## 3 Active Power Tracking Dynamic Characteristics of Wind-Turbine Cluster  
From the control design perspective, the blade pitch controller, the generator speed/torque controller and the yaw controller are working in coordination to make sure the WTGS functional. However, from the energy transfer perspective, a WTGS usually receives active power orders from the active power controller in the wind-farm level, and then adjusts the inside controllers to make sure that the generator power output catches up with the active power demand. In such a situation, the inner controllers can be regarded as WTGS inherent modules. The dynamic characteristic of power tracking for a WTGS can be reduced to a one-order inertial link from the active power demand to the generator power output, for which the input is the WTGS active power set-point from the wind farm and the output is the generator power output. The WTGS power tracking dynamic characteristic can be expressed as follows:  
\[P_{i}^{out}=\frac{1}{T_{i}s+1}P_{i}^{set}, \tag{10}\]  
where \(i\) represents the \(i\)-th wind turbine. The inertia time constant \(T_{i}\) can be easily obtained through the black box system identification and least square fit as shown in Figure 2.  
Figure 1: Wind speed—available power lookup table.  
Furthermore, the WTGS active power tracking dynamic characteristic in Equation (10) can be rewritten as a state-space model:  
\[\left\{\begin{array}{l}\dot{x}_{i}=-\frac{1}{T_{i}}x_{i}+\frac{1}{T_{i}}u_{i},\\ y_{i}=x_{i},\end{array}\right. \tag{11}\]  
in which the input \(u_{i}\) is the \(i\)-th wind turbine active power set-point \(P_{i}^{set}\), the state \(x_{i}\) and the output \(y_{i}\) are the \(i\)-th wind turbine generator active power output \(P_{i}^{out}\) tracking the active power set-point, respectively.  
Under such a circumstance, during the active power distributing process in a wind-turbine cluster, the cluster active power output is the combination of each inner wind-turbine active power, which is affected by the active power tracking dynamic characteristic of each WTGS in the cluster. Assuming that there are a total of \(n\) WTGSs in the cluster, the cluster active power output can be expressed as follows:  
\[\left\{\begin{array}{l}\dot{x}=Ax+Bu,\\ y=Cx,\end{array}\right. \tag{12}\]  
in which the input variable is defined as the active power set-point for each WTGS with \(u=\left[P_{1}^{set},P_{2}^{set},\ldots,P_{n}^{set}\right]^{\prime}\). The state is the active power output for each WTGS with \(x=\left[P_{1}^{out},P_{2}^{out},\ldots,P_{n}^{out}\right]^{\prime}\), and the output \(y\) is the output active power \(P_{cluster}^{out}\) for the specific wind-turbine cluster. The state space model parameters are given as follows:  
\[A =\begin{bmatrix}-\frac{1}{T_{1}}&&&\\ &-\frac{1}{T_{2}}&&\\ &&\ddots&\\ &&&-\frac{1}{T_{n}}\end{bmatrix}_{n\times n},\] \[B =\begin{bmatrix}\frac{1}{T_{1}}&&\\ &\frac{1}{T_{2}}&&\\ &&\ddots&\\ &&&\frac{1}{T_{n}}\end{bmatrix}_{n\times n}, C=\left[1,1,\ldots,1\right]_{1\times n}.\]  
Clearly, Equation (12) provides a feasible way to model the wind-turbine cluster active power tracking as a dynamic system, and the modern control method can be adopted to control the wind-turbine cluster active power dynamic process and optimize the cluster turbines operations. The wind-turbine cluster level active power control strategy will be introduced in the next section. In  
Figure 2: Wind turbine power tracking dynamic characteristics.  
order to simplify the notations, one WTGS is marked as a _unit_ and a wind-turbine cluster is marked as a _cluster_ briefly in the following analysis, and all the _power_ used later refers to the active power if not specially marked.