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# Effect of Turbulence on Fixed-Speed Wind Generators

Written By

Hengameh Kojooyan Jafari

Submitted: 07 March 2012 Published: 21 November 2012

DOI: 10.5772/51407

From the Edited Volume

Edited by Rupp Carriveau

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## 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

E1

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

E2
E3

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

E4

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

## 3. One-Mass Shaft Wind Station Model

Induction machine equation is

E5

Where, T m is the mechanical torque, T e 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.

 Generator Wind Turbine R s = .011Ω c 1 =.44 L s = .000054H c 2  = 125 L m = .00287H c 3 = 0 L ′ r = .000089H c 4 = 0 R ′ r = .0042 [Ω] c 5 = 0.1 J m =.5 to 20.26 [kgm2] c 6 = 6.94 p(#pole   pairs) = 2 c 7 = 16.5 P n = 2e6 [w] c 6 = 0.1 c 9 = -.002 R = 35 [m] A = πR2 [m2] ρ =1.2041 [kg/m3] v w = 6, 10, 13 [m/s] θ pitch = 0 [º]

## 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, J t is wind wheel inertia, J G 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 ω t can be obtained from equations (6) and (7) [1], [3], [4].

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

This model is described as equation (8).

E6
E7
E8
 υ 1/80 J G [kg.m2] .5 J t  [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.

E9
E10
E11
E12
E13

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

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 T m MathType@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 T m MathType@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]:

E14

Then:

E15

and

E16

or

E17

Thus, the new angular operation speed[2] is

E18
 υ w f s = 48 f s = 50 f s = 52 ω m [pu] T e [pu] ω m [pu] T e [pu] ω m [pu] T e [pu] 6 .96050 -.1157 1.0005 -.1064 1.0405 -.0974 10 .9621 -.5337 1.0021 -.491 1.0421 -.4493 13 .9631 -.7863 1.0035 -.8122 1.0439 -.8331

### Table 3.

Analytical MATLAB results for different frequencies.

 υ w f s = 48 f s = 50 f s = 52 ω m [pu] T e [pu] ω m [pu] T e [pu] ω m [pu] T e [pu] 6 .9606 -.1156 1.0006 -.1064 1.0406 -.0974 10 .9625 -.5163 1.0027 -.5137 1.0429 -.5086 13 .9738 -.7868 1.0043 -.8127 1.0448 -.8335

### Table 4.

Analytical MATLAB results for V sag = 10%.

 υ w f s = 48 f s = 50 f s = 52 ω m [pu] T e [pu] ω m [pu] T e [pu] ω m [pu] T e [pu] 6 .9607 -.1156 1.0007 -.1064 1.0407 -.0974 10 .9632 -.5163 1.0034 -.5136 1.0437 -.5085 13 .9648 -.7875 1.0054 -.8134 1.0461 -.8341

### Table 5.

Analytical MATLAB results for V sag = 20%

 υ w f s = 48 f s = 50 f s = 52 ω m [pu] T e [pu] ω m [pu] T e [pu] ω m [pu] T e [pu] 6 .9618 -.1153 1.0018 -.1061 1.0418 -.0971 10 .9681 -.5161 1.0088 -.5131 1.0494 -.5076 13 .9724 -.7927 1.0139 -.8181 1.0555 -.8382

### Table 6.

Analytical MATLAB results for V sag = 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.

 υ w f s = 48 f s = 50 f s = 52 ω m [pu] T e [pu] ω m [pu] T e [pu] ω m [pu] T e [pu] 6 .9619 -.1148 1.0019 -.1057 1.0418 -.0969 10 .9684 -.5179 1.0091 -.5134 1.0494 -.5076 13 .9724 -.7945 1.0147 -.8177 1.0559 -.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.

## 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.

 υ w f s = 48 f s = 50 f s = 52 ω m [pu] T e [pu] ω m [pu] T e [pu] ω m [pu] T e [pu] 6 .9624 -.1152 1.0024 -.106 1.0423 -.097 10 .9703 -.516 1.0111 -.5128 1.0519 -.5071 13 .9757 -.795 1.0176 -.8201 1.0595 -.8399

### Table 8.

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

 υ w f s = 48 f s = 50 f s = 52 ω m [pu] T e [pu] ω m [pu] T e [pu] ω m [pu] T e [pu] 6 .963 -.1151 1.003 -.1059 1.043 -.0969 10 .973 -.5159 1.014 -.5125 1.055 -.5066 13 .9799 -.7977 1.0223 -.8226 1.0648 -.842

### Table 9.

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

 υ w f s = 48 f s = 50 f s = 52 ω m [pu] T e [pu] ω m [pu] T e [pu] ω m [pu] T e [pu] 6 .9674 -.114 1.0074 -.1048 1.0474 -.0959 10 .9933 -.5146 1.0364 -.5096 1.0796 -.502 13 1.0248 -.8239 1.0474 -.8347 1.0917 -.85

### Table 10.

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

## 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.

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, L M 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, i d,qs and λ d,qs , the current and flux parameters, respectively, are derived from linear equations (20)–(23); they are sinusoidal because the voltage sources are sinusoidal.

E19

Where P is poles number, λ ds and λ qs are flux linkages and leakages, respectively, and i qs and i ds 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).

E20

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

E21
E22
E23

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

E24
E25
E26
E27
E28
E29

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.

E30
E31
E32
E33

## 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

C p = Power Coefficient

υ w = Wind speed

θ pitch = Pitch angle

T e = Electrical torque

T m = Mechanical torque

J = Inertia

ω m = Mechanical speed

C= Drag coefficient

ν= Gear box ration

R s = Stator resistance

L s  = Stator inductance

 L m = Mutual inductance

 L r = Rotor inductance

R r = Rotor resistance

p= Pole pairs

κ= Stiffness

λ r,s = Rotor and stator flux

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

i r,s = Rotor and stator current

v r,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.

## Acknowledgments

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. 1. Krause Paul C. 1986 Analysis of Electric Machinery MCGraw-Hill, Inc.
2. 2. Sunmper A. Gomis-Bellmunt O. Sudria-Andreu A. et al. 2009 Response of Fixed Speed Wind Turbines to System Frequency Disturbances ICEE Transaction on Power Systems 24 1 181 192
3. 3. Junyent-Ferre A. Gomis-Bellmunt O. Sunmper A. 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. 4. Lubosny Z 2003 Wind Turbine operation in electric power systems Springer publisher

Written By

Hengameh Kojooyan Jafari

Submitted: 07 March 2012 Published: 21 November 2012