APPENDIX I. LINEAR TRANSFORMATION TECHNIQUES

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INTRODUCTION

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Linear transformation techniques are used to enable the admittance matrix of any network to be found in a systematic manner. Consider, for the purpose of illustration, the network drawn in Fig. 1.1. Five steps are necessary to form the network admittance matrix by linear transformations.

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(i) Label the nodes in the original network. (ii) Number, in any order, the branches and branch admittances. (iii) Form the primitive network admittance matrix by inspection. This matrix relates the nodal injected currents to the node voltages of the primitive network. The primitive network is also drawn by inspection of the actual network. It consists of the unconnected branches of the original network with a current equal to the original branch current injected into the corresponding node of the primitive network. The voltages across the primitive network branches then equal those across the same branch in the actual network. The primitive network for Fig. 1.1. is shown in Fig. 1.2. The primitive admittance matrix relationship is

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(I.1.1)

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CYPR,MI

Off-diagonal terms are present where mutual coupling between branches is present. Form the connection matrix [C]. This relates the nodal voltages of the actual

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Figure 1.1 Actual connected network

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W'l *

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Figure 1.2 Primitive o r unconnected network

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network to the nodal voltages of the primitive network. By inspection of Fig. 1.1,

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v, - v b

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(I. 1.2)

v, = v, v, = v b v, = v,

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or in matrix form

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141-1

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(1.1.3)

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(v) The actual network admittance matrix which relates the nodal currents to the

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voltages by

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(1.1.4)

can now be derived from

CYobcl 3 x 3

=[CIT~[YPRIMl~[C1

3x5 5x5 5 x 3

(I.1.5)

which is a straightforward matrix multiplication.

L2.1

THREE-PHASE SYSTEM ANALYSIS

Discussion of the Frame of Reference

Sequence components have long been used to enable convenient examination of the balanced power system under both balanced and unbalanced loading conditions. The symmetrical component transformation is a general mathematical technique developed by Fortescue whereby any system of n vectors or quantities may be resolved, when n is prime, into n different symmetrical n phase systems . Any set of three-phase voltages or currents may therefore be transformed into three symmetrical systems of three vectors each. This, in itself, would not commend the method and the assumptions, which lead to the simplifying nature of symmetrical components, must be examined carefully. Consider, as an example, the series admittance of a three-phase transmission line, shown in Fig. 1.3, i.e. three mutually coupled coils. The admittance matrix relates the illustrated currents and voltages by

(1.2.1)

where

(1.2.2)

By the use of symmetrical component transformation the three coils of Fig. 1.3

(b) Figure I 3 Admittance representation of a three-phase series element: (a) series admittance element; (b) admittance matrix representation

can be replaced by three uncoupled coils. This enables each coil to be treated separately with a great simplification of the mathematics involved in the analysis. The transformed quantities (indicated by subscripts 012 for the zero, positive and negative sequences respectively) are related to the phase quantities by

CJ'012I

= [TsI-'.CJ'a,cI = [Ts1 =V

s I [Iabcl CYad.

(1.2.3) (1.2.4)

[IO 1 2 1

[ s . J'o 1 2 1 T I [:

(1.2.5)

where [T,] is the transformation matrix. The transformed voltages and currents are thus related by the transformed admittance matrix

ETJ -

yabtl

[z1*

(1.2.6)

Assuming that the element is balanced, we have

(1.2.7)

and a set of invariant matices [TIexist. Tranformation (1.2.6) will then yield a diagonal matrix LylOl2. In this case the mutually coupled three-phase system has been replaced by three uncoupled symmetrical systems. In addition, if the generation and loading are balanced, or may be assumed balanced, then only one system, the positive-sequence system, has any current flow and the other two sequences may be ignored. This is essentially the situation with the single-phase load flow. If the original phase admittance matrix [yab,] is in its natural unbalanced state then the transformed admittance matrix Cyol2] is full. Therefore, current flow of one sequence will give rise to voltages of all sequences, i.e. the equivalent circuits for the sequence networks are mutually coupled. In this case the problem of analysis is no simpler in sequence components than in the original phase components and symmetrical components should not be used. From the above considerations it is clear that the asymmetry inherent in all power systems cannot be studied with any simplification by using the symmetrical component frame of reference. Data in the symmetrical component frame should only be used when the network element is balanced, for example, synchronous generators. In general, however, such an assumption is not valid. Unsymmetrical interphase coupling exists in transmission lines and to a lesser extent in transformers and this results in coupling between the sequence networks. Furthermore, the phase shift introduced by transformer connections is difficult to represent in sequence component models. With the use of phase co-ordinates the following advantages become apparent.

Any system element maintains its identity. Features such as asymmetric impedances, mutual couplings between phases and between different system elements, and line transpositions are all readily considered. Transformer phase shifts present no problem.