Advanced Materials Research Online: 2014-06-18 ISSN: 1662-8985, Vols. 953-954, pp 384-388 doi:10.4028/www.scientific.net/amr.953-954.384 2014 Trans Tech Publications, Switzerland A Model of Wind Turbine s Flexibility Shaft Guodong Ding 1, a, Sanming Liu 1,b,Qinqin Chen 1,c Jianpeng Liu 2,d and Mingli Yang 1,e 1 No.690 Jiang chuan road Minhang district Shanghai 2 No.1318 Yicheng road Zhumadian city Henan province a 644526998@qq.com, b liusanmingxyx@163.com, c 975299144@qq.com Keywords: Wind turbines, Sub-synchronous oscillations, Flexibility shaft model Abstract. Wind power is a new technology compared with thermo power, the damage of sub-synchronous oscillations cloud bring to the shaft of wind turbine do not get enough notices. There are few studies on wind turbines sub-synchronous oscillation now, which are built on the model of rigid shaft. As we all know wind turbine shaft is unlike the shaft of stream turbine, there are couplings and gearbox in wind turbine which make the shaft flexible, so the studies of wind turbine sub-synchronous oscillations based on rigid shaft model are not reasonable, this paper introduces a new flexible shaft model for studying of wind turbine sub-synchronous oscillations. Flexible equipments, such as joints and gearbox are taken into consideration when the model are set up, so this model is far more scientific. Introduction Sub-synchronous oscillation problem appeared in the thermal power unit at first. There were two damage accidents of large turbine generator shaft in Mohave thermo station in America in less than one year between December 1970 and October 1971. The studies on the accidents showed that it is the inappropriate series capacitor that leaded to the accidents [1]. These two accidents attracted extensive attention of scientists of electric power worldwide, the study of sub-synchronous oscillations grow vigorously from then. The IEEE working group defined sub-synchronous oscillation as an abnormal working condition, energy exchanges significantly between unite shaft system and electric system on the one or more oscillation frequency below the inherent frequency in the condition, in which way cause damage to unite shaft [2,3]. The damage of sub-synchronous oscillations are reflected in three aspects as follow [4,5]. First, when a certain inherent frequency of wind turbine shafting is same or similar to the frequency of electric system, resonance occurred. This is a severe situation, in which the shaft can be damaged badly even the variation of excitation torque is tiny. Second, when a certain inherent frequency of wind turbine shafting is complementary or close to complementary with the frequency of electric system, there would produce the resonance phenomenon which may cause disastrous damages to unite shaft. Third, if wind turbine shafting experience unbalanced electrical excitation torque time to time would lead to torsional vibration, which would cause unite shafting fatigue failure, sequentially would shorten service life. All of these force us to try our best to forbid sub-synchronous oscillation happening, thus, it is significant necessary to study inherent torsion frequency of wind turbine shafting. The reality is that there are not enough studies on wind turbine sub-synchronous oscillation now, the only studies take sample by the way we research thermo power sub-synchronous oscillation, as we all know there are so many differences between wind power and thermo power, such as the structure of unite and the way connecting to the grid [6], for all that the current studies are unscientific and unbelievable. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (ID: 130.203.136.75, Pennsylvania State University, University Park, USA-12/05/16,17:14:52)
Advanced Materials Research Vols. 953-954 385 Modeling of wind turbine shafting We can further study sub-synchronous oscillation through inherent torsion frequency of shafting, the calculation of the inherent torsion frequency of shafting is based on the rigid shaft model now. rigid shaft model believe that low-speed shaft, gear box and high-speed shaft are connected rigidly,we can consider the model as two discs with a certain mass connect rigidly, the model is shown in the Fig.1.The modeling ideas are similar to how to create a model of steam turbine generator unite shafting [7]. However the shaft of wind turbine is more like flexibility because of the existing of coupler and gear box, furthermore horizontal and vertical disturbances created by the blades because of the random wind would act on the coupler and gear box though driving chain [8]. We should take all of these into consideration when we build a model of wind turbine shafting. Fig.1. Rigid shaft system model This article give a new way to create the model of flexibility shaft of wind turbine from transmission mechanic perspective. The model consider the low-speed shaft and high-speed shaft are flexible because of the existing of gear box, the model also believe that blades and generator rotor can rotate freely in a certain degree. The torque of shaft can be calculated by the formula(1).. (1) In the equations, Q---- Drive shaft torque; k----stiffness; B---- Damping; θ---- Angular displacement. We can set up the wind turbine drive system model by analyzing forces of the transmission system,we depart the drive system into several independent parts, such as shown in Fig.2., Fig.3. and Fig.4. First, we analyze the force of the system composed of blades and low-speed shaft, its force diagram is shown as Fig.2. Fig.2.Force diagram of rotor and low-speed system We can obtain the stress relation formula as following through analyzing force diagram.
386 Advanced Energy Technology In the equations, J R ----Moment of inertia of the blade; T A ----Rotor pneumatic torque; T 1 ----Low-speed shaft torque; θ R ----Turbine rotors angular displacement; θ 1 ----Low-speed shaft angular displacement; k 1 ----- Low-speed shaft stiffness; B R ---- Blade damping; B 1 ----- Low-speed shaft damping. In the formulas above, the resistance to the blades rotation only come from the air which is really small compared with the resistance to low-speed shaft which come from the friction of the low speed shaft and the gear box, so B 1 is far larger than B R, therefore, Equation 2 can be changed to: And when we put the formula(3) into the above equation can be obtained: The structure of gear box is very complex because the gears inside the gear box would mesh with each other, besides, the gear box also connect with low-speed and high-speed shaft in the transmission system of wind turbine. All of these make creating a model of gear box a difficult task. In order to alleviate the difficulty, I have done some simplifications of gearbox system while having no impact on the modeling accuracy. I assume that the low-speed shaft and the high-speed shaft have no quality, and the gear box is supported by rigid holder. The system shown in Fig.3. (2) (3) (4) (5) Fig.3. Equivalent dynamics model of gear box system and the dynamic equations of the model are: (6)
Advanced Materials Research Vols. 953-954 387 (7) In these equations, N-----gear box drive ratio; Tout ----- gearbox output torque; J1---- low-speed shaft rotational inertia; J2 ---- high-speed shaft rotational inertia; θ2---- high speed shaft angular displacement We can eliminate Tout and through combining the above three equations, the simplified formula is: And this is the dynamic model of gear box. The model of the system composed by high-speed shaft and generator rotor is shown in Fig.4. (8) (9) Fig.4. The model of the system composed by high-speed shaft and generator rotor Because of the similarity with the blades and low speed shaft system, the force equation can be derived in accordance with the previous analysis, the equations are: In the two equations, J G ---- Generator rotor inertia; θ G ---- Generator rotor angular displacement; T G ---- Generator rotor torque; B 2 ---- High-speed shaft damping. As the high-speed shaft rotational inertia is much smaller than the generator rotor inertia [9], so J 2 can be omitted, if we put equation (11) into equation (10), we obtain the formula as follow: We combine formula (5),(9) and (12) to a differential equations as follow: (10) (11) (12) We can get the shaft torsional vibration natural frequencies and mode shapes through solving the differential equations, then we can solve the non-homogeneous equation, if we know the boundary conditions, the continuity conditions of the adjacent shaft section and the initial conditions, at last the shafting sub-synchronous oscillation problem of wind turbines can be solved. (13)
388 Advanced Energy Technology Conclutions This flexibility shaft model can show the real working condition, and the model take the flexibility joint of the wind turbine shafting and blades oscillation into consideration, so the inherent torsion frequency we get from the model are more believable and reasonable. The shafting torsion inherent frequency is the key to analyzing wind turbine shafting sub-synchronous oscillation, if the frequency was wrong, then the work you did after that is vain. This article arise a new idea to set up a flexibility shafting model, and I believe this model would play a role in analyzing wind turbine shafting sub-synchronous oscillation in the future. My next work will move on to the simulation to prove this model is better than the model of rigid shaft. Acknowledgments This work was supported by National Natural Science Foundation of China, Grant No.11201267, and by Shanghai Natural Science Foundation of China, Grant No. 12ZR1411600. We would like to thank the reviewer for his valuable suggestions and the editor for his helpful assistance. References [1] Walker D N, Bowler C E J, Jackson R L,et al. Results of sub-synchronous resonance test at Mohave[J]. IEEE Transactions on Power Apparatus and Systems, 1975, 94 (5):1878-1889. [2] IEEE sub-synchronous resonance working group. Proposed terms and definitions for sub-synchronous oscillations [J]. IEEE Transaction on Power Apparatus and Systems 1980 99(2) :506-511. [3] IEEE Committee Report. Readers guide to sub-synchronous resonance [J].IEEE Transaction on Power Systems, 1992,7(1):150-157 [4] Shuiping Zhang, Shuhong Huang. Processings of the Chinese Society for Electrical Engineering[J].2000,20(11):10-16. [5] Shijie Cheng, Yijia Cao, Quanyuan Jiang. Theories and Methods of Electric System Sub-synchronous Oscillation[M]. Beijing. Science press, 2009. [6] Yongle Kong, Analysis of Sub-sychronous Oscillation Mechanism Resulted by Transmiting of Wind Power in Large Scale.[D].Beijing.Master Thesis of North China Electric Power University.2013. [7] Chao Liu,Dongrong Jiang,Xiaorong Xie,Liangyou Hong.Automation of Electric Power System[J].2010,34(15):19-22. [8] Yehang Ye. wind generator s monitoring and control[d].china Machine Press,2011 [9] Novak P, Ekelund T. Modeling and control of variable speed wind turbine drive - system dynamics [ J ].IEEE Control Systems, 1995, 8: (28 ) 37.
Advanced Energy Technology 10.4028/www.scientific.net/AMR.953-954 A Model of Wind Turbine s Flexibility Shaft 10.4028/www.scientific.net/AMR.953-954.384 DOI References [1] Walker D N, Bowler C E J, Jackson R L, et al. Results of sub-synchronous resonance test at Mohave[J]. IEEE Transactions on Power Apparatus and Systems, 1975, 94 (5): 1878-1889. 10.1109/T-PAS.1975.32034 [3] IEEE Committee Report. Readers guide to sub-synchronous resonance [J]. IEEE Transaction on Power Systems, 1992, 7(1): 150-157. 10.1109/59.141698