https://doi.org/10.6113/JPE.2019.19.1.89
ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718
Unbalance Control Strategy of Boost Type Three-Phase to Single-Phase Matrix Converters Based on Lyapunov Function
Yu-xiang Xu†, Hong-juan Ge*, and Hai Guo*
†,*Nanjing University of Aeronautics and Astronautics, Nanjing, China
Abstract
This paper analyzes the input side performance of a conventional three-phase to single-phase matrix converter (3-1MC). It also presents the input-side waveform quality under this topology. The suppression of low-frequency input current harmonics is studied using the 3-1MC plus capacitance compensation unit. The constraint between the modulation function of the output and compensation sides is analyzed, and the relations among the voltage utilization ratio and the output compensation capacitance, filter capacitors and other system parameters are deduced. For a 3-1MC without large-capacity energy storage, the system performance is susceptible to input voltage imbalance. This paper decouples the inner current of the 3-1MC using a Lyapunov function in the input positive and negative sequence bi-coordinate axes. Meanwhile, the outer loop adopts a voltage-weighted synthesis of the output and compensation sides as a cascade of control objects. Experiments show that this strategy suppresses the low-frequency input current harmonics caused by input voltage imbalance, and ensures that the system maintains good static and dynamic performances under input-unbalanced conditions. At the same time, the parameter selection and debugging methods are simple.
Key words: Lyapunov function, Matrix converter, Positive and negative sequence, Unbalance, Weighted synthesis
Manuscript received May 29, 2018; accepted Sep. 15, 2018
Recommended for publication by Associate Editor Liqiang Yuan.
†Corresponding Author: keyan_xyx@163.com Tel: +86-025-84896005, Nanjing Univ. of Aeronautics and Astronautics
*Nanjing University of Aeronautics and Astronautics, China
Ⅰ. INTRODUCTION
The matrix converter (MC) is a kind of “green converter” that has the advantages of low harmonic pollution, an adjustable input power factor, a bidirectional energy flow, and a high- power density. Based on these excellent attributes, it has received a lot of attention from scholars.
The three-phase to single-phase matrix converter (3-1MC) is a special form of the traditional M-phase to N-phase matrix converter. With the development of industrial technology, the 3-1MC has been used more and more as a special topology in matrix converter applications, such as battery charge and discharge management [2], micro-grid and distributed power grid interfaces [3], [4], wireless energy transmission [5], wind power [6], [7], and high-frequency induction heating [8]. The above research results show that the 3-1MC has good application prospects in single-phase power consumption. In this paper, the single-phase ripple power at the output side of the conventional 3-1MC topology directly affects the performance of the input side. The authors of [4], [9]-[12] propose a class of power compensation strategies for different types of 3-1MC topologies to suppress the impact of output pulsating power on system performance. The above methods make the double-frequency output ripple power form a loop on the output side and not couple to the input side. Although having no large-capacity DC side energy storage link brings a higher power density in practical applications, the input voltage imbalance and other anomalies directly affect the input and output performances of the matrix converter through the bidirectional switches, which results in input current and output voltage distortions. At present, research on control strategies for improving system performance under input imbalance is mainly focused on the three-phase to three-phase matrix converter (3-3MC) topology [13], [14]. The authors of [13] proposed an improved voltage synthesis control strategy to improve the quality of the 3-3MC input current and its output voltage waveforms. The authors of [14] proposed the use a proportional-integral resonance (PIR) regulator instead of the traditional proportional-integral (PI) regulator to suppress the low frequency ripple in the active power, which improved the quality of the input side current waveform. The authors of [15] proposed using a resonant controller to adjust the input current and active power values to eliminate the input current harmonics and to improve the output waveform. The unbalanced control strategy mentioned above covers matrix converters in the buck mode. However, the voltage boosting type is studies in this paper. Therefore, the voltage synthesis method described in [13] cannot be used to suppress the low-frequency harmonics in the input current. The parameter selection and discretization of the resonant converters in [14], [15] are more complex than a PI controller can handle. Since the 3-1MC can still be equivalent to a virtual rectification topology with a virtual inverter, the unbalanced control strategy of voltage-source pulse width modulation (PWM) rectifiers and inverters can provide ideas for the 3-1MC imbalance [16].
Therefore, in order to simultaneously determine the impact of output power pulsation and an unbalanced input on system performance, this paper studies the influence of the 3-1MC input imbalance on the input active and reactive powers, and derives the expressions of control equations under the positive and negative sequence double dq rotation axes based on the Lyapunov function method. Using the topologies in [10], [11] to eliminate the low-frequency harmonic components in the input current caused by unbalance, the use of the Lyapunov function theory is proposed to separate the active and reactive powers on the input side. The control improves the control effect of the single grid-side current feedback control strategy under a single rotating coordinate axis. Simultaneously, the system parameter setting is simplified, and the input and output performances under an unbalanced input are improved. Finally, the effectiveness and feasibility of the proposed control strategy are verified by simulation and experimental results.
Ⅱ. POWER COMPENSATION PRINCIPLE
A. Input Waveform Quality Analysis
A boost-type 3-1MC topology based on capacitance for power compensation was studied in [11], as shown in Fig. 1. The topology includes an input inductor L, a bidirectional switch Sij (i = a, b, c; j = u, v, w), a load RL, an output filter capacitor Cf, and a compensation capacitor Cc. The conventional 3-1MC topology does not include the compensation unit Cc or the bidirectional switches Sav, Sbv and Scv in Fig. 1. The output side single-phase voltage and fundamental current wave expression are expressed as:
(1)
Fig. 1. Pulsating power compensation unit of the 3-1MC topology.
where, 
denotes the output voltage angular frequency, while 
and 
denote the output voltage and current amplitude, respectively. 
is the phase difference between the output voltage and the current, 
is the output side current initial phase, and 
is the output side impedance.
The expression for the output power from the above equation is:
(2)where, 
and 
denote the constant and pulsating output powers, respectively.
Assuming the input is a symmetrical three-phase sinusoidal voltage, its expression is:
From the above equation, when the output is single-phase AC, in addition to the constant power in the output power, there is a pulsating power with twice the output frequency. Neglecting the system loss, according to the principle of the conservation of the instantaneous power of the input and output, in addition to the fundamental wave, the input three- phase current must contain harmonic components related to the output frequency. Ignoring the high-frequency harmonics in the input current, assume that the three-phase input current is:
where, 
denotes the amplitude of the fundamental current, 
denotes the initial phase of the fundamental wave current, 
and 
denote the initial phases of the harmonic currents of 
and 
in the input current, 
denote the amplitudes of the harmonics of 
and 
in the input current, and the harmonic current amplitude is assumed to satisfy: 
.
The expression for the input power 
can be obtained from Eqns. (3) and (4):
Ignoring the system loss, according to the principle that the input and output power are approximately equal, the frequency of the harmonics 
in the upper form are calculated according to the method of undetermined coefficients:
For the upper formula there is a harmonic in the input current of 
and 
, which is related to the input and output voltage frequencies 
and 
. In addition, the low frequency harmonics are hard to eliminate via a low pass filter.
B. Theoretical Derivation of Compensation Function Implementation
To facilitate the analysis of the compensation function implementation process of the 3-1MC, the topology in Fig. 1 is equivalent to the virtual AC-DC-AC circuit structure shown in Fig. 2. In addition, the expressions of the compensation-side current modulation function 
and the output-side current modulation function 
are separately set as:
Fig. 2. Equivalent AC-DC-AC topology of a 3-1MC.
where, 
denotes the modulation coefficient of the output-side current modulation function, and 
denotes the modulation coefficient of the compensation-side current modulation function.
is the initial phase of the compensation-side current modulation function.
The compensation side current 
and the voltage expressions 
are:
where, 
denotes the amplitude of the compensation side current, 
denotes the amplitude of the fundamental voltage, and 
is the capacitor reactance.
The compensation side and output total power expressions are:
If the input is the three-phase symmetry current, the input power 
is constant. It follows from (9) that the total output power 
should also be constant. In this case, the output fluctuating power 
is cancelled by the power 
of the compensation side.
C. System Voltage Utilization Analysis
The relation between the virtual bus voltage 
and the input voltage amplitude 
in the AC-DC-DC topology of the virtual rectification and virtual inversion (shown in Fig. 2) is given in (11):
The phase angle 
between the output voltage 
and the current 
is ignored, and the relationship between the voltage amplitude 
at the output side and the virtual bus voltage 
is:
To obtain the maximum voltage utilization of the system, the zero-vector time of the output virtual inverter stage is minimized. For the sake of simplifying the derivation process, the phase angle of the initial current 
in (7) is taken as zero. Then the current modulation function in (7) must satisfy the following constraint relationship:
(13)From (13), the maximum value of the current modulation coefficient 
at the output side is:
(14)Therefore, the 3-1MC voltage utilization 
expression shown in Fig. 1 is:
The system parameters are set as follows:
1) The single-phase load 
.
2) The output filter capacitor 
.
3) The compensation capacitor 
.
Figs. 3 and 4 show the relationships among 
and the system parameters.
Fig. 3. Effect of the output filter capacitor Cf on the system minimum voltage utilization 
.
Fig. 4. Influence of different compensation capacitors Cc on the system minimum voltage utilization .
From Figs. 3 and 4, it can be seen that the topology studied in this paper is a boost matrix converter. The minimum voltage utilization 
of the system is related to the compensation capacitor Cc, and it is affected by the output-side filter capacitor Cf. It can be seen from Fig. 3 that when the value of the output side filter capacitor Cf changes and the other system parameters remain unchanged, the minimum voltage utilization 
of the system decreases with an increase of the filter capacitor Cf at the same output frequency. It can be seen from Fig. 4 that when the value of the compensation side capacitor Cc changes and the other system parameters remain unchanged, the minimum voltage utilization 
of the system increases with an increase of the compensation side capacitor Cc at the same output frequency. When compared with the traditional 3-3MC voltage utilization rate of no higher than 0.866, the proposed 3-1MC has a very high step-up ratio, which is very suitable for single- phase power consumption with amplitude step-up requirements.
Ⅲ. 3-1MC CONTROL STRATEGY UNDER AN UNBALANCED INPUT
A. Insufficient Single Grid-Side Current Feedback Control Strategy
The 3-1MC inner loop control under a three-phase voltage balanced input often adopts the single grid-side current feedback control strategy, whose control block diagram is shown in Fig. 5. This control strategy adopts a single current inner loop control. After the output voltage amplitude 
is adjusted by the PI regulator, the dq axis current components 
and 
of the given inner loop are output, and double closed loop control under a balanced input is realized.
Fig. 5. Single grid-side current feedback loop control diagram of a 3-1MC.
When the input is a three-phase unbalanced voltage, assuming that the input is a three-phase asymmetrical current, this current only contains the fundamental component and no zero-sequence component. Then the input current is the sum of the positive sequence and negative sequence currents. As shown in (16):
where, 
denote the input current positive sequence components, 
denote the input current negative sequence components, and 
and 
denote input current positive and negative sequence amplitudes, respectively. 
and 
denote the initial phase of the positive and negative sequence components of the input current, respectively.
The expression of the abc-dq transformation of the three-phase to two-phase positive-sequence rotation axis transformation is:
(17)If the system adopts single grid-side current control, the input current positive sequence and negative sequence components are transformed under the positive sequence rotation dq coordinate axis as shown in (18) and (19), where the angular velocity of the positive and negative rotation dq axes is 
, the direction of rotation is reversed, and the initial positive and negative phases are 
and 
, respectively.
The total expression of the three-phase unbalanced current input in the positive-sequence dq rotation axis transformation is the sum of (18) and (19):
From the above equation, under the single grid-side current feedback control, due to the use of a PI regulator, it is impossible to implement static-free control of the AC component in (20). Therefore, a method based on the single grid-side current feedback control strategy cannot effectively inhibit an unbalanced input voltage on the input current.
The relationships among the virtual bus current 
and the input voltage, current positive sequence and negative sequence dq axis components are shown in (21):
where, 
is the input active power, 
and 
are the projection of the positive sequence component of the input voltage under the positive sequence dq rotation axis, 
and 
are the projection of the negative sequence component of the input voltage under the negative sequence dq rotation axis, 
are the projection of the positive sequence component of the input current under the positive sequence dq rotation axis, and 
are the projection of the negative sequence component of the input current under the negative sequence dq rotation axis. 
are the amount of DC and ripple in the virtual dc bus current idc.
The expressions of the output voltage 
and the compensation side voltage 
are shown in (22):
From (21) it can be seen that if the system unbalance is small, if 
, then 
. In addition, it is possible to ignore the effect of the virtual bus current twice the input frequency harmonics on the output voltage. When the imbalance is large, the harmonics contained in 
distort the output. If the above control strategy is still used at this time, (21) and (22) show that the output voltage contains the harmonics of 2ωi-ωo and 2ωi+ωo, respectively.
B. Method of Suppressing Double Frequency Pulsation in Input Active Power Under an Unbalanced Input
Assuming that input unbalanced voltage contains only positive and negative sequence components, the complex vector expression under the dq rotation coordinate of the input voltage and current is:
The relationships among the system apparent power 
, the active power 
, and the reactive power 
are:
where, 
is the conjugated plural form of 
. In addition:
where, 
and 
are the constant terms of the active power and reactive power, 
and 
are the steady and fluctuating amounts of active power, 
and 
are the steady and fluctuating amounts of reactive power, and 
and 
are the amplitudes of the active power cosine and sinusoidal ripple, respectively. 
and 
are the amplitudes of the reactive power cosine and sinusoidal ripple, respectively.
From (25), it can be seen that the pulsation component 
in the active power generates a double-frequency input ripple on the virtual dc bus. If the reactive power 
, the unity power factor cannot be achieved on the input side. Since the input and output of the system are directly connected, the input active power ripple also affects the output side. Therefore, it is impossible to reduce the influence of the unbalance on the output via absorbing harmonics by increasing the output filter capacitance in the PWM rectification control, since there is no large-capacity energy storage unit in the 3-1MC topology.
Therefore, in order to suppress the influence of the input active power secondary ripple on the input current and output voltage, the control system is designed to control the instantaneous active power 2nd harmonic component amplitude in (25) under the condition that the system output active power 
is determined by the output of the outer loop regulator. The values 
,
, and instantaneous reactive power dc component amplitude 
are zero. In this paper, independent control of the four control variables is implemented in the positive and negative sequence dq coordinate system of the input current, in order to suppress the double-frequency harmonics of the active power input at the grid side. The current decoupling inner loop control object expression is:
where, 
are the current commands for the input current positive and negative sequence components in the positive and negative sequence dq rotation axes.
C. Balanced Input Controller Design Based on a Lyapunov Function in the dq Axis Coordinates
The input voltage and current under a three-phase balanced input, and the expression in the dq axis coordinate system of the converter input voltage are:
where, 
and 
denote the components of the input current on the dq axis. 
and 
denote the components of the input voltage of the matrix converter on the dq axis.
The error functions 
and 
defined in the input three-phase current dq axis coordinate system are:
where, 
are the current dq axis component settings.
Take the positive definite Lyapunov function expression 
as:
The derivative of the above formula is as follows:
Substituting Equ. (27) into (30) yields:
(31)
In order to make the system stable, the Lyapunov function needs to be a positive definite function. The coefficients are chosen as 
. From the above formula, the expression 
is obtained. Then it is possible to derive:
From the above formula, the target control quantity expressions of 
and 
for the input side voltage of the matrix converter under the dq coordinate system are as follows:
D. Realization of a 3-1MC Control Strategy Based on a Lyapunov Function under an Unbalanced Input
According to Equ. (28), the current error expression under an unbalanced input on the input side positive and negative sequence axes is:
where, 
and 
are current error functions under the positive sequence axis, and 
and 
are current error functions under the negative sequence axis.
When the three-phase input is unbalanced, the positive and negative sequence dq axis voltages 
on the converter side are used as control variables, and the positive and negative sequence dq axis current components 
on the grid side are used as control objects. The 3-1MC input current is decoupled and the positive and negative sequence dq axis controllers independently control the dq axis component of the input current. This is done to control the input side current waveform and power factor. Since the current loop adopts a regular pattern based on a Lyapunov function, according to the feed-forward decoupling control law, and in combination with the expression of the target control quantity function under the equilibrium condition represented by (33), the conversion under the unbalanced condition can be obtained. The equations of the control commands for the positive and negative sequence voltages 
at the input side of the converter under unbalanced conditions are shown in (35) and (36):
The outer loop control often uses an output side voltage amplitude closed loop. However, this method requires a phase locked loop, which increases the difficulty and computational complexity. This study uses the output-side and compensation side voltage weighted synthesis as the feedback value, and the outer loop is output by the PI regulator as the constant component command value 
of the output active power. Its expression is shown in (37), and a 3-1MC control block diagram is shown in Fig. 6.
Fig. 6. Dual current decoupling closed-loop control strategy block diagram based on a Lyapunov function.
where, 
and 
are the proportional and integral parameters in the outer loop PI regulator.
Ⅳ. SIMULATION RESULTS
In order to verify the effectiveness of the proposed control strategy, simulations have been carried out using MATLAB/ Simulink. The system simulation parameters are set as follows:
1) Three-phase unbalanced voltage 
and 
.
2) Input frequency 
.
3) Input inductance parameters 
and 
.
4) Output voltage amplitude 
.
5) Output frequency 
.
6) Output filter parameter 
.
7) Switching frequency 
.
Fig. 7 shows simulation waveforms of the three-phase input current and output voltage using a single grid-side current feedback loop under an unbalanced input voltage. From the simulation waveforms, it can be seen that the three-phase input current waveform distortions are severe, and that the total harmonic distortion (THD) of the three-phase input current iabc are 8.12%, 8.67% and 8.18%. In addition, the unbalanced voltage also affects the output voltage uo. The THD of the output voltage uo is 4.62%.
Fig. 8 shows the dual-current decoupling based on a Lyapunov function to the inner loop, the output side voltage and the compensation side voltage weighted synthesis amount as outer loop simulation waveforms. According to Fig. 8(a), the input steady-state current waveforms at the input unbalanced voltage are good, and the phase difference between the output side and the compensation side satisfies the requirements of Equ. (10). When compared with the input current simulation results using single grid-side current feedback control (Fig. 7), the quality of the three-phase input current iabc and the output voltage uo waveforms under the proposed control strategy are improved, and the respective THD values of the input currents are improved. The THD values of the input currents iabc are: 4.63%, 3.74% and 3.55%. The THD of the output voltage uo is 4.16%.
Ⅴ. EXPERIMENTAL RESULTS AND ANALYSIS
In order to verify the effectiveness of the proposed control strategy, a prototype of the proposed 3-1MC based on four-step commutation is built with a DSP28335+CPLD as the main control core. The voltage and current samples in the prototype are measured by Hall sensors VSM025A and CSM025, the optocoupler is driven by TLP250, the power circuit is composed of 18 IGBTs of 1MBH60D-100, and the bidirectional switch is composed of two 1MBH60D-100. The system experiment parameters are set as follows:
1) Three-phase unbalanced voltages 
and 
.
2) Input frequency 
.
3) Input inductance parameters 
and 
.
4) Output voltage amplitude 
.
5) Output frequency 
.
6) Output filter parameter 
.
7) Switching frequency 
.
8) Four-step commutation time 
.
Fig. 9 shows steady-state experimental waveforms of the input currents iabc for the single grid-side current feedback control strategy under input three-phase voltage unbalanced conditions. From this figure, it can be seen that the control strategy cannot effectively suppress the influence of the unbalanced voltage on the input current and output voltage. These three-phase input current waveforms show obvious asymmetry and distortion. The THD values of the three-phase input current iabc under the single grid-side current feedback control strategy are: 11.8%, 13.18% and 12.19%. The THD value of the output voltage uo is 5.95%. This result is consistent with the simulation analysis in Fig. 7.
Fig. 10 shows three-phase steady-state input current and output voltage waveforms under the input unbalanced control strategy proposed in this paper. When compared with Fig. 9, the low-frequency harmonics in the three-phase input current are significantly suppressed and the input performance is improved. Additionally, the steady-state output is good. The THD values of the three-phase input current iabc under the control strategy proposed in this paper are: 7.71%, 6.92%, and 6.64%. The THD value of the output voltage uo is 5.48%.
Fig. 11 shows a transient process in which the system switches from a balanced input to unbalanced conditions under the unbalanced control strategy based on a Lyapunov function, in which the RMS of the phase C voltage uc is suddenly reduced from 18V to 12V. From this figure, it can be seen that the output voltage uo, and the input currents ib and ic can all reach a stable state quickly, and that the dynamic response process is very short and there is no obvious oscillation, which indicates that the control strategy proposed in this paper can effectively suppress unbalanced input current and output voltage.
Fig. 11. Switching from input voltage to unbalanced dynamic experimental waveforms based on Lyapunov function and dual current decoupling closed-loop strategy.
Ⅵ. CONCLUSION
This paper studied a 3-1MC topology, which can suppress the effect of output ripple on input performance. The principle of pulsating power absorption, the implementation process, and the effect on the system voltage transmission rate were analyzed in detail. For a 3-1MC without large capacity storage capacitors, input asymmetry can easily cause system performance degradation. This paper proposed a 3-1MC asymmetric control strategy based on a Lyapunov function in positive and negative sequence dual coordinates. Finally, a system simulation model was established and an experimental platform was built to verify the effectiveness and feasibility of the proposed control strategy. Experimental results showed that under an asymmetric input, when compared with the traditional single grid-side current feedback control strategy, the input and output performances based on the proposed control strategy were greatly improved and that the input side power factor was high. At the same time, experimental results of the transient response showed that the input and output response time was short and without obvious oscillation. The above results show that the proposed control strategy has the advantage of convenient debugging. They also show that it has the ability to eliminate the impact of output ripple power and input asymmetry on system performance, and improve the practicality of the studied topology.
ACKNOWLEDGMENT
This paper is funded by the following fund: The National Natural Science Foundation of China (Fund number U1233127); Huzhou Natural Science Fund Project (Fund number 2017YZ09).
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Yu-xiang Xu was born in Zhejiang, China, in 1981. He received his B.S. and M.S. degrees in Mechanical Engineering from the Zhejiang University of Technology (ZJUT), Hangzhou, China, in 2004 and 2008, respectively. He is presently working towards his Ph.D. degree in Electric Machines and Electric Apparatus from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China. His current research interests include matrix converters and digital control of power converters.
Hong-juan Ge was born in Jiangsu, China, in 1966. She received her B.S. and M.S. degrees in Electrical Engineering from Southeast University, Nanjing, China, in 1985 and 1988, respectively; and her Ph.D. degree in Electric Machines and Electric Apparatus from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2006. Her current research interests include space-vector control of PWM, AC-AC converters, and airworthiness technology.
Hai Guo received his M.S. degree from the Hefei University of Technology, Hefei, China, in 2009. He is presently working towards his Ph.D. degree in the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China. Since 2009, he has been working as an Associate Professor at the Anhui University of Science and Technology, Huainan, China. His current research interests include matrix converters, power electronics, and automatic control technologies.