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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.

Aron Pujana-Arrese IK4-IKERLAN, Spain
Co-authors:
Aron Pujana-Arrese (1) F P Asier Diaz de Corcuera (1) Pablo Fernandez (1) Jose M. Ezquerra (1) Aitor Milo (1) Joseba Landaluze (1)
(1) IK4-IKERLAN, Arrasate-Mondragón , Spain

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Abstract

Analysis of two interpolation methods of H infinity controllers for robust wind turbine control

Introduction

This paper presents the design process of two collective pitch angle controllers based on different interpolation methods of three Linear Time Invariant (LTI) H infinity controllers to improve the generator speed regulation in the above rated power production zone. These interpolation methods develop a gain-scheduling of the controller’s state space matrices by polynomials approximations or solving a Linear Matrix Inequalities (LMI) system. These controllers are designed for the 5 MW wind turbine defined in the Upwind European project developed in GH Bladed. An exhaustive analysis is carried out to compare fatigue and some extreme load reduction with these controllers.

Approach

Over the last few years, the interpolation of LTI controllers is an important task in the control systems design for non-linear applications. In literature, the interpolation is commonly applied to low ordered LTI controllers and it could be divided into two approaches [1]: Gain Scheduling approach and Linear Paramater Varying (LPV) approach. The first one uses the family of linear models extracted from the non-linear model to design LTI controllers in different operating points to finally interpolate the controllers. On the other hand, the LPV approach needs LPV models of the plant to design LPV controls for the specified model. The presented work in this paper is focused in the first approach and it analyses two different methods to interpolate LTI controllers. The methods presented to develop the interpolation of LTI controllers are based on gain-scheduling of the state space matrices LTI controllers by polynomials approximations or solving Linear Matrix Inequalities (LMI) systems. In the first one, the stability is not guaranteed in the control design and it has to be demonstrated with simulations in the non-linear closed loop system. However, the stability in the second method is guaranteed because it is considered in the formulation of the LMI system.

The adaptability of the presented controllers, which varies their behaviour according to the different operating points in wind turbines non-linear systems, improves the closed loop performance compared to Linear Time Invariant control techniques. Furthermore, the presented controllers are designed by using high ordered (55th order) family of linear plants extracted from the high reliability wind turbine non-linear model developed in GH Bladed. Inherently, the high order of the plants involves high order of controllers when their design is based on the H infinity norm reduction, so the interpolation of high ordered controllers (9th order) involves an important contribution in the LTI control interpolation field due to the mathematical calculation convergence problems.


Main body of abstract

Fig. 1 shows a general view of the control scenario. The baseline collective pitch H infinity control used to regulate the generator speed in this paper, which is presented in [3], is called CAR and it is designed to work in all operating points in the above rated zone. Three collective pitch H infinity controllers are designed to optimize the closed loop performance in three operating points in the above rated zone: C13, C19 and C25 for winds near to 13 m/s, 19 m/s and 25 m/s respectively. Two methods to interpolate the state-space represented C13, C19 and C25 controllers are proposed without losing the stability and performance in all trajectories of the above rated zone. The objective of this interpolation is to improve the output sensitivity function of this control loop increasing its bandwidth and reducing its peak in all operating points of the controlled zone.


Fig 1. Control scenarios

The main control structure included in the ‘Upwind’ wind turbine control system for the above rated zone is shown in Fig. 2. The C1 baseline control strategy consists of two multi input single output (MISO) H infinity controllers and it is explained in [3]. The first MISO controller is a multi-objective collective pitch angle controller (MISO H infinity PA) which keeps the generator speed at the nominal value and mitigates the wind effect in the tower fore-aft first mode. The second one (MISO H infinity GT) is another multi-objective generator torque controller which mitigates the wind effect in the drive train mode and also reduces the wind effect in the tower side to side first mode. The generator speed regulator channel in the MISO H infinity PA controller is deactivated when the C2 and C3 strategies are activated, and the collective pitch signal to regulate the generator speed is calculated from the controller block (INTPA) based on the interpolation of three H infinity controllers. The two controllers based on two interpolation methods are included in the INTPA block and the scheduling parameter p is obtained from the pitch control signal after being adapted in the Scheduling Parameter Calculator (SPC) block. The interpolation methods used in C2 and C3 control strategies are:
C2: The interpolation of the LTI controllers is based on the gain-scheduling of the coefficients of the LTI controller’s state-space matrices. This method is commonly used in the literature [4], however, the stability in all trajectories of the above rated zone is not guaranteed in the control design. The controller is Linear Fractional Transformation (LFT) represented.
C3: This is a more sophisticated interpolation method. The controller included in the INT PA block is based on solving a Linear Matrix Inequalities (LMI) system [5] to represent it in a new gain scheduled scenario, where the LTI controllers are interpolated according to a gain matrix and guaranteeing the stability in all points of the parameter trajectory considered in the design process.

Fig 2. Control structure for the above rated zone.

Regarding to the simulation results in GH Bladed, the regulation of the generator speed with the C2 and C3 strategies is improved compared to the C1 in the above rated zone. Fig. 3 shows the generator speed signal for a power production wind with a mean speed of 19 m/s with the three control strategies. Fig. 4 shows the statistical analysis, where STD is the standard deviation, of the generator speed signal with the three control strategies C1, C2 and C3 for odd production winds speeds with means from 3 m/s to 25 m/s. This statistical analysis confirms the better regulation of the generator speed in the above rated zone (winds from 11 m/s to 25 m/s) with the C2 and C3 control strategies. From the load analysis point of view, the fatigue loads are not very affected by the improving of the output sensitivity function in the generator speed regulation control loop due to gain-scheduling techniques. On the other hand, the extreme loads in different wind conditions are improved due to the faster response of the pitch angle with C2 and C3.


Fig 3. Generator speed for a power production wind ( mean speed of 19 m/s).


Fig 4. Statistical analysis of the generator speed.


Conclusion

This paper presents two methods to interpolate robust H infinity controllers applied to the regulation of the generator speed in the above rated power production zone of wind turbines. The use of interpolated H infinity controllers improves the response of the collective pitch control loop in this control zone due to the controller adaptability to the wind turbine operating points. The two methods presented to develop the interpolation are based on the gain-scheduling of the controller’s state space matrices by polynomials approximations or solving Linear Matrix Inequalities (LMI) systems. In the first one, the stability is not guaranteed in the control design and it has to be demonstrated with simulations in the non-linear closed loop. However, the stability in the second method is guaranteed because it is considered in the formulation of the LMI system.

The designed controllers with these two methods are discretized with a sample time of 0.01 seconds and they are validated in MATLAB/Simulink before being included in the wind turbine external control code. This control code works with the wind turbine non-linear model developed in GH Bladed software package.

The generator speed regulation is improved with the gain-scheduled controllers in the above rated zone and, inherently, the electric power production is regulated with more accuracy near the nominal value of 5 MW.

In terms of load mitigation, the use of these gain-scheduled controllers considerably reduces the loads in different components of the wind turbine, mainly in the extreme load cases where the fast response of the pitch angle is crucial.



Learning objectives
The LTI controller’s interpolation via LMIs guarantees the stability in the control design process and the interpolation process is considered like a global task of the control system instead of a particular task of each state space matrices coefficient.

The proposed interpolation methods can be perfectly applied to high ordered robust controllers designed from a wind turbine high reliability model developed in GH Bladed.



References
[1] Rugh WJ and Shamma JS. Research on gain scheduling. Automatica, 2000; vol. 36, no. 10; pp. 1401-1425.

[2] Jonkman, J., Butterfield, S., Musial, W. and G.Scott, 2009. Definition of a 5 MW Reference Wind Turbine for Offshore System Development. NREL.

[3] Díaz de Corcuera A, Pujana-Arrese A, Ezquerra J M, Segurola E and Landaluze J 2012 H infinity Based Control for Load Mitigation in Wind Turbines Energies vol 5 pp 938-967

[4] Díaz de Corcuera A, Pujana-Arrese A, Ezquerra J M, Segurola E and Landaluze J 2012 LPV Model of Wind Turbines from GH Bladed’s Linear Models. 26th European Conference on Modelling and Simulation, ECMS2012, May 29th-June 1st, Koblenz (Germany).

[5] Bianchi FD and Sanchez Peña RS. Interpolation for gain-scheduled control with guarantees. Automatica,2011; vol. 47, pp 239-243.