A Systematic Method for Designing a PR Controller and Active Damping of the LCL Filter for Single-Phase Grid-Connected PV Inverters

ABSTRACT:

The Proportional Resonant (PR) current controller provides gains at a certain frequency (resonant frequency) and eliminates steady state errors. Therefore, the PR controller can be successfully applied to single grid-connected PV inverter current control. On the contrary, a PI controller has steady-state errors and limited disturbance rejection capability. Compared with the L- and LC filters, the LCL filter has excellent harmonic suppression capability, but the inherent resonant peak of the LCL filter may introduce instability in the whole system. Therefore, damping must be introduced to improve the control of the system.

PV INVERTER

Considering the controller and the LCL filter active damping as a whole system makes the controller design method more complex. In fact, their frequency responses may affect each other. The traditional trial-and-error procedure is too time-consuming and the design process is inefficient. This paper provides a detailed analysis of the frequency response influence between the PR controller and the LCL filter regarded as a whole system.

LCL FILTER

In addition, the paper presents a systematic method for designing controller parameters and the capacitor current feedback coefficient factor of LCL filter active-damping. The new method relies on meeting the stable margins of the system. Moreover, the paper also clarifies the impact of the grid on the inverter output current. Numerical simulation and a 3 kW laboratory setup assessed the feasibility and effectiveness of the proposed method.

 KEYWORDS:

  1. Single phase
  2. Grid-connected
  3. LCL filter
  4. Active damping
  5. Proportional resonant (PR) controller

SOFTWARE: MATLAB/SIMULINK

CIRCUIT DIAGRAM:

 

 Figure 1. Two-stage single-phase PV system with LCL-filter control scheme.

EXPECTED SIMULATION RESULTS:

 

Figure 2. Grid voltage and injected current at full load with nominal parameters: simulation results. (a) Grid voltage sag; (b) grid voltage swell.

Figure 3. Grid voltage and injected current at full load with inductor L1 variation: simulation results. (a) Inductor L1 increased by 20%: grid voltage sag; (b) Inductor L1 increased by 20%: grid voltage swell; (c) Inductor L1 decreased by 20%: grid voltage sag; (b) Inductor L1 decreased by 20%: grid voltage swell.

Figure 4. Grid voltage and injected current at full load with inductor L2 variation: simulation results. (a) Inductor L2 increased by 150%: grid voltage sag; (b) inductor L2 increased by 150%: grid voltage swell; (c) inductor L2 decreased by 20%: grid voltage sag; (b) inductor L2 decreased by 20%: grid voltage swell.

Figure 5. Grid voltage and injected current at full load with capacitor C variation: simulation results. (a) Capacitor C increased by 20%: grid voltage sag; (b) capacitor C increased by 20%: grid voltage swell; (c) capacitor C decreased by 20%: grid voltage sag; (b) capacitor C decreased by 20%: grid voltage swell.

CONCLUSION:

The stability analysis of the system composed by a PR controller and an LCL filter together is not easy: the frequency responses may affect each other and the PR controller design becomes complex. The traditional method based on trial-and-error procedures, is too time-consuming, and the design process is inefficient. This paper provides a detailed analysis of the frequency response influence between the PR controller and the LCL filter.

PR CONTROLLER

In addition, the paper presents a systematic design method for the PR controller parameters and the capacitor current feedback coefficient, used in the active damping of the LCL filter. Using the new parameters, a numerical simulation shows that the system meets the requirements of stable margins and current tracking steady-state error. The robustness of the current controller is verified through several experimental tests carried out on a 3 kW platform varying the system parameters.

INDUCTOR

The Bode diagrams of the system varying inductor, capacitor, and grid impedance values confirmed that the controller parameters enhance robustness against the system parameters variation. Moreover, the system remains stable even in case of grid voltage fluctuation. Both the simulation and the experimental results assess the validity of the proposed design method.

REFERENCES:

  1. Carrasco, J.M.; Franquelo, L.G.; Bialasiewicz, J.T.; Galvan, E.; Guisado, R.C.P.; Prats, A.M.; Leon, J.I.; Moreno-Alfonso, N. Power-electronic systems for the grid integration of renewable energy sources: A survey. IEEE Trans. Ind. Electron. 2006, 53, 1002–1016.
  2. Wessels, C.; Dannehl, J.; Fuchs, F.W. Active Damping of LCL-Filter Resonance based on Virtual Resistor for PWM Rectifiers—Stability Analysis with Different Filter Parameters. In Proceedings of the 2008 IEEE Power Electronics Specialists Conference, Rhodes, Greece, 15–19 June 2008; pp. 3532–3538.
  3. Castilla, M.; Miret, J.; Matas, J.; de Vicuna, L.G.; Guerrero, J.M. Control design guidelines for single-phase grid-connected photovoltaic inverters with damped resonant harmonic compensators. IEEE Trans. Ind. Electron. 2009, 56, 4492–4501.
  4. Yi, L.; Zhengming, Z.; Fanbo, H.; Sizhao, L.; Lu, Y. An Improved Virtual Resistance Damping Method for Grid-Connected Inverters with LCL Filters. In Proceedings of the 2011 IEEE Energy Conversion Congress and Exposition (ECCE 2011), Phoenix, AZ, USA, 17–22 September 2011; pp. 3816–3822.
  5. Parker, S.G.; McGrath, B.P.; Holmes, D.G. Regions of Active Damping Control for LCL Filters. In Proceedings of the Energy Conversion Congress and Exposition (ECCE), Raleigh, NC, USA, 15–20 September 2012; pp. 53–60.

An Adaptive Proportional Resonant Controller for Single Phase PV Grid Connected Inverter Based on Band-Pass Filter Technique

ABSTRACT:  

This paper being an adaptive proportional resonant (PR) controller for single phase grid connected inverter that modify its control parameters to grid impedance change. Forth order band bass filter is method and then combine with the adaptive system for on-line detection of any variations in the resonance frequency. The estimated frequency is then prepared by mathematical signal processing operation to identify the variations in the grid impedance. For the on–line tuning of the PR parameters, a look-up table technique is apply and its parameters are linked with the measure impedance values. Simulation results based on MATLAB environment clearly verify the effectiveness of the proposed control scheme for 2 kW grid connected inverter system.

KEYWORDS:
  1. Adaptive Proportional Resonant Controller
  2. Grid Impedance Estimation
  3. LCL Filter
  4. Look-up Table

 SOFTWARE: MATLAB/SIMULINK

 BLOCK DIAGRAM:

Fig. 1. Block diagram of the proposed adaptive PR controller.

 EXPECTED SIMULATION RESULTS:

Fig. 2. Simulation result of emulated grid voltage.

Fig.3. FFT analysis of grid current. (a) APR controller. (b).PR controller.

Fig. 4. Online adaptation of the APR control parameters.

Fig. 5. Grid voltage and current waveforms under changeable grid

impedance with the proposed control strategy.

 CONCLUSION:

 A new control strategy based on an adaptive proportional resonant (APR) controller has been grown and successfully proved on a simulated 2 kW single phase grid tide PV inverter. A fourth order Sallen-Key band pass filter tailored to the system to taking the harmonic components around the resonant frequency has been execute. data signal processing method was employed in order to provide the controller with signals compare to the variable grid impedance. A large low level of current total harmonic distortion (THD) is reach in comparison with conventional PR controller and submission with IEEE929-Standard has been show.

REFERENCES:

[1] S. Kouro, J. I. Leon, D. Vinnikov, and L. G. Franquelo, “Grid-Connected Photovoltaic Systems: An Overview of Recent Research and Emerging PV Converter Technology,” IEEE Industrial Electronics Magazine, vol. 9, pp. 47-61, 2015.

[2] “IEEE Recommended Practice for Utility Interface of Photovoltaic (PV) Systems,” in IEEE Std 929-2000, ed, 2000.

[3] “IEEE Draft Application Guide for IEEE Standard 1547, Interconnecting Distributed Resources With Electric Power Systems,” in IEEE Unapproved Draft Std P1547.2/D11, Sept 2008, ed, 2008, p. 1.

[4] H. M. El-Deeb, A. Elserougi, A. S. Abdel-Khalik, S. Ahmed, and A. M. Massoud, “An adaptive PR controller for inverter-based distribution generation with active damped LCL filter,” in 2013 7th IEEE GCC Conference and Exhibition (GCC), 2013, pp. 462-467.

[5] W. L. Chen and J. S. Lin, “One-Dimensional Optimization for Proportional-Resonant Controller Design Against the Change in Source Impedance and Solar Irradiation in PV Systems,” IEEE Transactions on Industrial Electronics, vol. 61, pp. 1845-1854, 2014.

A New Design Method of an LCL Filter Applied in Active DC-Traction Substations

ABSTRACT:

This paper concentrates on the LCL filter with damping resistance intended to connect the shunt active power filter of an active DC-traction substation to the point of common coupling with the transmission grid. In order to find design conditions and conceive a design algorithm, attention is directed to the transfer functions related to currents and the associated frequency response. The mathematical foundation of the design method is based on the meeting the requirements related to the significant attenuation of the high-frequency switching current, concurrently with the unalterated flow of the current that needs to be compensated by active filtering. It is pointed out that there are practical limitations and a compromise must be made between the two requirements. To quantify the extent to which the harmonics to be compensated are influenced by imposing the magnitude response at both highest harmonic frequency to be compensated and switching frequency, a performance indicator is defined. As an additional design criterion, the damping power losses are taken into consideration. The validity and effectiveness of the proposed method are proved by simulation results and experimental tests on a laboratory test bench of small scale reproducing the specific conditions of a DC-traction substation with six-pulse diode rectifier.

KEYWORDS:

  1. DC-traction substations
  2. LCL filter
  3. Passive damping
  4. Regeneration
  5. Shunt active power filters

SOFTWARE: MATLAB/SIMULINK

BLOCK DIAGRAM:

Fig. 1. Block diagram of the active DC-traction substation.

EXPECTED SIMULATION RESULTS:

Fig. 2. Voltages and currents in the TT’s primary in traction regime

Fig. 3. LCL filter input current.

Fig. 4. Current flowing through the capacitor of the interface filter.

Fig. 5. Harmonic spectra of the LCL filter input current (black bars) and output current (yellow bars) for harmonic order k[1, 37].

Fig. 6. Voltages and currents upstream of PCC during the operation in traction regime.

Fig. 7. Succesive traction (filtering) and braking (regeneration) regimes: (a)

phase voltage (blue line) and supply current (green line); (b) DC-capacitor

voltage (black line) and DC-line voltage (red line).

 CONCLUSION:

A new design method of an LCL filter with damping resistance intended to couple the three-phase VSI of an active DC-traction substation to the power supply has been proposed in this paper. The following elements of originality are outlined.

1) The theoretical substantiation is based on the frequency response from transfer functions related to currents, taking into account the existence of the series damping resistances.

2) The expressed amplitude response and resonance frequency highlight their dependence on only pairs L2Cf and RdCf, It is a very important finding for the conceived design algorithm.

3) The expression of the power losses in the damping resistances is highlighted and an equivalent resistance is introduced as a quantitative indicator of them.

4) By considering the switching frequency as main parameter and taking into consideration the frequency of the highest order harmonic to be compensated, the design algorithm is based on the imposition of the associated attenuations.

5) In the substantiation of the design algorithm, a detailed analysis is performed on the existence of physical-sense solutions, providing the domain in which the values of the parameters must be located.

6) As a large number of parameters values sets can be obtained, a new performance indicator (MPI) is proposed, to quantify the extent to which the harmonics to be compensated are influenced.

The analysis and the simulation results achieved for an active DC-traction substation with six-pulse diode rectifier and LCL coupling filter have indicated that the proposed method is valid and effective. The experimental tests conducted in a laboratory test bench of small scale reproducing the specific conditions of a DC-traction substation illustrate good performance of the system for active filtering and regeneration connected to the power supply by the passive damped LCL filter.

The design proposal can be applied in any three-phase LCL-filter-based shunt active power filter.

REFERENCES:

[1] A. Ghoshal and V. John, “Active damping of LCL filter at low switching to resonance frequency ratio,” IET Power Electron., vol. 8, no. 4, pp. 574–582, 2015.

[2] G. E. Mejia Ruiz, N. Munoz, and J. B. Cano, “Modeling, analysis and design procedure of LCL filter for grid connected converters,” in Proc. 2015 IEEE Workshop Power Electron. and Power Quality Appl.  (PEPQA), pp. 1–6.

[3] M. Hanif, V. Khadkikar, W. Xiao, and J. L. Kirtley, “Two degrees of freedom active damping technique for filter-based grid connected PV systems,” IEEE Trans. Ind. Electron., vol. 61, no. 6, pp. 2795–2803, June 2014.

[4] X. Wang, F. Blaabjerg, and P. C. Loh, “Grid-current-feedback active damping for LCL resonance in grid-connected voltage source converters,” IEEE Trans. Power Electron., vol. 31, pp. 213–223, 2016.

[5] W. Xia, J. Kang, “Stability of LCL-filtered grid-connected inverters with capacitor current feedback active damping considering controller time delays,” J. Mod. Power Syst. Clean Energy, vol. 5, no. 4, pp. 584– 598, July 2017.

Simultaneous Microgrid Voltage and Current Harmonics Compensation Using Coordinated Control of Dual-Interfacing-Converters

 

ABSTRACT

The growing installation of distributed generation (DG) units in low voltage distribution systems has popularized the concept of nonlinear load harmonic current compensation using multi functional DG interfacing converters. It is analyzed in this paper that the compensation of local load harmonic current using a single DG interfacing converter may cause the amplification of supply voltage harmonics to sensitive loads, particularly when the main grid voltage is highly distorted. To address this limitation, unlike the operation of conventional unified power quality conditioners (UPQC) with series converter, a new simultaneous supply voltage and grid current harmonic compensation strategy is proposed using coordinated control of two shunt interfacing converters. Specifically, the first converter is responsible for local load supply voltage harmonic suppression. The second converter is used to mitigate the harmonic current produced by the interaction between the first interfacing converter and the local nonlinear load. To realize a simple control of parallel converters, a modified hybrid voltage and current controller is also developed in the paper. By using this proposed controller, the grid voltage phase-locked loop and the detection of the load current and the supply voltage harmonics are unnecessary for both interfacing converters. Thus, the computational load of interfacing converters can be significantly reduced. Simulated and experimental results are captured to validate the performance of the proposed topology and the control strategy.

KEYWORDS:

  1. Parallel converters
  2. Active power filter
  3. Dynamic voltage restorer
  4. LCL filter
  5. Resonance; power quality
  6. Harmonic detection
  7. Phase-locked loop.

SOFTWARE: MATLAB/SIMULINK

CIRCUIT DIAGRAM:

Fig. 1. Diagram of the proposed topology.

EXPECTED SIMULATION RESULTS:

Fig. 2. Only the local load harmonic current is compensated. (From upper to lower: 𝑉𝑠𝑢𝑝𝑝𝑙𝑦, 𝐼𝑔, 𝐼2, 𝐼𝐿𝑜𝑎𝑑)

Fig. 3. The harmonic spectrum of grid current 𝐼𝑔 in Fig. 11.

Fig. 4. The harmonic spectrum of supply voltage 𝑉𝑠𝑢𝑝𝑝𝑙𝑦 in Fig. 11.

Fig. 5. Only the supply voltage harmonic component is compensated. (From upper to lower: 𝑉𝑠𝑢𝑝𝑝𝑙𝑦, 𝐼𝑔, 𝐼2, 𝐼𝐿𝑜𝑎𝑑)

Fig. 6. The harmonic spectrum of grid current 𝐼𝑔 in Fig. 14.

Fig. 7. The harmonic spectrum of supply voltage 𝑉𝑠𝑢𝑝𝑝𝑙𝑦 in Fig. 14.

CONCLUSION

When a single multi-functional interfacing converter is adopted to compensate the harmonic current from local nonlinear loads, the quality of supply voltage to local load can hardly be improved at the same time, particular when the main grid voltage is distorted. This paper discusses a novel coordinated voltage and current controller for dual-converter system in which the local load is directly connected to the shunt capacitor of the first converter. With the configuration, the quality of supply voltage can be enhanced via a direct closed-loop harmonic voltage control of filter capacitor voltage. At the same time, the harmonic current caused by the nonlinear load and the first converter is compensated by the second converter. Thus, the quality of the grid current and the supply voltage are both significantly improved. To reduce the computational load of DG interfacing converter, the coordinated voltage and current control without using load current/supply voltage harmonic extractions or phase-lock loops is developed to realize to coordinated control of parallel converters.

REFERENCES

  • Singh, K. AI-Haddad, A. Chandra, “A review of active filters for power quality improvement,” IEEE Trans. Ind. Electron., vol. 46, no. 5, pp. 960 – 971, May. 1999.
  • Acuna, L. Moran, M. Rivera, J. Dixon, and J. Rodriguez, “Improved active power filter performance for renewable power generation systems,” IEEE Trans. Power Electron., vol. 29, no.2, pp. 687-694, Feb. 2013.
  • W. Li, F. Blaabjerg, D. M. Vilathgamuwa, and P. C. Loh, “Design and Comparison of High Performance Stationary-Frame Controllers for DVR Implementation,” IEEE Trans. Power Electron., vol. 22, pp. 602-612, Mar. 2007.
  • Meyer, R. W. DeDoncker, Y. W. Li, and F. Blaabjerg, “Optimized Control Strategy for a Medium-Voltage DVR – Theoretical Investigations and Experimental Results,” IEEE Trans. Power Electron., vol. 23, pp. 2746-2754, Nov. 2008.
  • Blaabjerg, Z. Chen, and S. B. Kjaer, “Power electronics as efficient interface in dispersed power generation systems,” IEEE Trans. Power Electron., vol. 19, pp. 1184-1194, Sep. 2004.

 

Deadbeat Weighted Average Current Control with Corrective Feed-forward Compensation for Microgrid Converters with Non-Standard LCL Filter

ABSTRACT

Microgrid converters are required to have the capability of both grid-tied mode and islanding mode operation. For this dual-mode operation, large shunt capacitors are often used in the interfacing converter output LCL filter, as it can help to stabilize supply voltage and to reduce switching ripple pollutions to sensitive loads during autonomous islanding operation. At the same time, this modification causes a few challenges, including the low frequency harmonic distortions, the steady-state tracking errors and the slow dynamic response, to the line current regulation during grid-tied operation. To overcome these drawbacks, a modified weighted average current controller is developed. First, to realize a fast line current response, a deadbeat control of weighted average current is developed based on a reduced-order virtual filter plant. Second, a grid voltage feed-forward term is added to the weighted average current reference to mitigate the steady-state line current tracking errors. Note that this compensation term is directly added to the current reference, thus, it is very well decoupled from the closed-loop current regulator. In addition, it can be seen that the low-order line current harmonics caused by grid voltage distortion is inherently compensated by this proposed corrective feed-forward control.

KEYWORDS:

  1. Virtual filter
  2. Deadbeat control
  3. Weighted average current control
  4. Active damping
  5. LCL filter
  6. Microgrid

SOFTWARE: MATLAB/SIMULINK

CIRCUIT DIAGRAMS:

Fig. 1. Diagram of a grid-tied converter controlled by conventional weighted average current feedback.

Fig. 2. Diagram of the proposed control deadbeat scheme with weighted average current feedback and line current tracking error compensation.

EXPECTED SIMULATION RESULTS:

Fig.3. Performance of the system using the proposed deadbeat control method (compensation term is activated in 0.5sec). (from top to bottom: (1) grid voltage Vgrid; (2) line current I2 ; (3) output current I1 ; (4) current tracking errors ( Iref-I2).)

Fig. 4. Performance of the system using the proposed deadbeat control method and the method in [14]. (from top to bottom: (1) grid voltageVgrid ; (2) line current I2; (3) output current I1 ; (4) current tracking errors ( ).)

Fig. 5. Performance of the system using the proposed method, operating in a distorted grid with grid impedance variation

Fig. 6. Performance of the system using the proposed deadbeat control method with feed-forward control. Grid frequency changes from 50Hz to 50.15Hz at 0.2sec. (from top to bottom: (1) Grid voltage Vgrid; (2) line current I2 ; (3) output current I1; (4) weighted average current I12 .)

Fig. 7. Performance of the system using the proposed deadbeat control method but without feed-forward control. Grid frequency changes from 50Hz to 50.15Hz at 0.2sec. (from top to bottom: (1) Grid voltage Vgrid ; (2) line current I2; (3) output current I1; (4) weighted average current I12 .)

Fig. 8. Performance of the system using the PI control for weighted average current regulation, with feed-forward control. Grid frequency changes from 50Hz to 50.15Hz at 0.2sec. (from top to bottom: (1) Grid voltage Vgrid ; (2) line current I2; (3) output current I1; (4) weighted average current I12.)

CONCLUSION

An enhanced current controller is proposed in this paper. The research work of this paper is summarized here as:

1) In order to realize rapid control of converter current, the deadbeat control is applied to regulate the weighted average current based on a virtual filter plant.

2) The feed-forward compensator is developed to mitigate the steady-state fundamental current tracking errors caused by conventional weighted average current control.

3) The frequency-selective capacitor leg current estimation is proposed and the corresponding compensation term can be used to increase the robustness of the converter against grid harmonic distortions. The design and implementation of this compensator are highly decoupled from the closed-loop deadbeat current regulator. Thus, both the current regulator and the compensator can be independently designed.

REFERENCES

  • Blaabjerg, Z. Chen, and S. B. Kjaer, ―Power electronics as efficient interface in dispersed power generation systems,‖ IEEE Trans. Power Electron., vol. 19, no. 5, pp. 1184-1194, May. 2004.
  • Rocabert, A. Luna, F. Blaabjerg, and P. Rodriguez, ―Control of power converters in AC microgrids,‖ IEEE Trans. Power Electron., vol. 27, no. 11, pp. 4734–4749, Nov. 2012.
  • W. Li, D. M. Vilathgamuwa and P. C. Loh, ―Design, analysis and real-time testing of a controller for multibus microgrid system,‖ IEEE Trans. Power Electron., vol. 19, pp. 1195-1204, Sep. 2004.
  • M. Guerrero, L. G. Vicuna, J. Matas, M. Castilla, and J. Miret, ―A wireless controller to enhance dynamic performance of parallel inverters in distributed generation systems,‖ IEEE Trans. Power Electron., vol. 19, no. 4, pp. 1205-1213, Sep, 2004.

New Control Strategy for Three-Phase Grid-Connected LCL Inverters without a Phase-Locked Loop

ABSTRACT:

 The three-phase synchronous reference structure phase-locked loop (SRF-PLL) is widely used for synchronization use in power systems. In this paper, a new control strategy for three-phase grid-connected LCL inverters without a PLL is given. According to the new strategy, a current reference can be produce by using the immediate power control scheme and the proposed positive-sequence voltage detector.

CONTROL STRATEGY

Through logical search, it is determined that a high-quality grid current can be created by introducing the new control strategy. In addition, a kind of independent control for reactive power can be produce under unbalanced and distorted grid conditions. Finally, the excellent work of the planned control strategy is confirm by means of simulation and experimental results.

KEYWORDS:

1.Control strategy

2.Grid-connected inverters

3. Instantaneous power control scheme

4.LCL filter

5.Positive-sequence voltage detector

 SOFTWARE: MATLAB/SIMULINK

BLOCK DIAGRAM:

image001

Fig. 1. Block diagram of the control system with LCL filter.

 EXPECTED SIMULATION RESULTS:

 image002

 Fig. 2. Simulation results of the proposed control system. (a) Generated current reference signals. (b) A-phase grid voltage and three-phase current. (c) Actual active and reactive powers.

image003

Fig. 3. Experimental results of the positive-sequence voltage detector under actual grid operating conditions. (a) Utility voltage and the detected positive-sequence signals. (b) Harmonic spectrum of the utility voltage. (c) Harmonic spectrum of the detected positive-sequence signals.

image004

Fig. 4. Experimental results of a step in the reactive power reference. (a) A-phase grid voltage and three-phase current. (b) A-phase grid voltage and A-phase current.

 CONCLUSION:

A new control form for three-phase grid-connected voltage source inverters (VSI) with an LCL-filter is planned. By using the immediate power control scheme and the planned positive-sequence voltage detector, the current reference can be indirectly generated, which avoids the complex PLL.

PLL

The performance of the proposed system for three-phase grid-connected VSIs is display via simulation results, which show a significant growth in both the steady state and transient action. The same action is experimentally confirmed. The fast dynamic response to a reference step is not troubled by the inclusion of additional control loops. Good work is guaranteed even under unbalanced and distorted grid voltages.

REFERENCES:

[1] X. Wang, J. M. Guerrero, F. Blaabjerg, and Z. Chen, “A review of power electronics based microgrids,” Journal of Power Electronics, Vol. 12, No. 1, pp. 181-192, Jan. 2012.

[2] S. Peng, A. Luo, Y. Chen, and Z. Lv, “Dual-loop power control for single-phase grid-connected converters with LCL filters,” Journal of Power Electronics, Vol. 11, No. 4, pp. 456-463, July. 2011.

[3] F. Blaabjerg, R. Teodorescu, M. Liserre, and A. V. Timbus, “Overview of control and grid synchronization for distributed power generation systems,” IEEE Trans. Ind. Electron., Vol. 53, No. 5, pp. 1398-1409, Oct. 2006.

[4] R. Inzunza, T. Sumiya, Y. Fujii, and E. Ikawa, “Parallel connection of grid-connected LCL inverters for MW-scaled photovoltaic systems,” in Proc. IEEE IPEC, pp. 1988-1993, 2010.

[5] T. Noguchi, H. Tomiki, S. Kondo, and I. Takahashi, “Direct power control of PWM converter without power-source voltage sensors,” IEEE Trans. Ind. Appl., Vol. 34, No. 3, pp. 473-479, Mar./ Jun. 1998.