waveforms of grid voltage
e
_{a}
and gridside current
i
_{a}
are shown in
Fig. 14
with the different control strategy.
The grid voltage e_{a} and the gridside i_{a} with different control strategy
Fig. 14
shows that complex current controller has a better characteristic in reducing the harmonic distortion of the gridside current.
During the experiment, the given
V
_{dc}
= 300V, at
t
= 2s,
V
_{dc}
= 450V and then reduced to 300V again. The corresponding DClink voltage, current in dq coordinates are shown in
Fig. 15
. Sudden loading and unloading experimental results are shown in
Fig. 16
.
Waveforms with a changement on the DClink voltage, CH1: given DClink voltage; CH2: real DClink voltage; CH3: current of d coordinate; CH4: current of q coordinate
Waveforms with a changement on the load, CH1: given DClink voltage; CH2: real DClink voltage; CH3: current of d coordinate; CH4: current of q coordinate
When a normal PI current controller was adopted, the dynamic waveforms of gridside current
i
_{a}
, the DClink voltage
V
_{dc}
and the current in dq coordinates are shown in
Fig. 17
with the same experimental processes (the given
V
_{dc}
= 300V, at
t
=2s,
V
_{dc}
=450V and then reduced to 300V again), which is consistent with the simulation result shown in
Fig. 10(a)
and
Fig. 11(a)
.
Waveforms by PI controller CH1: gridside current; CH2: real DClink voltage; CH3: current of d coordinate; CH4: current of q coordinate
When the load changed suddenly, the dynamic grid voltage
e
_{a}
and current
i
_{a}
are shown in
Fig.18
as well the dynamic DClink voltage, which verify the dynamic performance of this kind of control scheme for threelevel PWM rectifier.
Experimental waveforms in dynamic state, CH1: DClink voltage ; CH2: gridside current i_{a}; CH3 grid voltage e_{a}
6. Conclusions
Some research focusing on the decoupling control have been carried out for the threelevel voltage source PWM rectifier considering a low switching frequency. Firstly, a novel complex model was established for the threelevel PWM rectifier, and the influence resulted from low switching frequency was considered based on the detailed analysis on the influence on crosscoupling between in dq current components and performance of the normal current controller.
A complex current controller was proposed for the threelevel PWM rectifier along with its parameter design, and the comparisons have been conducted between the normal PI current controller and the complex one.The whole control scheme was also given based on the microprocessors DSP and FPGA, and experiments were carried out to verify the effectiveness of the designed control system.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (51207091/E070303) and the HuJiang Foundation of China (B14002/D14002) and the Shanghai Natual Science Foundation.
BIO
Qingqing YUAN She received her B.S., M.S. and Ph.D from the China University of Mining and Technology, China, in 2009, 2011, and 2014 respectively. Since 2014, she has been with the School of OpticalElectrical and Computer Engineering, University of ShangHai for Science and Technology, China, where she is currently a teacher. her current research interests include the modeling and control of high power drives with a low switching frequency.
Kun XIA He received his B.S., M.S. and Ph.D from the Hefei University of Technology, China, in 2002, 2004, and 2007 respectively. Since 2007, He is working as a teacher in Electrical Department, School of OpticalElectrical and Computer engineering, University of Shanghai for Science and Technology (USST) from July 2007 up to now. He is acting as the department dean and the leader of the electric professor group. He focused his main research work on electric engineering including motor control and new energy technology application.
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