Power system inertia is falling as more energy is supplied by renewable generators, and there are concerns about the frequency controls required to guarantee satisfactory system performance. The majority of research into the negative effect of low inertia has focused on poor dynamic response following major disturbances, when the transient frequency dip can become unacceptable. However, another important practical concernkeeping average frequency deviations within acceptable limits -was mainly out of the sight of the research community. In this manuscript we present a method for finding the frequency probability density function (PDF) for a given power system. We pass from an initial stochastic dynamic model to deterministic equations for the frequency PDF, which are analyzed to uncover key system parameters influencing frequency deviations. We show that system inertia has little effect on the frequency PDF, making virtual inertia services insufficient for keeping frequency close to nominal under ambient load fluctuations. We establish, that aggregate system droop and deadband width are the only parameters that have major influence on the average frequency deviations, suggesting that energy storage might be an excellent solution for tight frequency regulation. We also show that changing the governor deadband width does not significantly affect generator movement..

appealing because the application of voltage can turn on or off the instability at any time and therefore may represent a new way to study thin film instabilities. Recently, a similar electrochemical approach has been utilized to withdraw liquid metal from microchannels. [28,29] Electrochemical processes can pattern solid metals and other surfaces in a process called electrochemical micromachining. [30] Electrochemical micromachining locally dissolves and shapes metals through an anodic reaction (i.e., oxidation). In contrast, our approach removes only the surface metal oxide via a localized cathodic reaction (i.e., reduction) and therefore results in inconsequential loss of material while allowing the metal to flow to a new location.The patterning approach described here relates to the classical dewetting phenomenon of polymeric films, [31] which are typically trapped in an energetically unfavorable state (i.e., a film) until the temperature exceeds a sufficiently high value to enable the polymer to flow. In contrast, the method presented here induces destabilization of films using electrochemistry [6,32,33] at constant temperature. We call this technique "recapillarity" because the reduction of the oxide induces capillary action. [27,34] The studies here suggest a new way to pattern liquid metals and to study instabilities. Figure 1 depicts the in-plane patterning of liquid metal using localized electrochemistry. To demonstrate the principle, we spread a drop of EGaIn ( Figure 1a) on a glass slide to create a thin film. A copper wire contacts the edge of the film, thereby making the EGaIn the working cathode. On a separate copper wire, a small drop of electrolyte (i.e., water or salt solution) wets the tip of the wire after dipping it in a reservoir of electrolyte. Contacting this suspended drop with the metallic film completes the electrochemical circuit. Application of a reductive bias to the film removes locally the oxide skin, which causes capillary withdrawal of the metal as shown schematically in Figure 1a. We drew the letters "NCSU" (cf. Figure 1b) across the film by guiding the tip across the surface by hand. We occasionally re-dipped the tip into the electrolyte to replenish it, similar to an old fashioned ink-well pen. Although thin films of the metal are generally rough with some pinholes (cf. inset of Figure 1c), we added liquid metal to the patterned metal film to render a smooth, reflective surface (Figure 1c). Video S1 (Supporting Information) illustrates this process and Figure S1 (Supporting Information) shows an American flag composed of liquid metal produced by strategically localizing the electrochemical reduction of the oxide by the electrode "scribe."We sought to quantify and understand the primary variables that control the velocity of withdrawal of the film during reduction. To improve repeatability, we built a spreader (cf., Figure 2a(i)) to smear drops of EGaIn (≈0.1 mL) onto a glass slide cleaned with an oxygen plasma. The spreader consists of a glass tube "roller" (diameter ≈12 mm) h...

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