Rational Modeling of Nonhomogeneous Systems

Abstract

Nonhomogeneous transmission systems occur in several practical configurations such as nonideally transposed overhead lines, cross-bonded cables, and river crossings of overhead lines. In the past, a compact and efficient representation of nonhomogeneous system (NhS) was only possible in the frequency domain as the chain matrix (matrix transfer function) was used to obtain an equivalent nodal admittance matrix. Time-domain representation of a NhS demands explicit representation of each homogeneous section, thereby not being an efficient solution. This work proposes to represent nonhomogeneous systems using a rational approximation which allows for a compact and accurate time-domain realization. In this approach, we exploit the fact that a NhS can be seen as a particular case of a frequency-dependent network equivalent. The frequency dependence of the NhS is included via the rational modeling of the admittance matrix using the so-called Vector Fitting algorithm. Two test cases are considered to illustrate the gain of the proposed solution. The first one is the modeling of a nonideally transposed transmission line, and the second one is a case of river crossing. To asses the accuracy of the modeling, the results are compared against the ones obtained either using the Numerical Laplace Transform and PSCAD.

Appendices

Appendix 1: Overhead Line Conductor Data

For the 500-kV lines, each phase has a four-conductor bundle with a 0.5-m spacing, phase conductors are Rail, and ground wires are 3/8” EHS. The coordinates for the centers of phase conductors bundle and for the ground wires are given as follows:

$$\begin{aligned} x&= \left[ -4.5 \quad 0 \quad 4.5 \quad -3.5 \quad 3.5\right] \nonumber \\ y&= \left[ 11 \quad 15.5 \quad 11 \quad 26 \quad 26 \right] \end{aligned}$$

(27)

Appendix 2: State Space Formulation

For a first-order proper pole-residue model with input $u$ and output $y$, we may write in the frequency domain:

$$\begin{aligned} y=\left( \frac{r}{s-a}+d\right) \, u \end{aligned}$$

(28)

in the time domain (28) can be written as

$$\begin{aligned} \dot{x}&= a\, x + r\, u\nonumber \\ y&= r\, x+d\,u \end{aligned}$$

(29)

a discrete time model of (29) can be obtained using either trapezoidal rule integration or recursive convolution. The expressions are shown in (30).

$$\begin{aligned} x{\left( n\right) }&= \alpha \, x{\left( n-1\right) } +\left( \alpha \lambda +\mu \right) \,u{\left( n-1\right) }\nonumber \\ y{\left( n\right) }&= x{\left( n\right) } +\left( \lambda +d\right) \,u{\left( n\right) } \end{aligned}$$

(30)

If the trapezoidal integration rule is applied, then the coefficients $\alpha$, $\lambda$ and $\mu$ are given by

$$\begin{aligned} \alpha =\frac{2+a\,\Delta t}{2-a\,\Delta t} \qquad \lambda =\mu =\frac{r\,\Delta t}{2-a\,\Delta t} \end{aligned}$$

(31)

and in the case where recursive convolutions are used, we have

$$\begin{aligned} \alpha&= \exp \left( a\,\Delta t\right) \qquad \lambda = -\frac{r}{a}\left( 1+\frac{1-\alpha }{a\,\Delta t}\right) \nonumber \\ \quad \mu&= \frac{r}{a}\left( \alpha +\frac{1-\alpha }{a\,\Delta t}\right) \end{aligned}$$

(32)