Inferring Power System Dynamics from Synchrophasor Data using Gaussian Processes

The paper derives the modal decoupling of the swing dynamics under uniform damping (D = γM), defines y = V^T M^{1/2} θ and x = V^T M^{-1/2} p, and obtains N SISO ODEs ÿ_i + γ ẏ_i + λ_i y_i = x_i (S1) together with the impulse response h_i(t) = (a_i e^{c_i t} + b_i e^{d_i t}) u(t) for the output ẏ_i (eq. (10)) . With whitened inputs x(t) (eq. (11)), the paper proves the auto-covariance E[ẏ_i(t+τ) ẏ_i(t)] = α_ii (1/(2γ)) [h_i(τ) + h_i(−τ)] (eq. (12)) and the cross-covariance E[ẏ_i(t+τ) ẏ_j(t)] = α_ij [k_ij(τ) u(τ) + k_ji(−τ) u(−τ)] with k_ij(τ) = a_ij e^{c_i τ} + b_ij e^{d_i τ}, where a_ij and b_ij are given in closed form (eq. (13)) . Finally, it maps back to bus speeds: E[ω(t+τ) ω^T(t)] = M^{-1/2} V E[ẏ(t+τ) ẏ(t)^T] V^T M^{-1/2} (eq. (14)) . The candidate solution reproduces exactly these steps, computes the same impulse responses and convolution integrals, and arrives at the same kernels and back-transform, including the explicit formulas for a_ij and b_ij. Hence both are correct and essentially the same proof, with the model giving more algebraic detail while fully aligning with the paper.