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We present a physics-informed Gaussian Process Regression (GPR) model topredict the phase angle, angular speed, and wind mechanical power from alimited number of measurements. In the traditional data-driven GPR method, theform of the Gaussian Process covariance matrix is assumed and its parametersare found from measurements. In the physics-informed GPR, we treat unknownvariables (including wind speed and mechanical power) as a random process andcompute the covariance matrix from the resulting stochastic power gridequations. We demonstrate that the physics-informed GPR method is significantlymore accurate than the standard data-driven one for immediate forecasting ofgenerators' angular velocity and phase angle. We also show that thephysics-informed GPR provides accurate predictions of the unobserved windmechanical power, phase angle, or angular velocity when measurements from onlyone of these variables are available. The immediate forecast of observedvariables and predictions of unobserved variables can be used for effectivelymanaging power grids (electricity market clearing, regulation actions) andearly detection of abnormal behavior and faults. The physics-based GPR forecasttime horizon depends on the combination of input (wind power, load, etc.)correlation time and characteristic (relaxation) time of the power grid and canbe extended to short and medium-range times.

Physics-informed Machine Learning Method for Forecasting and Uncertainty Quantification of Partially Observed and Unobserved States in Power Grids