POWER SYSTEM TRANSIENTS*

The traveling-wave concept

As is well known, many physical phenomena are expressed mathematically as a second-order partial differential equation.

Electrical transients associated with a wave-propagation characteristic are mathematically represented by a hyperbolic partial differential equation; Therefore, solving electrical transients necessitates the solution of the differential equation with the given initial and boundary conditions.

D’Alembert gave the earliest solution of the partial differential equation for the case of a vibrating string in 1750. At the same time, Bernoulli found a solution that was quite different from D’Alembert’s solution.

Bernoulli’s solution is based on the eigenfunction and is comparable to the Fourier series.

Traveling-wave concepts and theories have been well-developed since D’Alembert’s solution.

Allievi first applied the theory to the field of hydraulic engineering and established the general theory and idea of a graphical method, which was a direct application of the traveling-wave concept to engineering fields.

At a later stage, Bewley developed the traveling-wave theory and its application to various electrical transients. The propagation of the traveling wave has been well analyzed using the modified Heavisine transform and the Sylvester theorem, which is the same as the eigenvalue theory of matrix algebra by Hayashi.

The graphical method developed by Allievi has been applied to the analysis of a water hammer by Schnyder, Bergeron, and Angus. This was originally called the Schnyder–Bergeron method in the electrical engineering field. The name of the method has since been abbreviated, and it is nowadays called the Bergeron method, although Schnyder originated it. The detail and application of the graphical method are well described by Parmakian.

The graphical method corresponds to the method of characteristics to solve Maxwell’s equations mathematically.

Similarly, the lattice diagram method based on the traveling-wave concept was developed by Bewley to solve electrical transients.

At a later stage, both the graphical method and the lattice diagram method were implemented on a digital computer for calculating electrical transients. This technique is generally called the “traveling wave technique” or the “time domain method.”

The numerical Fourier transform appeared in the electrical engineering field in the late 1950s, although the basis of the method was given by Bernoulli in 1750. Gibbs’ phenomena and instability in a transform process, which are the inherent nature of the discrete Fourier transform, were greatly reduced with the development of the modified Fourier (Laplace) transform. At a later stage, the modified Fourier/Laplace transform was applied to transient calculations by various authors.

Since the modified Fourier/Laplace transform provides high accuracy for obtaining a time solution, the analysis of a partial differential equation is rather easy in the frequency domain, and the implementation of the fast Fourier transform procedure into the modified Fourier/Laplace transform greatly improves computational efficiency. The method has become one of the most accurate and efficient computer techniques for transient calculations.

The EMTP was based on the Schnyder–Bergeron method of traveling-wave analysis in a hydraulic system, known as a water hammer. The Schnyder–Bergeron method was introduced to the field of electrical transients by Frey and Althammer.

The method was incorporated with a nodal analysis method by representing all of the circuit elements by a lumped resistance and the current source by Dommel; this is the origin of the EMTP(Electromagnetic Transient Program).

*POWER SYSTEM TRANSIENTS, Second Edition, Akihiro Ametani, Naoto Nagaoka, Yoshihiro Baba, Teruo Ohno, Koichi Yamabuki; 2017

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