  We present a physics-informed Gaussian Process Regression (GPR) model to
predict the phase angle, angular speed, and wind mechanical power from a
limited number of measurements. In the traditional data-driven GPR method, the
form of the Gaussian Process covariance matrix is assumed and its parameters
are found from measurements. In the physics-informed GPR, we treat unknown
variables (including wind speed and mechanical power) as a random process and
compute the covariance matrix from the resulting stochastic power grid
equations. We demonstrate that the physics-informed GPR method is significantly
more accurate than the standard data-driven one for immediate forecasting of
generators' angular velocity and phase angle. We also show that the
physics-informed GPR provides accurate predictions of the unobserved wind
mechanical power, phase angle, or angular velocity when measurements from only
one of these variables are available. The immediate forecast of observed
variables and predictions of unobserved variables can be used for effectively
managing power grids (electricity market clearing, regulation actions) and
early detection of abnormal behavior and faults. The physics-based GPR forecast
time horizon depends on the combination of input (wind power, load, etc.)
correlation time and characteristic (relaxation) time of the power grid and can
be extended to short and medium-range times.
