48
Computer Modelling of Electrical Power Systems, Second Edition. J Arrillaga and N R Watson. © 2001 John Wiley & Sons, Ltd. Published 2001 by John Wiley & Sons, Ltd. 2 TRANSMISSION SYSTEMS 2.1 Introduction The conventional power transmission system is a complex network of passive compo- nents, mainly transmission lines and transformers, and its behaviour is commonly assessed using equivalent circuits consisting of inductance, capacitance and resistance. This chapter deals with the derivation of these equivalent circuits and with the formation of the system admittance matrix relating the current and voltage at every node of the transmission system. Among the many alternative ways of describing transmission systems to comply with Kirchhoff's laws, two methods, mesh and nodal analysis, are normally used. The latter has been found to be particularly suitable for digital computer work, and is almost exclusively used for routine network calculations. The nodal approach has the following advantages: The numbering of nodes, performed directly from a system diagram, is very simple. Data preparation is easy. The number of variables and equations is usually less than with the mesh method for power networks. Network crossover branches present no difficulty. Parallel branches do not increase the number of variables or equations. Node voltages are available directly from the solution, and branch currents are easily calculated. Off-nominal transformer taps can easily be represented. 2.2 Linear Transformation Techniques Linear transformation techniques are used to enable the admittance matrix of any network to be found in a systematic manner. Consider, for the purposes of illustration, the network drawn in Figure 2.1. Five steps are necessary to form the network admittance matrix by linear transfor- mation, i.e. (i) Label the nodes in the original network.

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Page 1: Computer Modelling of Electrical Power Systems (Arrillaga/Computer Modelling of Electrical Power Systems) || Transmission Systems

Computer Modelling of Electrical Power Systems, Second Edition. J Arrillaga and N R Watson.© 2001 John Wiley & Sons, Ltd. Published 2001 by John Wiley & Sons, Ltd.

2 TRANSMISSION SYSTEMS

2.1 Introduction

The conventional power transmission system is a complex network of passive compo­nents, mainly transmission lines and transformers, and its behaviour is commonly assessed using equivalent circuits consisting of inductance, capacitance and resistance.

This chapter deals with the derivation of these equivalent circuits and with the formation of the system admittance matrix relating the current and voltage at every node of the transmission system.

Among the many alternative ways of describing transmission systems to comply with Kirchhoff's laws, two methods, mesh and nodal analysis, are normally used. The latter has been found to be particularly suitable for digital computer work, and is almost exclusively used for routine network calculations.

The nodal approach has the following advantages:

• The numbering of nodes, performed directly from a system diagram, is very simple.

• Data preparation is easy.

• The number of variables and equations is usually less than with the mesh method for power networks.

• Network crossover branches present no difficulty.

• Parallel branches do not increase the number of variables or equations.

• Node voltages are available directly from the solution, and branch currents are easily calculated.

• Off-nominal transformer taps can easily be represented.

2.2 Linear Transformation Techniques

Linear transformation techniques are used to enable the admittance matrix of any network to be found in a systematic manner. Consider, for the purposes of illustration, the network drawn in Figure 2.1.

Five steps are necessary to form the network admittance matrix by linear transfor­mation, i.e.

(i) Label the nodes in the original network.

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6 2 TRANSMISSION SYSTEMS

Figure 2.1 Actual connected 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 Figure 2.1 is shown in Figure 2.2. The primitive admittance matrix relationship is:

yll

Yn

y33

y44

Yss

[fPRIM]

(2.1)

Off-diagonal terms are present where mutual coupling between branches is present.

(iv) Form the connection matrix [C].

Figure 2.2 Primitive or unconnected network

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2.3 BASIC SINGLE-PHASE MODELLING 7

This relates the nodal voltages of the actual network to the nodal voltages of the primitive network. By inspection of Figure 2.1,

or in matrix form

V, = Va- Vb,

V2 = Vb- Vc,

V3 = Va,

v4 = vb,

1 -1

1 -1

1

1

1

[C]

(2.2)

(2.3)

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

can now be derived from

[Yabc] = [C]T 3x3 3x5

[YPRIM] 5x5

which is a straightforward matrix multiplication.

[C] 5 X 3 '

2.3 Basic Single-phase Modelling

(2.4)

(2.5)

Under perfectly balanced conditions, transmission plant can be represented by single­phase models, the most extensively used being the equivalent-n circuit.

2.3.1 Transmission lines

In the case of a transmission line, the total resistance and inductive reactance of the line is included in the series arm of the n-equivalent and the total capacitance to neutral is divided between its shunt arms.

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8 2 TRANSMISSION SYSTEMS

lp Ysc- Yoc/2

Is/ l l

vP lvo/2 Y0/2 vs

T T

Figure 2.3 Transformer equivalent circuit

2.3.2 Transformer on nominal ratio

The equivalent-IT model of a transformer is illustrated in Figure 2.3, where Yoe is the reciprocal of Zoe (magnetizing impedance) and Ysc is the reciprocal of Zse (leakage impedance). Zsc and Zoe are obtained from the standard short-circuit and open-circuit tests.

This yields the following matrix equation:

Gls = f------Ys_e ---+---Y_se_+_Yo_c_/2--\

[!J -Yse + Yoc/2 Yse

where

Ysc is the short-circuit or leakage admittance,

Yoe is the open-circuit or magnetizing admittance.

lvJ, [!J

(2.6)

The use of a three-terminal network is restricted to the single-phase representation and cannot be used as a building block for modelling three-phase transformer banks.

The magnetizing admittances are usually removed from the transformer model and added later as small, shunt-connected admittances at the transformer terminals. In the per unit system, the model of the single-phase transformer can then be reduced to a lumped leakage admittance between the primary and secondary busbars.

2.3.3 Off-nominal transformer tap representation

A transformer with turns ratio a interconnecting two nodes i, k can be represented by an ideal transformer in series with the nominal transformer leakage admittance as shown in Figure 2.4(a).

If the transformer is on nominal tap (a = 1), the nodal equations for the network branch in the per unit system are:

In this case, l;k = - h;.

l;k = Yik Vi - Yik Vk

h; = Y;kVk- Y;kVi

(2.7)

(2.8)

For an off-nominal tap setting and letting the voltage on the k side of the ideal transformer be V1, we can write

vi Vr = -, (2.9)

a

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a:1

2.3 BASIC SINGLE-PHASE MODELLING

Y;k k

1' l;k

(a) (b)

Figure 2.4 Transformer with off-nominal tap setting

h; = Y;k(Vk- Vt),

hi l;k = --.

a

Eliminating V1 between Equations (2.9) and (2.1 0), we obtain

Yik h; = Y;kVk- -Vi,

a Yik Yik

I;k = --vk + 2 vj. a a

9

(2.10)

(2.11)

(2.12)

(2.13)

A simple equivalent IT circuit can be deduced from Equations (2.12) and (2.13), the elements of which can be incorporated into the admittance matrix. This circuit is illustrated in Figure 2.4(b ).

The equivalent circuit of Figure 2.4(b) has to be used with care in banks containing delta-connected windings. In a star-delta bank of single-phase transformer units, for example, with nominal turns ratio, a value of 1.0 per unit voltage on each leg of the star winding produces under balanced conditions 1.732 per unit voltage on each leg of the delta winding (rated line to neutral voltage as base). The structure of the bank requires in the per unit representation an effective tapping at ,J3 nominal turns ratio on the delta side, i.e. a = 1. 732.

For a delta-delta or star-delta transformer with taps on the star winding, the equiv­alent circuit of Figure 2.4(b) would have to be modified to allow for effective taps to be represented on each side. The equivalent-circuit model of the single-phase unit can be derived by considering a delta-delta transformer as comprising a delta-star transformer connected in series (back-to-back) via a zero-impedance link to a star delta transformer, i.e. star windings in series. Both neutrals are solidly earthed. The leakage impedance of each transformer would be half the impedance of the equiva­lent delta-delta transformer. An equivalent per unit representation of this coupling is shown in Figure 2.5. Solving this circuit for terminal currents:

I' (V'- V")y ~=- = ----

a a

= (Vp/a- V,j fJ)y = LV - _21_ Vs a a 2 P afJ '

(2.14)

I' y y -Is = fj = afJ Vp - (32 Vs, (2.15)

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10

or in matrix form:

2 TRANSMISSION SYSTEMS

G]s = 1-----Y_/ a_2 --+---y_/ a_f3-l

[!J -y/af3 Y/f32 ~. 5]

(2.16)

These admittance parameters form the primitive network for the coupling between a primary and secondary coil.

2.3.4 Phase-shifting representation

To cope with phase-shifting, the transformer of Figure 2.5 has to be provided with a complex turns ratio. Moreover, the in variance of the product VI* across the ideal transformer requires a distinction to be made between the turns ratios for current and voltage, i.e.

or

V: I* = - V' I'* p p ,

Vp =(a+ jb)V' = aV',

I'* I*=----p a+ jb'

I' I' I------­P- a- jb- a*·

Thus, the circuit of Figure 2.5 has two different turns ratios, i.e.

av =a+ jb for the voltages,

and ai =a- jb for the currents.

Solving the modified circuit for terminal currents:

I' (V'- V")y Ip=-=----

ai ai

(2.17)

Figure 2.5 Basic equivalent circuit in p.u. for coupling between primary and secondary coils with both primary and secondary off-nominal tap ratios of a and f3

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2.4 THREE-PHASE SYSTEM ANALYSIS 11

(2.18)

Thus, the general single-phase admittance of a transformer including phase shifting is:

y y -- --

[y] = OI;Oiv 01;{3

(2.19) y y

--

01vf3 f32

It should be noted that, although an equivalent lattice network similar to that in Figure 2.5 could be constructed, it is no longer a bilinear network as can be seen from the asymmetry of y in Equation (2.19). The equivalent circuit of a single-phase phase­shifting transformer is thus of limited value and the transformer is best represented analytically by its admittance matrix.

2.4 Three-phase System Analysis

2.4.1 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'[l]. 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 that 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 Figure 2.6, i.e. three mutually coupled coils. The admittance matrix relates the illustrated currents and voltages by

where

and

[!abe] = [Yabe][Vabc],

Uabe] = Uahfe]T,

[Vabe] = [VaVbVe]T,

Yaa Yah Yae

[Yabc] = Yba ybb Ybe

Yea Yeb Ycc

(2.20)

(2.21)

(2.22)

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12 2 TRANSMISSION SYSTEMS

Yaa Yab Yac

Yba ybb Ybc

(b) Yea Ycb Ycc

Figure 2.6 Admittance representation of a three-phase series element (a) series admittance element; (b) admittance matrix representation

By the use of the symmetrical components transformation the three coils of Figure 2.6 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

[Vo12l = [Tsr 1[Vabc], (2.23)

[/ 012l = [Tsr 1 Uabc]

= [Tsr 1[YabcHTs][Vo!2],

where [Tsl is the transformation matrix.

(2.24)

(2.25)

The transformed voltages and currents are thus related by the transformed admittance matrix,

[fo12l = [Tsr 1[Yabc][Ts]. (2.26)

Assuming that the element is balanced, we have

Yaa = Ybb = Ycc'

Yab = Ybc = Yea'

Yba = Ycb = Yac' (2.27)

and a set of invariant matrices [T] exists. Transformation (2.26) will then yield a diagonal matrix [yo12l

In this case, the mutually coupled three-phase system has been replaced by three uncoupled symmetrical systems. In addition, if the generation and loading may be

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2.4 THREE-PHASE SYSTEM ANALYSIS 13

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 [Yabcl is in its natural unbalanced state, then the transformed admittance matrix [yo12] 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:

(i) Any system element maintains its identity.

(ii) Features such as asymmetric impedances, mutual couplings between phases and between different system elements, and line transpositions are all readily considered.

(iii) Transformer phase shifts present no problem.

2.4.2 The use of compound admittances

When analysing three-phase networks, where the three nodes at a busbar are always associated together in their interconnections, the graphical representation of the network is greatly simplified by means of compound admittances, a concept which is based on the use of matrix quantities to represent the admittances of the network.

The laws and equations of ordinary networks are all valid for compound networks by simply replacing single quantities by appropriate matrices [2].

Consider six mutually coupled single admittances, the primitive network of which is illustrated in Figure 2. 7.

Figure 2.7 Primitive network of six coupled admittances

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14 2 TRANSMISSION SYSTEMS

The primitive admittance matrix relates the nodal injected currents to the branch voltages as follows:

II Y11 Y12 Yl3 Yl4 YIS Y16 V1

I2 Y21 Y22 Y23 Y24 Y2s Y26 v2

I3 Y31 Y32 Y33 Y34 Y3s Y36 V3

I4 Y41 Y42 Y43 Y44 Y4s Y46 v4 (2.28)

Is Ys1 Ys2 Ys3 Ys4 Yss Ys6 Vs

I6 Y61 Y62 Y63 Y64 Y6s Y66 v6

6 X 1 6x6 6 X 1

Partitioning Equation (2.28) into 3 x 3 matrices and 3 x 1 vectors, the equation becomes:

[Yaa] [Yab] ~~ (2.29) ~· ] [Yba] [Ybb] ]

where [Ia] = [/] h I3]T,

[h] = [/4 Is h]T,

Y11 Y12 Yl3 Y44 Y4s Y46

[Yaa] = Y21 Y22 Y23 Ys4 Yss Ys6

Y31 Y32 Y33 Y64 Y6s Y66 (2.30)

Yl4 YIS Yl6 Y41 Y42 Y43

Y24 Y2s Y26 Ys1 Ys2 Ys3

Y34 Y3s Y36 Y61 Y62 Y63

Graphically, we represent this partltwning as grouping the six coils into two compound coils (a) and (b), each composed of three individual admittances. This is illustrated in Figure 2.8.

On examination of [Yab] and [YbaL it can be seen that

[Yba] = [Yab]T,

if, and only if, Yik = Yki for i = 1 to 3 and k = 4 to 6. That is if, and only if, the couplings between the two groups of admittances are bilateral.

In this case, Equation (2.29) may be written

~­~

(2.31)

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2.4 THREE-PHASE SYSTEM ANALYSIS 15

Figure 2.8 Two coupled compound admittances

Figure 2.9 Sample network represented by single admittances

Figure 2.10 Sample network represented by compound admittances

The primitive network for any number of compound admittances is formed in exactly the same manner as for single admittances, except in that all quantities are matrices of the same order as the compound admittances.

The actual admittance matrix of any network composed of the compound admittances can be formed by the usual method of linear transformation; the elements of the connection matrix are now n x n identity matrices where n is the dimension of the compound admittances.

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16 2 TRANSMISSION SYSTEMS

(a)

Y11 Y12 Y13

Y21 Y22 Y23

Y31 Y32 Y3s

Y44 Y45 Y4s

Y54 Y55 Y56

Ys4 Ys5 Yss

Y77 Y7s Y?e

Ys? Yss Yse

Ye? Yes Yes

(b)

(c)

(d)

Figure 2.11 Primitive networks and corresponding admittance matrices. (a) Primitive network using single admittances; (b) Primitive admittance matrix; (c) Primitive network using compound admittances; (d) Primitive admittance matrix

If the connection matrix of any network can be partitioned into identity elements of equal dimensions greater than one, the use of compound admittances is advantageous.

As an example, consider the network shown in Figures 2.9 and 2.1 0, which repre­sents a simple line section. The admittance matrix will be derived using single and compound admittances to show the simple correspondence. The primitive networks and associated admittance matrices are drawn in Figure 2.11. The connection matrices for the single and compound networks are illustrated by Equations (2.32) and (2.33),

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2.4 THREE-PHASE SYSTEM ANALYSIS 17

respectively.

-1 1

-1 1

-1 1

1

1 (2.32)

1

1

1

1

(2.33)

The exact equivalence, with appropriate matrix partitioning, is clear. The network admittance matrix is given by the linear transformation equation,

[YNooE] = [C]T[YPRIM][C].

This matrix multiplication can be executed using the full matrices or in partitioned form. The result in partitioned form is

[YA] + [Ys] -[YA] [YNooE] = f------+--------1

-[YA] [YA] + [Yc]

2.4.3 Rules for forming the admittance matrix of simple networks

The method of linear transformation may be used to obtain the admittance matrix of any network. For the special case of networks where there is no mutual coupling, simple rules may be used to form the admittance matrix by inspection. These rules, which apply to compound networks with no mutual coupling between the compound admittances, may be stated as follows:

(a) Any diagonal term is the sum of the individual branch admittances connected to the node corresponding to that term.

(b) Any off-diagonal term is the negated sum of the branch admittances which are connected between the two corresponding nodes.

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18 2 TRANSMISSION SYSTEMS

2.4.4 Network subdivision

To enable the transmission system to be modelled in a systematic, logical and conve­nient manner, the system must be subdivided into more manageable units. These units, called subsystems, are defined as follows: A subsystem is the unit into which any part of the system may be divided such that no subsystem has any mutual couplings between its constituent branches and those of the rest of the system. This definition ensures that the subsystems may be combined in an extremely straightforward manner.

The system is first subdivided into the most convenient subsystems consistent with the definition above.

The most convenient unit for a subsystem is a single network element. In previous sections, the nodal admittance matrix representation of all common elements has been derived.

The subsystem unit is retained for input data organization. The data for any subsystem is input as a complete unit, the subsystem admittance matrix is formulated and stored and then all subsystems are combined to form the total system admittance matrix.

2.5 Three-phase Models of Transmission Lines

Transmission line parameters are calculated from the line geometrical characteristics. The calculated parameters are expressed as a series impedance and shunt admittance per unit length of line. The effects of ground currents and earth wires are included in the calculation of these parameters [3,4].

2.5.1 Series impedance

A three-phase transmission line with a ground wire is illustrated in Figure 2.12(a). The following equations can be written for phase a

Va - V~ = I a (!?a + j wlu) + h (j wlub) + lc (j wluc) (2. 34)

+ Jwluglg - Jwlunln + Vn,

Vn = ln(R, + jwL.,)- la}wlna- hJwLnb- fc}Wlnc- lg}wL.,8 , (2.35)

and substituting

In =fa + h + fc + fg,

Va- V~ = la(Ra + Jwlu) + hJwlub + lc}wluc

+ Jwluglg - Jwlun (Ia + h + lc + 18 ) + Vn.

Regrouping and substituting for Vn, i.e.

~Va = Va- V~

= la(Ra + Jwlu- Jwlun + R, +}win- Jwlna)

+ h(Jwlub - Jwlun + R, + JwLn - Jwlnb)

+ lc(Jwluc- Jwlun + Rn +}win- }wine)

+ lg(Jwlug- Jwlun + R, + JwLn -}wing).

(2.36)

(2.37)

(2.38)

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Figure 2.12 (a) Three-phase transmission series impedance equivalent; (b) Three-phase trans­mission shunt impedance equivalent

or

L). Va = I a (Ra + jw~ - 2jw~n + Rn + Jwl.n)

+ h (J W~b - J w4n - J W~n + R, + J win )

+ lc(JW~c - jwi-r;, - jw~n + R, + }win)

+ lg(Jw~g- jwLg11 - jw~11 + R11 + Jwl.n), (2.39)

(2.40)

and writing similar equations for the other phases, the following matrix equation results:

L). Va Zaa-n Zab-n Zac-n Zag-n I a

L). vb Zba-n zbb-n Zbc-n Zbg-n h

L). Vc Zca-n Zcb-n Zcc-n ' Zcg-n lc (2.41)

' ' ' -------------------------L--------'

L). Vg Zga-n Zgb-n Zgc-n Zgg-n lg

Since we are interested only in the performance of the phase conductors, it is more convenient to use a three-conductor equivalent for the transmission line. This

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20 2 TRANSMISSION SYSTEMS

is achieved by writing matrix Equation (2.41) in partitioned form as follows:

From Equation (2.42)

Do Vabe = ZAlabe + Zslg,

D. Vg = Zclabe + Znlg.

(2.42)

(2.43)

(2.44)

From Equations (2.42) and (2.43), and assuming that the ground wire is at zero potential:

(2.45)

where

z~a-n z~b-n z~e-n

zba-n zbb-n zbe-n (2.46)

z~a-n z~b-n z~e-n

2.5.2 Shunt admittance

With reference to Figure 2.12(b), the potentials of the line conductors are related to the conductor charges by the matrix Equation [3]:

Faa Pab Pae Fag

= Pba pbb Pbe Pbg

X (2.47) Pea Feb Pee Peg

Pga Pgb Pge Pgg

Similar considerations as for the series impedance matrix lead to

(2.48)

where P~be is a 3 x 3 matrix which includes the effects of the ground wire. The capacitance matrix of the transmission line of Figure 2.12 is given by

Caa -Cab -Cae

Cl ,_,

abe=Pabe= -Cba ebb -Cbe (2.49)

-Cea -Ceb Cee

The series impedance and shunt admittance lumped-11 model representation of the three-phase line is shown in Figure 2.13(a) and its matrix equivalent is illustrated

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2.5 THREE-PHASE MODELS OF TRANSMISSION LINES 21

Half shunt Yaa admittance ik

Y bb ik Half shunt

yff admittance

(a)

(b)

(c)

Figure 2.13 Lumped-rr model of a short-three phase line series impedance. (a) Full circuit representation; (b) Matrix equivalent; (c) Using three-phase compound admittances

in Figure 2.13(b ). These two matrices can be represented by compound admittances, (Figure 2.13(c)), as described earlier.

Following the rules developed for the formation of the admittance matrix using the compound concept, the nodal injected currents of Figure 2.13(c) can be related to the nodal voltages by the equation:

D!IJ1

= r-[_z_r_1 _+_[_r_J/_2-+-_-_[z_J_-_1 _____,

~ -[z]- 1 [Zr1 + [Y]/2

6 X ] 6x6

[E2J §]·

6 X J

(2.50)

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22 2 TRANSMISSION SYSTEMS

This forms the element admittance matrix representation for the short line between busbars i and k in terms of 3 x 3 matrix quantities.

This representation may not be accurate enough for electrically long lines. The physical length at which a line is no longer electrically short depends on the wavelength; therefore, if harmonic frequencies are being considered, this physical length may be quite small. Using transmission line and wave propagation theory, more exact models may be derived [5,6]. However, for normal mains frequency analysis, it is considered sufficient to model a long line as a series of two or three nominal-rr sections.

2.5.3 Equivalent rr model

For long lines a number of nominal rr models are connected in series to improve the accuracy of voltages and currents, which are affected by standing wave effects. For example, a three-section rr model provides an accuracy to 1.2% for a quarter wavelength line (a quarter wavelength corresponds with 1500 and 1250 km at 50 and 60 Hz, respectively).

As the frequency increases, the number of nominal rr sections to maintain a particular accuracy increases proportionally, e.g. a 300 km line requires 30 nominal rr sections to maintain the 1.2% accuracy for the 50th harmonic. However, near resonance the accuracy departs significantly from an acceptable value.

The computational effort can be greatly reduced and the accuracy improved with the use of an equivalent rr model derived from the solution of the second order linear differential equations describing wave propagation along transmission lines [5].

The solution of the wave equations at a distance x from the sending end of the line is:

V(x) = exp( -yx)Vi + exp(yx)Vr,

l(x) = (Z')- 1y[exp( -yx)Vi- exp(yx)Vr],

(2.51)

(2.52)

where y = ~ = a + j fJ is the propagation constant, Z' = r + j2rr f L is the series impedance per unit length, Y' = g + j2rr f C is the shunt admittance per unit length, and Vi and Vr the forward and reverse travelling voltages, respectively.

Depending on the problem in hand, e.g. if the evaluation of terminal quantities only is required, it is more convenient to formulate a solution using two-port matrix equations. This leads to the equivalent rr model, shown in Figure 2.14, where

Z = Z0 sinh(yl), (2.53)

y 1 = y 2 = _!_ cosh ( yl) - l = _!_ tanh ( yl ) Z0 sinh(yl) Z0 2 '

(2.54)

and (2.55)

is the characteristic impedance of the line. Due to the standing wave effect of voltages and currents on transmission lines, the

maximum values of these are likely to occur at points other than at the receiving end or sending end bus bars. These local maxima could result in insulation damage, overheating

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2.5 THREE-PHASE MODELS OF TRANSMISSION LINES 23

Figure 2.14 The equivalent n model of a long transmission line

or electromagnetic interference. It is thus important to calculate the maximum values of currents and voltages along a line and the points at which these occur.

In the case of multiconductor transmission lines, the nominal n series impedance and shunt admittance matrices per unit distance, [Z'] and [Y'] respectively, are square and their size is fixed by the number of mutually coupled conductors. The derivation of the equivalent n model for harmonic penetration studies from the nominal n matrices is similar to that of the single phase lines, except that it involves the evaluation of hyperbolic functions of the propagation constant which is now a matrix:

[y] = ([Z'][Y'])I/2. (2.56)

There is no direct way of calculating sinh or tanh of a matrix, thus a method using eigenvalues and eigenvectors, called modal analysis, is employed [6] that leads to the following expressions for the series and shunt components of the equivalent n circuit [7]:

[ sinh(yl)] [Z]EPM = l[Z'][M] yl [Mr 1, (2.57)

where l is the transmission line length, [Z]EPM is the equivalent n series impedance matrix, [M] is the matrix of normalized eigenvectors,

sinh(y1l) 0 0

Y1l

0 sinh(y2l)

0 [ sin~iyl)] Y2l (2.58)

0 0 sinh(y1l)

yjl

and YJ is the jth eigenvalue for j /3 mutually coupled circuits. Similarly

~[Y]EPM = ~l[M] [tan~;~;2)] [Mr1[Y'], (2.59)

where [Y]EPM is the equivalent rr shunt admittance matrix. Computer derivation of the correction factors for conversion from the nominal rr to

the equivalent rr model, and their incorporation into the series impedance and shunt

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24 2 TRANSMISSION SYSTEMS

Calculate equivalent PI series impedance and

shunt admittance matrices

I I I I Calculate the Check eigen-

Calculate the Calculate diagonal correction

Formmatrix eigenvalues and system

matrices of factors and product [ Y'][Z')

eigenvectors solution for hyperbolic apply to give

and form [M) acceptable eigenvalue [Z)EPM and

calculate [M)-1 accuracy functions [YJEPM

Figure 2.15 Structure diagram for calculation of the equivalent 77: model

admittance matrices, is carried out as indicated in the structure diagram of Figure 2.15. The LR2 algorithm of Wilkinson and Reinsch [8] is used for accurate calculations in the derivation of the eigenvalues and eigenvectors.

2.5.4 Mutually coupled three-phase lines

When two or more transmission lines occupy the same right of way for a considerable length, the electrostatic and electromagnetic coupling between those lines must be taken into account.

Consider the simplest case of two mutually coupled three-phase lines. The two coupled lines are considered to form one subsystem composed of four system busbars. The coupled lines are illustrated in Figure 2.16, where each element is a 3 x 3 compound admittance and all voltages and currents are 3 x 1 vectors.

[VA)

L ~~~--_J~~~~~

t

J Figure 2.16 Two coupled three-phase lines

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2.5 THREE-PHASE MODELS OF TRANSMISSION LINES 25

The coupled series elements represent the electromagnetic coupling while the coupled shunt elements represent the capacitive or electrostatic coupling. These coupling param­eters are lumped in a similar way to the standard line parameters.

With the admittances labelled as in Figure 2.16, and applying the rules of linear transformation for compound networks, the admittance matrix for the subsystem is defined as follows:

yll + y33 yl2 + y34 -Y11 -YI2

yT12 + yT34 Y22+Y44 _yT12 -Yn

Ic -Yll -YI2 Y11 + Yss Y12 + Ys6 Vc

Io _yTl2 -Y22 yTI2 + YI6 y22 + y66

12 X 1 12 X 12 12 X 1 (2.60)

It is assumed here that the mutual coupling is bilateral. Therefore Y21 = YT 12, etc. The subsystem may be redrawn as in Figure 2.17. The pairs of coupled 3 x 3

compound admittances are now represented as a 6 x 6 compound admittance. The matrix representation is also shown. Following this representation and the labelling of

y11 y12

y1; y22

6x6

[~;] 6x1

y33 y34 Yss Yse

Y3! [Ys1l

Ys~ [Ys2l

y44 6x6 Yee 6x6 6x1

(a)

[ ~;] [Zsl [ ~~] " /

[~;] [Ys2l [~~] (b)

Figure 2.17 A 6 x 6 compound admittance representation of two coupled three-phase lines: (a) 6 x 6 matrix representation; (b) 6 x 6 compound admittance representation

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26 2 TRANSMISSION SYSTEMS

the admittance blocks in the figure, the admittance matrix may be written in terms of the 6 x 6 compound coils as

[ ~~] [Zsr1 + [Ysd -[Zsr1 [ ~~] u~J -[Zsr 1 [Zsr 1 + [Ys2J [ ~~]

(2.61)

12 X 1 12 X 12 12 X 1

This is clearly identical to Equation (2.60) with the appropriate matrix partitioning. The representation of Figure 2.17 is more concise and the formation of

Equation (2.61) from this representation is straightforward, being exactly similar to that which results from the use of 3 x 3 compound admittances for the normal single three-phase line.

The data that must be available to enable coupled lines to be treated in a similar manner to single lines are the series impedance and shunt admittance matrices. These matrices are of order 3 x 3 for a single line, 6 x 6 for two coupled lines, 9 x 9 for three and 12 x 12 for four coupled lines.

Once the matrices [Z5 ] and [Y5 ] are available, the admittance matrix for the subsystem is formed by application of Equation (2.61)

When all the busbars of the coupled lines are distinct, the subsystem may be combined directly into the system admittance matrix. However, if the busbars are not distinct then the admittance matrix as derived from Equation (2.61) must be modified. This is considered in the following section.

2.5.5 Consideration of terminal connections

The admittance matrix as derived above must be reduced if there are different elements in the subsystem connected to the same busbar. As an example, consider two parallel transmission lines as illustrated in Figure 2.18.

The admittance matrix derived previously related the currents and voltages at the four busbars AI, A2, Bl and B2. This relationship is given by

(2.62)

The nodal injected current at busbar A, /A, is given by

(2.63)

similarly (2.64)

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2.5 THREE-PHASE MODELS OF TRANSMISSION LINES 27

Busbar Q) ls1 Busbar ® ~

A1~------------------------------~B1

A2~------------------------------~B2

Figure 2.18 Mutually coupled parallel transmission lines

Also, from inspection of Figure 2.18

VA= VAl = VA2. VB= VB! = VB2· (2.65)

The required matrix equation relates the nodal injected currents, fA and fs, to the voltages at these busbars. This is readily derived from Equation (2.62) and the condi­tions specified above. This is simply a matter of adding appropriate rows and columns, and yields

(2.66)

This matrix [YAs] is the required nodal admittance matrix for the subsystem. It should be noted that the matrix in Equation (2.62) must be retained as it is required

in the calculation of the individual line power flows.

2.5.6 Shunt elements

Shunt reactors and capacitors are used in a power system for reactive power control. The data for these elements are usually given in terms of their rated MVA and rated kV; the equivalent phase admittance in p.u. is calculated from these data.

Consider, as an example, a three-phase capacitor bank shown in Figure 2.19. A similar triple representation to that for a line section is illustrated. The final two forms are the most compact and will be used exclusively from this point on.

1 1/jXc

1/jXc

-~ -'- _._ 1/jXc

I ~ Figure 2.19 Representation of a shunt capacitor bank

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28 2 TRANSMISSION SYSTEMS

[V;]

Figure 2.20 Graphic representation of series capacitor bank between nodes i and k

The admittance matrix for shunt elements is usually diagonal as there is normally no coupling between the components of each phase. This matrix is then incorporated directly into the system admittance matrix, contributing only to the self-admittance of the particular bus.

2.5.7 Series elements

Any element connected directly between two buses may be considered a series element. Series elements are often taken as being a section in a line sectionalization which is described later in the chapter.

A typical example is the series capacitor bank which is usually taken as uncou­pled, i.e. the admittance matrix is diagonal. This can be represented graphically as in Figure 2.20.

The admittance matrix for the subsystem can be written by inspection as:

[YsE] -[YsE] [Y] = (2.67)

-[YsE] [YsE]

2.5.8 Line sectionalization

A line may be divided into sections to account for features such as the following:

• Transposition of line conductors.

• Change of type of supporting towers.

• Variation of soil permitivity.

• Improvement of line representation (series of two or more equivalent-JT networks).

• Series capacitors for line compensation.

• Lumping of series elements not central to a particular study.

An example of a line divided into a number of sections is shown in Figure 2.21.

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2.5 THREE-PHASE MODELS OF TRANSMISSION LINES 29

The network of Figure 2.21 is considered to form a single subsystem. The resultant admittance matrix between bus A and bus B may be derived by finding, for each section, the ABCD or transmission parameters, then combining these by matrix multiplications to give the resultant transmission parameters. These are then converted to the required admittance parameters.

This procedure involves an extension of the usual two-port network theory to multi­two-port networks. Currents and voltages are new matrix quantities and are defined in Figure 2.22. The ABCD matrix parameters are also shown.

Bus A

abc Phases

P1 P2 Section I Section I

I 1 I I I I I I

f Transposition

2 I I I

J

L

I Change of configuration

Section 3

Section 4

P3 I P4

I t I I I I

I

I I

I

' I I

Series c/;

Section 5

Section 6

P5tP6 Bus 8 I I . 1 1 Sect1on I I 7 I I I I

abc

Figure 2.21 Example of a transmission line divided into sections

Figure 2.22 Two-port network transmiSSiOn parameters. (a) Normal two-port network; (b) Transmission parameters; (c) Multi-two-port network; (d) Matrix transmission parameters

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30 2 TRANSMISSION SYSTEMS

Figure 2.23 Sample system to illustrate line sectionalization. (a) System single line diagram; (b) system redrawn to illustrate line sectionalization

The dimensions of the parameter matrices correspond to those of the section being considered, i.e. 3, 6, 9, or 12 for 1, 2, 3 or 4 mutually coupled three-phase elements, respectively. All sections must contain the same number of mutually coupled three­phase elements, ensuring that all the parameter matrices are of the same order and that the matrix multiplications are executable. To illustrate this feature, consider the example of Figure 2.23.

Features of interest

(a) As a matter of programming convenience, an ideal transformer is created and included in Section 1.

(b) The dotted coupling represents coupling which is zero. It is included to ensure correct dimensionality of all matrices.

(c) In the p.u. system, the mutual coupling between the 220 kV and 66 kV lines is expressed to a voltage base given by the geometric mean of the base line-neutral voltages of the two parallel circuits.

In Table 2.1, [u] is the unit matrix, [0] is a matrix of zeros, and all other matrices have been defined in their respective sections.

Once the resultant ABCD parameters have been found, the equivalent nodal admit­tance matrix for the subsystem can be calculated from the following equation.

[D][Br 1 [C)- [D][Br 1[A] [Y] = (2.68)

[Br 1 -[Br1[A]

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2.6 EVALUATION OF OVERHEAD LINE PARAMETERS

Table 2.1 ABCD parameter matrices for the common section types

[u] + [Z][Y]/2 -[Z] Transmission line

[Y]{[u] + [Z][Y]/4} -{[u] + [Z][Y]/2}

-[Ysp ]-1 [Yssl Transformer

[u] [0] Shunt element

[YsHl -[u]

[u] -[YsE]- 1

Series element [0] -[u]

Note: All the above matrices have dimensions corresponding to the number of coupled three-phase elements in the section.

2.6 Evaluation of Overhead Line Parameters

31

The lumped series impedance matrix [Z] of a transmission line consists of three compo­nents, while the shunt admittance matrix [Y] contains one.

where

[Z] = [Ze] + [Zg] + [Zc],

[Y] = [Yg].

[Zc] is the internal impedance of the conductors (Q·krn- 1),

(2.69)

(2.70)

[Zg] is the impedance due to the physical geometry (shown in Figure 2.24) of the conductor's arrangement (Q·krn- 1),

[Ze] is the earth return path impedance (Q·km- 1), and

[Yg] is the admittance due to the physical geometry of the conductor (Q- 1-km- 1).

In multiconductor transmission all primitive matrices (the admittance matrices of the unconnected branches of the original network components) are symmetric and, therefore, the functions that define the elements need only be evaluated for elements on or above the leading diagonal.

2.6.1 Earth impedance matrix [Ze]

The impedance due to the earth path varies with frequency in a non-linear fashion. The solution of this problem, under idealized conditions, has been given in the form of either an infinite integral or an infinite series [9].

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32

Y;+ )j

2 TRANSMISSION SYSTEMS

0 i'

j'

0 ulmageof J ooodocto"

Figure 2.24 Conductor and its image

As the need arises to calculate ground impedances for a wide spectrum of frequen­cies, the tendency is to select simple formulations aiming at a reduction in computing time, while maintaining a reasonable level of accuracy.

Consequently, what was originally a heuristic approach [10] is becoming the more favoured alternative, particularly at high frequencies.

Based on Carson's work, the ground impedance can be concisely expressed as

Ze = lOOOJ(r, e)(Q·km- 1),

where

Ze

J(r, e)

E [Ze], W/La = -{P(r, e)+ jQ(r, e)},

lT

= fi!J5_ D;J, v p

Dij = 2 y; for i = j,

for i =f:. j,

(x·-x·) = arctan -'--1

Yi + YJ for i =f:. j,

= 0 fori= j,

= 2:rrf(rad·s- 1 ),

f =frequency (Hz),

y; =height of conductor i(m),

(2.71)

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2.6 EVALUATION OF OVERHEAD LINE PARAMETERS 33

x; - Xj =horizontal distance between conductors i and j (m),

!-La =permeability of free space= 7n X w-7 (H·m- 1 ),

p = earth resistivity (Q·m).

Carson's solution to Equation (2.71) is defined by eight different infinite series which converge quickly for problems related to transmission line parameter calculation, but the number of required computations increases with frequency and separation of the conductors.

More recent literature has described closed form formulations for the numerical evaluation of line-ground loops, based on the concept of a mirroring surface beneath the earth at a certain depth. The most popular complex penetration model which has had more appeal is that of C. Dubanton [11], due to its simplicity and high degree of accuracy for the whole frequency span for which Carson's equations are valid.

Dubanton's formulae for the evaluation of the self- and mutual impedances of conductors i and j are

z.. = jWf.-Lo x In (2(y; + p)) " 2n r; '

(2.72)

zij = jWf.-Lo X In ( J2(Y; + p) ) , 2TC j(yi- yj)2 +(Xi- Xj)Z

(2.73)

where p = 1/ J jwf.1,0 a is the complex depth below the earth at which the mirroring surface is located.

An alternative and very simple formulation has been recently proposed by Acha [12], which for the purpose of harmonic penetration yields accurate solutions when compared with those obtained using Carson's equations.

2.6.2 Geometrical impedance matrix [Zg] and admittance matrix [Yg]

If the conductors and the earth are assumed to be equipotential surfaces, the geometrical impedance can be formulated in terms of potential coefficients theory.

The self-potential coefficient 1/Jii for the ith conductor and the mutual potential coefficient 1/lij between the ith and jth conductors are defined as follows,

1/lii =In (2"y;/ri), (2.74)

(2.75)

where r; is the radius of the ith conductor (m) while the other variables are as defined earlier.

For bundle conductors, r; is replaced by the Geometrical Mean Radius (GMRi), given by

where

RBundle = radius of bundle,

n = number of conductors in bundle.

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34 2 TRANSMISSION SYSTEMS

The use of GMR ignores proximity effects, hence it is valid only if the subconductor is much smaller than the spacing between phases of the line.

Potential coefficients depend entirely on the physical arrangement of the conductors and need only be evaluated once.

For practical purposes the air is assumed to have zero conductance and

[Zg] = jwK'[l/r] Q·km- 1, (2.76)

where [ 1/r] is a matrix of potential coefficients and K' = 2 x 1 o-4 .

The lumped shunt admittance parameters [Y] are completely defined by the inverse relation of the potential coefficients matrix, i.e.

(2.77)

where Ea =permittivity of free space= 8.857 X IQ-12 (F·m-l ). As [Zg] and [Yg] are linear functions of frequency, they need only be evaluated once

and scaled for other frequencies.

2.6.3 Conductor impedance matrix [Zc]

This term accounts for the internal impedance of the conductors. Both resistance and inductance have a non-linear frequency dependence. Current tends to flow on the surface of the conductor, this skin effect increases with frequency and needs to be computed at each frequency. An accurate result for a homogeneous non-ferrous conductor of annular cross-section involves the evaluation of long equations based on the solution of Bessel functions,

where

jwfJ.,o 1 lo(Xe)N~(xi)- No(xe)l~(xi)

Zc = ~ Xe l~(xe)N~(xi)- N~(xe)l~(xi) '

Xe = j y' }Wt£oacre,

Xi = jy'jwfJ.,oacr;,

re =external radius of the conductor (m),

r; =internal radius of the conductor (m),

J 0 = Bessel function of the first kind and zero order,

J~ = derivative of the Bessel function of the first kind and zero order,

N 0 = Bessel function of the second kind and zero order,

N~ = derivative of the Bessel function of the second kind and zero order,

(2.78)

ac =conductivity of the conductor material at the average conductor temperature.

The Bessel functions and their derivatives are solved, within a specified accuracy, by means of their associated infinite series. Convergence problems are frequently encoun­tered at high frequencies and low ratios of conductor thickness to external radius, i.e. (re- r;)/re, necessitating the use of asymptotic expansions.

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2.6 EVALUATION OF OVERHEAD LINE PARAMETERS 35

A new closed form solution has been proposed based on the concept of complex penetration [10]; unfortunately errors of up to 6.6% occur in the region of low order harmonic frequencies.

To overcome the difficulties of slow convergence of the Bessel function approach and the inaccuracy of the complex penetration method at relatively low frequency, an alternative approach based upon curve fitting to the Bessel function formula has been proposed by Acha [12].

Lewis and Tuttle [13] presented a practical method for calculating the skin effect resistance ratio by approximating ACSR (aluminium conductor steel reinforced) conductors to uniform tubes having the same inside and outside diameters as the aluminium conductors, see Figure 2.25(a). Figure 2.25(b) illustrates the skin effect ratio for different models and various tube ratios for ACSR conductors. Skin effect

(a)

(b)

Steel strand core

Aluminium strands

5.0 .-------------------.,

<i cr:-o

4.0

<i 3.0 cr:'"

2.0

I I

I

tlr= 1.0

Breuer I

I I

I

'\...I

I I

I

I

I I

I

tlr= 0.4 I

I

tlr=0.2 I 1

I I I I

I I

I / I I I

~V 1/ /// I/ I

I; / ;j /

/, / / h / l/ /_ .....

50 100 150 200 250 300

Figure 2.25 ACSR hollow conductor: (a) conductor geometry; (b) skin effect resistance for different models

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36 2 TRANSMISSION SYSTEMS

modelling is important for long lines. Although the series resistance of a transmission line is typically a small component of the series impedance, it dominates its value at resonances.

2.6.4 Series impedance approximation for electromagnetic transients

Based on Semlyen's complex depth of penetration concept, PSCAD/EMTDC uses the following expressions for the self and mutual series impedance parameters of the line (on the assumption that De « D;j in Figure 2.24):

Z;j = jWJLo (In (D;~) +~In (1 + 4De(Y; + ;j +De))) Q.m-1, (2.79) 2:n: d,1 2 Dij

jwJLo ( (D;;) 0.3565 PJL coth- 1(0.777RcM)) _1 Z;; = -- In - + -- + Q-m , (2.80) 2:n: r; :n:J?l:: 2:n:~

where

Rc and ~ are the conductive and external conductor radii,

M = j jwJLo, Pc

De = ftfi· Pc = conductor resistivity (Q·m),

Pg ground resistivity (Q·m).

2.7 Underground and Submarine Cables [14]

A unified solution similar to that of overhead transmission is difficult for underground cables because of the great variety in their construction and layouts.

The cross-section of a cable, although extremely complex, can be simplified to that of Figure 2.26 and its series per unit length harmonic impedance is calculated by the following set of loop equations.

where

z~1

z~ore-outside

Z ~ore- insulation

Z ~heath- inside

[J is the sum of the following three component impedances,

is the internal impedance of the core with the return path

outside the core,

is the impedance of the insulation surrounding the core,

is the internal impedance of the sheath with the return path

inside the sheath.

(2.81)

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2.7 UNDERGROUND AND SUBMARINE CABLES [14] 37

Loop 1

Loop 2

Loop 3

Figure 2.26 Cable cross-section

Similarly

z;2 = z~heath-outside + z~heathjarmour-insulation + z~rmour-inside• (2.82)

and

z~3 = z:.rmour-outside + z:.rmourjearth-insulation + z~arth-inside· (2.83)

The coupling impedances Z'12 = z;, and Z23 = Z32 are negative because of opposing current directions (/ 2 in negative direction in loop 1, and I 3 in negative direction in loop 2), i.e.

where

Z~z = z;I = -Z~heath-mutual•

z;3 = Z~z = -Z~rmour-mutual•

z~heath-mutual is the mutual impedance (per unit length) of the tubular sheath

between inside loop 1 and the outside loop 2.

(2.84)

(2.85)

z:.rmour-mutual is the mutual impedance (per unit length) of the tubular armouc

between the inside loop 2 and the outside loop 3.

Finally, z;3 = Z3 1 = 0 because loop 1 and loop 3 have no common branch. The impedances of the insulation are given by

, . f..L ( routside) Zinsulation = )W-2 ln --

lf rinside (2.86)

where

f..L is the permeability of insulation in H-m- 1,

routside is the outside radius of insulation,

rinside is the inside radius of insulation.

If there is no insulation between the armour and earth, then z;nsulation = 0.

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38 2 TRANSMISSION SYSTEMS

The internal impedances and the mutual impedance of a tubular conductor are a function of frequency, and can be derived from Bessel and Kelvin functions.

with

z;ube-inside = Jlwfl Uo( J}mq)K l ( J}mr) + Ko( J}mq)I l ( J}mr)], (2.87a) 2nmqD

z;ube-outside = Jlwfl Uo( J}mr)KI ( J}mq) + Ko( J}mr)/1 ( J}mq)], (2.87b) · 2nmrD

Z' - __ w_fi __ tube-mutual - 2JTmqmrD' (2.87c)

(2.87d)

where

with

q = inside radius,

r = outside radius,

87! X 10-4 f fir K= I '

q s == -,

r

Rdc

R~c = d.c. resistance in Q·km- 1.

(2.88)

(2.89)

(2.90)

(2.91)

The only remaining term is Z~arth-inside in Equation (2.83) which is the earth return impedance for underground cables, or the sea return impedance for submarine cables. The earth return impedance can be calculated approximately with Equation (2.87a) by letting the outside radius go to infinity. This approach, also used by Bianchi and Luoni [15] to find the sea return impedance is quite acceptable considering the fact that sea resistivity and other input parameters are not known accurately.

Equation (2.81) is not in a form compatible with the solution used for overhead conductors, where the voltages with respect to local ground and the actual currents in the conductors are used as variables. Equation (2.81) can easily be brought into such a form by introducing the appropriate terminal conditions, namely with

V1 = Vcore- Vsheath• /1 = lcore.

V 2 = V sheath - V armour• I 2 = I core + I sheath,

and V 3 = Varmour• I 3 = I core + I sheath + 1 armour·

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2.8 THREE-PHASE MODELS OF TRANSFORMERS

Equation (2.81) can be rewritten as

where

[ dVcore/dx l [Z~c Z~,

dVsheath/dx = Z~c Z~,

dVarmour/dx Z~c Z~s

[ I core l

!sheath ,

I armour

z~c = Z'11 + 2Z'12 + z;2 + 2z;3 + z;3,

z~, = z~c = Z'12 + z;2 + 2Z;3 + z;3,

z~a = z~c = z~a = z~, = z;3 + z;3,

z~, = z;2 + 2z;3 + z;3,

z~a = z;3.

39

(2.92)

Because a good approximation for many cables having bonding between the sheath and the armour, and the armour earthed to the sea, is V sheath = V armour = 0, the system can be reduced to

-dVcore/dx = Zlcore, (2.93)

where Z is a reduction of the impedance matrix of Equation (2.92). Similarly, for each cable the per unit length harmonic admittance is:

[ dl I/dx l [}we;

- d/2/dx = 0

d/3/dx 0

0

(2.94) }we; 0

where c; = 2n£0 £r/ln(r/q). Therefore, when converted to core, sheath and armour quantities,

-Y'1 0 l Y' + Y; -Y'

Y' Y' + 2Y' - 2 2 3

[ dlcore/dx l [ Y'1

- dlsheath/dx = -Y',

dl armour/ dx 0

[ V core l

V sheath ,

Varmour

(2.95)

where Y; = jwli. If, as before, Vsheath = Varmour = 0, Equation (2.95) reduces to

(2.96)

Therefore, for frequencies of interest, the cable per unit length harmonic impedance, Z', and admittance, Y', are calculated with both the zero and positive sequence values being equal to the Z in Equation (2.93), and the Y' in Equation (2.96), respectively.

In the absence of rigorous computer models, such as described above, power compa­nies often use approximations to the skin effect by means of correction factors. Typical corrections used by the NGC (UK) and EDF (France) are given in Table 2.2.

2.8 Three-phase Models of Transformers

The inherent assumption that the transformer is a balanced three-phase device is justi­fied in the majority of practical situations, and traditionally, three-phase transformers are represented by their equivalent sequence networks.

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40 2 TRANSMISSION SYSTEMS

Table 2.2 Corrections for skin effect in cables

Company Voltage (kV) Harmonic order Resistance

NGC 400, 275 (Based on 2.5 h c 1.5 0.74R1 (0.267 + 1.073.Jh) sq.in. conductor at 5 in. spacing between centres)

132 h c 2.35 R1 (0.187 + 0.532.Jh) EDF 400, 225 hc2 0.74R1 (0.267 + 1.073.Jh)

150, 90 hc2 R1 (0.187 + 0.532.Jh)

More recently, however, methods have been developed [3,4] to enable all three-phase transformer connections to be accurately modelled in phase co-ordinates. In phase co­ordinates, no assumptions are necessary although physically justifiable assumptions are still used in order to simplify the model. The primitive admittance matrix, used as a basis for the phase co-ordinate transformer model is derived from the primi­tive or unconnected network for the transformer windings and the method of linear transformation enables the admittance matrix of the actual connected network to be found.

2.8.1 Primitive admittance model of three-phase transformers

Many three-phase transformers are wound on a common core and all windings are, therefore, coupled to all other windings. Therefore, in general, a basic two-winding three-phase transformer has a primitive or unconnected network consisting of six coupled coils. If a tertiary winding is also present the primitive network consists of nine coupled coils. The basic two-winding transformer shown in Figure 2.27 is now considered, the addition of further windings being a simple but cumbersome extension of the method.

The primitive network, Figure 2.28, can be represented by the primitive admittance matrix which has the following general form:

yll y12 yl3 Y14 Y,s Y16

y21 y22 y23 y24 Y2s y26

y3i y32 y33 y34 Y3s y36 = (2.97)

y4i y42 y43 y44 Y4s y46

Ys, Ys2 Ys3 Ys4 Yss Ys6

y6i y62 y63 y64 Y6s y66

The elements of matrix [Y] can be measured directly, i.e. by energizing coil i and short-circuiting all other coils, column i of [Y] can be calculated from Yki = h/Vi.

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2.8 THREE-PHASE MODELS OF TRANSFORMERS 41

Figure 2.27 Diagrammatic representation of two-winding transformer

Figure 2.28 Primitive network of two-winding transformer. Six coupled coil primitive network. (Note the dotted coupling represents parasitic coupling between phases)

Considering the reciprocal nature of the mutual couplings in Equation (2.97), 21 short circuit measurements would be necessary to complete the admittance matrix. Such a detailed representation is seldom required.

By assuming that the flux paths are symmetrically distributed between all windings, Equation (2.97) may be simplified to Equation (2.98).

Yp Y' m Y' m -Ym Y" m Y" m

Y' m Yp Y' m Y" m -Ym Y" m

Y' m Y' m Yp Y" m Y" m -Ym

-Ym Y" m Y" m Ys Y"' m ym

m

Y" m -Ym Y" m Y"' m Ys Y"' m

Y" m Y" m -Ym ym m Y"' m Ys

where

y~ is the mutual admittance between primary coils;

y~ is the mutual admittance between primary and secondary coils

on different cores;

y::; is the mutual admittance between secondary coils.

(2.98)

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42 2 TRANSMISSION SYSTEMS

where Yp; = y!af, Ysj = y!fiy and Mij = yla;fi; for i = 1 ,2 or 3 and j = 4,5 or 6

Figure 2.29 Primitive network

For three separate single-phase units, all the primed values are effectively zero. In three-phase units, the primed values, representing parasitic interphase coupling, do have a noticeable effect. This effect can be interpreted through the symmetrical component equivalent circuits.

If the values in Equation (2.98) are available, then this representation of the primitive network should be used. If interphase coupling can be ignored, the coupling between a primary and a secondary coil is modelled as for the single-phase unit, giving rise to the primitive network of Figure 2.29.

The new admittance matrix equation is

Yp1 M14

Ypz Mzs

Yp3 M36 (2.99)

M41 Ys4

Msz Yss

M63 Ys6

2.8.2 Models for common transformer connections

The network admittance matrix for any two-winding three-phase transformer can now be formed by the method of linear transformation.

As a simple example, consider the formation of the admittance matrix for a star-star connection with both neutrals solidly earthed in the absence of interphase mutuals. This example is chosen as it is the simplest computationally.

The connection matrix is derived from consideration of the actual connected network. For the star-star transformer, illustrated in Figure 2.30, the connection matrix [C) relating the branch voltages (i.e. voltages of the primitive network) to the node voltages

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2.8 THREE-PHASE MODELS OF TRANSFORMERS 43

Figure 2.30 Network connection diagram for three-phase star-star transformer

(i.e. voltages of the actual network) is a 6 x 6 identity matrix, i.e.

1

1

1 (2.100)

1

1

1

The nodal admittance matrix [Y]NODE is given by:

[Y]NODE = [C]'[Y]PRIM[C]. (2.101)

Substituting for [ C] yields: [Y]NODE = [Y]PRIM· (2.102)

Let us now consider the Wye G-Delta connection, illustrated in Figure 2.31. The following connection can be written by inspection between the primitive branch volt­ages and the node voltages:

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0 (2.103)

0 0 0 1 -1 0

0 0 0 0 1 -1

0 0 0 -1 0 1

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44 2 TRANSMISSION SYSTEMS

Figure 2.31 Network connection diagram of Wye G-Delta transformer

or

[V]branch = [ C][V]node' (2.104)

we can also write

[V]NODE = [C]'[V]PRIM[C], (2.105)

and using [Y]PRIM from Equation (2.98)

Yv y~ y~ -(Ym + y:;,) (Ym + y:;,) 0 a

y~ Yv y~ 0 -(ym + y:;,) (Ym + y:;,) b

y~ y~ Yv (Ym + y:;,) 0 -(Ym + y:;,) c

[Y]NODE = -(Ym + y:;,) 0 (Ym + y:;,) 2(y,- y~) -(y,- y~) -(y,- y~) A

(Ym + y:;,) -(Ym + y:;,) 0 -(y,- y~) 2(y,- y~) -(y,- y~) B

0 (Ym + y:;,) -(Ym + y:;,) -(y,- y~) -(y,- y~) 2(y,- y~) c

(2.106) Moreover, if the primitive admittances are expressed in per unit, with both the

primary and secondary voltages being one per unit, the Wye-Delta transformer model must include an effective turns ratio of ,J3. The upper right and lower left quadrants of matrix (2.106) must be divided by ,J3 and the lower right quadrant by 3.

In the particular case of three-single phase transformer units connected in Wye G­Delta all the y' and y" terms will disappear. Ignoring off-nominal taps (but keeping in mind the effective ,J3 turns ratio in per unit) the nodal admittance matrix equation

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2.8 THREE-PHASE MODELS OF TRANSFORMERS 45

relating the nodal currents to the nodal voltages is:

y -yj./3 yj./3

y -yj./3 y./3

y yj./3 -yj./3

-yj./3 yj./3 h I -3y I -3y

yj./3 -yj./3 I -3y h I -3y

yj./3 -yj./3 I -3y I -3y ~y (2.107)

where Y is the transformer leakage admittance in p.u. An equivalent circuit can be drawn, corresponding to this admittance model of the transformer, as illustrated in Figure 2.32.

The large shunt admittances to earth from the nodes of the star connection are apparent in the equivalent circuit. These shunts are typically around 10 p.u. (for a 10% leakage reactance transformer).

The models for the other common connections can be derived following a similar procedure.

In general, any two-winding three-phase transformer may be represented using two coupled compound coils. The network and admittance matrix for this representation is illustrated in Figure 2.33.

It should be noted that

as the coupling between the two compound coils is bilateral. Often, because more detailed information is not required, the parameters of all three

phases are assumed balanced. In this case, the common three-phase connections are found to be modelled by three basic submatrices.

y!./3 '--------:::.tA

y/3

Primary Secondary

Figure 2.32 Equivalent circuit for star-delta transformer

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46 2 TRANSMISSION SYSTEMS

fE ~ Ypp Yps

Ysp Yss s s

[/p] [I.J

" )I

[V,l1 [Vpp] [V •• l rv,]

Figure 2.33 Two-winding three-phase transformer as two coupled compound coils

Table 2.3 Characteristic submatrices used in forming the transformer admittance matrices

Transformer Self- Mutual connection admittance admittance

Bus P BusS Yvv Yss Yps• Ysp WyeG WyeG yl yl -Y1 WyeG Wye yl yll/3 -Yll/3 WyeG Delta yl yll ylll Wye Wye yll/3 yll/3 -Yll/3 Wye Delta yll/3 yll ylll

Delta Delta yll yll -Y11

The submatrices, [Ypp] [Yps] etc., are given in Table 2.3 for the common connections, where

2Yt -yt -yt -yt Yt

-yt 2yt -yt ,Y111= -Yt Yt

-yt -Yt 2yt Yt -Yt

Finally, these submatrices must be modified to account for off-nominal tap ratio as follows:

(i) Divide the self-admittance of the primary by a2 .

(ii) Divide the self-admittance of the secondary by {32•

(iii) Divide the mutual admittance matrices by af3.

It should be noted that in the p.u. system, a delta winding has an off-nominal tap of ../3. For transformers with ungrounded Wye connections, or with neutral connected

through an impedance, an extra coil is added to the primitive network for each unearthed neutral and the primitive admittance matrix increases in dimension. By noting that the injected current in the neutral is zero, these extra terms can be eliminated from the connected network admittance matrix [16].

Once the admittance matrix has been formed for a particular connection it represents a simple subsystem composed of the two busbars interconnected by the transformer.

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2.8 THREE-PHASE MODELS OF TRANSFORMERS

2.8.3 Three-phase transformer models with independent phase tap control

47

Disregarding interphase mutual couplings, the per unit primitive admittance matrix in terms of the transformer leakage admittance (yti) is

Ytl Ytl --a2

I a1

Ytz Ytz --a2

2 a2

Yt3 Yt3 - --

[Ypriml = a2 a3 3

Ytl Ytl --

a1

Ytz Ytz --

az

Yt3 Yt3 --

a3

where a1. a2 and a3 are the off-nominal taps on windings 1, 2 and 3, respectively. In addition, any windings connected in delta will, because of the per unit system, have an effective tap of ,J3.

The nodal admittance matrix for the transformer windings is:

where [C] is the connection (windings to nodes) matrix. As an example, [Ynode] for a star-delta transformer with earthed neutral is as follows:

Yu -yu Ytl az

I v'3at .J3al

Ytz -ytz Ytz --;;r

2 .J3az .J3az

Yt3 Yt3 -Yt3 az 3 .J3a3 .J3a3

[Ynode] = -Ytl Yt3 Ytl + Yt3 -Ytl -Yt3

--

.J3al .J3a3 3 3 3

Ytl -Ytz -yu Ytz + Ytl -Ytz -- --.J3al v'3az .J3 3 3

Ytz -Yt3 -Yt3 -ytz Ytz + Yt3 --

.J3az .J3a3 3 3 3

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48 2 TRANSMISSION SYSTEMS

2.8.4 Sequence components modelling of three-phase transformers

In most cases, lack of data will prevent the use of the general model based on the prim­itive admittance matrix and will justify the conventional approach in terms of symmet­rical components. Let us now derive the general sequence components equivalent circuits and the assumptions introduced in order to arrive at the conventional models.

With reference to the Wye G-Delta common-core transformer of Figure 2.31, repre­sented by Equation (2.1 06), and partitioning this matrix to separate self and mutual elements, the following transformations apply:

Primary side:

where

and a = ei2rr/3 .

Therefore

Secondary side:

Yr p y-1

Yo12 = s y:n y:n

1

1

1

Yr + 2y:n p -

Yo12- 0

0

y:n y:n Yp y:n y:n Yr

1 1

a2 a

a a2

0 0

Yr- y:n 0 (2.108)

0 Yr- y:n

The delta connection on the secondary side introduces an effective ,J3 turns ratio and the sequence components admittance matrix is

2(Ys - y;;:) -(Ys - y;;:) -(Ys - y;;:) s 1y-

YoJ2 = 3 s I -(Ys - y;;:) 2(y5 - y;;:) -(Ys - y;;:)

-(Ys - y;;:) -(Ys - y;;:) 2(y5 - y;;:)

0 0 0

0 Ys - Y;;: 0 (2.109)

0 0 Ys - Y;;:

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2.8 THREE-PHASE MODELS OF TRANSFORMERS 49

Mutual terms:

The mutual admittance submatrix of Equation (2.106), modified for effective turns ratio, is transformed as follows:

yM - ,J3T_, 012- 3 s

0

0

0

-(Ym + Y~) (Ym + Y~)

0 -(ym + y~)

(Ym + Y~) 0

0

-(ym + y:;)L-30°

0

0

(Ym + Y~)

-(ym + y~)

0

0

-(Ym + y~)L30o

Recombining the sequence components submatrices yields:

Yp- y:r, -(Ym + y;;;)L30

Yp- y:r, -(Ym + y;;;)L-30

Yp + 2y:r,

= -(Ym + y;;;)L30 Ys- Y;;;

-(Ym + y;;;)L-30 Ys- y;;;

(2.110)

0

0

0

0

0

0

(2.111) Equation (2.111) can be represented by the three sequence networks of Figures 2.34,

2.35 and 2.36, respectively. In general, therefore, the three sequence impedances are different on a common-core

transformer. The complexity of these equivalent models is normally eliminated by the following

simplifications:

• The 30° phase shifts of Wye-Delta connections are ignored.

Wye G Delta

Figure 2.34 Zero-sequence node admittance model for a common-core grounded Wye-Delta transformer (3) (©1982 IEEE)

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50 2 TRANSMISSION SYSTEMS

Figure 2.35 Positive-sequence node admittance model for a common-core grounded Wye-Delta transformer (3) (©1982 IEEE)

WyeG Delta

Figure 2.36 Negative-sequence node admittance model for a common-core grounded Wye-Delta transformer (3) (©1982 IEEE)

Table 2.4 Typical symmetrical-component models for the six most common connections of three-phase transformers (3) (© 1982 IEEE)

Bus P Bus Q Pos Seq Neg Seq Zero Seq

Wye G Wye G p Zsc Q o--uu.ure

p Zsc Q p Zsc Q o--uu.ure o--uu.ure

~~~

Wye G Wye p Zsc 0 p Zsc 0 p Zsc 0 o--uu.ure o--uu.ure ~ •

WyeG Delta

Wye Wye P. Zsc 0 p Zsc 0 p Zsc 0 o--uu.ure o--uu.ure ~ •

~~~

Wye Delta p Zsc 0 p Zsc 0 p Zsc 0 o--uu.ure o--uu.ure ~ •

~~~

Delta Delta p Zsc 0 p Zsc 0 p Zsc 0 o--uu.ure o--uu.ure ~ •

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2.10 REFERENCES 51

• The interphase mutuals admittances are assumed equal, i.e. y~ = y~ = y;;:. These are all zero with uncoupled single-phase units.

• The differences (yp - Ym) and (Ys - Ym) are very small and are, therefore, ignored.

With these simplifications, Table 2.4 illustrates the sequence impedance models of three-phase transformers in conventional steady-state balanced transmission system studies.

2.9 Formation of the System Admittance Matrix

It has been shown that the element (and subsystem) admittance matrices can be manipu­lated efficiently if the three nodes at the bus bar are associated together. This association proves equally helpful when forming the admittance matrix for the total system.

The subsystem, as defined in section 2.4, may have common busbars with other subsystems, but may not have mutual couplings terms to the branches of other subsys­tems. Therefore, the subsystem admittance matrices can be combined to form the overall system admittance matrix as follows:

• The self-admittance of any busbar is the sum of all the individual self-admittance matrices at that busbar.

• The mutual admittance between any two busbars is the sum of the individual mutual admittance matrices from all the subsystems containing those two nodes.

2.10 References

I. Clarke, E, (1943). Circuit Analysis of a. c. Power Systems, Vol. 1, John Wiley and Sons, New York.

2. Kron, G, (republished in 1965). Tensor Analysis of Networks, MacDonald, London. 3. Chen, M S and Dillon, WE, (1974). Power-system modelling, Proceedings of the IEEE,

62, (7), pp. 901. 4. Laughton, M A, ( 1968). Analysis of unbalanced polyphase networks by the method of phase

co-ordinates, Part I. System representation in phase frame of reference, Proceedings of the lEE, 115, (8), pp. 1163-1172.

5. Kimbark, E W, (1950). Electrical Transmission of Power and Signals, John Wiley and Sons, New York.

6. Wedepohl, L M and Wasley, R G, (1966). Wave propagation in multiconductor overhead lines, Proceedings of the lEE, 113, (4), pp. 627-632.

7. Bowman, K J and McNamee, J M, (1964). Development of equivalent pi and T matrix circuits for long untransposed transmission lines, IEEE Transactions on Power Apparatus and Systems, PAS-84, pp. 625-632.

8. Wilkinson, J Hand Reinsch, (1971). Handbook for Automatic Computations, Vol. Jl, Linear Algebra, Springer-Verlag, Berlin.

9. Carson, J R, (1926). Wave propagation in overhead wires with ground return, Bell Systems Technical Journal, 5, pp. 539-556.

10. Deri, A, Tevan, G, Semlyen, A and Castanheira, A, (1981). The complex ground return plane, a simplified model for homogeneous and multi-layer earth return, IEEE Transactions on Power Apparatus and Systems, PAS-100, pp. 3686-3693.

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52 2 TRANSMISSION SYSTEMS

11. Semlyen, A and Deri, A, (1985). Time domain modelling of frequency dependent three­phase transmission line impedance, IEEE Transactions on Power Apparatus and Systems, PAS-104, pp. 1549-1555.

12. Acha, E, (1988). Modelling of power system transformers in the complex conjugate harmonic space, Ph.D. thesis, University of Canterbury, New Zealand.

13. Lewis, VA and Tuttle, P D, (1958). The resistance and reactance of aluminium conductors steel-reinforced, IEEE Transactions on Power Apparatus and Systems, PAS-77, pp. 1189-1215.

14. Dommel, H W, (1978). Line constants of overhead lines and underground cables, Course E.E. 553 notes, University of British Columbia.

15. Bianchi, G and Luoni, G, (1976). Induced currents and losses in single-core submarine cables, IEEE Transactions on Power Apparatus and Systems, PAS-95, pp. 49-58.

16. Dillon, WE and Chen, M S, (1972). Transformer modelling in unbalanced three-phase networks, IEEE Summer Power Meeting, Vancouver.