Analysis of Grid Tied Inverter with Proportional Resonant Regulator

Transcription

1 Volume 114 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Analysis of Grid Tied Inverter with Proportional Resonant Regulator R.S. Ravi Sankar 1, Dr. S V Jaya ram Kumar 2, A.Venkatesh 3 satya_ravi2001@yahoo.com 1, svjkumar101@rediffmaiil.com 2 venky.allam4003@gmail.com 3 April 14-15, 2017 Abstract The main intension of this work is the Stability Analysis of the Inverter Connected to Grid coupled with LCL-filter with inverter and grid side current control techniques are analyzed using Nyquist Stability Criterion and the stability region of system cannot be grantee when filter values are not designed properly. The d-axis small signal modeling is used to find out the output impedances of both the inverter and grid side currents with Proportional Resonant (PR) Current Controller. From the stability analysis, the Inverter current control is stable then the grid current control. The Grid tied Inverter, Stability region with inverter current control is analyzed by Bode Plots. The effectiveness of the control technique is observed in Matlab Simulation results. Key Words: Nyquist Stability Criteria (NSC), Proportional Resonant (PR) Controller, Small Signal Modelling, Return Ratio Matrix. 1 Introduction: In recent days utilization of renewable energy sources are increases because of deficiency in conventional energy sources. These are like Solar systems, Wind power generation depending 293

2 on the time. This power we can use it as Isolated system or grid Connected Mode, the different control techniques are proposed for the grid which are presented in [1]. The main problem raised with the synchronization is the system reliable and system stable. To know the stability of the system is main task to achieve continuous and uninterrupted operation. Controllers have the much impact on stability of the system i.e. without changing the system parameters changing the parameters of controller to achieve the stable operating point. Stability is determined by conventional graphical methods like Bode plots, Pole Zero placement, Root Locus and Nyquist Criterion. The impedance based stability criterion introduced by the Middlebrook and which can be improved by Feng [2], the connection between stability of cascaded system and transfer function is presented in [3]. The system stability is analyzed in d-channel small-signal modelling is applied to inverter connected to grid by Generalized Nyquist Stability Criterion through study the interactions between power electronic converters and filters [4]. The stability of grid tied inverter coupled with LCL-filter with PI controller is assessed both inverter and grid side currents [5,6]. This paper focusing on analysis of grid tied inverter with an LCL -Filter and PR- regulator, Section II explains about the Output Impedance calculation of grid tied inverter, Section III presents calculation of Eigen values and stability analysis s, Section IV Projects the MATLAB Simulation Results. 2 Calculation of output Impedance model of 3-Ø Inverter Connected to Grid. Fig.1, Shows the 3-Ø inverter connected to grid with LCL Filter having the inductive parameters are L1& L2, Resistive of inductance are R1 &R2 Filter capacitance is Cf and damping resistance is Rd respectively, Grid side impedance is Zg. Three phase inverter T1 T3 T5 Grid Connected Inverter Grid i 1 d L 1 L 2 i 2d v + dc _ a b c i 1 a L 1 LCL Filter R R 1 2 L 2 i 2 a Z g e a e b e c O R d d U d dc v d T4 T6 T2 R C f d PCC D d ȗ dc C f v cd Fig.1: 3-Ø Grid tied Inverter Fig.2: Reduced Small Signal Modeling of 3-Ø Grid tied Inverter d-axis Circuit The output impedance is represented by second order matrix. Given by 294

3 Where is the output impedance of grid tied inverter in d-axis circuit, is the output impedance of grid tied inverter in q- channel and are the cross coupled impedance. If neglect cross-coupling terms and resistance of the filter. The reduced small signal modeling of 3-Ø grid tied inverter shown in Fig.2. The Ratio between the Inverter current to duty cycle of the inverter is given in Eqa.2. The ratio between inverter duty cycle to the grid current is given in Eqa.3. The ratio between grid voltage to the inverter current is given in Eqa.4 Inverter output Impedance is given in Eqa.5. Block diagram representation of grid tied inverter with controlling with inverter current and grid current controls are given Fig.3. & Fig.4. V d V d -1/Z oo (s) ȋ * 1d + - G iv (s) F i (s) F m (s) G id1 (s) Z c (s) + Z L2 (s) - ȋ1d ȗ dc ȋ2d ȋ * 2d + - d d F i (s) F m (s) G id2 (s) + + ȋ 2d ȋ 2d Fig.3: Inverter Current Control Fig.4: Grid Current Control technique Where Fi(s) is Out to Input relation of PR Current Controller Here is proportional gain constant, is resonant gain constant 295

4 is natural frequency of the system or grid frequency in Radiance/Sec Fm(s) is the out to input relation of PWM Converter Here is gain of PWM Converter By using d-channel small signal modelling grid tied inverter with inverter current control shown in Fig.3. Variations in Inverter and grid currents are derived when : from Fig.3 are given in equation (8)& (10). Combining equations (8) & (9) get equation (11) Where is, the gain of the open loop system with the inverter side current. By substituting (11) in (10), Output impedance of closed loop inverter current control system is derived as: Where From the Fig.4, Output impedance of closed loop system is given in Eqa.17. Where Tm2(s), The open loop gain of the above system is given in Eqa

5 Inverter I1dq(s) PCC Zgdq(s) Grid Idq(s) Vdq(s) Yoc(s) Edq(s) Fig.5: Inverter tied Grid Equivalent Circuit 3 Stability Analysis through Eigen Value of the System 3.1 Eigen Value Calculation of Grid tied Inverter The output impedance model of the inverter tied grid equivalent circuit is shown in Fig.5 The System parameters, impedance matrix of the grid is and output inverter admittance matrix is and the system variables are voltage matrix at Point of Common Coupling, voltage matrix of grid is and current matrix of grid is. The expression for grid impedance is given in equation (20), The impedance matrix of grid is represented in (21) Here, is far greater than,. So, neglect the terms, in the above equation. The diagonal elements in the d & q channel impedance matrix are equal because these satisfies symmetry property. 297

6 The grid current of the system is given in equation (23) Where is the return -ratio matrix of the System. It is given in equation (24) From the equation (21), Replacing the terms in equation (24), we get the equation (25) Here observed that the order of the return ratio matrix two and it has two Eigen values are given below is Substitute equation (20) in (26), (27) to get the (28),(29). According to NSC, the system is stable if the number of encirclements by nyquiest plot of the with (-1, j0) is equal to the number of right hand side poles of. Table.1: System Parameters Parameter Grid specifications DC link voltage Filter inductance L2 Filter inductance L1 Filter Capacitance Cf Damping resistor Rd Grid inductance Grid Resistance Switching frequency Value 230V,50Hz 600V 1.2mH 1mH 14.1μF 5Ώ 1mH 1 Ώ 5kHz 298

7 Table.2. Variation of the Filter Parameters Parameters Change L1 L2 Cf Rd L1 L2 Cf Rd Lg (mh) (mh) (μf) (Ohms) (mh) Fig.6.Nyquist Plot of inverter Current control Fig.7. Nyquist Plot of Grid Current control Fig.8. Pole Zero Plot of inverter Current Control Fig.9. Pole Zero Plot of Grid Current Control 3.2 Two-current control techniques are provided for assessment of stability of the 3-Ø inverter connected to grid From Return Ratio Matrix, the Eigen values are derived for inverter side and grid side current control, Nyquiest plot and Pole-Zero plots of both current control techniques are presented 299

8 with respect to the are shown in Fig6 &7 and Fig.8 & 9. Respectively.From the NSC, Nyquiest plots are not encircling the (-1, j0) and from the pole zero plots all poles are in the left side of the S-plane for converter side current control which makes the system stable. For grid, current control two poles placed in the right side of the S- Plane which makes the system unstable. III.3. Stability Margin with Inverter Current Control Invert current Control of the system is stable. Because it is obey the NSC and sufficient stability region is provided only when phase Margin PM>30 0 and gain margin GM>6dB. Stability region of system is calculated neglect the grid resistance and determine the Eigen values of the system which can be l(s) is given in Eqa.30. is inductance of the grid. The Nyquist plots for equation (30) with the values presented in Table.1, the Point of Intersecting M which is corresponding to the point in the bode plot as A shown in the Fig.11. Phase and Gain margin at that point is 45 0 and 22.5 db, which are more than the specified Values above. So the Converter Side Current Control has sufficient stability margin with the above-mentioned Parameters in Table-I Fig.10.Nyquist plot Fig.11.Bode Plot From the Fig.12. even though if there is change filter inductance L1 is effecting the stable point, Fig.13. As the filter inductance L2 Increases, The Stable point is moving towards the low frequency range. Further increases the system becomes unstable. 300

9 Fig.12.Change in Filter Inductance L1 Fig.13.Change in Filter Inductance L2 Fig.14.Change in Filter Capacitance Fig.15. Change in Damping Resistance From Fig.14. as filter capacitance increases stable point is moving towards the low frequency range and causes to increases the phase margin causes the system more stable. Fig.15. Damping resistance increases the stable point move towards high frequency range causes to increase the phase margin. further increases system becomes unstable. 4 Simulation Results Fig.16. Representing the Grid voltage and Current Wave forms with inverter Current control is Stable and Fig.17.showes the grid voltage and current wave forms with grid current control is unstable. Fig.16. Inverter Current Control Fig.17. Grid Current Control 301

10 5 Conclusion In this Paper Stability analysis is done using Nyquist Stability criteria applied to the Return Ratio Matrix. It is calculated for grid tied inverter with inverter and grid current control technique from the small signal modeling ind-axis circuit. This analysis shows the inverter current control technique is more stable compared to the grid Current Control, it has sufficient stability region. Stability analysis is verified by Matlab- Simulation. References [1] M. P. Kazmierkowski and L. Malesani, Current control techniques for three-phase voltage source PWM converters: A survey, IEEE Trans. Ind. Electron., vol. 45, no. 5, pp , Oct [2] X. Feng, Z. Ye, K. Xing, F. C. Lee, and D. Borojevic, Individual load impedance specification for a stable DC distributed power system, in Proc. IEEE APEC 99, pp , [3] S. D. Sudhoff, S. F. Glover, P. T. Lamm, D. H. Schmucker, and D. E. Delisle, Admittance space stability analysis of power electronic systems, IEEE Trans. Aerosp. Electron. Syst., Vol. 36, No. 3, pp , Jul [4] R. Burgos, D. Boroyevich, F. Wang, K.Karimi, and G. Francis, On the ac stability of high power factor three phase rectifiers, in Proc. IEEE Energy Convers. Congr. Expo., pp , Sep [5] X. Q. Li, X. J. Wu, Y. W. Geng, and Q. Zhang, Stability Analysis of Grid-Connected Inverters with an LCL Filter Considering Grid Impedance, in Journal of Power Electronics, Vol.13, No.5, sept [6] A. MacFarlane and I. Postlethwaite, The generalized Nyquist stability criterion and multivariable loci, in Int. Jour of Control, Vol. 25, pp , Jan

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