# Coordinated Control of Virtual Power Plants to Improve Power System Short-Term Dynamics
## 4 Case Study
### Monte Carlo Analysis  
This section discusses the robustness of the proposed control approach concerning the perturbations such as a three-phase fault at bus 7 occurring at \(t=1\) s and cleared after 100 ms. The stochastic perturbations are the power variations of wind and solar power plants with respect to the forecast of wind speed/solar irradiance, and 15% of the loads in the whole system. For each scenario, 500 Monte Carlo time domain simulations are solved. Figure 6 shows the frequency trajectories for the system that VPP without frequency control on DERs and without ESS. The trajectories of the frequencies for Modes 1 to 3 are shown in Figure 7, whereas the trajectories of the frequency for Modes 4 to 6 with three different transfer functions \(H(s)\) are shown in Figure 8. The considered transfer functions are the most commonly used controllers such as proportional gain, PI controller and lead-lag controller. For all controllers, the parameters the lead to the best dynamic performance are selected through a trial-and-error approach. For simplicity, and since the size of all DERs and the ESS is similar, \(K_{i}=1\), \(\forall i=1,\ldots,n\). Finally, Table 1 shows the mean frequency, \(\mu_{\text{COI}}\), as well as the standard deviation of the frequency \(\sigma_{\text{COI}}\), calculated at \(t=50\) s, for the six control modes and the three transfer functions \(H(s)\) considered in this case study. The comparison of Figures 6-8 indicates that the VPP without DER frequency control and without ESSs leads to the largest frequency deviation in the system. Please note that the results shown in Figure 6 can be also viewed as the worst-case scenario representing the complete outage of the communication network in the VPP. The comparison of the trajectories shown in Figures 7 and 8 as well as of the results given in Table 1 indicates that the VPP control modes that include the proposed coordinated approach control (Modes 4-6) have an overall better performance than the strategies with no co  
Figure 5: Modified WSCC 9-bus, 3-machine system with a VPP. It is interesting to note that for Mode 4 (see Figures 8,d,g,), the standard deviation slightly increases for \(10<t<30\) s because the ESS is running at its maximum output that loses its capability to regulate the frequency. Coordinating the DERs, i.e., Mode 5, to provide extra frequency support can help to solve this problem as shown in Figures 8,e,h. However, this leads to a larger frequency deviation during the initial transients (\(0<t<10\) s). Finally, Mode 6, where a timer is used to delay the action of DERs, shows the best dynamic performance (see Figures 8,f,i). The results shown in Figure 8 are obtained with the best set of parameters for each controller, namely proportional gain, lead-lag controller, and PI controller. For the PI controller, a small dead-band on the frequency deviation \(\Delta\omega_{\text{VOC}}\). Although all controllers perform well, they have different performance depending on the mode. The PI controller performs best if both ESS and DERs regulate the frequency. However, if only the ESS is