Fundamental Limitations on the Control of Lossless Systems

In this letter we derive fundamental limitations on the levels of $H_{2}$ and $H_{\infty {}}$ performance that can be achieved when controlling lossless systems. The results are applied to the swing equation power system model, where it is shown that the fundamental limit on the $H_{2}$ norm scales with the inverse of the harmonic mean of the inertias in the system. This indicates that power systems may see a degradation in performance as more renewables are integrated, further motivating the need for new control solutions to aid the energy transition.