Engineering » Energy Engineering » "Advances in Wind Power", book edited by Rupp Carriveau, ISBN 978-953-51-0863-4, Published: November 21, 2012 under CC BY 3.0 license

Chapter 2

Effect of Turbulence on Fixed-Speed Wind Generators

By Hengameh Kojooyan Jafari
DOI: 10.5772/51407

Article top

Overview

Curve of 
							
 
  
   C
   p
  
  
 


						 for different tip speed ratios 
							
 λ


						
					.
Figure 1. Curve of C p for different tip speed ratios λ .
Electrical torque (nonlinear and linear) versus speed (slip).
Figure 2. Electrical torque (nonlinear and linear) versus speed (slip).
Mechanical and linear electrical torque versus slip.
Figure 3. Mechanical and linear electrical torque versus slip.
Mechanical and electrical torque versus frequency curves per unit with V
						sag = 10%
Figure 4. Mechanical and electrical torque versus frequency curves per unit with V sag = 10%
Mechanical and electrical torque versus frequency per unit with V
						sag = 20%.
Figure 5. Mechanical and electrical torque versus frequency per unit with V sag = 20%.
Mechanical and electrical torque versus frequency per unit with V
						sag = 50%.
Figure 6. Mechanical and electrical torque versus frequency per unit with V sag = 50%.
Electrical torque when 
							
						= 48 and 
							
						= 6m/s.
Figure 7. Electrical torque when = 48 and = 6m/s.
Electrical torque when 
							
 
  
   f
   s
  
  
 


						= 50 and 
							
 
  
   υ
   w
  
  
 

						= 6m/s.
Figure 8. Electrical torque when f s = 50 and υ w = 6m/s.
Electrical torque when 
							
 
  
   f
   s
  
  
 


						= 52 and 
							
 
  
   υ
   w
  
  
 

						= 6m/s.
Figure 9. Electrical torque when f s = 52 and υ w = 6m/s.
Electrical torque when 
							
 
  
   f
   s
  
  
 


						= 48 and 
							
 
  
   υ
   w
  
  
 

						= 10m/s.
Figure 10. Electrical torque when f s = 48 and υ w = 10m/s.
Electrical torque when 
							
 
  
   f
   s
  
  
 


						= 50 and 
							
 
  
   υ
   w
  
  
 

						= 10m/s.
Figure 11. Electrical torque when f s = 50 and υ w = 10m/s.
Electrical torque when 
							
 
  
   f
   s
  
  
 


						= 52 and 
							
 
  
   υ
   w
  
  
 

						= 10m/s.
Figure 12. Electrical torque when f s = 52 and υ w = 10m/s.
Electrical torque when 
							
 
  
   f
   s
  
  
 


						= 48 and 
							
 
  
   υ
   w
  
  
 

						= 13m/s.
Figure 13. Electrical torque when f s = 48 and υ w = 13m/s.
Electrical torque when 
							
 
  
   f
   s
  
  
 


						= 50 and 
							
 
  
   υ
   w
  
  
 

						= 13m/s.
Figure 14. Electrical torque when f s = 50 and υ w = 13m/s.
Electrical torque when 
							
 
  
   f
   s
  
  
 


						= 52 and 
							
 
  
   υ
   w
  
  
 

						= 13m/s.
Figure 15. Electrical torque when f s = 52 and υ w = 13m/s.
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 10% and 
							

 
  
   υ
   w
  
  
 


						 = 6m/s, 
							
 
  
   f
   s
  
  
 


						=48
Figure 16. Torque-time in per unit while V sag = 10% and υ w = 6m/s, f s =48
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 10% and 
							

 
  
   υ
   w
  
  
 


						 = 10m/s, 
							
 
  
   f
   s
  
  
 


						= 48
Figure 17. Torque-time in per unit while V sag = 10% and υ w = 10m/s, f s = 48
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 10% and 
							

 
  
   υ
   w
  
  
 


						 = 13m/s, 
							
 
  
   f
   s
  
  
 


						= 48
Figure 18. Torque-time in per unit while V sag = 10% and υ w = 13m/s, f s = 48
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 20% and 
							

 
  
   υ
   w
  
  
 


						 = 6m/s, 
							
 
  
   f
   s
  
  
 


						= 48
Figure 19. Torque-time in per unit while V sag = 20% and υ w = 6m/s, f s = 48
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 20% and 
							

 
  
   υ
   w
  
  
 


						 = 10m/s, 
							
 
  
   f
   s
  
  
 


						= 48
Figure 20. Torque-time in per unit while V sag = 20% and υ w = 10m/s, f s = 48
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 20% and 
							

 
  
   υ
   w
  
  
 


						 = 13m/s, 
							
 
  
   f
   s
  
  
 


						= 48
Figure 21. Torque-time in per unit while V sag = 20% and υ w = 13m/s, f s = 48
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 50% and 
							

 
  
   υ
   w
  
  
 


						 = 6m/s, 
							
 
  
   f
   s
  
  
 


						= 48
Figure 22. Torque-time in per unit while V sag = 50% and υ w = 6m/s, f s = 48
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 50% and 
							

 
  
   υ
   w
  
  
 


						 = 10m/s, 
							
 
  
   f
   s
  
  
 


						= 48
Figure 23. Torque-time in per unit while V sag = 50% and υ w = 10m/s, f s = 48
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 50% and 
							

 
  
   υ
   w
  
  
 


						 = 13m/s, 
							
 
  
   f
   s
  
  
 


						= 48
Figure 24. Torque-time in per unit while V sag = 50% and υ w = 13m/s, f s = 48
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 10% and 
							

 
  
   υ
   w
  
  
 


						 = 6m/s, 
							
 
  
   f
   s
  
  
 


						= 50
Figure 25. Torque-time in per unit while V sag = 10% and υ w = 6m/s, f s = 50
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 10% and 
							

 
  
   υ
   w
  
  
 


						 = 10m/s, 
							
 
  
   f
   s
  
  
 


						= 50
Figure 26. Torque-time in per unit while V sag = 10% and υ w = 10m/s, f s = 50
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 10% and 
							

 
  
   υ
   w
  
  
 


						 = 13m/s, 
							
 
  
   f
   s
  
  
 


						= 50
Figure 27. Torque-time in per unit while V sag = 10% and υ w = 13m/s, f s = 50
Torque-time in per unit while 
														
 
  
   V
   
    sag
   
  
  
 

						= 20% and 
							

 
  
   υ
   w
  
  
 


						 = 6m/s, 
							
 
  
   f
   s
  
  
 


						= 50
Figure 28. Torque-time in per unit while V sag = 20% and υ w = 6m/s, f s = 50
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 20% and 
							
 
  
   υ
   w
  
  
 

						 = 10m/s, 
							
 
  
   f
   s
  
  
 


						= 50
Figure 29. Torque-time in per unit while V sag = 20% and υ w = 10m/s, f s = 50
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 20% and 
							
 
  
   υ
   w
  
  
 

						 = 13m/s, 
							
 
  
   f
   s
  
  
 


						= 50
Figure 30. Torque-time in per unit while V sag = 20% and υ w = 13m/s, f s = 50
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 50% and 
							
 
  
   υ
   w
  
  
 

						 = 6m/s, 
 
  
   f
   s
  
  
 


						= 50
Figure 31. Torque-time in per unit while V sag = 50% and υ w = 6m/s, f s = 50
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 50% and 
							
 
  
   υ
   w
  
  
 

						 = 10m/s, 
							
 
  
   f
   s
  
  
 


						= 50
Figure 32. Torque-time in per unit while V sag = 50% and υ w = 10m/s, f s = 50
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 50% and 
							
 
  
   υ
   w
  
  
 

						 = 13m/s, 
							
 
  
   f
   s
  
  
 


						= 50 in new simulation of wind generator
Figure 33. Torque-time in per unit while V sag = 50% and υ w = 13m/s, f s = 50 in new simulation of wind generator
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						=10% and 
							
 
  
   υ
   w
  
  
 

						 = 6m/s, 
							
 
  
   f
   s
  
  
 


						= 52
Figure 34. Torque-time in per unit while V sag =10% and υ w = 6m/s, f s = 52
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						=10% and 
							
 
  
   υ
   w
  
  
 

						 = 10m/s, 
							
 
  
   f
   s
  
  
 


						= 52
Figure 35. Torque-time in per unit while V sag =10% and υ w = 10m/s, f s = 52
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 10% and 
							
 
  
   υ
   w
  
  
 

						 = 13m/s, 
							
 
  
   f
   s
  
  
 


						= 52
Figure 36. Torque-time in per unit while V sag = 10% and υ w = 13m/s, f s = 52
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 20% and 
							
 
  
   υ
   w
  
  
 

						 = 6m/s, 
							
						= 52
Figure 37. Torque-time in per unit while V sag = 20% and υ w = 6m/s, = 52
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 20% and 
							
 
  
   υ
   w
  
  
 

						 = 10m/s, 
							
 
  
   f
   s
  
  
 


						= 52
Figure 38. Torque-time in per unit while V sag = 20% and υ w = 10m/s, f s = 52
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 20% and 
							
 
  
   υ
   w
  
  
 

						 = 13m/s, 
							
 
  
   f
   s
  
  
 


						= 52
Figure 39. Torque-time in per unit while V sag = 20% and υ w = 13m/s, f s = 52
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 50% and 
							
 
  
   υ
   w
  
  
 

						 = 6m/s, 
							
 
  
   f
   s
  
  
 


						= 52
Figure 40. Torque-time in per unit while V sag = 50% and υ w = 6m/s, f s = 52
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 50% and 
							
 
  
   υ
   w
  
  
 

						 = 10m/s, 
							
 
  
   f
   s
  
  
 


						= 52
Figure 41. Torque-time in per unit while V sag = 50% and υ w = 10m/s, f s = 52
Torque-time in per unit while 
							
 
  
   V
   
    sag
   
  
  
 

						= 50% and 
							
 
  
   υ
   w
  
  
 

						 = 13m/s, 
							

 
  
   f
   s
  
  
 

						= 52 in new simulation of wind generator
Figure 42. Torque-time in per unit while V sag = 50% and υ w = 13m/s, f s = 52 in new simulation of wind generator
Induction machine Model in 
							
 
  dqo
 

						 system
Figure 43. Induction machine Model in dqo system

Effect of Turbulence on Fixed-Speed Wind Generators

Hengameh Kojooyan Jafari1

1. Introduction

The influence of wind energy connection to the grid has increased greatly and turbulence or unreliable characteristics of wind energy are expected to produce frequency and voltage changes in power systems and protection system equipment. To prevent these changes, it is necessary to study the working point change due to turbulence. In other papers, the voltage and transient stability analysis have been studied during and after turbulence [2] and the impact of WTGs (wind turbine generators) on the system frequency, inertia response of different wind turbine technologies, and comparison between inertia response of single-fed and doubly-fed induction generators have been examined. Moreover study of the frequency change alone was conducted using Dig-SILENT simulator for FSWTs (fast-speed wind turbines) with one-mass shaft model [2].

In this chapter both frequency and grid voltage sag change are presented with MATLAB analytically and also by SIMULINK simulation in FSWTs with one- and two-mass shaft turbine models to compare both results and a new simulation of induction machine without limiter and switch blocks is presented as a new work. The first part of study is frequency change effect on wind station by SIMULINK that shows opposite direction of torque change in comparison with previous studies with Dig-SILENT. The second part of study is effect of frequency and voltage sag change on wind station torque due to turbulence in new simulation of induction generator that is new idea.

2. Wind turbine model

The equation of wind turbine power is

(1)

where ρMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYbaa@37A9@ is air density, AMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeaaaa@36AF@ is area of turbine, CpMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaWgaaWcbaGaamiCaaqabaaaaa@37D2@ is power coefficient and υwMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew8a1naaBaaaleaacaWG3baabeaaaaa@38D8@ is wind speed.

The C p curve and equation are shown in Fig. 1 and given by equation (2) and (3)

(2)
(3)

where θpitch is blade pitch angle, λ is the tip speed ratio described by equation (4). The parameters are given in Table 1.

(4)

where R is blade radius.

media/image8.jpg

Figure 1.

Curve of Cp for different tip speed ratios λ .

The curve of Fig.1 has positive slope before Cp max and it has negative slope after Cp max.

3. One-Mass Shaft Wind Station Model

Induction machine equation is

(5)

Where, Tm is the mechanical torque, Te is the generator torque, C is the system drag coefficient and J is the total inertia.

Table 1 shows the parameters of the one-mass shaft turbine model and induction generator.

GeneratorWind Turbine
Rs = .011Ω c1=.44
Ls = .000054H c2 = 125
Lm = .00287H c3 = 0
Lr = .000089H c4 = 0
Rr = .0042 [Ω] c5 = 0.1
Jm =.5 to 20.26 [kgm2] c6 = 6.94
p(#polepairs) = 2 c7 = 16.5
Pn = 2e6 [w] c6 = 0.1
c9 = -.002
R = 35 [m]
A = πR2 [m2]
ρ =1.2041 [kg/m3]
vw = 6, 10, 13 [m/s]
θpitch = 0 [º]

Table 1.

Parameters of one- mass shaft turbine model and generator.

4. Two-Mass Shaft Induction Machine Model

This model is used to investigate the effect of the drive train or two-mass shaft, i.e., the masses of the machine and the shaft, according to the equation (8) [3], [4]. In this equation, Jt is wind wheel inertia, JG is gear box inertia and generator’s rotor inertia connected through the elastic turbine shaft with a κ as an angular stiffness coefficient and C as an angular damping coefficient.

The angular shaft speed ωt can be obtained from equations (6) and (7) [1], [3], [4].

TGMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaam4raaqabaaaaa@37BA@ is the torque of the machine, TtMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaamiDaaqabaaaaa@37E7@ is the turbine torque, δtMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBaaaleaacaWG0baabeaaaaa@38B3@ is the angular turbine shaft angle, δGMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBaaaleaacaWGhbaabeaaaaa@3886@ is the angular generator shaft angle, νMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUbaa@37A1@ is the inverse of the gear box ratio and JGMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadQeadaWgaaWcbaGaam4raaqabaaaaa@37B0@ and JtMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadQeadaWgaaWcbaGaamiDaaqabaaaaa@37DD@ are the inertia of the machine shaft and turbine shaft, respectively.

The Parameters, defined above, are given in Table 2.

This model is described as equation (8).

(6)
(7)
(8)
υ 1/80
JG [kg.m2].5
Jt [kg.m2]1
C [Nm/rad2]1e6
κ [Nm/rad]6e7

Table 2.

Parameters of two-mass shaft model.

5. Induction Machine and Kloss Theory

In a single-fed induction machine, the torque angular speed curve of equation (12) [1] is nonlinear, but by using the Kloss equation (13), equations (9), (10), and (11), this curve is linearly modified [1], [2] as shown in Fig. 2. Therefore, the effect of frequency changes in wind power stations can be derived precisely by equation (12) and approximately using equation (13), as shown in Figs. 26.

(9)
(10)
(11)
(12)
(13)
media/image89.png

Figure 2.

Electrical torque (nonlinear and linear) versus speed (slip).

Equations (11) and (12) are given in per unit, but the associated resistances are in ohms.

media/image90.png

Figure 3.

Mechanical and linear electrical torque versus slip.

media/image91.jpeg

Figure 4.

Mechanical and electrical torque versus frequency curves per unit with V sag = 10%

.
media/image92.jpeg

Figure 5.

Mechanical and electrical torque versus frequency per unit with V sag = 20%.

media/image93.png

Figure 6.

Mechanical and electrical torque versus frequency per unit with V sag = 50%.

Figs. 3, 4, 5, and 6 illustrate that for lower wind speeds of 6 and 10 m/s, as the synchronous frequency f s and V sag change, the T e and TmMathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaamyBaaqabaaaaa@37E1@ values of the rotor change in the same direction as the frequency of the network, as shown in Tables III, IV, V, and VI. These figures and tables give the results for V sag = 0% (i.e., only the frequency changes), 10%, 20%, and 50%. However, for a higher wind speed of 13 m/s, as f s and V sag change, the T e and TmMathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaamyBaaqabaaaaa@37E1@ values of the rotor change in the opposite direction to the changes in the frequency of the network.

For small changes in the slip according to the Kloss approach in equation (13), the torque changes as follows [2]:

(14)

Then:

(15)

and

(16)

or

(17)

Thus, the new angular operation speed[2] is

(18)
υw fs= 48 fs = 50 fs= 52
ωm[pu] Te[pu] ωm[pu] Te[pu] ωm[pu] Te[pu]
6.96050-.11571.0005-.10641.0405-.0974
10.9621-.53371.0021-.4911.0421-.4493
13.9631-.78631.0035-.81221.0439-.8331

Table 3.

Analytical MATLAB results for different frequencies.

υw fs = 48 fs = 50 fs = 52
ωm[pu] Te[pu] ωm[pu] Te[pu] ωm[pu] Te[pu]
6.9606-.11561.0006-.10641.0406-.0974
10.9625-.51631.0027-.51371.0429-.5086
13.9738-.78681.0043-.81271.0448-.8335

Table 4.

Analytical MATLAB results for Vsag = 10%.

υw fs= 48 fs= 50 fs= 52
ωm[pu] Te[pu] ωm[pu] Te[pu] ωm[pu] Te[pu]
6.9607-.11561.0007-.10641.0407-.0974
10.9632-.51631.0034-.51361.0437-.5085
13.9648-.78751.0054-.81341.0461-.8341

Table 5.

Analytical MATLAB results for Vsag = 20%

υw fs= 48 fs= 50 fs= 52
ωm[pu] Te[pu] ωm[pu] Te[pu] ωm[pu] Te[pu]
6.9618-.11531.0018-.10611.0418-.0971
10.9681-.51611.0088-.51311.0494-.5076
13.9724-.79271.0139-.81811.0555-.8382

Table 6.

Analytical MATLAB results for Vsag = 50%

6. Simulation of wind generator with frequency change

During turbulence and changes in the grid frequency, the torque speed (slip) curves change in such a way that as the frequency increases, the torque is increased at low wind speeds; 6 and 10 m/s, in contrast to Fig. 6 and decreases at a high speed of 13 m/s [2], as shown in Table 7 and Figs. 715.

υw fs = 48 fs = 50 fs = 52
ωm[pu] Te[pu] ωm[pu] Te[pu] ωm[pu] Te[pu]
6.9619-.11481.0019-.10571.0418-.0969
10.9684-.51791.0091-.51341.0494-.5076
13.9724-.79451.0147-.81771.0559-.8373

Table 7.

Simulink simulation results for one- and two-mass shaft models

Figs. 715 show the electrical torque and mechanical speed of the induction machine for the one- and two-mass shaft turbine models at wind speeds of 6, 10, and 13 m/s to validate Table 7.

media/image107.jpeg

Figure 7.

Electrical torque when = 48 and = 6m/s.

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Figure 8.

Electrical torque when fs = 50 and υw = 6m/s.

media/image112.jpeg

Figure 9.

Electrical torque when fs = 52 and υw = 6m/s.

media/image114.jpeg

Figure 10.

Electrical torque when fs = 48 and υw = 10m/s.

media/image116.jpeg

Figure 11.

Electrical torque when fs = 50 and υw = 10m/s.

media/image118.jpeg

Figure 12.

Electrical torque when fs = 52 and υw = 10m/s.

media/image120.jpeg

Figure 13.

Electrical torque when fs = 48 and υw = 13m/s.

media/image122.jpeg

Figure 14.

Electrical torque when fs = 50 and υw = 13m/s.

media/image124.jpeg

Figure 15.

Electrical torque when fs = 52 and υw = 13m/s.

7. Simulation of wind station with one-mass and two-mass shaft turbine models

The results of simulations of a simple grid, fixed-speed induction machine, and one-mass and two-mass shaft turbines are given in Tables 8 -10 and Figs. 1642. For an induction wind generator using the induction block in SIMULINK with high voltage sag i.e. 50% with frequencies 50 and 52 and equal to 13, C p becomes negative, and the results are unrealistic. Then results of 50% voltage sag are realistic in new simulation of induction machine in Tables 8 -10.

υw fs= 48 fs= 50 fs= 52
ωm[pu] Te[pu] ωm[pu] Te[pu] ωm[pu] Te[pu]
6.9624-.11521.0024-.1061.0423-.097
10.9703-.5161.0111-.51281.0519-.5071
13.9757-.7951.0176-.82011.0595-.8399

Table 8.

Simulation results by SIMULINK for one and two mass shaft model for Vsag = 10%

υw fs= 48 fs= 50 fs= 52
ωm[pu] Te[pu] ωm[pu] Te[pu] ωm[pu] Te[pu]
6.963-.11511.003-.10591.043-.0969
10.973-.51591.014-.51251.055-.5066
13.9799-.79771.0223-.82261.0648-.842

Table 9.

Simulation results by SIMULINK for one and two mass shaft model for Vsag = 20%

υw fs= 48 fs= 50 fs= 52
ωm[pu] Te[pu] ωm[pu] Te[pu] ωm[pu] Te[pu]
6.9674-.1141.0074-.10481.0474-.0959
10.9933-.51461.0364-.50961.0796-.502
131.0248-.82391.0474-.83471.0917-.85

Table 10.

Simulation results by SIMULINK for one and two mass shaft model for Vsag = 50%

media/image127.jpeg

Figure 16.

Torque-time in per unit while Vsag = 10% and υw = 6m/s, fs =48

media/image128.jpeg

Figure 17.

Torque-time in per unit while Vsag = 10% and υw = 10m/s, fs = 48

media/image130.jpeg

Figure 18.

Torque-time in per unit while Vsag = 10% and υw = 13m/s, fs = 48

media/image132.jpeg

Figure 19.

Torque-time in per unit while Vsag = 20% and υw = 6m/s, fs = 48

media/image134.jpeg

Figure 20.

Torque-time in per unit while Vsag = 20% and υw = 10m/s, fs = 48

media/image136.jpeg

Figure 21.

Torque-time in per unit while Vsag = 20% and υw = 13m/s, fs = 48

media/image138.jpeg

Figure 22.

Torque-time in per unit while Vsag = 50% and υw = 6m/s, fs = 48

media/image140.jpeg

Figure 23.

Torque-time in per unit while Vsag = 50% and υw = 10m/s, fs = 48

media/image142.jpeg

Figure 24.

Torque-time in per unit while Vsag = 50% and υw = 13m/s, fs = 48

media/image144.jpeg

Figure 25.

Torque-time in per unit while Vsag = 10% and υw = 6m/s, fs = 50

media/image146.jpeg

Figure 26.

Torque-time in per unit while Vsag = 10% and υw = 10m/s, fs = 50

media/image148.jpeg

Figure 27.

Torque-time in per unit while Vsag = 10% and υw = 13m/s, fs = 50

media/image150.jpeg

Figure 28.

Torque-time in per unit while Vsag = 20% and υw = 6m/s, fs = 50

media/image152.jpeg

Figure 29.

Torque-time in per unit while Vsag = 20% and υw = 10m/s, fs = 50

media/image154.jpeg

Figure 30.

Torque-time in per unit while Vsag = 20% and υw = 13m/s, fs = 50

media/image156.jpeg

Figure 31.

Torque-time in per unit while Vsag = 50% and υw = 6m/s, fs = 50

media/image158.jpeg

Figure 32.

Torque-time in per unit while Vsag = 50% and υw = 10m/s, fs = 50

media/image160.jpeg

Figure 33.

Torque-time in per unit while Vsag = 50% and υw = 13m/s, fs = 50 in new simulation of wind generator

media/image162.jpeg

Figure 34.

Torque-time in per unit while Vsag =10% and υw = 6m/s, fs = 52

media/image164.jpeg

Figure 35.

Torque-time in per unit while Vsag =10% and υw = 10m/s, fs = 52

media/image166.jpeg

Figure 36.

Torque-time in per unit while Vsag = 10% and υw = 13m/s, fs = 52

media/image168.jpeg

Figure 37.

Torque-time in per unit while Vsag = 20% and υw = 6m/s, = 52

media/image170.jpeg

Figure 38.

Torque-time in per unit while Vsag = 20% and υw = 10m/s, fs = 52

media/image172.jpeg

Figure 39.

Torque-time in per unit while Vsag = 20% and υw = 13m/s, fs = 52

media/image174.jpeg

Figure 40.

Torque-time in per unit while Vsag = 50% and υw = 6m/s, fs = 52

media/image176.jpeg

Figure 41.

Torque-time in per unit while Vsag = 50% and υw = 10m/s, fs = 52

media/image178.jpeg

Figure 42.

Torque-time in per unit while Vsag = 50% and υw = 13m/s, fs = 52 in new simulation of wind generator

8. New Simulation of Induction Machine

Figs. 33 and 42 show the results of new simulation of the induction machine model illustrated in Fig. 43 [1]. The new simulation, which has no limiters and switches, is used because at high grid voltage drop-down or sag, the Simulink induction model does not yield realistic results.

media/image180.png

Figure 43.

Induction machine Model in dqo system

The new simulation of induction machine is in dqo system and synchronous reference frame simulation on the stator side; n (Transfer coefficient) is assumed to be 1. Circuit theory is used in this simulation, and it does not have saturation and switch blocks, unlike the MATLAB–SIMULINK Induction block. In Fig. 43, LM is the magnetic mutual inductance, and r and L are the ohm resistance and self-inductance of the dqo circuits, respectively. The machine torque is given by equation (19). In this equation, id,qs and λd,qs , the current and flux parameters, respectively, are derived from linear equations (20)–(23); they are sinusoidal because the voltage sources are sinusoidal.

(19)

Where P is poles number, λds and λqs are flux linkages and leakages, respectively, and iqs and ids are stator currents in q and d circuits of dqo system, respectively.

Then i matrix produced by the λ matrix is given by equation (20).

(20)

where the inductance matrix parameters are given by (21), (22), (23).

(21)
(22)
(23)

The linkage and leakage fluxes are given by (24) to (29).

(24)
(25)
(26)
(27)
(28)
(29)

To create the torque in equation (19), it is necessary to determine the currents in equations (30)–(33) from the stator and rotor currents by using current meters.

(30)
(31)
(32)
(33)

9. Conclusion

As frequency changes and voltage sag occurs because of turbulence in wind stations in ride-through faults, the system’s set point changes. The theoretical and simulation results results are similar for one mass shaft and two mass shaft turbine models. At lower wind speeds; 6 and 10 m/s, the directions of the changes in the new working point are the same as those of the frequency changes. At a higher wind speed; 13 m/s, the directions of these changes are opposite to the direction of the frequency changes. Simulation results of high grid voltage sag with SIMULINK induction block has error and new simulation of wind induction generator in synchronous reference frame is presented without error and in 50% voltage sag, new simulation of wind generator model has higher precision than that in 10% and 20% voltage sags; however, this model can simulate wind generator turbulence with voltage sags higher than 50%. Although results of new simulation of induction machine with wind turbine for 50% voltage sag and frequencies 50 and 52 have been presented in this chapter.

10. Nomenclature

P= Generator power

ρ= Air density

A= Turbine rotor area

Cp= Power Coefficient

υw= Wind speed

θpitch= Pitch angle

Te= Electrical torque

Tm= Mechanical torque

J = Inertia

ωm= Mechanical speed

C= Drag coefficient

ν= Gear box ration

R= Blade radius

Rs = Stator resistance

Ls = Stator inductance

Lm = Mutual inductance

Lr = Rotor inductance

Rr = Rotor resistance

p= Pole pairs

κ= Stiffness

λr,s= Rotor and stator flux

Kr,s= Rotor and stator park transformation in synchronous reference frame

ir,s= Rotor and stator current

vr,s= Rotor and stator voltage

11. Future Work

The new simulation of induction generator will be tested by new innovative rain turbine theory and model of the author.

Acknowledgements

I appreciate Dr. Oriol Gomis Bellmunt for conceptualization, Discussions and new information and Dr. Andreas Sumper for discussions about first part of chapter, with special thanks to Dr. Joaquin Pedra for checking reference frame and starting point in new simulation of induction machine.

References

1 - Paul C. Krause, 1986 Analysis of Electric Machinery MCGraw-Hill, Inc.
2 - A. Sunmper, O. Gomis-Bellmunt, A. Sudria-Andreu, et al. 2009 Response of Fixed Speed Wind Turbines to System Frequency Disturbances ICEE Transaction on Power Systems 24 1 181 192
3 - A. Junyent-Ferre, O. Gomis-Bellmunt, A. Sunmper, et al. 2010 Modeling and control of the doubly fed induction generator wind turbine Simulation modeling practice and theory journal of ELSEVIER 1365 1381
4 - Z Lubosny, 2003 Wind Turbine operation in electric power systems Springer publisher