# Fixed-Time Optimization of Perturbed Multi-Agent Systems under the Resource Constraints
## 4 Simulation  
This section considers a scenario of distributed cooperative power generation in a wind farm. The wind farm contains many wind turbines, and the centralized wind farm scheduling method has the problems of complex communication links and poor communication fault tolerance. The distributed generation distribution control method only requires the establishment of communication links among the WTGs to ensure that the WTGs form a fully connected topology, which is robust to topology switching. In this section, the effectiveness of the algorithm (16) is tested on the distributed cooperative power tracking of the Princess Amalia wind farm in the Netherlands, and the detailed configuration of which can be found in [20]. The wind farm contains 60 wind turbines with a rated power of 2 MW. Considering the wake effect, the power cost function of the wind turbines is affected by the power of the neighboring wind turbines upwind. For a given wind direction, the wake effect for the Princess Amalia wind farm is shown in Figure 1. Take turbine 24 as an example; its generation cost is a function of its own power and the power of turbines 15 and 16. Considering the decision variable \(x\) as the wind turbine power of the wind farm, the generation cost function of the \(i\) wind turbine is \(f_{i}=f_{i,i}(x_{i})+\sum_{j\in\mathcal{N}_{j}}f_{i,j}(x_{i},x_{j})\), where \(f_{i,i}(x_{i})\) is the own generation cost of the \(i\) wind turbine, \(f_{i,j}(x_{i},x_{j})\) is the generation cost of the \(i\)th wind turbine affected by the tail flow effect of the \(j\)th wind turbine. The wind turbine's own generation cost is a quadratic cost curve, i.e., \(f_{i,i}(x_{i})=c_{2,i}x_{i}^{2}+c_{1,i}x_{i}+c_{0,i}\). Based on the main characteristics of the wake effect, the generation cost under the influence of adjacent turbines is assumed to be \(f_{i,j}(x_{j})=k_{i,j}x_{i}x_{j},k_{i,j}\geq 0\). Note that the local generation cost \(f_{i}\) of the turbine under this assumption may be a non-convex function. The generation cost of turbine 24 is \(f_{24}=c_{2,24}x_{24}^{2}+c_{1,24}x_{24}+c_{0,24}+k_{24,15}x_{15}x_{24}+k_{24,1 6}x_{16}x_{24}\). The parameters of the wind turbine cost function are as follows: \(c_{2,i}\in[4.0,6.0]\), \(c_{1,i}\in[2.0,3.0]\), \(c_{0,i}\in[95.0,110.0]\), \(i=1,2,\cdots,60\), and the effect coefficient of the wake effect \(k_{i,j}=0.1\). In this setting, the total generation cost of the wind farm \(f(x)=\sum_{i=1}^{60}f_{i}\) is a strongly convex function. Additionally, consider the matching perturbation \(\omega_{i}(t)=0.5\sin(0.1 it)+W_{Gi}\) for the \(i\)th wind turbine, where \(W_{Gi}\) denotes zero-mean Gaussian noise with a variance of \(0.1\). Taking the initial value \(x_{i}(0)\in[0.8,1.8]\), the total power demand of the wind farm is \(D=78\) MW and is cut to \(68\) MW at \(t=2\) s and increased to \(83\) MW at \(t=4\) s. Note that the algorithm in this paper need not satisfy \(\sum_{i=1}^{60}x_{i}(0)=D\). Set the parameters of the algorithm (16) to \([\kappa_{0},\kappa_{1},\kappa_{2},\xi_{1},\xi_{2},\zeta_{1},\zeta_{2},\alpha, \mu,\
u]=[10,10,10,10,10,100,100,1.5,0.5,1.5]\), the fixed convergence time can be calculated as \(T_{3}=1.3856\) s.  
The convergence curve for each wind turbine is shown in Figure 2, and the total power variation curve of the wind farm is shown in Figure 3. It is shown that the global resource constraint \(\sum_{i=1}^{60}x_{i}=D\) is satisfied in \(T_{3}\) for \(t<2\). Moreover, the proposed algorithm maintains a quick response to the power demand variations at \(t=2\) s and \(t=4\) s. The total power generation cost of the wind farm is shown in Figure 4. Under different global resource demands, the total power generation cost of the wind farm converges to the global optimal solution as labeled in Figure 4. The convergence trajectories of sliding manifolds \(s_{i}\) are shown in Figure 5, where all the sliding manifolds are reached in fixed-time and the reaching time is shorter than the estimated one.