## 4 Active Power Model Predictive Control of Wind-Turbine Cluster
### Initial Test  
Based on the above discussion, an MPC strategy is constructed for the active power distribution of the wind-turbine cluster. As follows, the proposed strategy is tested preliminarily under a four-unit cluster to verify the feasibility. During simulations, wind turbines are differently adjusted to represent the dynamic characteristic difference. Then, the inertia time constants of simulated wind turbines are identified though the _ident_ system identification toolbox in _Matlab_. The inertia time constant \(T\) for four simulated wind turbines are identified as \(T_{1}=0.09\) s, \(T_{2}=0.125\) s, \(T_{3}=0.20\) s and \(T_{4}=0.3333\) s, respectively. A conventional active power control strategy is used to compare the effectiveness, which can be expressed as:  
\[P_{i}^{PD}=\frac{P_{cluster}^{set}}{P_{cluster}^{wi}}P_{i}^{wi}. \tag{18}\]  
The conventional strategy is also referred to as the proportional distribution (PD) method, which calculates the output coefficient for the wind farm and each unit to guarantee that the controlled power output meets the power demand. The compared output figures are shown as follows.  
In Figure 3, the black curve is the cluster power set-point, the blue curve represents the cluster power output under the PD strategy, while the red curve is with respect to the power output under the MPC strategy. In Figure 4, the black curves represent the available power, while the blue curves represent the power output for each unit respectively, which are corresponding to constraint (17). Clearly, the cluster power output through MPC is much more stable than the conventional PD strategy, and the output power tracking performance has also been improved significantly.  
Although the power output under MPC fluctuates around at about \(60\,s\), but the amplitude is not dramatic and the output returns to normal within a short time period. However, the power set-point for each unit in Figure 4 shows a little deficiency in the current control strategy. The inertial time constant for each unit introduced in the previous paragraph shows that unit 1 has the best dynamic characteristic and fastest power set-point tracking ability, and unit 4 has the worst dynamic characteristic in contrast. Such characteristics can be clearly reflected in Figure 4. In particular, units 1 and 2 contribute most of their available capacity to support the cluster power output for tracking the power set-point. The power contributed by units 3 and 4 are not very much in the low-load phase, and increase their power outputs only when the cluster power set-point is large. This leads to a serious problem for the power regulation flexibility of the cluster, i.e., how to coordinate the small-inertia units and large-inertia ones for better power demand tracking with enough active power regulation flexibility left?  
In fact, for a sensible cluster power control strategy, small-inertia units should play a leading part when the cluster power demand changes rapidly to achieve fast tracking of cluster power set-point.  
When the quick tracking and adjustment is completed, some power output will be handed over from the small-inertia turbines so as to maintain the adequate power regulation capacity of the cluster, which should be filled by the less rapid units. In such a way, there could be enough margin left for the future cluster power set-point changes for quicker tracking and adjustment. To realize this idea, a simple but efficient strategy for MPC with time-varying constraints will be proposed in the next section.