Line Side Converters in Wind Power Applications
6.7 Control Method for Input-Output Harmonic Elimination
of the PWM Converter under Unbalanced Operating Conditions Recently, Stankovic and Chen17 proposed a generalized method for input–output harmonic elimination of the three-phase PWM converters under unbalanced oper-ating conditions. The method is related to harmonic elimination of the PWM Boost type converters under severe fault conditions. Under severe fault conditions both line voltages and line impedances have to be considered unbalanced.
The circuit in Fig. 6.5 is analyzed with unbalanced line voltages and unbalanced line impedances. It is assumed that the converter is lossless. Harmonic elimination
can be achieved by generating unbalanced reference commands for three line cur-rents under unbalanced voltages and impedances. The following equations are derived for unbalanced grid voltages U1, U2, U3 and unbalanced line impedances z1, z2, and z3. From the circuit shown in Fig. 6.5 it follows, grid voltages, line currents, line impedances, synthesized voltages at the input of the converter, apparent power and switching functions, respectively, represented as phasors.
Equation (6.30) represents the condition for the second harmonic elimination.
Synthesized voltages Vs1, Vs2and Vs3can be expressed as, Vs1= SW1
where Vdcis the dc voltage.
By substituting Eqs. (6.31) to (6.33) into Eqs. (6.25) to (6.27) the following set of equations is obtained,
SW1I1+ SW2I2+ SW3I3= 0. (6.39) For given input power, S grid voltages, U1, U2, U3 and line impedances z1, z2
and z3, line currents, I1, I2and I3, can be obtained from the above set of equations.
By multiplying Eqs. (6.34) to (6.36) by I1, I2and I3, respectively, and adding them up the following equation is obtained,
U1I1+ U2I2+ U3I3= −z1I12− z2I22− z3I32, + Vdc
2√
2(SW1I1+ SW2I2+ SW3I3). (6.40) The set of six equations with six unknowns, Eqs. (6.34) to (6.39), reduces to three equations with three unknowns.
By substituting Eq. (6.39) into (6.40) the following equation is obtained, U1I1+ U2I2+ U3I3= −z1I12− z2I22− z3I32, (6.41)
I1= −I2− I3, (6.42)
S*= −(U1*I1+ U2*I2+ U3*I3). (6.43) Equations (6.41) to (6.43) represent a set of three equations with three unknowns.
By substituting Eq. (6.42) into Eqs. (6.41) and (6.43), the following set of equa-tions is obtained and given by,
U1(−I2− I3)+ U2I2+ U3I3= −z1(−I2− I3)2− z2I22− z3I32, (6.44) S*= −(−U1*I2− U1*I3+ U2*I2+ U3*I3). (6.45) Equation (6.44) can be simplified as,
I2(U2− U1)+ I3(U3− U1)= −(z1+ z2)I22− (z1+ z3)I32− 2z1I2I3. (6.46) From Eq. (6.43) current, I2, can be expressed as,
I2= −S*− I3(U3*− U1*) U2*− U1*
. (6.47)
Finally by substituting Eq. (6.47) into Eq. (6.46),
−S*− I3(U3*− U1*) U2*− U1*
(U2− U1)+ I3(U3− U1)
= − (z1+ z2)S*2+ 2S*I3(U3*− U1*)+ I32(U3*− U1*)2 (U2*− U1*)2
− (z1+ z2)I32− 2z1−S*− I3(U3*− U1*) U2*− U1*
I3, (6.48)
Currents I2and I1can be obtained from Eqs. (6.47) and (6.42).
Equations (6.42), (6.47) and (6.49) represent the steady state solution for line currents under both unbalanced grid voltages and unbalanced line impedances. An analytical solution represented by Eq. (6.49) always exists unless all the coefficients of the quadratic equations are equal to zero.
Critical Evaluation
The analytical solution that has been obtained is general. The only constraint that exists, as far as the level of unbalance is concerned, is governed by constraints of the operation of the PWM Converter itself.
The proposed generalized method for input–output harmonic elimination is valid if and only if Ui, zi = 0, where i = 1, 2, 3. In other words the solution exists for all levels of unbalance in line voltages and impedances, except for cases where both voltage and impedance in the same phase are equal to zero. Therefore, the maximum level of voltage imbalance with balanced line impedances, for which the proposed solution is still valid is given as,
U1= 0 U2 = U3= 0
z1 = z2= z3= 0.
The maximum level of imbalance in both line voltages and impedances for which the proposed solution is still valid is given as,
U1= 0 U2= U3= 0
z1= 0 z2= z3= 0.
Based on the analysis of the open loop configuration presented above, a feed forward control method is proposed. The line voltages as well as line impedances
Fig. 6.9. Control of a line side converter under unbalanced operating conditions.
have to be measured as shown in Fig. 6.9. In Fig. 6.9, block “unbalanced detector”
is used to measure unbalanced voltages and unbalanced impedances. Based on this information and a dc bus error, reference currents are calculated (block “calculate I1, I2, I3) according to Eqs. (6.42), (6.47) and (6.49) which become reference signals for the hysteresis controller16 shown in Fig. 6.9. Only one PI controller is utilized, which has been shown to be sufficient for good regulation. The proposed control method is shown in more detail in Fig. 6.9.
6.8 Examples
Wind turbine with two PWM converters shown in Fig. 6.10 was simulated in Simulink under balanced and unbalanced operating conditions.
Example 1
In this example control method shown in Fig. 6.8, under balanced grid voltages was used in simulation.
UTILITY GRID
Fig. 6.10. Wind turbine with two PWM converters.
The following parameters were used in simulation:
3 kW PM synchronous generator The rotor speed wm =77.78 rad/s The electromagnetic torque TB= 30 Nm dc link capacitor C= 1 mF
Line inductances L= 10 mH
The grid voltages are balanced van= vbn= vcn = 220V The power P = 2318 W, Q = 0.
Figures 6.11 to 6.17 show rotor speed, electromagnetic torque, stator voltages, stator currents, dc link voltage, grid side voltages and line currents for one wind speed.
0.05 0.1 0.15 0.2 0.25 0.3
75
74 76 77 78 79 80
Rotor speed wm
t(s)
wm(rad/s)
Fig. 6.11. Rotor speed for one wind speed (Example 1).
-70 -50 -30 -10
Electromagnetic torque Te
t(s)
Te(N*m)
0.05 0.1 0.15 0.2 0.25 0.3
Fig. 6.12. Electromagnetic torque for one wind speed (Example 1).
0.05 0.055 0.06 0.065 0.07 0.075 0.08
-200 -400
-600
-800 200 0 400 600 800
t(s)
Vab(V)
Stator Voltage Vab
Fig. 6.13. Stator voltage for one wind speed (Example 1).
0 100 200 300 400 500 600 700 800
Vdc
Vdc(V)
t(s)
0.05 0.1 0.15 0.2 0.25 0.3
Fig. 6.14. DC link voltage for one wind speed (Example 1).
0.05 0.1 0.15 0.2 0.25 0.3
-20 -15 -10 -5 0 5 10 15
20 Stator current Iabc
t(s)
Iabc(A)
Fig. 6.15. Stator currents for one wind speed (Example 1).
Example 2
In this example the wind turbine system shown in Fig. 6.10 was simulated under unbalanced grid voltages. The control method shown in Fig. 6.9 was used in the simulation.
The following parameters were also used in the simulation:
3 kW PM synchronous generator The rotor speed wm = 77.78 rad/s
Grid side voltage (Line-ground)
0.05 0.1 0.15 0.2 0.25 0.3
-100
-200
-300 0 100 200 300
t(s)
Vgrid-abc(V)
Fig. 6.16. Grid side voltages for one wind speed (Example 1).
0.05 0.1 0.15 0.2 0.25 0.3
-10 -8 -6 -4 -2 0 2 4 6 8
10 Grid side current
t(s)
Igrid-abc(V)
Fig. 6.17. Grid side currents for one wind speed (Example 1).
The electromagnetic torque TB= 30 Nm dc link capacitor C= 1 mF
Line inductances L= 10 mH
The grid voltages are unbalanced van= 150 0vbn −120 = vcn= 220V 120 The power P = 2318 W.
Figures 6.18 to 6.24 show rotor speed, electromagnetic torque, stator voltages, stator currents, dc link voltage, unbalanced grid voltages and line currents for one wind speed. In spite of unbalance in grid voltages, line currents do not contain low order harmonics. This was achieved by using the control method shown in Fig. 6.9.
0.05 0.1 0.15 0.2 0.25 0.3
74 75 76 77 78 79 80
Rotor speed wm
t(s)
wm(rad/s)
Fig. 6.18. Rotor speed for one wind change (Example 2).
0.05 0.1 0.15 0.2 0.25 0.3
-70 -50 -30 -10
Electromagnetic torque Te
t(s)
Te(N*m)
Fig. 6.19. Electromagnetic torque for one wind speed (Example 2).
0.05 0.055 0.06 0.065 0.07 0.075 0.08 -800
-600 -400 -200 0 200 400 600 800
Stator Voltage Vab
t(s)
Vab(V)
Fig. 6.20. Stator voltages for one wind speed (Example 2).
0.1
0.05 0.15 0.2 0.25 0.3
-20 -15 -10 -5 0 5 10 15 20
Stator current Iabc
t(s)
Iabc(A)
Fig. 6.21. Stator currents for one wind speed (Example 2).
0.1
0.05 0.15 0.2 0.25 0.3
t(s) 0
100 200 300 400 500 600 700
800 Vdc
Vdc(V)
Fig. 6.22. DC link voltage for one wind speed (Example 2).
Grid side voltage (Line-ground)
0.05 0.1 0.15 0.2 0.25 0.3
-300 -200 -100 0 100 200 300
t(s)
Vgrid-abc(V)
Fig. 6.23. Grid side voltage for one wind speed (Example 2).
0.05 0.1 0.15 0.2 0.25 0.3 -10
-8 -6 -4 -2 0 2 4 6 8
10 Grid side current
t(s)
Igrid-abc(V)
Fig. 6.24. Grid side currents for one wind speed (Example 2).
Example 3
In this example the wind turbine system shown in Fig. 6.10 is simulated. The grid voltages are unbalanced and the variable speed of a wind turbine is incorporated.
The grid voltages are unbalanced.
van = 150 0vbn −120 = vcn= 220 V 120.
Figures 6.25 to 6.30 show rotor speed, electromagnetic torque, stator voltage, stator currents, unbalanced grid voltages and line currents. In spite of unbalanced
0.05 0.1 0.15 0.2 0.25 0.3
65 70 75
80 Rotor speed wm
t(s)
wm(rad/s)
Fig. 6.25. Rotor speed (Example 3).
0.05 0.1 0.15 0.2 0.22 0.25 0.3 -70
-50 -30 -10
Electromagnetic torque Te
t(s)
Te(N*m)
Fig. 6.26. Electromagnetic torque (Example 3).
0.05 0.055 0.06 0.065 0.07 0.075 0.08
-200
-400
-600
-800 0 200 400 600 800
Stator Voltage Vab
t(s)
Vab(V)
Fig. 6.27. Stator voltage (Example 3).
0.05 0.1 0.15 0.2 0.22 0.25 0.3 -20
-15 -10 -5 0 5 10 15 20
Stator current Iabc
t(s)
Iabc(A)
Fig. 6.28. Stator currents (Example 3).
Grid side voltage (Line-ground)
200
100
0 300
0.05 0.1 0.15 0.2 0.25 0.3
t(s) -100
-200
-300
Vgrid-abc(V)
Fig. 6.29. Grid side voltages (Example 3).
0.05 0.1 0.15 0.2 0.22 0.25 0.3 -10
-8 -6 -4 -2 0 2 4 6 8 10
Grid side current
t(s)
Igrid-abc(V)
Fig. 6.30. Grid side currents (Example 3).
voltages, line currents do not contain low order harmonics which was achieved by using the control method shown in Fig. 6.9.
6.9 Concluding Remarks
In this chapter, operation of a line side converter used in variable-speed wind energy conversion systems under balanced and unbalanced grid voltages was analyzed.
Control methods under balanced and unbalanced grid voltages were described and simulated. It has been shown that the PWM line side converter can operate under unbalanced grid voltages without injecting harmonic currents into the grid.
References
1. U.S Department of Energy, “20% Wind energy by 2030 report,” July 2008, http://www1.eere.
energy.gov/windandhydro/pdfs/41869.pdf.
2. J.M. Carrasco, L.G. Franquelo, J.T. Bialasiewicz, E. Galvan, R.C.P. Guisado, A.M. Prats, J.I.
Leon and N. Moreno-Alfonso, “Power-electronic systems for the grid integration of renewable energy source: A survey, IEEE Trans. Industrial Electronics 53 (2006) 1002–1016.
3. M. Chinchilla, S. Arnaltes and J.C. Burgos, “Control of permanent-magnet generator applied to variable-speed wind-energy systems connected to the grid,” IEEE Trans. Energy Conversion 21 (2006) 130–135.
4. A. Grauers, “Efficiency of three wind energy generator systems,” IEEE Trans. Energy Conversion 11 (1996) 650–657.
5. T. Ahmed, M. Nakaoka and K. Nishida, “Advanced control of a boost AC-DC PWM rectifier for variable-speed induction generator,” Applied Power Electronics Conf. and Exposition, March 2006.
6. J.-I. Jang, Y.-S. Kim and D.-C. Lee, “Active and reactive power control of DFIG for wind energy conversion under unbalanced grid voltage,” Power Electronics and Motion Control Conf. (2006).
7. T. S¨urgevil and E. Akpınar, “Modeling of a 5-kw wind energy conversion system with induction generator and comparison with experimental results,” Int. J. Renewable Energy 30 (2004) 913–
929.
8. I. Schiemenz and M. Stiebler, “Control of a permanent magnet synchronous generator used in a variable speed wind energy system,” IEEE Int. Electric Machines and Drive Conf. (2001).
9. G. Johnson, Wind Energy Systems (Prentice-Hall, Inc., 1985).
10. S. Morimoto, T. Nakamura and Y. Takeda, “Power maximization control of variable-speed wind generation system using permanent magnet synchronous generator,” Electrical Engineering in Japan 150 (2005) 1573–1579.
11. A.V. Stankovic and K. Chen, “A new control method for input-output harmonic elimination of the PWM boost type rectifier under extreme unbalanced operating conditions,” IEEE Trans.
Industrial Electronics 56 (2009) 2420–2430.
12. A.V. Stankovic and T.A. Lipo, “A novel generalized control method for input output harmonic elimination of the PWM boost type rectifier under simultaneous unbalanced input voltages and input impedances,” 32nd Annual Power Electronics Specialists Conf. (2001).
13. A.V. Stankovic and T.A. Lipo, “A novel control method for input output harmonic elimination of the PWM boost type rectifiers under unbalanced operating conditions,” IEEE APEC 2000 (2000), pp. 413–419.
14. A.V. Stankovic and T.A. Lipo, “A novel control method for input output harmonic elimination of the PWM boost type rectifiers under unbalanced operating conditions,” IEEE Trans. Power Electronics 16 (2001) 603–611.
15. D. Schreiber, “State of the art of variable speed wind turbines,” 11th Int. Symp. Power Elec-tronics — Ee 2001, Novi Sad, Yugoslavia.
16. A. V. Stankovic and T. A. Lipo, “A novel control method for input-output harmonic elimination of the PWM boost type rectifier under unbalanced operating conditions,” IEEE Trans. Power Electronics 16 (2001) 603–611.
17. J.W. Dixon and B.T. Ooi, “Indirect current control of a unity power factor sinusoidal current boost type three-phase rectifier,” IEEE Trans. Industry Applications 35 (1988) 508–515.
18. T.A. Lipo, “Recent progress and development of solid state AC motor drives,” IEEE Trans. on Power Electronics 3 (1988) 105–117.
19. J.W. Wilson, “The forced-commutated inverter as a regenerative rectifier,” IEEE Trans. Industry Applications IA-14 (1978) 335–340.
20. D.M. Brod and D.W. Novotny, “Current Control of VSI-PWM Inverters,” IEEE Trans. Industry Applications IA-21 (1984) 769–775.
21. L. Moran, P.D. Ziogas and G. Joos, “Design aspects of synchronous pwm rectifier- inverter system under unbalanced input voltages conditions,” IEEE Trans. Industry Applications 28 (1992) 1286–
1293.
22. A.V. Stankovic, “Unbalanced operation of three-phase boost type rectifiers,” in Handbook of Automotive Power Electronics Motor Drives (CRC Press, 2005).
23. D. Schreiber, “State of the art of variable speed wind turbines,” 11th Int. Symp. Power Elec-tronics — Ee 2001, Novi Sad, Yugoslavia.
24. R.C. Bansal, “Three-phase self-excited induction generator: An overview,” IEEE Trans. Energy Conversion 20 (2005) 292–299.
25. http://www.gwec.net/index.php?id=153.