“…The figure shows that the 𝜔 m1 and 𝛿 4 waveforms, which are oscillating at the frequency related to the mode 𝜆 1,2 , settle in a much shorter time under the decentralized optimal controller. Figure 15 is given to show the 𝜔 m1 waveform under the same fault at a higher fictitious clearing time of 5 s. It is evident from the figure that the decentralized optimal controller can significantly reduce the FIGURE 11 Decentralized optimal controller performance after applying a 20% step increase to MMC1 reference AC-side active power at t = 10 s: (yellow) with decentralized optimal controller only, (blue) with MMC1 droop only, (red) with both decentralized optimal controller and MMC1 droop FIGURE 12 Decentralized optimal controller performance after applying a 20% step increase to MMC1 reference zero-sequence energy sum at t = 10 s: (yellow) with decentralized optimal controller only, (blue) with MMC1 droop only, (red) with both decentralized optimal controller and MMC1 droop FIGURE 13 Decentralized optimal controller performance under the worst-case perturbation scenario, x 0,worst , applied at t = 10 s FIGURE 14 Decentralized optimal controller performance under the three-phase short circuit fault at bus B2 applied at t = 10 s and cleared after 500 ms: (yellow) with decentralized optimal controller only, (blue) with PSS1 only, (red) with both decentralized optimal controller and and (green) without both decentralized optimal controller and PSS1 𝜔 m1 fluctuations after the fault and improve the grid stability compared with the PSS1.…”