Overhead Lines
R, L, G, and C are the line parameter matrices expressed in per unit length
2.2.4 Solution Techniques
The techniques developed to solve the equations of a multiconductor frequency-depen-dent overhead line can be classifi ed into two main categories: modal-domain techniques and phase-domain techniques. An overview of the main approaches is presented in the following [28].
2.2.4.1 Modal-Domain Techniques
Overhead line equations can be solved by introducing a new reference frame:
=
ph v m
V T V (2.34a)
=
ph i m
I T I (2.34b)
where the subscripts “ph” and “m” refer to the original phase quantities and the new modal quantities. Matrices Tv and Ti are calculated through an eigenvalue/eigenvector problem such that the products YZ and ZY are diagonalized
−1 =
v v
T ZYT Λ (2.35a)
−1 =
i i
T YZT Λ (2.35b)
L being a diagonal matrix.
Thus, the line equations in modal quantities become
− m = v−1 i m
d dx
V T ZT I (2.36a)
FIGURE 2.7
Equivalent circuit for time-domain simulations.
k +
Zc Zc
–
+ m
–
vk(t) Ik(t) Im(t) vm(t)
ik(t) im(t)
−d m = −1 m
dx i v
I T YT V (2.36b)
It can be proved that [Tv]−1 = [Ti]T (superscript T denotes transposed) and that the prod-ucts Tv−1ZTi (= Zm) and Ti−1YTv (=Ym) are diagonal [15,16].
The solution of a line in modal quantities can be then expressed in a similar manner as in Equation 2.32. The solution in time domain is obtained again by using convolution, as in Equation 2.33.
However, since both Tv and Ti are frequency dependent, a new convolution is needed to obtain line variables in phase quantities:
= ∗
ph( )t v( )t m( )t
v T v (2.37a)
= ∗
ph( )t i( )t m( )t
i T i (2.37b)
The procedure to solve the equations of a multiconductor frequency-dependent overhead line in the time domain involves in each time step the following:
1. Transformation from phase-domain terminal voltages to modal domain.
2. Solution of the line equations using modal quantities and calculation of (past history) current sources.
3. Transformation of current sources to phase-domain quantities.
Figure 2.8 shows a schematic diagram of the solution of overhead line equations in the modal domain.
Two approaches have been used for the solution of the line equations in modal quanti-ties: constant and frequency-dependent transformation matrices.
1. The modal decomposition is made by using a constant real transformation matrix T calculated at a user-specifi ed frequency, being the imaginary part usually dis-carded. This has been the traditional approach for many years. It has an obvious advantage, as it simplifi es the problem of passing from modal quantities to phase quantities and reduces the number of convolutions to be calculated in the time domain, since Tv and Ti are real and constant. Differences between methods in the time-domain implementation, based on this approach, differ from the way in which the characteristic admittance function Yc and the propagation function H
FIGURE 2.8
Transformations between phase-domain and modal-domain quantities.
Phase
domain Modal domain Phase
domain Linear
transformations Tv–1, Ti–1
Linear transformations
Tv, Ti
of each mode are represented. The characteristic admittance function is in general very smooth and can be easily synthesized with RC networks. To evaluate the convolution that involves the propagation function, several alternatives have been proposed: weighting functions [29], exponential recursive convolution [30,31], lin-ear recursive convolution [32], and modifi ed recursive convolution [33,34]. The work presented in [35] uses the constant Clarke’s transformation matrix for pass-ing from model domain to phase domain, and represents the frequency depen-dence of uncoupled line modes by a cascade of synthetic π-circuits.
2. The frequency dependence of the modal transformation matrix can be very signifi cant for some untransposed multicircuit lines. An accurate time-domain solution using a modal-domain technique requires then frequency-dependent transformation matri-ces. This can, in principle, be achieved by carrying out the transformation between modal- and phase-domain quantities as a time-domain convolution, with modal parameters and transformation matrix elements fi tted with rational functions [36–38].
Although working for cables, see [36], it has been found that for overhead lines, the elements of the transformation matrix cannot be always accurately fi tted with stable poles only [38]. This problem is overcome by the phase-domain approaches.
2.2.4.2 Phase-Domain Techniques
Some problems associated with frequency-dependent transformation matrices could be avoided by performing the transient calculation of an overhead line directly with phase quantities. A summary of the main approaches is presented in the following.
1. Phase-domain numerical convolution: Initial phase-domain techniques were based on a direct numerical convolution in the time domain [39,40]. However, these approaches are time consuming in simulations involving many time steps.
This drawback was partially solved in [41] by applying linear recursive convolu-tion to the tail porconvolu-tion of the impulse responses.
2. z-Domain approaches: An effi cient approach is based on the use of two-sided recursions (TSR), as presented in [42]. The basic input–output in the frequency domain is usually expressed as follows:
=
( )s ( ) ( )s s
y H u (2.38)
Taking into account the rational approximation of H(s), Equation 2.38 becomes
= −1
( )s ( ) ( ) ( )s s s
y D N u (2.39)
D(s) and N(s) being polynomial matrices. From this equation one can derive
=
( ) ( )s s ( ) ( )s s
D y N u (2.40)
This relation can be solved in the time domain using two convolutions:
− −
= =
∑
kn0D yk r k =∑
kn0N uk r k (2.41)The identifi cation of both side coeffi cients can be made using a frequency-domain fi tting [42].
A more powerful implementation of the TSR, known as ARMA (Auto-Regressive Moving Average) model, was presented in [43,44] by explicitly introducing modal time delays in Equation 2.41.
3. s-Domain approaches: A third approach is based on s-domain fi tting with ratio-nal functions and recursive convolutions in the time domain. Two main aspects are issued: how to obtain the symmetric admittance matrix, Y, and how to update the current source vectors. These tasks imply the fi tting of Yc(ω) and H(ω).
The elements of Yc(ω) are smooth functions and can be easily fi tted. However, the fi tting of H(ω) is more diffi cult since its elements may contain different time delays from individual modal contributions; in particular, the time delay of the ground mode differs from those of the aerial modes. Some works consider a single time delay for each element of H(ω) [45,46]. However, a very high order fi t-ting can be necessary for the propagation matrix in the case of lines with a high ground resistivity, as an oscillating behavior can result in the frequency domain due to the uncompensated parts of the time delays. This problem can be solved by including modal time delays in the phase domain. Several line models have been developed on this basis, using polar decomposition [47], expanding H(ω) as a linear combination of the natural propagation modes with idempotent coef-fi cient matrices [48], or calculating unknown residues once the poles and time delays have been precalculated from the modes, in the so-called universal line model (ULM) [49]. For a discussion on the advantages and limitations of these models see [50].
4. Nonhomogeneous models: Overhead line parameters associated with external electromagnetic fi elds are frequency independent. That is, the series impedance matrix Z is a full matrix that can be split up as follows:
ω = loss ω + ω ext
( ) ( ) j
Z Z L (2.42)
where
= + ωΔ
loss j
Z R L (2.43)
Elements of Lext are frequency independent and related to the external fl ux, while elements of R and ΔL are frequency dependent and related to the internal fl ux.
Finally, the elements of the shunt admittance matrix, Y(ω) = jωC, depend on the capacitances, which can be assumed frequency independent. Taking into account this behavior, frequency-dependent effects can be separated, and a line section can be represented as shown in Figure 2.9 [48,51].
Modeling Zloss as lumped has advantages, since their elements can be synthe-sized in phase quantities, and limitations, since a line has to be divided into sec-tions to reproduce the distributed nature of parameters.
Some models based on a phase-domain approach have been implemented in some EMT programs [43,49,50].
2.2.4.3 Alternate Techniques
1. Finite differences models: In this type of models the set of partial differential equations (Equation 2.1) are converted to an equivalent set of ordinary differential equations.
This new set is discretized with respect to the distance and time by fi nite differences and solved sequentially along the time [52]. It has been shown that these models have advantages over those described above when the line has to be discretized, for instance in the presence of incident external fi elds or/and corona effect.
2. Frequency-domain solution: The line equations are solved in the frequency domain directly and going to the time domain when the solution is found. This is done through the use of Fast Fourier transform (FFT)-based routines such as the numerical Laplace transform [53].