## 3 Wind System Control at Variable Wind Speeds from 3 [m/s] up to \(V_{MAX}\) [m/s]
### Linearized System Analysis and Control  
Equation (3) is transformed to a more common shape under the assumption \(\beta=0\), as in  
\[\frac{dn}{dt}=\frac{1}{\overline{f}n}(P_{TV}(n,V)-P_{GE}(n,f)). \tag{14}\]  
Define \(n\) to be a state variable and further rewrite the state space model  
\[ \begin{array}{l}\frac{dn(t)}{dt}=g(n(t),f(t),V(t)),\\ y(t)=h(n(t),f(t),V(t))=n(t),\end{array}  \tag{15}\]  
where \(g(. )\) is the state equation and \(h(. )\) is the output equation. A small signal analysis around a constant operating point is next performed. Let this point be \(\mathcal{P}(V_{0},n_{0},f_{0})\). Using the first order linearization around \(\mathcal{P}\), we can write  
\[ \begin{array}{l}\frac{d\Delta n(t)}{dt}=\left.\
abla_{n}g\right|_{V_{0},n_{ 0},f_{0}}\Delta n+\left.\
abla_{f}g\right|_{V_{0},n_{0},f_{0}}\Delta f+\left. \
abla_{V}g\right|_{V_{0},n_{0},f_{0}}\Delta V,\\ \Delta y(t)=\left.\
abla_{n}h\right|_{V_{0},n_{0},f_{0}}\Delta n+\left.\
abla_ {f}h\right|_{V_{0},n_{0},f_{0}}\Delta f+\left.\
abla_{V}h\right|_{V_{0},n_{0}, f_{0}}\Delta V=\Delta n(t),\end{array}  \tag{16}\]  
where we denote with \(\
abla\) the Jacobian of the functions \(g,h\) with respect to their arguments at the operating point \(\mathcal{P}(V_{0},n_{0},f_{0})\) and \(\Delta\) means the small variation of the argument about the linearization value. Subsequently, the variation notation with \(\Delta\) will be discarded and we use only absolute notation \(f,V,n\). The variables' roles are as follows. \(f\) is the control input, \(V\) is the disturbance input, \(n\) is the control output. In the Laplace transform domain of the variable \"\(s\)\", the following transfer functions result: \(H_{f\to n}(s)=K_{f}/(1+sT)\) and \(H_{V\to n}(s)=K_{v}/\left(1+sT\right)\). Remark 5: _Linearization around several points \(\mathcal{P}(V_{0},\ n_{0},\ f_{0})\) reveals crucial observations that have critical impact upon the control design. We classify two main cases: Case 1. When linearization is performed at the MPP by imposing the operating point constraint \(n_{0}=3.8269f_{0}\), we obtain that \(K_{f}>0,K_{v}<0\) and \(T<0\). Case 2. When the linearization is performed at arbitrary point \(\mathcal{P}(V_{0},n_{0},f_{0})\), we observe the following. (a) When \(n_{0}>3.8269f_{0}\) (operation to the right of the MPPT on the \(P_{TV}\) curve), we have \(K_{f}<0,K_{v}>0\) and \(T>0\), making for an open-loop stable transfer function; (b) when \(n_{0}<3.8269f_{0}\) (operation to the left of the MPPT on the \(P_{TV}\) curve), we have \(K_{f}>0,K_{v}<0\) and \(T<0\), making for an open-loop unstable transfer function._  
With the nonlinear process as the one in (12), there are several tradeoffs to be accounted for, when linear control is to be used: robustness under the time varying process nature, stability and performance. When operating at variable wind speeds, the design choices become more stringent. For a reasonable ramp-like variation of the wind speed value \(V(t)\), a double integrator should be included in the controller (according to the internal model principle); however, this reduces the stability margin. Two options are considered: Option (1) The feedback controller is a proportional-integral (PI) controller with the transfer function \(C(s)=(p_{1}s+p_{0})/s\). With the closed-loop transfer function from MAS setpoint, the controlled MAS is \((K_{f}p_{1}s+K_{f}p_{0})/(Ts^{2}+(1+K_{f}p_{1})s+K_{f}\ p_{0})\) with the characteristic polynomial of the closed loop being \(Ts^{2}+(1+K_{f}p_{1})s+K_{f}\ p_{0}\), which allows for flexible and robust design. However, there will be nonzero steady state trackingerror with respect to ramp-like disturbance \(V(t)\) as well as for ramp-like MAS reference input imposed by \(V(t)\) via linear dependence, based on the MPP principle. For both Case 1 and Case 2b in Remark 5, there exist negative values of \(p_{1}<0,p_{0}<0\) that will make the closed loop stable. For Case 2a in Remark 5, again, a suitable combination of \(p_{1},p_{0}\) can be found to stabilize the closed loop.