# Adaptive Pitch Controller of a Large-Scale Wind Turbine Using Multi-Objective Optimization
## 3 Control Methodology
### Description of the Control Blocks  
#### 3.2.1 Gain Scheduling PI Control  
Gain scheduling control consists of pre-setting a controller for various operating points, and subsequently updating its parameters based on these previous designs according to the operating point where the plant is working. There are different gain scheduling schemes in the literature. One of the possible strategies is equivalent to several controllers working in parallel where only the output of one of them is chosen depending on the operating conditions [61]. A very simple scheme of gain scheduling PI control is shown in Figure 4, where there are six previously designed controllers running in parallel. The most appropriate one is selected based on the disturbance d, whereas the other five controllers are configured in integral tracking mode. The other five controllers not selected are configured in the integral tracking mode. Using a similar scheme in this work, an adaptive PI control is designed. It is composed of six PI controllers tuned for each linear model obtained in the previous section. The value of the wind speed determines the PI controller that must be selected.  
In order to assure a bumpless transfer between controllers without sudden changes in the control signal, all the PI controllers work in the integral tracking mode updating their integral action so that the output of the unselected controllers matches with that of the active controller. It is similar to the mechanism used to achieve anti-windup and bumpless transfer between automatic/manual modes in PID controllers [62; 63]. The equation of a non-interactive PI controller is as follows:  
\[\text{u}(\text{s})=\text{K}_{\text{p}}\bigg{(}\text{r}(\text{s})-\text{y}( \text{s})+\frac{\text{r}(\text{s})-\text{y}(\text{s})}{\text{T}_{\text{i}} \text{s}}\bigg{)} \tag{17}\]  
where u(s) is the control signal, r(s) is the reference, y(s) is the controlled signal, Kp is the proportional gain, and T\({}_{\text{i}}\) is the integral time constant. This continuous control law is  
Figure 4: Adaptive proportional–integral (PI) control scheme.  
discretized using the Tustin approximation. The proportional P(k) and integral I(k) actions of this implementation in the k-th iteration are detailed below:  
\[\begin{array}{c}\mathrm{P(k)=K_{p}\;e(k)}\\ \mathrm{I(k)=I(k-1)+K_{pi}\;(e(k)+e(k-1))}\end{array} \tag{18}\]  
where e(k) is the error signal r(k) \(-\) y(k). The constant K\({}_{pi}\) is given by:  
\[\begin{array}{c}\mathrm{K_{pi}=K_{p}\frac{h}{2\cdot T_{i}}}\end{array} \tag{19}\]  
where h is the sample time (with a value of 0.02 s in this work). The control signal is the sum of these two actions P(k) and I(k). Additionally, as shown in Figure 4, there can be an extra control action from the input f. In this work, this input is used to add the feedforward action FF(k) to compensate for the wind speed changes, and the action FA(k) to reduce the tower f-a displacements. Therefore, the control signal is given by u(k) = P(k) + I(k) + FF(k) + FA(k).  
To cope with the input constraints of the process, an anti-windup mechanism is implemented using an actuator constraint model that considers the process input saturations. When the final control signal u(k) is out of its limits, this mechanism updates the integral term I(k), constraining u(k) to the exceeded limit.  
The mechanism is shown in (20). In addition, to ensure the tracking of the rest of the controllers to the final control signal u(k), it must be considered that the variables P(k) and I(k) are vector signals of six elements, one for each controller of the adaptive control. As a result, the signal I(k) is updated in the else statement in such a way that the rest of the non-active controllers follow the final signal u(k):  
\[\begin{array}{c}\mathrm{u(k)=P(k)+I(k)+FF(k)+FA(k)}\\ \mathrm{if\;u(k)>u_{max}}\\ \mathrm{I(k)=u_{max}-P(k)-FF(k)-FA(k)}\\ \mathrm{else\;ifu(k)<u_{min}}\\ \mathrm{I(k)=u_{min}-P(k)-FF(k)-FA(k)}\\ \mathrm{else}\\ \mathrm{I(k)=u(k)-P(k)-FF(k)-FA(k)}\\ \mathrm{end} \tag{20}\]