  Real-time state estimation and forecasting is critical for efficient
operation of power grids. In this paper, a physics-informed Gaussian process
regression (PhI-GPR) method is presented and used for probabilistic forecasting
and estimating the phase angle, angular speed, and wind mechanical power of a
three-generator power grid system using sparse measurements. In standard
data-driven Gaussian process regression (GPR), parameterized models for the
prior statistics are fit by maximizing the marginal likelihood of observed
data, whereas in PhI-GPR, we compute the prior statistics by solving stochastic
equations governing power grid dynamics. The short-term forecast of a power
grid system dominated by wind generation is complicated by the stochastic
nature of the wind and the resulting uncertain mechanical wind power. Here, we
assume that the power-grid dynamic is governed by the swing equations, and we
treat the unknown terms in the swing equations (specifically, the mechanical
wind power) as random processes, which turns these equations into stochastic
differential equations. We solve these equations for the mean and variance of
the power grid system using the Monte Carlo simulations method. We demonstrate
that the proposed PhI-GPR method can accurately forecast and estimate both
observed and unobserved states, including the mean behavior and associated
uncertainty. For observed states, we show that PhI-GPR provides a forecast
comparable to the standard data-driven GPR, with both forecasts being
significantly more accurate than the autoregressive integrated moving average
(ARIMA) forecast. We also show that the ARIMA forecast is much more sensitive
to observation frequency and measurement errors than the PhI-GPR forecast.
