
Research Article


The Analysis and Implementation of Proportional Resonant Control Algorithmbased Pwm Inverter Imbalanced Gridconnected


Zhang Jun Li,
Li Yu Ren,
Wang Peng,
Liang Bo
and
FU Long Fei


ABSTRACT

The mathematical model of Pulse Width Modulation (PWM) inverter was established. According to the symmetrical components theory, the phenomena of 2nd ripple in the DC side voltage of PWM inverter fed by imbalanced grid voltage was analyzed. The control performance degradation of traditional control methods was caused by the presence of negative sequence components. The Proportional Resonant (PR) control algorithm was proposed in detail in this paper. The improved PR control algorithm was applied to control PWM inverter fed by imbalanced grid voltage. The improved PR controller could effectively reduce the 2nd ripple in PWM inverter DC side voltage and could reduce harmonic content of PWM inverter AC side current. Thereby the PR controller proposed could improve the control performance of the PWM inverter. The effectiveness of the proportional resonant control algorithm proposed was demonstrated by the simulation results in MATLAB.





Received:
October 27, 2014; Accepted: February 18, 2015;
Published: March 05, 2015 

INTRODUCTION Due to many advantages such as sinusoidal input current, power factor being controlled, smaller capacitors in dc side achieving high quality output voltage, active power and reactive power flowing bidirectional and being adjusted independently, etc., threephase PWM inverter gridconnected has been widely employed in a few of fields of motor fourquadrant operation, new energy generation systems, flexible AC transmission system and other industrial domains (Fujita et al., 1993; Jung et al., 1996; Malesani et al., 1996; Shiraishi et al., 2004; Suh and Lipo, 2004).
Zhan et al. (2005) investigated QRDCL softswitching 3dimensional hysteresis current control for power quality compensator and the easy time sequence matching simplifies the system control. De Paula et al. (2005) presented a time domain methodology for cable modeling able to represent the cable parameters variation due to skin effect in this broad range of frequencies. It included frequencydependent cable earthreturn model, allowing the computation of the zerosequence currents generated by the commonmode voltage produced by the inverter. Mihalache (2005) detailed the multiloop method, positive and negative sequence controllers were used to regulate the fundamental currents and resonant controllers tuned to each harmonic from third to thirteenth remove the low frequency components. Oshikata et al. (2007) proposed the suitable switching sequence for partial resonance and it was required to achieve full ZVS on the main switches after partial resonance. De Paula et al. (2008) proposed a time domain methodology for cable modeling, which could reproduce accurately the wave propagation and reflection phenomena, thus showing to be very appropriate to transient overvoltage studies in PWM motor drives. It included a new alternative to represent the frequencydependent cable earthreturn path. Borage et al. (2009) studied the characteristics of an ADC controlled LCLT resonant converter operating at the resonant frequency. In four operating modes, different conditions during the device switching was created. The mode boundaries are obtained and plotted on the DQ plane. A region on the DQ plane is identified for the converter design. Hsu and Chao (2009) discussed the trend in ac adapters for notebook computers which aims to offer smaller size, higher efficiency and lower price along with a highly integrated BCM PFC and QR PWM combo controller. A builtin THD optimizer improved THD at light load and the controller rapidly protected this cyclebycycle sampling in case of output over voltage.
Nouri and Ghasemzadeh (2011) proposed a FDG with a new control scheme for the purpose of contributing to power generation and harmonic compensation under nonideal source voltages. The proportional resonant controllers were used for reference signal tracking. Cetin and Hava (2012) investigated the CMV/CMC properties of the two and threelevel VSIs for various PWM methods and it provided a comprehensive leakage current evaluation. Chen et al. (2013) presented a dual power control scheme to control the active and reactive power independently under the unbalanced power grid voltage conditions. Chen et al. (2013), the proportional resonant controller was in the outer loop suppresses the DClink ripple and the PI repetitive control was applied in the inner loop to eliminate active and reactive power oscillation. Yang et al. (2013) proposed an easy method to design the PR regulator parameters separately, which can guarantee the stability of the system and the performance of the inner current loop. Russi et al. (2013) provided a mathematical expression to determine the occurrence of softswitching for a general topology of ZVT converters with auxiliary resonant voltage source. The proposed conditions was useful for analysis and design, allowing evaluating operation sequence, total commutation time, resistive losses and other important parameters to describe the converter performance. Xu et al. (2013) proposed a novel compensation system consisting of a threephase three windings V/v transformer, passive filters and a threephase VSC. Xu et al. (2013), the harmonic suppression theory, the negative sequence current compensation principle, a novel reference current detection algorithm and the quasi proportional resonance were presented. In this study, the mathematical model of PWM inverter fed by imbalanced grid voltage was established. According to the analysis via symmetrical components theory, there was 2nd ripple content in the DC side voltage and plenty of harmonic content in the AC side currents of PWM inverter. The proportional resonant control strategy was described in this study and a more indepth study was conducted in it. The PR control strategy was applied to control PWM inverter fed by imbalanced grid voltage. The method of adjusting its resonant frequency to two times of the fundamental frequency could effectively reduce the harmonic content in PWM inverter DC side voltage. Compared with the traditional control methods, the control method proposed in this study could effectively reduce the harmonic content of PWM inverter while it did not affect the dynamic response of PWM inverter. The control method proposed could improve the power factor of the inverter and it could be extended to the relevant applications of the inverter. METHODOLOGY
Mathematical model of the PWM inverter gridconnected: The schematic diagram of the voltage source PWM inverter is shown Fig. 1 (Shiraishi et al., 2004; Suh and Lipo, 2004; Mihalache, 2005).
 Fig. 1:  Schematic of gridconnected PWM inverter 
In Fig. 1, u_{a}, u_{b} and u_{c} is the instantaneous voltage value of a, b, cphase of the grid, respectively, v_{a}, v_{b} and v_{c} is the instantaneous voltage value of a, b, cphase of the inverter ac side, respectively, i_{a}, i_{b} and i_{c} is the instantaneous current value of aphase, bphase, cphase of the grid respectively, L is the inductance value of line reactor, R is the line resistance, C is the capacitance of the dc side and u_{dc} is its both ends voltage, R_{L} is the load resistance. According to Kirchhoff's voltage theorem, the voltage equation of voltage source PWM inverter in threephase stationary coordinate can be obtained as Eq. 1:
The change rates of active and reactive power in αβ coordinates can be given in Eq. 2: The Eq. 3 can be described as:
The distribution of inverter voltage vectors and its sectors are shown in Fig. 2 (De Paula et al., 2008; Soltanzadeh et al., 2014). There are six active voltage vectors and two zero voltage vectors in Fig. 2 and the subscript of the voltage vector represents the switching mode of phase a, b and c, respectively.
 Fig. 2:  Distribution of inverter voltage vectors and its sectors 
Table 1:  Switching states and voltage vectors of inverter 

The switching states and the corresponding voltage vector values of PWM inverter gridconnected in twophase stationary αβ coordinate are shown in Table 1, where S_{a}, S_{b} and S_{c} represents the switching status of phase a, b, c, respectively and S_{k} = 1 represents the upper switch in phase k turning on and the under switch turning off, S_{k} = 0 denotes the upper switch in phase k turning off and the under switch turning on, where k = a, b, c; V_{i} indicates the inverter voltage vector in i_{th} sector, where i = I, II,..., XII.
When the threephase grid voltage is imbalanced, there are positive sequence component, negative sequence components and zero sequence components in the electrical variables of the PWM inverter fed by the grid according to the symmetrical components theory (Pan et al., 2014; Xu et al., 2014; Zeng et al., 2014). For the threephase threewire PWM inverter fed by the grid, there is no zero sequence components without neutral wire. Only the positive sequence component and negative sequence component were considered. Coordinate transformation relationship between the electrical variables was shown in Eq. 4: The threephase grid voltages could be decomposed into positive and negative sequence preamble section, as Eq. 5: In Eq. 5, ω is the angular frequency of the grid voltage. The superscripts p and n denote positive sequence component and negative sequence component respectively. The subscript + and – denote positive and negative synchronous rotating coordinate system, respectively. The variables in negative sequence synchronous rotating coordinate system was transformed into the ones in the positive synchronous rotating coordinate system as shown in Eq. 6: In Eq. 6, It could be seen that in the condition of imbalanced grid voltage, the grid voltage was transformed into the synchronous rotating coordinate system. The grid voltage included the positive sequence DC component and negative sequence two times fundamental frequency harmonic fluctuations AC component: As can be seen from Eq. 7, when the grid voltage was imbalance, the positive sequence component and the negative sequence component in synchronous rotating coordinate system of the PWM inverter fed by the grid were given in Eq. 8: The complex power, active power and reactive power of the PWM inverter fed by the grid were given in Eq. 9: In Eq. 9, P_{0} and Q_{0}, respectively, was the average value of P and Q. P_{c2} and P_{s2}, respectively, was the amplitude of the two times fundamental frequency fluctuation component in P. Q_{c2} and Q_{s2}, respectively, was the amplitude of the two times fundamental frequency fluctuation component in Q. P_{0}, Q_{0}, P_{c2}, P_{s2}, Q_{c2} and Q_{s2} were defined in Eq. 10: Analysis of proportional resonant control algorithm: The zero static error control of the AC input signal could be achieved via Proportional Resonant (PR) controller. The PR controller could be applied to control PWM inverter fed by the grid, could regulate the current in twophase stationary coordinate system. It could simplify the coordinate transformation of control process, could eliminate the current error in twophase stationary coordinate system. It could eliminate coupling relationship between the current in daxis and qaxis component and it could ignore the disturbance role of the grid voltage to the PWM inverter (Karimi et al., 2014; Qu and Zhao, 2014). PR controller: PR controller, namely the proportion of resonant controller, was composted with the proportion section and the resonant section (PokryszkoDragan et al., 2014; Bai et al., 2014). PR controller could achieve no static error control to the sine signal. The transfer function of ideal PR controller was given in Eq. 11:
where, K_{p} is the proportional coefficient, K_{R} is the resonance coefficient, ω_{0} is the resonant frequency. The integral section of the PR controller was also known as the generalized integrator, which could integrate the amplitude of the sine signal in the resonant frequency points. For the input signal with the same frequency Msin(ωt+φ), the response analysis in time domain was as below. Laplace transform of the input signal was as Eq. 12:
After the expression was as Eq. 13:
Laplace transform of tsinωt was in Eq. 14: Laplace transform of tcosωt was in Eq. 15: The antiLaplace transform of Eq. 13 was as Eq. 16: From the above equation, when φ = 0, the output signal of the resonant controller was:
which was the same phase with the input signal and the amplitude linear increasing with time. At that time φ = 90°, the output signal of the resonance controller was as:
When the time was slightly longer, the value close to cos(ωt). From the overall look of the resonator (or socalled generalized integrator), it was by the time increments with the error signal. As shown in Fig. 3, the integral part of PR controller in the resonant frequency point could achieve infinite gain and was with little attenuation in addition to the resonant frequency. Therefore, in order to selectively compensate the harmonic, PR controller could be used as a rightangle filter.
QuasiPR controller: As shown in section above, compared with PI controller, PR controller could achieve zero steadystate error and selectively improve the ability of anti disturbances from grid voltage (Chu et al., 2014; Bao et al., 2014).
 Fig. 3:  Bode plot of ideal proportional resonant controller 
But in actual system applications, the achievement of PR controller implied two main problems. The one was that PR controller was not easy to be achieved for the accuracy restriction of the analog system component parameters and digital system and the other was that as PR controller gain at nonfundamental frequency points was very small, it could not effectively suppress harmonics from the grid while an offset in grid frequency.
Therefore, on the basis of PR controller, a quasiPR controller easy to implement was proposed. The quasiPR controller could either maintain high gain as PR controller but also could effectively reduce the impact on inverter output inductor current while the grid frequency offset. The transfer function of QuasiPR controller was in Eq. 17: It could be seen from the Bode plot of PR controller that the amplitudefrequency characteristics of PR controller in the fundamental frequency was A(ω_{0}) = 60dB and the margin of phase angle was infinite. Basically, it could achieve zero steadystate error but also had good steady margin and transient performance. RESULTS In order to validate the effectiveness of the proposed proportional resonant controllerbased PWM inverter at unbalanced grid voltages, simulation results were given in this section. The asymmetric threephase grid voltage drop was studied, the PWM simulation system parameters were shown in Table 2.
The simulation model of PWM inverter at asymmetric grid voltages was built in MATLAB software and its system characteristics was analyzed. The proposed method of proportional resonant control was compared with conventional control methods. The asymmetric threephase grid voltages were shown in Fig. 4, in which the voltage amplitude of phasea is 100 V, that of phaseb is 130 V and that of phasec is 120 V. When the threephase PWM inverter was fed with the asymmetric grid given in Fig. 4, its threephase currents were shown in Fig. 5. It can be seen that the threephase currents is also asymmetric. In the traditional control methods, the direct axis current or crossaxis current of PWM inverter AC side currents via rotating coordinate transformation was not controlled. There was always the presence of the 2nd ripple in PWM inverter DC side voltage. The proposed proportional resonant control strategy could effectively reduce the 2nd ripple in PWM inverter DC side voltage as it could control direct axis current and quadrature axis current in PWM inverter AC side currents.
Table 2:  Circuit parameters of PWM inverter 

 Fig. 4:  The imbalanced grid voltages 
 Fig. 5:  The imbalanced grid currents 
 Fig. 6:  The current of proportional resonant control algorithmbased inverter 
The curves of directaxis current and the quadrature axis current in PWM inverter AC side currents were shown in Fig. 6. The curves of PWM inverter DC side voltage based on proportional resonant controller was shown in Fig. 7. It could be seen in Fig. 47 that, the PWM inverter DC side voltage had been effectively controlled and the amplitude of 2nd ripple in PWM inverter DC side voltage was effectively controlled.
DISCUSSION The mathematical model of PWM inverter at unbalanced grid voltage was established in this study. The AC side current and DC voltage in PWM inverter was studied in depth via the symmetrical components theory.
 Fig. 7:  The DC voltage of proportional resonant control algorithmbased inverter 
The 2nd ripple in the DC voltage and the harmonic content in the AC side current could not be effectively reduced while the PWM inverter was controlled by traditional control methods. In traditional control methods, the direct axis current could be controlled but the quadrature axis current could not be effectively controlled. The direct axis current and the quadrature axis current could not effectively controlled simultaneously, so that there were the 2nd ripple in the DC voltage and plenty of harmonic content in the AC side current. The voltage or current in resonance point could be effectively controlled in proportional resonant control algorithm. The working principle of proportional resonant control strategy was analyzed and improved in this study. The proposed control strategy was applied to controlling PWM inverter at unbalanced grid voltage. Compared to the references (Martin et al., 2013; Wang et al., 2014; Karimi et al., 2014), it can be seen that the directaxis component and quadratureaxis component in the AC side current of the PWM inverter could be controlled in the proposed control strategy. The 2nd ripple in the DC voltage and the harmonic content in the AC side current could be effectively reduced while the PWM inverter was controlled by the proportional resonant control algorithm. The effectiveness of the proportion of the resonant control strategy applied to controlling PWM inverter at unbalanced grid voltage was demonstrated by the simulation results in Fig. 47.
CONCLUSION According to the analysis of the symmetrical components theory, the 2nd ripple emerged in the DC side voltage of PWM inverter at asymmetric grid voltages. The 2nd ripple would cause overvoltage in DC side capacitor but also would increase the substantial harmonic content in the PWM inverter AC side current. The control performance of the PWM inverter would be weakened and the grid source would be polluted. In this paper, the improved proportional resonant control strategy was applied to control PWM inverter, which could effectively reduce the 2nd ripple in PWM inverter DC side voltage. The effectiveness and the strong robustness of the proposed control strategy were verified by simulation results in this study. ACKNOWLEDGMENTS Project Supported by Baoji University of Arts and Sciences (GK1506). Project Supported by Baoji City Science and Technology Bureau (2013R12). Project Supported by National Natural Science Foundation of China (51207002).

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