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and by choosing the proper transformation matrix [Tu],equation (111.3) can be changed to
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where [A] is now a diagonal matrix. This diagonalisation is a well defined procedure in matrix algebra; the elements of [A] are the eigenvalues of the matrix product [Z]*[Y'], and the transformation matrix [Tu] is the matrix of eigenvectors of that matrix product. Equation (111.4) can be diagonalised as well, with the same diagonal matrix [A], i.e. (111.7) but the transformation matrix for currents differs from that used for voltages (in contrast to symmetrical components):
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where the subscript 't' indicates a transposed matrix. With the diagonalised equations(III.6) and (111.7), an m-phase line can now be studied as if it consisted of m single-phase lines, similar to the symmetrical component approach, except that the zero-, positive- and negative-sequence networks now become the mode 1, mode 2 and mode 3 networks. The modal series impedance and shunt admittance are not directly available but must be computed from
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C Y'modcI = [Til - ' .CY']* [Tu]
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may with both modal matrices being diagonal. [Ymodc] no longer be purely imaginary even though only shunt capacitance is modelled. This will depend on how the transformation matrices were normalised. For steady-state analysis at one particular and have frequency, this causes no problems. Once Zscrics Yshun, been calculated for each mode, the representation in phase quantities is easily obtained by transforming back, with
= [Tu] *CZscrics-modcI *[Til -
(111.1 la)
becoming the values of the equivalent-n model which will accurately represent the untransposed line. In expanded form the following are expressions for the series impedance and shunt admittance of the equivalent-n model:
zcz'l~IMl[T]cMl-' sinh y l
Calculate equivalent P I series impedonce and shunt admittonce matrices
Farm matrix product t Y '1 [Z'I
Colculote the eigenvalues and eigenvectors and form [MI calculate [ M I - '
solution for occepta ble occuracy
diogonol matrices of hyperbolic eigenvol ue
correction factors ond apply to give
Figure 111.1 Structure diagram for calculation of the equivalent-n model
where I is the transmission line length, [Z],,, is the equivalent-n series impedance matrix, [MIis the matrix of normalised eigenvectors, and
sinh,, I - ...
and y j is the jth eigenvalue for j/3 mutually coupled circuits. Similarly
where [Y]EpM is the equivalent-x shunt admittance matrix. Computer derivation of the correction factors for conversion from the nominal-n to the equivalent-x model, and their incorporation into the series impedance and shunt admittance matrices, is carried out as indicated in the structure diagram of Fig. 111.1. The LR2 algorithm of Wilkinson and Reinscht is used with due regard for accurate calculations in the derivation of the eigenvalues and eigenvectors.
'J. H. Wilkinson and C. Reinsch, (1971). 'Handbook for Automatic Computations' Vol. I1 (Linear Algebra) Springer-Verlag, Berlin.
Basic to the computer modelling of power system transients is the numerical integration of the set of differential equations involved. Many books have been written on the numerical solution of ordinary differential equations, but this appendix is restricted to the techniques in common use for the dynamic simulation of power system behaviour. It is therefore appropriate to start by identifying and defining the properties required from the numerical integration method in the context of power system analysis.
This property is limited by two main causes, i.e. round-off and truncation errors. Round-off error occurs while performing arithmetic operations and is due to the inability of the computer to represent numbers exactly. A word length of 48 bits is normally sufficient for scientific work and is certainly acceptable for transient stability analysis. When the stability studies are carried out on computers with a 32-bit word length, it is necessary to use double precision on certain areas of the storage to maintain adequate accuracy. The difference between the true and calculated results is mainly determined by the truncation error, which is due to the numerical method chosen. The true solution at any one point can be expressed as a Taylor series based on some initial point and by substituting these into the formulae, the order of accuracy can be found from the lowest power of step length (h) which has a nonzero coefficient. In general terms, the truncation error T(h)of a method using a step length h is given by
T(h)= O(hP+ l )
where superscript p represents the order of accuracy of the method. The true solution y(t,) at t , is thus
y(t,) = y ,
+ O W + + E,