Delegates are invited to meet and discuss with the poster presenters in this topic directly after the session 'Whole-life foundation and structure integrity' taking place on Wednesday, 12 March 2014 at 14:15-15:45. The meet-the-authors will take place in the poster area.

**Erik MÃ¶llerstrÃ¶m**Halmstad University, Sweden

Co-authors:

Erik MÃ¶llerstrÃ¶m (1)

^{F}

^{P}Fredric Ottermo (1) Jonny Hylander (1) Hans Bernhoff (2)

(1) Halmstad University, Halmstad, Sweden (2) Uppsala University, Uppsala, Sweden

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## Presenter's biography

*Biographies are supplied directly by presenters at EWEA 2014 and are published here unedited*

Erik MÃ¶llerstrÃ¶m is a PhD-student in Energy Technology at Halmstad University in Sweden. He does his research on vertical wind turbine dynamics in collaboration with Uppsala University. He started in 2012 and has so far been studying the effect regarding natural frequencies upon adding guy wires to a vertical axis wind turbine. He received his MSc in Environmental Engineering at Lund University in 2008.

## Abstract

**Avoidance of resonances in a semi-guy-wired vertical axis wind turbine
****Introduction**

An innovative 200 kW VAWT, owned by Uppsala University has been the subject of research in a variety of fields. Its laminated wooden tower was from the start free standing but are now supported by three guy-wires. The construction may, therefore, be described as semi-guy wired. Instability due to eigen frequencies are of concern regarding wind turbines as well as most other tall structures. In this article the focus is on resonance in the guy wires and how the eigen frequency of the wires is affected by the wind load.

**Approach**

An analytical model is derived for determining the first mode eigen frequency for the guy wire at different wind speeds that the turbine is exposed to. The model is based on the equation describing the eigen frequency of a vibrating string but with a new variable added to the tension factor (pre-tension) compensating the extra tension added by the wind load for a windward guy wire in line with the wind direction. Beam theory is used for deriving the expression for the deviation at the height of guy wire attachment point of the tower with guy wires. The deviation is then used to acquire the tension added by the wind load. If this tension is subtracted instead of added to the pre-tension the eigen frequency of the leeward wire in line with wind direction is attained. These eigen frequencies for the windward and leeward wires will create a span growing with the wind speed and housing the eigen frequencies for any wire independent of angle towards wind direction.

The model is verified using FEM software and Campbell diagrams showing how the eigen frequency of the wire will behave for different rotational speeds is produced. Then the model is used for setting up a diagram (here called an EA-T diagram) which for a resonance free wire and a pre-determined effective spring force on the tower allows comparing different combinations of wire setup regarding nominal axial stiffness, inclination angle and pre-tension of the wire. The diagram is based on the 200kW-VAWT owned by Uppsala University but the same kind of diagram can easily be used for other guy wired structures.

**Main body of abstract**

Resonance analysis is performed for a 200 kW VAWT today owned by Uppsala University which is the subject of research in a variety of fields. This particular VAWT has a direct drive permanent magnet synchronous generator which is mounted at the bottom of the tower and connected to the rotor by a steel shaft. Furthermore, it has a tower made out of laminated wood which from the start was free standing but after two years was complemented with support from three guy-wires. The construction may therefore be described as semi-guy wired.

When designing a wind turbine tower, every possible potential resonance needs to be addressed. The tower has several different modes of eigen frequency oscillations but for the case of a typical wind turbine it is mainly the first mode that is of interest since this frequency may coincide with a dynamic load of the turbine. The dynamic loads of significance are imbalance in the main shaft and/or rotor as well as aerodynamical loads of the passing blades. The frequencies of these loads are 1P and 3P respectively, where P is the rotational speed of the turbine. Earlier work [6] by the main author has derived an analytical model for describing the eigen frequency of a tower attached to the ground as well as supported by guy wires.

In the case of a guyed tower the eigen frequencies of the wires also becomes of significance. Resonance of the guy wires will produce amplified motions that in worst case can destroy the entire tower. Just as for the tower it is mainly the first mode of eigen frequency that is of interest since it may be excited by the tower or blade passing frequencies. Since the eigen frequency of a wire depends on the tension, the wind load that deviates the tower and thus stretches or slackens the wire will be of significance, making the eigen frequency vary with wind speeds. An analytical model has been derived with the purpose of describing how the eigen frequency of the wire depends on the wind load. The FEM-based software SolidWorks Simulation is used to verify the model. The frequency span of the wire with current setup can be seen illustrated with green lines in figure 1. It is desirable to keep the frequency span above the 3P load for rated rotational speed, as seen illustrated with purple lines in figure 1. This can be done by changing the wire setup, that is length, size and pre-tension of the wire.

Using the analytical model, a diagram is assembled showing how to combine the wire size, inclination angle and pre-tension for an eigen frequency range over the 3P load for nominal rotational speed and for a certain effective spring force acting on the tower. This diagram, here called an EA-T diagram, may be used as a quick tool for comparing resonance free wire setups for a certain effect on the tower and a similar diagram can be used for other guy wired structures. The EA-T diagram for the 200kW VAWT can be seen in figure 2.

**Conclusion**

The derived analytical model describing the eigen frequency of the guy wire of a ground attached tower exposed to significant wind loads can be used for finding the approximate range of eigen frequencies for the three wires for a given wind speed. Using this expression, an EA-T diagram can be assembled showing how to combine nominal axial stiffness of the wire with pre-tension and inclination angle for a wire with eigen frequency range above the 3P load for nominal rotational speed and for a certain effective spring force acting on the tower. The EA-T diagram constitutes a simple and quick tool for comparing different possible resonance free wire setups having the same effect on the tower.

As can be seen in figure 1 with blue and green lines the guy wire and tower eigen frequency is excited within the operational range by the 3P load somewhere around 17 rpm and 23 rpm respectively. Programming the operational routine for passing these rotational speeds would have a negative impact on the efficiency of the turbine and furthermore programming might be difficult since there will be an insecurity regarding the wire frequency since an unplanned decrease of the pre-tension will lower it. If using guy wires it would be preferable to keep them stiff to avoid resonance for all modes. This is done easiest by adding more tension to the wires making the pre-tension 34 tonnes as seen with purple lines in figure 1. However, if re-planning the entire guy wire installment using figure 2, a resonance free wire could for the same effective spring force k Ìƒ and by using the same wire only altering the inclination angle to 40Â°, be achieved with a much more moderate pre-tension of â‰¥17 tonnes.

If designing a new tower it is desirable to have a total resonance free operational range without using guy wires. This can be achieved by designing the freestanding tower to be soft (3P resonance below and 1P above the operational range) or stiff (3P and 1P above operational range).

**Learning objectives**

How the eigen frequency of a guy wire supporting a wind turbine is affected by the wind load.

How to find different options of resonance free wire setups with a certain effective spring force acting on the wind turbine tower.

**References**

[1] S. Eriksson, H. Bernhoff, and M. Leijon. Evaluation of different turbine concepts for wind power. Renewable and Sustainable Energy Reviews, 12(5): 1419â€“1434, 2008.

[2] F. Ottermo, H. Bernhoff. Resonances and Aerodynamic Damping of a Vertical Axis Wind Turbine. Wind Engineering, ISSN: 0309-524X, May 2012.

[3] F. Ottermo, H. Bernhoff. An upper size of vertical axis wind turbines. Wind Energy, DOI: 10.1002/we.1655, Accepted 25 June 2013.

[4] G.J.M. Darrieus. Turbine having its rotating shaft transverse to the flow of the current. US Patent 1.835.018, December 1931.

[5] J. Kjellin. Vertical Axis Wind Turbines: Electrical System and Experimental Results. Uppsala Universitet. ISBN 978-91-554-8496-5.

[6] E. MÃ¶llerstrÃ¶m, F. Ottermo, J. Hylander, H. Bernhoff. Eigen frequencies of a vertical axis wind turbine tower made of laminated wood and the effect upon attaching guy wires. Submitted to Wind Engineering in April 2013.

[7] S. Eriksson, H. Bernhoff. Generator-Damped Torsional Vibrations of a Vertical Axis Wind Turbine. Wind Engineering, 29(5): 449-461, 2005.

[8] T.G. Carne, Guy Cable Design and Damping for Vertical Axis Wind Turbines. Sandia national Laboratories energy report. SAND80-2669, 1981, Albuquerque, New Mexico

[9] T.G. Carne, Guy Cable and Foundation Design Techniques. Proceedings of the Vertical Axis Wind Turbine (VAWT) Design Technology Seimar for Industry. SAND80-0984, 1981, Albuquerque, New Mexico

[10] C. Jerry Wong, Michael D. Miller. Guidelines for Electrical Transmission Line Structural Loading, ASCE Manuals and Reports on Engineering Practice No. 74. ASCE. ISBN: 978-0-7844-1035-6.

[11] Hugh D. Young, Roger A. Freedman. Sears and Zemanskyâ€™s University physics: With modern physics. Pearson Addison Wesley. ISBN: 0321501314, 9780321501318.

[12] J. Kjellin, S. Eriksson, H. Bernhoff. Tip Speed ratio control of a 200 kW VAWT with synchronous generator and variable DC-voltage. Open access, energies, ISSN 1996-1073.

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