## 5 Time Varying Constraints  
To make sure the small time-scale units taking the leading part in the fast tracking of power set-points, a time-varying constraint is applied instead of (17), which can be expressed as:  
\[P_{i}^{lb}(t)\leq P_{i}^{set}\leq\min\left\{P_{i}^{ub}(t),P_{i}^{ext}\right\}, \tag{19}\]  
\[P_{i}^{lb}(t)=\begin{cases}\left[(1-a)-b\times\dfrac{t_{0}-t}{t_{0}}\right] \times P_{i}^{PD}&,&t<t_{0},\\ (1-a)\times P_{i}^{PD}&,&t\geq t_{0},\end{cases}\]  
Figure 4: Active power output of each wind turbine under step power demand, sub-figures a, b, c, d are the active power output for wind turbine 1, 2, 3, 4 respectively.  
Figure 3: Active power output of wind-turbine cluster under step power demand.  
\[P_{i}^{ub}(t)=\begin{cases}\left[(1+a)+b\times\dfrac{t_{0}-t}{t_{0}}\right]\times P _{i}^{PD}&,&t<t_{0},\\ (1+a)\times P_{i}^{PD}&,&t\geq t_{0},\end{cases}\]  
in which \(a\) and \(b\) are constants satisfying \(0\leq a+b\leq 1\), \(P_{i}^{PD}\) is the \(i\)-th unit power set-point under a conventional PD strategy, \(t\) is the time length since the cluster power set-point changes to current time step, and \(t_{0}\) is the constraints contraction time. Constants in (19) can be adjusted according to the practical requirements. Set \(P_{i}^{PD}\) constant as 1 and the time varying constraints can be shown as in Figure 5.  
Once the cluster set-point changes, the trigger time is reset to record the change time and stays constant in the next period. The difference between the current time and the trigger time is the calculate time \(t\). Once the cluster power set-point is received with step changes, the \(i\)-th turbine power is set within a constrained available interval \([(1-a-b)\times P_{i}^{PD},(1+a+b)\times P_{i}^{PD}]\) and gradually converges over time, after \(t_{0}\) seconds, the available interval converges to its smallest range that the upper bound is \(a\) higher than the proportional distribution method and the lower bound is \(a\) lower. The gap between the upper and lower boundaries is used to overcome the power fluctuations caused by inaccurate models and inaccurate wind speed. In other words, the fast time-scale units will take the leading part when the cluster power set-point changes to meet the new set-point, and the feasible power set-point of each unit will gradually reduce so that the fast time scale units will gradually hand over the excessive power to the less rapid units until the power output for each turbine restored to a minimum bound close to the proportional distribution after \(t_{0}\) seconds. In addition, \(a+b\) is set within 0 and 1 to ensure the lower bound for one unit power output is always above zero and all units participate in the cluster power production. In addition, due to the different inertia time constant between units, the cluster power output may fluctuate during the process when restraining the units' available interval, the equality constraint in (16) is changed with a small extra range as follows:  
\[(1-c)\times P_{cluster}^{set}\leq\sum_{i=1}^{n}P_{i}^{set}\leq(1+c)\times P_ {cluster}^{set}. \tag{20}\]  
Thus, a time-varying constrained MPC is established for the active power distribution of the wind-turbine cluster. In the following tests, constants in the time varying constraints are set as \(a=10\%,b=40\%,c=2\%\), and \(t_{0}\) is 30. Considering that the wake effect is included during the available power calculation, the proposed strategy can be used for a waked wind farm although the detailed wake effect is not modeled and explained. In other words, the proposed strategy doesn't take  
Figure 5: Time varying constraints.  
wake effect into consideration during controller design. Moreover, the proposed controller will be tested under several scenarios in the next section.