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.
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 bi-directional and being adjusted independently, etc., three-phase PWM inverter grid-connected has been widely employed in a few of fields of motor four-quadrant 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 soft-switching 3-dimensional 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 frequency-dependent cable earth-return model, allowing the computation of the zero-sequence currents generated by the common-mode voltage produced by the inverter. Mihalache (2005) detailed the multi-loop 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 frequency-dependent cable earth-return path. Borage et al. (2009) studied the characteristics of an ADC controlled LCL-T 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 D-Q plane. A region on the D-Q 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 built-in THD optimizer improved THD at light load and the controller rapidly protected this cycle-by-cycle 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 non-ideal 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 three-level 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 DC-link 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 soft-switching 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 three-phase three windings V/v transformer, passive filters and a three-phase 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 in-depth 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 grid-connected: 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 grid-connected PWM inverter |
In Fig. 1, ua, ub and uc is the instantaneous voltage value of a, b, c-phase of the grid, respectively, va, vb and vc is the instantaneous voltage value of a, b, c-phase of the inverter ac side, respectively, ia, ib and ic is the instantaneous current value of a-phase, b-phase, c-phase 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 udc is its both ends voltage, RL is the load resistance. According to Kirchhoff's voltage theorem, the voltage equation of voltage source PWM inverter in three-phase stationary coordinate can be obtained as Eq. 1:
| (1) |
The change rates of active and reactive power in αβ coordinates can be given in Eq. 2:
| (2) |
The Eq. 3 can be described as:
| (3) |
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 grid-connected in two-phase stationary αβ coordinate are shown in Table 1, where Sa, Sb and Sc represents the switching status of phase a, b, c, respectively and Sk = 1 represents the upper switch in phase k turning on and the under switch turning off, Sk = 0 denotes the upper switch in phase k turning off and the under switch turning on, where k = a, b, c; Vi indicates the inverter voltage vector in ith sector, where i = I, II,..., XII.
When the three-phase 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 three-phase three-wire 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:
| (4) |
The three-phase grid voltages could be decomposed into positive and negative sequence preamble section, as Eq. 5:
| (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:
| (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:
| (7) |
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:
| (8) |
The complex power, active power and reactive power of the PWM inverter fed by the grid were given in Eq. 9:
| (9) |
In Eq. 9, P0 and Q0, respectively, was the average value of P and Q. Pc2 and Ps2, respectively, was the amplitude of the two times fundamental frequency fluctuation component in P. Qc2 and Qs2, respectively, was the amplitude of the two times fundamental frequency fluctuation component in Q. P0, Q0, Pc2, Ps2, Qc2 and Qs2 were defined in Eq. 10:
| (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 two-phase stationary coordinate system. It could simplify the coordinate transformation of control process, could eliminate the current error in two-phase stationary coordinate system. It could eliminate coupling relationship between the current in d-axis and q-axis 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 (Pokryszko-Dragan 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:
| (11) |
where, Kp is the proportional coefficient, KR is the resonance coefficient, ω0 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:
| (12) |
After
| (13) |
Laplace transform of tsinωt was in Eq. 14:
| (14) |
Laplace transform of tcosωt was in Eq. 15:
| (15) |
The anti-Laplace transform of Eq. 13 was as Eq. 16:
| (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 so-called 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 right-angle filter.
Quasi-PR controller: As shown in section above, compared with PI controller, PR controller could achieve zero steady-state 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 non-fundamental 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 quasi-PR controller easy to implement was proposed. The quasi-PR 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 Quasi-PR controller was in Eq. 17:
| (17) |
It could be seen from the Bode plot of PR controller that the amplitude-frequency characteristics of PR controller in the fundamental frequency was A(ω0) = 60dB and the margin of phase angle was infinite. Basically, it could achieve zero steady-state error but also had good steady margin and transient performance.
RESULTS
In order to validate the effectiveness of the proposed proportional resonant controller-based PWM inverter at unbalanced grid voltages, simulation results were given in this section. The asymmetric three-phase 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 three-phase grid voltages were shown in Fig. 4, in which the voltage amplitude of phase-a is 100 V, that of phase-b is 130 V and that of phase-c is 120 V. When the three-phase PWM inverter was fed with the asymmetric grid given in Fig. 4, its three-phase currents were shown in Fig. 5. It can be seen that the three-phase currents is also asymmetric. In the traditional control methods, the direct axis current or cross-axis 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 algorithm-based inverter |
The curves of direct-axis 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. 4-7 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 algorithm-based 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 direct-axis component and quadrature-axis 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. 4-7.
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 over-voltage 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 (2013R1-2). Project Supported by National Natural Science Foundation of China (51207002).