A Novel Subspace Decomposition with Rotational Invariance Technique to Estimate Low-Frequency Oscillatory Modes of the Power Grid

Algorithm 2

Computational algorithm of TLS-ESPRIT.

(1)	Initially, signal vector from (3) is used to form a correlation matrix .
(2)	Defines MKLT of for a specified data matrix as
(3)	Signal and noise subspaces can be obtained by disintegrating the i.e. , where represents signal subspace.
(4)	The sub-space of the signal is then decomposed into two smaller subspaces (M-1) as dimension where unstaggered and staggered subspaces, i.e., and describe in the same way as and .
(5)	These could be mapped as: . and are related as: Substituting and in gives:
(6)	The correlation between both the subspaces provided by Algorithm 2
	The eigenvalues are diagonal elements of , where . columns represent the eigenvectors of .
(7)	Therefore, the frequency is given by where is the phase of .
(8)	The TLS presents the exact response by minimizing the Frobenius norm of the true subspace and estimated subspace by reducing the errors [20]
(9)	Compute the matrix of right singular vectors of
(10)	The matrix separation can be done using in compliance with gives
(11)	The singular values are computed using of the matrix
(12)	Algorithm 2 gives the frequency estimates using Algorithm 2.