Abstract
Longitudinal power systems can transmit a certain amount of power independently of their lengths. This nature is widely different from one-machine-infinite-bus systems. This paper clarifies from where the nature comes. First, we linearize swing equations of a longitudinal system, and show that its coefficient matrix is essentially a band matrix. Furthermore, the matrix is cyclic. Namely, if you shift all elements of a row by one to the right, then you get the next row. Utilizing this feature, we solve a difference equation consisting of the elements to get analytical expressions for the eigenvalues and eigenvectors of the system. The eigenvectors do not change with increase in the power transmission if there is no load at all intermediate buses. Contrarily, several eigenvectors rapidly change their shapes if there are some loads at the buses. Two eigenvectors of low frequency modes lastly change to a couple of conjugate complex vectors. Simultaneously, one eigenvalue comes to have positive real part, which means that the corresponding low frequency mode oscillately diverges. These changes determine the transmission capacity of the system. However, the changes are brought by the row elements of the matrix. Since the elements do not depend on the system length, the transmission capacity is also independent of the length.
