## 1. Introduction

The demand for energy world wide is increasing every day. And in these "green times" renewable energy is a hot topic all over the world. Wind energy is currently one of the most popular energy sectors. The growth in the wind power industry has been tremendous over the last decade, its been increasing every year and it is nowadays one of the most promising sources for renewable energy. Since the early 1990s wind power has enjoyed a renewed interest, particularly in the European Union where the annual growth rate is about 20%. It is also a growing interest in offshore wind turbines, either bottom fixed or floating. Offshore wind is higher and less turbulent than the conditions we find onshore. In order to sustain this growth in interest and industry, wind turbine performance must continue to be improved. The wind turbines are getting bigger and bigger which in turn leads to larger torques and loads on critical parts of the structure. This calls for a multi-objective control approach, which means we want to achieve several control objectives at the same time. E.g. maximize the power output while mitigating any unwanted oscillations in critical parts of the wind turbine structure. One of the major reasons the wind turbine is a challenging task to control is due to the nonlinearity in the relationship between turning wind into power. The power extracted from the wind is proportional to the cube of the wind speed.

A wind turbines power production capability is often presented in relation to wind speed, as shown in Fig 1. From the figure we see that the power capability vs. wind speed is divided into four regions of operation. Region I is the start up phase. As the wind accelerates beyond the cut-in speed, we enter region II. A common control strategy in this region is to keep the pitch angle constant while controlling the generator torque. At the point where the wind speed is higher than the rated wind speed of the turbine (rated speed), we enter region III. In this region the torque is kept constant and the controlling parameter is the pitching angle. This is the region we are concerned with in this paper, i.e. the above rated wind speed conditions. The last region is the shutting-down phase (cut-out).

The expression for power produced by the wind is given by ([1]):

The dimensionless tip-speed ratio (TSR)

where

where

### 1.1. Modeling and control of wind turbines

#### 1.1.1. Modeling

Whenever we are dealing with control of a wind turbine generating system (WTGS), the turbine becomes a critical part of the discussion. There are many ways to model it, this can be done by simple one mass models ([2]-[3]) or multiple mass models ([4]). Several advanced wind turbine simulation softwares have emerged during the last decade. HAWC2 ([5]), Cp-Lambda ([6]) and FAST ([7]) are a few examples. They are developed at RISØ in Denmark, POLI-Wind in Italy and NREL in the US, respectively. In these codes the turbine and structure are considered as complex flexible mechanisms, and uses the finite-element-method (FEM) multibody approach. An aero-servo-elastic model is introduced, which consists of aerodynamic forces from the wind, the servo dynamics from the different actuators and the elasticity in the different joints and the structure. Both FAST and HAWC2 can simulate offshore and onshore cases while Cp-Lambda is limited to the onshore case.

#### 1.1.2. Control

Recently, linear controllers have been extensively used for power regulation through the control of blade pitch angle in wind turbine systems e.g. [8]-[15]. However, the performance of these linear controllers are limited by the highly nonlinear characteristics of wind turbines. Advanced control is one research area where such improvement can be achieved.

### 1.2. Outline of chapter

The paper starts with describing the wind turbine model in section 2. Section 3 deals with control design of the wind turbine system. The system is formulated on a generalized form and the LMI constraints for

## 2. Wind turbine model

The floating wind turbine model used for the simulation is a 5MW turbine with three blades, which is an upscaled version of Statoils 2.3MW Hywind turbine situated off the Norwegian west coast. The turbine is specified by the National Renewable Energy Laboratory (NREL), more information about the turbine specifications can be found in ([16]). Table 1 shows some of the basic facts of the turbine.

The simulation scenario is for above rated wind speed conditions, this means the turbine is operating in region III (see Fig. 1). In this region the major objectives are to maintain the turbines stability, calculate the collective pitch angle in order to prevent large oscillations in the drive train and in the tower and try to keep the rotor and generator at their rated speeds. If we can achieve this, then we keep the power output smooth. The model is obtained from the wind turbine simulation software FAST (Fatigue, Aerodynamics, Structures, and Turbulence). More information about the software can be found in the user’s manual ([7]). FAST has two different forms of operation or analysis modes (Fig. 2). The first analysis mode is time-marching of the nonlinear equations of motion - which is, simulation. During simulation, wind turbine aerodynamic and structural response to wind-inflow conditions are determined in time. Outputs of simulation include time-series data on the aerodynamic loads as well as loads and deflections of the structural parts of the wind turbine. The second form of analysis provided in FAST is linearization. FAST has the capability of extracting linearized representations of the complete nonlinear aeroelastic wind turbine model.

Three degrees of freedom are chosen, and they are; rotor, generator and tower dynamics. If desirable, the model can easily be expanded to include more degrees of freedom. The form of the linear model obtained from the software is stated in (5).

The bar over the matrices in the state space system tells us that these are the average matrices for the wind turbine. FAST calculates the matrices for a state space system at each desired azimuth angle and gives out the average values. The state space system is strictly proper, i.e.

The nonlinear model is linearized at a wind speed of

## 3. Control design

The control purpose of the

### 3.1. System representation

Fig. 3 shows the output feedback control scheme.

where

where

The closed loop system is given in (8) with the states

where

The closed loop transfer functions from

### 3.2. H 2 Control

The closed loop

Since

it is verified that

### 3.3. H ∝ Control

The closed loop

In order to formulate the

The first crucial steps in obtaining the BRL are shown next. Firstly we remove the roots by squaring both sides of the inequality sign. Secondly we collect everything in one integral expression and lastly we do the trick where we add and subtract the Lyapunov function to the inequality. We can do this because we know that

Now we insert both the expression for the signal of interest

### 3.4. Change of variables

For the multi-objective case we want to create a feedback controller

where

The new matrices

where

as can be inferred from the identity

Now we are ready to convert the nonlinear matrix inequalities into LMIs. This is done by performing congruence transformation with

where

In order for (28) to be true the following relationship must hold

This relationship can be solved by utilizing the singular value decomposition (SVD). We know that

### 3.5. LMI region

An LMI region is any convex subset

for fixed real matrices

Also, here we need to include the change of variables. This is done in (36) where

As an example we define the desired region

where the closed loop eigenvalues may be placed.

From this we can find that the matrices

All the constraints in (24) and the constraint in (36) are subjected to the minimization of the objective function given in (20). They need to be solved in terms of

Finally, the controller matrices can be found by the following relationship

From the aforementioned expressions we are able to solve the mixed

## 4. Simulation

All calculations and simulations are carried out in MatLab/Simulink ([19]) interfaced with YALMIP ([20]). The solver which is used for the LMI calculation is SeDuMi ([21]). FAST comes with a Simulink template which can be changed how ever the user may see fit. As described earlier the simulation scenario is for the above rated wind speed situation. We want to mitigate oscillations in the drive train and dampen tower movement while maintaining the rotors rated rotational speed. Satisfactory simulation results were found with

The norms obtained from the optimization are shown in (45) and the closed loop poles are shown in Fig. (5).

In order to find the performance measures we argue that there are no oscillations in the drive train if the position of the generator and the rotor are the same. In other words we must try to keep

As a consequence of having better performance the pitching activity has heavily increased, see Fig. 10. The pitching activity lies around

The controller is tested on the fully nonlinear system, where the only degrees of freedom which are left out are yaw and translational surge. The state space system of the controller is shown in the appendix. The plots included in this chapter are the same ones that were used as feedback to the controller. That is, rotor rotational speed (Fig. 7), generator rotational speed (Fig. 8) and tower fore-aft displacement (Fig. 9). The simulation results show a comparison between our simulations and simulations done with FAST’s baseline controller. The baseline controller is a gain scheduled PI controller and is indicated on the plots as the blue line. The controller proposed in this chapter is indicated with the red line.

## 5. Conclusion

In this chapter we have introduced a nonlinear model of an offshore floating wind turbine using the commercial wind turbine software FAST. By the use of its embedded routines a linear model is extracted. The linear model used in this chapter is built up with relatively few degrees of freedom. More degrees of freedom can easily be added to the model, depending on the control objectives. In our approach this relatively simple model serves its purpose in testing advanced control routines on an offshore floating wind turbine system. On the basis of this linear model a mixed