The mathematical model of a three phase PWM converter under the stationary
αβ
reference frame is deduced and constructed based on a ProportionalResonant (PR) regulator, which can replace trigonometric function calculation, Park transformation, realtime detection of a Phase Locked Loop and feedforward decoupling with the proposed accurate calculation of the inductance voltage vector. To avoid the parallel resonance of the LCL topology, the active damping method of the proportional capacitorcurrent feedback is employed. As to current vector error elimination, an optimized PR controller of the inner current loop is proposed with the zeropole matching (ZPM) and cancellation method to configure the regulator. The impacts on system’s characteristics and stability margin caused by the PR controller and control parameter variations in the innercurrent loop are analyzed, and the correlations among active damping feedback coefficient, sampling and transport delay, and system robustness have been established. An equivalent model of the inner current loop is studied via the polezero locus along with the pole placement method and frequency response characteristics. Then, the parameter values of the control system are chosen according to their decisive roles and performance indicators. Finally, simulation and experimental results obtained while adopting the proposed method illustrated its feasibility and effectiveness, and the inner current loop achieved zero static error tracking with a good dynamic response and steadystate performance.
I. INTRODUCTION
Sinusoidal current regulation with low total harmonic distortion (THD) of a 3phase bidirectional power flowing PWM voltage source converter (VSC) is an aspect of paramount importance to obtain good performance in many different applications, such as active power filters, motor drives, variablespeed wind turbines and photovoltaic inverters
[1]

[8]
. As a result, the research and development of various methods in the field of current regulators have been put forward to realize good power quality for VSCs. Over the past few years, pulse width modulation (PWM) and hysteresis controls with their optimization have been popularly adopted due to their advanced features
[5]

[9]
. Hysteresis modulation introduces a minor error to the average current and offers much better dynamic current tracking than the PWM method. However, the inherent disadvantage is its variable switching frequency, which makes it rather difficult to design a power filter for harmonics attenuation and to reduce the switching loss for efficiency improvement
[5]
. Some researchers have proposed a nonlinear sliding mode variable structure approach based on the adaptive band hysteresis theory, and a lot of effort has been done to achieve a fixed switching frequency. However, obtaining a highfrequency chatter free switching frequency on the sliding surface requires prior knowledge of the upper bounds of the system uncertainties
[6]
. Space vector modulation (SVM) is a widely used PWM strategy due to its constant switching frequency, and because its chosen switching sequence can be easily implemented with Park transformation
[7]
,
[8]
. Meanwhile, this voltage oriented control divides alternating current into two parts, active and reactive segments, and then controls them separately with linear PI regulators, which achieves a good dynamic response by the inner current control loop with accurate decoupling calculations and proportionalintegral regulating
[9]
. Some alternative control algorithms have been proposed such as instance predictive control
[10]
, direct power control (DPC)
[11]
,
[12]
, deadbeat control
[13]

[15]
, etc. The main principle of the DPC method is to directly regulate the instantaneous power instead of using the inner current loop, which will produce serious power ripples and a variable switching frequency along with predefined onoff forms and hysteresis comparators. In addition, the main shortcoming of the deadbeat strategy is its request for a small sampling period so as to achieve satisfactory performance. In terms of the inner currentloop controllers, there are some approaches to improve system characteristics of the PWM converter. A great deal of effort has been expended to tackle the above problems for tracking error cancellation. PI and PR approaches have been widely adopted in linear tuning strategies. The control of the VSC can be implemented under different reference frames, such as the
abc
,
αβ
or
dq
reference frames. Detailed information on the reference frames for the VSC is introduced in
[4]
. There is strong coupling between the
d
and
q
axes for a PI regulator, and sinusoidal values lead difficulty in precisely tracking the reference AC signals without a static error
[16]
,
[17]
. For solving this issue, a PR controller is introduced into the inner current loop
[18]

[22]
. Thus, the steady state error of a specific frequency can be completely removed since this ideal PR controller provides an infinite gain and a 180° phase shift at the points where the stability problem is produced. Nevertheless, unlike the PI controller, the PR controller is generally complicated to design for whole control systems with stability
[22]
.
In addition, the harmonics generated by converters having an impact on other grid connected utilities and devices are limited by IEEE 5192014, which presents limits for the THD of currents. Therefore, it is important to do filter studies for eliminating current harmonics
[23]

[25]
. A third order LCLfilter is quite suitable for gridtied VSCs with good performance in current ripple attenuation even with small inductances. In addition, an LCL filter exhibits an undesired resonance effect that causes stability problems. A passive damping method has good reliability and simplicity, but it leads to a lot of power dissipation
[26]
. These problems can be addressed by adopting the capacitor current feedback strategy of active damping, since it has better robustness to external disturbances than passive damping when the line current includes lots of harmonics
[27]
,
[28]
. However, the damping regulator introduces a voltage component into the VSC modulation index reference that is difficult to achieve for the traditional DPC. This is due to the fact that the PWM modulation is not directly controlled via the inner current loop. Thus, it can be tackled through a collaboration between the deadbeat predictive power control approach and an optimized PR regulator under the stationary
αβ
reference frame with the SVM. This is done to obtain a high dynamic instantaneous active and reactive power track with a variable onoff frequency prevention and a relatively satisfactory performance. Normally, the system’s stability margin and characteristics can be illustrated by the means of the steadystate error, crossover frequency, phase margin (PM), and gain margin (GM)
[17]
. As a result, the above mentioned methods can be employed for the system parameter and robustness design of the current controller and the capacitorcurrentfeedback activedamping.
First, this paper analyzes the mathematical model of a threephase grid connected VSC under the stationary
αβ
reference frame, and then presents an accurate calculation of the inductance voltage vector approach with a predictive power control outer loop. The impact of parameter variations on an innercurrent loop adopting an active damping PR regulator on the system dynamic and static performance is mechanismly studied. Furthermore, the stability problem and optimal parameter configuration of the control block diagram correctly addressed by considering the polezero assignment method of the frequency domain response are also reviewed and discussed. A specific and effective approach for the control of an active damping LCLtopologybased VSC with a PR controller is proposed. Finally, simulation and experimental results are presented in a systematic way to verify the theoretical concepts and implementation is discussed throughout the whole presentation.
II. CALCULATION METHOD OF THE REFERENCE VOLTAGE
A gridconnected 3phase VSC adopting an LCLfilter is illustrated in
Fig. 1
, where
L_{g}
,
L_{r}
and
C_{f}
are the gridside inductor, converterside inductor and branch capacitor respectively, and the VSC typically consists of a 3phase IGBT bridge and a DC side electrolytic capacitor.
Topology diagram of threephase PWM converter.
The following equation could be defined by
Kirchhoff Law
when system is under the rectifier mode.
Where
V_{dc}
and
i_{dc}
are the DClink voltage and current,
R_{L}
is the equivalent resistance considering the line impedance and series resistance of the inductors
L_{g}
and
L_{r}
,
V_{L}
is inductance voltage vector, and
V_{k}
is the converter average voltage vector. Thus, the corresponding inductance voltage vector can be represented as:
To achieve the real parameter values of the series resistance, capacitor and reactor, a Precision Impedance Analyzer WK6500B is employed. Thus, the following expression can be written, considering (1) and (2), as:
To realize the actual current tracking its reference, the vector direction of
di/dt
should be approximately the same as
–ΔI
. This is equivalent to seeking the corresponding
V_{k}
with the smallest included angle between
ΔI
and
V_{Lk}
, as shown in
Fig. 2
.
Distribution of inductance voltage vectors for different switching states.
The equivalent control model for the VSC is generally achieved by neglecting the influence of the series resistance
R
[8]
,
[25]
,
[27]
. However, through the observation of (3) and
Fig. 2
of the inductance voltage vectors distribution, the series resistance directly impacts the calculation accuracy for the reference inductance voltage vector selection, and the resistance value for underestimating or overestimating has large effects on the integral absolute error although it is relatively small, as described Dr. Ana Vidal and Dr. Alejandro G. Yepes at the University of Vigo
[29]
. The function
IAE
should be defined to identify the underestimated and overestimated cases. The error
ε(k, t)
between both the curves and the integral absolute error
IAE(k)
are calculated.
IAE
is decided through the area covered between the two curves
i^{R=R’}
and
i^{R≠R’}
. Moreover, the sign of the error area is calculated via the integral error
IE(k)
.
By this means, the weighted
IAE
(
WIAE
) can be defined as:
As in the above analysis, the cost function
IAE
illustrates considerable influence on accurate reference vector selection, and the area between
i^{R=R’}
and
i^{R≠R’}
is relatively less when overestimating than when underestimating. Thus, a LCR analyzer WK6500B for precise vector computation in the VSC control strategy, which characterizes components up to 120 MHz with 0.05% basic measurement accuracy, can generally satisfy the requirement. And equation (4) and (5) along with
Fig. 3
theoretically verified whole analysis.
Effects of WIAE for R’ = R, R’ > R and R’ < R.
Define the inductance voltage vector
V_{Lc}
, which is composed by the adjacent vectors
V_{Li}
and
V_{Lj}
, is in the same direction, as
ΔI
in
Fig. 4
. Assume the parameter
Γ
as follows.
Distribution of voltage vector V_{Lc} and reference axis Y_{II}.
Define
T_{des}
as the vector composition period of
V_{Li}
and
V_{Lj}
for
ΔI
elimination.
The reference inductance voltage vector can be expressed as follows.
In order to calculate the parameter
Γ
, the reference axes
Y_{S}
are needed. They are located in six sectors and are perpendicular to axes a, b and c, as shown in
Fig. 5
. Assume that
Y_{II}(ΔI)
,
Y_{II}(V_{Lc})
and
Y_{II}(U)
are projections of
ΔI, V_{Lc}
and
U
on the reference axis
Y_{II}
. Since
ΔI
and
V_{Lc}
are in the same direction, the parameter
Γ
can be calculated via (7) and (8).
Distribution of reference axis Y_{S}.
Therefore, if the reference frame of a certain sector is confirmed, the parameter
Γ
of this sector will be obtained.
Each sector corresponds to a pair of adjacent inductance voltage vectors
V_{Li}
and
V_{Lj}
as shown in
Fig. 6
. Assume that
ΔI
is located in the region
S
, the reference axis
Y_{S}
should be used to calculate
Γ
.
Location of ΔI and distribution of V_{Lc1}, V_{Lc2} and V_{Lc3}.
The following vector and variables are defined.
When
ΔI
is located in section
II
, the projection value can be obtained as
and
which is substituted into (10) for the calculation of
Γ
. Thus,
Γ
with all of the conditions is obtained as shown in
Table I
by parity of reasoning.
COMPUTATION OF EACH SECTOR’S Γ
COMPUTATION OF EACH SECTOR’S Γ
In addition, as to the outer control loop, according to the instantaneous power theory and the equivalent coordinate transformation, the system’s instantaneous power
S
can be derived by multiplication between the voltage vector and the current vector conjugate, and it is expressed as follows:
Define
v_{αβ}
= [
v_{α} v_{β}
]
^{T}
,
u_{αβ}
= [
u_{α} u_{β}
]
^{T}
and
i_{αβ}
= [
i_{α} i_{β}
]
^{T}
as the average voltage vector of the VSC, and the line voltage and current vectors in the stationary
αβ
frame, respectively.
Under the
α–β
plane, the instantaneous active and reactive power model of the threephase VSC is defined as follows.
Assume that the line voltage
u_{αβ}
during instant
T
_{s}
is ideally invariant (
u_{αβ}
(
k+1
)=
u_{αβ}
(
k
)). Consequently, the algebraic iterative expression of the instantaneous powers during 2 continuous sampling periods is observed as:
To achieve the aim of the predictive control, the controlled object is adjusted to track its dynamical given values by the next period as shown in the following expression.
Accordingly, the converter average voltage vector is expressed as follows:
The reference input current in the
αβ
plane by the transformations from (14)(16) can be performed by means of the following relation:
Therefore, the topology of the proposed current control strategy adopting a PR regulator under the stationary
αβ
frame for a PWM converter is shown in
Fig. 7
.
Deadbeat predictive instantaneous power control strategy with PR controller based on inductance voltage vector under αβ frame.
III. ROBUSTNESS DESIGN OF THE CURRENT REGULATOR
The transfer function under the
αβ
plane of a PI regulator achieved by a positive or negativesequence is computed via the usage of the frequency shift adopting internal model control, and the optimized state transformation or frequency domain approach of a PR regulator is presented, which directly regulates the overall current, including both the positive and negative components under stationary
αβ
coordinates
[21]
,
[30]
. An ideal PR regulator has an infinite gain and a 180° phase shift at the fundamental frequency
ω_{0}
, and it has little phase shift and gain except for
ω_{0}
. In order to solve the stability problems caused by an infinite gain, an optimized PR regulator can be used in practical implementations with a finite gain, but high enough to achieve little static error
[18]
,
[19]
. The improved PR controller is in the following form:
Where
K_{P}
and
K_{R}
are the proportional and resonant coefficients,
ω_{c}
is represents the bandwidth at 3 dB which can enhance the robustness and stability in the presence of line frequency variations.
 A. Model and Control of the System
For the inner current control loop, the grid side inductor current
i_{g}
is compared with its reference
i_{g}
^{*}
and produces an error into the PR regulator. This generates a given current value
i_{c}
^{*}
. The capacitor current
i_{c}
is fed back to adapt
i_{c}
^{*}
, and along with the proportion loop for active damping to produce the modulation reference for the converter. Therefore, the control block diagram can be derived as follows.
In terms of Mason's Rule and the system model, the input current expression can be introduced as:
By (19), the gain of
G_{PR}
(
s
)·
G_{D}
(
s
)·
G_{VSC}
(
s
)·
G_{LCL}
(
s
) at the control system’s resonant frequency is much larger than 1. Thus, the first item approximation of (19) is 1, and the grid voltage disturbance is approximately 0. The transfer function of the system current open loop is derived as:
Instability will be led to by a time delay of the calculation and the PWM loop
[31]
,
[32]
. In addition, the firstorder delay is on behalf of the
ZOH
of the VSC current. In addition, the delay
T_{d}
is considered to represent a digital implementation of the computational delay. The Taylor series expansion is used for modeling approximations of the two firstorder inertial elements.
Control block diagram of inner current loop.
For a three phase PWM converter with synchronous sampling of the input current and with a sampling period
T_{s}
, the total time delay including the sampling and transport delay is generally given by Σ
T_{i}
=
1.5T_{s}
.
The active damping technique using capacitor current feedback in
Fig. 7
is a type of control algorithm rather than physical elements. After the introduction of a second order oscillation element by the active damping gain
K_{w}
to suppress resonance peak, its Laplace transfer function becomes:
The system order is added greatly after employing a PR regulator. Thus, the equivalent expression of (20) can be written in the form of (23).
The damping ratio can be written as:
With the damping ratio
ζ
increasing gradually from 0 to 0.9, the resonance peak of the transfer function G
_{LCL}
(s) changes from 30dB to disappearance, so as to effectively restrain the amplification effect at the resonance frequency point, which is illustrated in
Fig. 9
. Since a larger
K_{w}
leads to the controller’s saturation and stability, the parameters are chosen as
ξ
=0.707 and
K_{w}
=1.36 based on the good convergence performance of the secondorder optimal theory.
Bode frequency characteristics corresponding to variable damping coefficient.
The controller’s bandwidth
ω_{c}
reflects the ability to track the input signal. Therefore, the system should have a larger bandwidth in order to enhance the dynamic response characteristics
[20]
,
[21]
. However, high frequency interference noise such as the switching frequency affects the system’s stability when
ω_{c}
increases. The change rule for this is depicted in
Fig. 10
. Since the VSC is required to run well while the grid fundamental frequency fluctuates between 49.5 Hz and 50.5 Hz
[17]
,
ω_{c}
=
2πΔf
= π rad/s is defined to get a lot of gain in whole operating frequency scale with a related maximum frequency variation of
Δf
= 0.5 Hz.
Frequency response of resonant term for variation in ω_{c}.
 B. Influence of System Parameters on the Root Locus
The sixorder system in (23) is a little too complex to get a practical solution. It shows that there are two zeros and the Bode diagram in
Fig. 9
shows that the general derivation LCL topology behaves like L at frequencies lower than the approximate resonant frequency
[29]
. Compared with the original system, it is easier to calculate with polezero cancellation from six to four. As illustrated in
Fig. 11
, the control target is to configure
P_{1}
and
P_{2}
as the dominant poles of a second order underdamped linear element with
ζ
=0.707, and the distance of
P_{3}
and
P_{4}
off the imaginary axis is 5 times bigger than that of the dominant poles. This ensures stability with enough of a margin for system parameter variations. As shown in the following analysis, parametercorresponding polezero locus for the closedloop transfer function is employed for optimal parameter configuration.
Polezero locus of inner current loop corresponding to variable parameter value. Arrows show the variation of pole placement. (a) With variable K_{w}, ΣT_{i} and L. (b) With variable K_{P} and K_{R}.
From
Fig. 11
(a), when the equivalent inductance
L
is increasing, the system’s poles distribution moves off the imaginary axis although its impact on the polezero locus of the system is relatively less. This improves the system stability. Thus, a good robustness to reactor value variations can be achieved. When the active damping coefficient
K_{w}
increases, the root locus of the dominant poles
P_{1}
and
P_{2}
moves from instability in the right half plane to a stable state with damp increasing, and then into the instable region again after the critical value is reached with
K_{w}
=1.4. This is in accordance with the optimum value mentioned above. Correspondingly, the poles
P_{3}
and
P_{4}
lean to shift far off of the imaginary axis with the system damp increasing and good resonance peak suppression. Meanwhile, the poles generated from the LCLfilter and PR controller shift quickly near the imaginary axis while the total delay Σ
T_{i}
increases. It can be seen that the parameters scale that leads to system instability is extended with a time delay. This makes the controller and poles hard to configure. From
Fig. 11
(b),
K_{P}
and
K_{R}
should be determined via the predicted placement of
P_{1}
and
P_{2}
so that the damping ratio and inherent frequency are involved. When
K_{P}
increases, the dominant poles shift off of the imaginary axis and then near it, when the system’s damp changes from ‘large’ to ‘small.’ This has almost as much influence as
K_{w}
.
K_{R}
has less influence on the root locus than
K_{P}
. This leads to the poles shifting near the imaginary axis. Thus, its damping is decreased. It is wellknown that
K_{P}
is directly related to the dynamic performance of regulator. However,
K_{R}
mainly decides its gain at a specific frequency and adjusts its bandwidth close to the resonance. According to the polezero locus, a parameters scale that meets the system stability and performance requirements can be obtained.
Based on the principle discussed above, the rules of the parameters’ influence are illustrated, and their optimal ranges are derived. The frequency response of the system Bode and Nyquist diagram with selected optimal parameters is depicted in
Fig. 12
. In terms of the Nyquist theory, the control system is for stability since its trajectory does not encircle the critical point (
1, j0
)
[33]
. Compared with the original continuous sdomain model, they have an amplitude margin of
h
= 20lg
G
(
jω_{gs}
) = 9.7
dB
and a phase margin of
γ
=
π
+ ∠
G
(
jω_{cs}
) = 30.2° at the cross over frequency. In addition, the corresponding frequency of the dominant poles at 3dB is about 220Hz, which is far enough from the fundamental frequency. This is good for the system’s characteristics. Consequently, the phase and amplitude margin guarantees the control system’s stability.
Frequency response performance of current loop. (a) Overview of Nyquist diagram. (b) Bode plot diagram.
 C. Digital Implementation of the Current Regulator
Generally, the current regulator is analyzed under the continuous time domain and it is necessary to adopt an appropriate discretization approach in the digital control system, such as the Tustin (bilinear transformation), firstorder hold, and impulse invariance to guarantee the discrete domain of current controller can accurately match its continuous model
[14]
,
[31]
,
[33]
. In addition, an inappropriate discretization method leads to the zeros and poles shifting in terms of the system’s transfer function, which may deteriorate controller’s stability and decrease its tracking precision.
During the discretization procedure, the impacts of aliasing distortion in the impulse invariance method are indistinguishable from the original continuous model. In addition, the time lagging response and frequency shift in the Tustin method lead to an unsatisfied frequency response as a result of the continuous sdomain transfer function
[33]
. In order to improve the signal tracking precision in a practical system, the ideal digital implementation approach cannot cause a gain attenuation or phase shift at the resonant frequency. To solve the above problems during the controller’s discretization, the zeropole matching (ZPM) discipline is predicted as follows:
There are proportional and resonant parts for a PR controller. The proportional part is linear with a small constant coefficient
K_{p}
. Thus, the controller’s discretization derivation about the resonant segment is as follows.
Where,
and
Define
and
, and according to the zeropole matching rule shown in (25), the discretization result of the PR controller using the ZPM method is derived as follows:
Define
P
=
e
^{p1Ts}
and
Q
=
e
^{p2Ts}
, and then the transfer function in the z domain can be obtained.
From (28), the differential expression to be coded for the discretetime implementation can be written as:
Equation (29) is implemented by using the digital implementation structure shown in
Fig. 13
.
Digital implementation diagram of controller.
As shown in
Fig. 14
, the discretization results using the zeropole matching method can offer reasonable positions for the system’s resonant peaks and a similar frequency response as the continuous transfer function. When the Tustin method is applied, due to its simple calculation, the results are quite unsatisfactory. The resonant frequency shifts from 50 Hz to 49.02 Hz, and phase response has a delay of −23.5°, which is harmful to the regulator stability. Define
ω_{A}
as the angular frequency of the continuous time domain and
ω_{D}
as the the angular frequency of discretetime domain, so the transformation mechanism from continuous to the discrete domain via Tustin is written as following substitution.
Discretization results adopting Tustin and ZPM method with 4 kHz sample frequency.
The expression
is obtained, and when the sampling frequency is quite fast, the approximate equation
ω_{A}≈ω_{D}
can be reached theoretically. Thus, the ZPM method has better performance due to the accurate zero/pole transformation and mapping between the two domains without frequency aliasing distortion and phase shift.
IV. SIMULATION AND EXPERIMENTAL RESULTS
 A. Simulation Results
In this section, the results of the described mathematical analysis and the effectiveness of the suggested strategy have been verified by means of numerical simulations. The system’s parameters are listed in
Table II
.
PARAMETERS OF ACTIVEDAMPING LCL VSC PROTOTYPE
PARAMETERS OF ACTIVEDAMPING LCL VSC PROTOTYPE
Fig. 15
and
Fig. 16
show the ACside current trajectory presented in this paper compared with the traditional SVM. The three marked sections of the trajectory mean: [I] the rectifier state, [II] the inverter state, and [III] the state that limits the amplitude of the current. The dynamic tracking performance of the proposed solution is much better than that of the traditional SVM.
The trajectory of the current vector I under traditional DPCSVM.
The trajectory of the current vector I under proposed method.
It can be observed from
Figs. 17

20
, that the instantaneous power and the DC voltage both track each reference and include good stability and little static error. The input current has an almost sinusoidal waveform (THD = 2.38%) and is synchronous to the grid voltage. Thus, the unity power factor running of the VSC is successfully implemented with a reactive power that is approximately zero.
Waveforms of converter side current and voltage. (a) Current and voltage of converter side. (b) Line voltage vs converter side current.
Waveforms of grid side Aphase voltage and current.
Current spectrum of phase A.
DCLink voltage dynamic response.
 B. Experimental Verification
The considered gridconnected VSC prototype with a LCLfilter has been experimentally tested according to the design strategy proposed in this paper. The parameters of the system are reported in
Table II
and a PR regulator is designed as analyzed in Section III.
The voltage and current waveforms of phase A both at the VSC and the grid sides are shown in
Fig. 21
and
Fig. 22
, respectively. The harmonics component analysis of the full load is given in
Fig. 23
. The current THD analysis at different loads by adopting the modified PR regulator with active damping is compared to that obtained by adopting the traditional PI controller with normal DPCSVM, as shown in
Fig. 24
.
Fig. 25
shows the experimental waveforms of the DC voltage and line current transient response with less over shoot of the DC bus reference voltage track. The reference for the active power has been varied in five steps with a constant zero reactive power. Each of them has a width of 100ms. The given and measured power components are plotted in
Fig. 26
with good dynamic track performance. It can be concluded that the proposed algorithm can improve the quality of the grid current, and that the robustness of the controller is verified through experimental results in accordance with the simulation results.
Experimental waveforms of converter side current and voltage. (a) Current and voltage of converter side. (b) Line voltage vs converter side current.
Experimental waveforms of line side voltage and current.
Experimental input current spectrum of phase A.
THD comparison chart of grid side phase current.
Experimental waveforms of DC voltage transient response. (a) DC voltage step from 93V to 106V. (b) DC voltage step from 93V to 118V. (c) Zoomin graphic of Fig. 25(a).
Dynamic response of reference and measured power value.
V. CONCLUSION
This paper presented a theoretical analysis of the inductance voltage vector control strategy based on a Proportional Resonant regulator under the stationary
αβ
frame and the stability robustness approach for a threephase VSC with respect to an optimum parameter match. To achieve an appropriate inductance voltage vector for each sector with accurate calculations on the current error to modulate the PWM converter, the judgment rule of sector selection and the optimal switch state by the geometrical distribution model is established. This can replace the trigonometric function calculation, Park transformation, realtime detection of the PLL and feedforward decoupling used in the traditional space vector modulation. The proposed strategy with an active damping LCL filter achieves tracking errors of the dynamic instantaneous power via the application of the required inductance voltage vector under the stationary
αβ
coordinate during each switching period. In addition, the PR controller is designed and discretized by means of the frequency response of the zeropole assignment matching and cancellation with an indepth performance influence analysis, which is frequently encountered when considering a system’s robustness in the presence of parameter fluctuations and regulator digital discretization of a currentcontrolled VSC. The methodology to analyze and enhance the transient response of the inner current loop, based on the study of the system’s transfer function locus through pole–zero placement, is also discussed. The optimal gain margin and phase margin are configured to get a rapid and nonoscillating transient response. Finally, the feasibility and effectiveness of the proposed design method are verified by the means of simulation and experimental results on a laboratory PWM converter prototype.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (50977063), and the National High Technology Research and Development Program of China (863 Program) (2013BAG01B00, 2011AA11A279).
BIO
Qiang Sun was born in Shandong Province, China. He received his B.S. and M.S. degrees from the Tianjin University of Technology, Tianjin, China, in 2007 and 2010, respectively. He is presently working towards his Ph.D. degree in the School of Electrical Engineering and Automation, Tianjin University, Tianjin, China. In 2010, he was with Samsung ElectroMechanics Co., Tianjin, China, where he worked on the research and development of electromechanical system applications. Since 2011, he has been with SIEMENS Electrical Drives Ltd., Tianjin, China, where he is working on the research and development of power electronics. His current research interests include power electronics technology, EMC mechanisms and control theory for power converters, electric vehicles and power quality.
Kexin Wei was born in Tianjin, China. He received his B.S. and M.S. degrees from Tianjin University, Tianjin, China, in 1978 and 1988, respectively. He was a Visiting Scholar at the Delft Technology University, Delft, Netherlands, and at Texas A&M University, College Station, Texas, in 1994 and 2000, respectively. Since 1997, he has been a Professor in the School of Electrical Engineering, Tianjin University of Technology, Tianjin, China, and a parttime Professor at Tianjin University. He is presently working as the Head of the Tianjin Key Laboratory of Control Theory and Applications in Complicated Systems and as the Vicepresident of the Tianjin University of Technology. His current research interests include power electronics technology and control theory for power converters, electric vehicles and power quality.
Chenghai Gao was born in Henan Province, China. He received his B.S. degree from the Xi’an University of Architecture Science, Xi’an, China, in 1991; and his M.S. degree in the School of Electrical Engineering and Automation, Tianjin University, Tianjin, China, in 2010. He is presently working towards his Ph.D. degree in the School of Electrical Engineering and Automation, Tianjin University. Since 1995, he has been with SIEMENS Electrical Drives Ltd., Tianjin, China, where he is working on the research and development of power electronics as a R&D manager. His current research interests include power electronics technology and control theory for power converters and power quality.
Shasha Wang was born in Shandong Province, China. She received her B.S. degree from Dezhou University, Dezhou, China, in 2008; and her M.S. degree from the Tianjin University of Technology, Tianjin, China, in 2011. Since 2011, she has been with the R&D Department, Tianjin EV Energies Co., Ltd., Tianjin, China, where she is working on BMS development for electric vehicles. Her current research interests include BMS technology and battery SOC characteristics.
Bin Liang was born in Tianjin, China. She received her B.S. degree from the Tianjin University of Technology, Tianjin, China, in 2004; and her M.S. and Ph.D. degrees from Tianjin University, Tianjin, China, in 2006 and 2013, respectively. She is presently working as a Lecturer in the School of Electrical Engineering, Tianjin University of Technology. Her current research interests include power electronics technology and EMC mechanisms.
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