Delegates are invited to meet and discuss with the poster presenters in this topic directly after the session 'Advanced control concepts' taking place on Wednesday, 12 March 2014 at 11:15-12:45. The meet-the-authors will take place in the poster area.

**David Thompson**University of Strathclyde, United Kingdom

Co-authors:

David Thompson (1)

^{F}

^{P}William Leithead (1)

(1) University of Strathclyde, Glasgow, United Kingdom

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## Presenter's biography

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

David Thompson is a final year PhD student conducting research in wind turbine blade loads and up-scaling at the University of Strathclyde as a part of the Supergen Wind Energy Technologies Consortium. He completed his MEng in Mechanical Engineering at Strathclyde in 2009.

## Abstract

**A gain-scheduled controller for improved power regulation of very large wind turbines
****Introduction**

Variable speed pitch regulated horizontal axis wind turbines exhibit highly non-linear characteristics at high wind speeds presenting a challenge for power regulation control [1]. As turbine size increases and natural frequencies of the blades and tower decrease, these nonlinearities have an impact on performance. A method for improving controller performance for very large turbines is presented. The performance of the improved controller is assessed.

**Approach**

As wind turbines increase in size their blade and tower natural frequencies reduce. Wind speed dependant non-linear effects are present in the dynamics at low frequencies particularly for very large wind turbines. By directly counteracting these nonlinearities, improved control is possible.

A low frequency pole exists in the plant dynamics in above rated wind speeds which varies with the wind speed [2]. By designing a controller with a zero which also varies with wind speed, the effect of this low frequency pole can be neutralised. However, nonlinear controllers with varying poles rather than zeros generally demonstrate better performance. Hence, instead of implementing a controller with a varying zero, a controller with several varying poles is preferred. These poles vary as a function of wind speed. The increase in order of the controller is required to ensure that a close approximation to the zero as its frequency changes with wind speed.

A continuous time model of the controller is constructed in MATLAB/Simulink and a discrete time model in C for implementation on a Bladed simulation. Prior to coding the gain-scheduled controller in C it is required that the model be restructured to remove the algebraic loops. This requires that the wind speed varying functions are limited to being linear. The Supergen 5MW exemplar wind Turbine for is used for all models and simulations.

Both continuous and discrete-time implementations are presented in this paper together with analysis and simulation results. Improved controller performance in above-rated wind speeds and linearized dynamics at low frequencies is demonstrated.

**Main body of abstract**

A controller with a varying zero is used to account for non-linearities present due to a pole which varies with wind speed in the frequency range below approximately 0.1rad/s. To counteract this pole, a similar zero can be included in the controller as in (1) where α(ν) is a function of wind speed ν.

(1)

An equivalent transfer function with varying poles is shown in (2). This is an approximation of (1) but the accuracy of representation can be improved by increasing the number of poles. A fourth order transfer function is sufficiently accurate here.

(2)

αν1, αν2, αν3 and αν4 are the values of α(ν) at wind speeds ν1, ν2, ν3 and ν4. ψ1(ν), ψ2(ν) and ψ3(ν) are functions which cancel corresponding values of αν at a given wind speed. k will normally be chosen so that (s+k) is equal to (s+αν1) to reduce the order by one. For example, at wind speed ν2, the values of ψn(v)(n=1,2,3) is such that (s+αν1), (s+αν3) and (s+αν4) are cancelled.

The correct implementation of a controller with varying poles [3] is shown in figure 1; the equivalent transfer function is that of (3). In this implementation, an is a function of the output, y.

Figure 1 – Continuous-time implementation of gain-scheduled controller

(3)

In above rated wind speeds, the output, y, is linearly proportional to pitch demand which is indicative of wind speed and so is an appropriate value on which to base scheduled gains.

The controller is applied to the Supergen 5MW exemplar turbine with the following values for the parameters, where ν is wind speed and ψn is a parameter of (2):

Table 1 – Controller parameters for a selection of wind speeds

From this table, functions for the varying parameters an(y) in (3) are derived.

The above controller is modelled in MATLAB/Simulink and Bladed. The controller for use in Bladed is a discrete version of the above controller written in C.

For implementation in discrete time, the structure of the controller has to be rearranged to remove any algebraic loops. The structure of the discrete controller and the equivalent difference equations are shown in figure 2 and (4).

Figure 2 – Discrete time implementation

(4)

a(y), b(y) and c(y) are linear functions of the controller output, y. T is the time step in seconds. It is found that, for functions a(y) and c(y), higher order functions offered no performance improvements over a linear approximation. For b(y), however, a single linear function is not sufficient. It is therefore necessary to have two functions, ba(y) and bb(y), and switch between them. These functions are switched at an appropriate value of y to maintain the accuracy of representation.

For a simulation with mean wind speed of 16m/s and turbulence intensity of 10%, a reduction in the standard deviation of generator speed of 51.5% from 3.12 to 1.51 is achieved. A plot of generator speed vs. time is shown in figure 3 for the standard controller and the gain-scheduled controller.

Figure 3 – Generator speed vs time for simulation at 16m/s mean wind speed and 10% turbulence intensity

The effect of neutralising the low frequency pole can be seen in figure 4 and figure 5. In figure 4, the gain-scheduled controller is not implemented and the transmittances are different at low frequency for different wind speeds. In figure 5 the gain response for different wind speeds at frequencies blow 0.1rad/s are much more closely matched.

Figure 4 – Bode plot of pitch demand to generator speed for baseline controller

Figure 5 – Bode plot of pitch demand to generator speed for gain-scheduled controller

**Conclusion**

Wind turbines have steadily grown in size over recent years and as it is increasingly desirable for turbines to be developed for offshore use, this trend is expected to continue. As wind turbines increase in size, the lower natural frequencies of blades and tower can become limiting factors on the performance of the controller. These limits must be addressed if the trend of growing turbine blade length, tower height and power output is to continue.

By adjusting the poles of the controller as the wind speed varies in above rated conditions, the non-linear effects in the low frequency range can be counteracted. A representation of the controller that meets this requirement to counteract this effect and enables the algebraic loop inherent in discrete control to be removed is presented. The representation requires a zero, whose frequency varies with wind speed, to be replaced by some poles, whose frequencies also vary. For the example presented here, the controller has 3varying poles. Their frequencies are chosen so that the transmittance is a very similar to that of the single varying zero.

When applied to a 5MW turbine model, this gain-scheduled controller provides a significant improvement in controller performance as assessed through a reduction of almost 50% in the fluctuations in generator speed at high wind speed. Additionally, Bode plots of the transmittance from pitch demand to generator speed are now similar at low frequencies for all wind speeds.

The design of the controller presented can be applied to turbines of any size .Hence, they are suitable for ever larger offshore machines.

**Learning objectives**

A controller is presented which improves control performance and allows for larger wind turbines by accounting for the effects of non linearities at low frequencies.

**References**

[1] S. Dominguez and W. E. Leithead, “Size related performance limitations on wind turbine control performance”, Proceedings of the European Wind Energy Conference, 2006.

[2] A. Stock, A. Chatzopoulos, H. Yi and D. Thompson, “Improved Holistic Controllers for Wind Turbines”, Department of Electronic and Electrical Engineering, University of Strathclyde, SUPERGEN Wind Energy Technologies Consortium Report 2012.

[3] D.J. Leith and W. E. Leithead, “Appropriate realization of gain-scheduled controllers with application to wind turbine regulation”, International Journal of Control, vol. 65 pp. 223-248, 1996

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