Equivalent circuit in actual physical units

The conventional equivalent circuit of a single-phase two-winding transformer is shown in Figure 4.3(a).

Where all quantities shown are in physical units of volts, amps and ohms, and Z_H = R_H + jX_H, R_H = HV winding resistance, X_H = HV winding leakage reactance. Z_L = R_L + jX_L, R_L = LV winding resistance, X_L = LV winding leakage reactance. R_I = resistance representing core iron losses which are assumed to vary with the square of applied HV winding voltage. X_M = core magnetising reactance referred to the HV side and representing the rms value of the magnetising current. N_H and N_L are the actual number of turns of HV and LV windings, respectively. N_H(nominal) and N_L(nominal) are HV and LV winding number of turns at nominal tap positions, respectively.

From basic transformer theory and since the same flux will thread both HV and LV windings, the induced voltage per turn on the HV and LV windings are equal, hence

     (4.1a)

In the derivation of the equivalent circuit, we will initially neglect the ampere-turns or MMF due to the exciting shunt branch that represents the very small core iron and magnetising losses. The HV winding and LV winding MMFs, for the current directions shown in Figure 4.3(a) are related by

     (4.1b)

Also, the voltage drop equations for the transformer HV and LV windings can be written as follows:

     (4.2a)

     (4.2b)

Using Equations (4.1a) and (4.2b), Equation (4.2a) can be rewritten as

     (4.3)

Substituting Equation (4.1b) into Equation (4.3), we have

     (4.4a)

or

     (4.4b)

where

     (4.4c)

Z_HL is the equivalent transformer leakage impedance referred to the HV side and is the LV impedance referred to the HV side. Figure 4.3(b) is drawn using Equation (4.4a) ignoring the shunt exciting branch. The LV voltage V_L and LV current I_L are referred to the HV side as N_VHV_L and I_L/N_HL, respectively. The exciting shunt branch can be reinserted between on the input HV terminals. Alternatively, Z_H could have been referred to the LV side and the exciting branch inserted similarly. The exciting impedance is much bigger than the leakage impedances of either winding by almost a factor of 400–1. However, modern short-circuit, power flow and transient stability analyses practice includes the iron loss branch as well as the magnetising branch.

Equivalent circuit in per unit

In many types of steady state analysis, we are interested in the transformer equivalent circuit where all quantities are in per unit on some defined base quantities. Let the base quantities be defined as

     (4.5a)

     (4.5b)

In Equation (4.5b), the HV and LV windings’ nominal tap positions are chosen to correspond to the HV and LV base voltages of the network sections to which the windings are connected.

Let us now define the following per-unit quantities:

     (4.6a)

     (4.6b)

     (4.6c)

Dividing Equation (4.3) by S_(B) and using Equations (4.5) and (4.6), we obtain

which can be rewritten as

     (4.7a)

where the pu tap ratios of the HV and LV windings are given by

     (4.7b)

and

     (4.7c)

We will now assume that the impedance of each winding is proportional to the square of the number of turns of the winding. We know from basic transformer theory that this is generally valid for the winding inductive reactance but not for the winding resistance. However, because the resistance is generally much smaller than the reactance, this approximation is accepted. Thus, we define Z_H(nominal) as the pu HV winding impedance at nominal HV winding tap position and Z_L(nominai) as the pu LV winding impedance at nominal LV winding tap position. Therefore, for each winding, we have

     (4.8a)

     (4.8b)

Substituting Equation (4.8) into Equation (4.7a), we obtain

     (4.9)

The general pu equivalent circuit shown in Figure 4.4(a) represents Equation (4.9).

In practice, we are more interested in deriving an equivalent where both impedances appear on one side of the ideal transformer. This can be done once we have derived the relationship between the HV and LV pu currents from Equation (4.1b) as follows:

which, using Equation (4.5b), can be written as

or, using Equations (4.7b) and (4.7c), reduces to

     (4.10)

Substituting Equation (4.10) into Equation (4.9) and rearranging, we obtain the following two equations

     (4.11a)

and

     (4.11b)

where

     (4.11c)

     (4.11d)

     (4.11e)

     (4.11f)

Figure 4.4(b) and (c) represent Equations (4.11a) and (4.11b), respectively. Both equivalents are identical in terms of representing the general equivalent circuit of an off-nominal ratio transformer. However, in practice, it is more convenient to use one or the other depending on the actual transformer characteristics. That is, Figure 4.4(b) would be used to represent a transformer with the following characteristics:

Figure 4.4(c), however, would be used to represent a transformer with the following characteristics:

We will make a number of important practical observations that apply to Figure 4.4(b) and (c). Considering Figure 4.4(b), and in the case of a tap-changer located on the HV winding with a fixed nominal or off-nominal ratio on the LV winding, then only t_H changes but Z_LH does not change with HV tap position. Similar observation applies to Figure 4.4(c). In both circuits, the variable tap ratio of the ideal transformer, t_HL or t_LH, is located on the side equipped with the on-load tap-changer. That is, it is located away from the transformer impedance which has a value that correspond to the fixed nominal or off-nominal ratio of the other side. It should be noted that the location of the variable tap ratio is a purely arbitrary choice and it could have been placed on the impedance side provided that Equations (4.11a) and (4.11b) continue to be satisfied.

A nominal-ratio-transformer is one where the chosen base voltages of the network sections to which the transformer is connected are the same as the transformer turns ratio or rated voltages. For such a transformer, Equation (4.7) reduces to t_H = 1 and t_L = 1, hence the ideal transformers shown in Figure 4.4(b) and (c) can be eliminated as shown in Figure 4.4(d).

However, for many transformers used in power systems, the transformer voltage ratio is not the same as the ratio of base or rated voltages of the associated network sections. Therefore, the choice of voltage base quantities for the network has to be made independent of the transformer actual voltage ratio or turns ratio. The most common reason for this is that a very large number of transformers used in power systems are equipped with tap-changers, on the HV or LV winding, whose function is to vary the number of turns of their associated winding. In addition, some distribution transformers, in particular, can have one winding with a fixed off-nominal ratio.

To ensure a proper understanding of the concepts introduced in deriving Figure 4.4, we will present a number of practical cases. In the first case, consider a single-phase 11 kV/0.25 kV distribution transformer interconnecting an 11 kV network and a 0.25 kV network. The transformation ratio or turns ratio of 11–0.25 kV corresponds exactly to the network base voltages. Therefore, Figure 4.4(d) represents such a transformer. We also note that with the transformer on open circuit, V_H(pu) = V_L(pu) but with the transformer on-load, I_H(pu) = −I_L(pu).

In the second case, we consider a 10 kV/0.25 kV single-phase transformer interconnecting an 11 kV network and a 0.25 kV network. The HV winding is clearly a fixed off-nominal-ratio winding. The pu tap ratio t_HL in Figure 4.4(b) is calculated as

In the third case, we consider a transformer equipped with an on-load tap-changer on the HV side and we will also consider the effect on the transformer impedance. Consider a 239 MVA, 231 kV/21.5 kV single-phase transformer (this is actually one of a 717 MVA, 400 kV/21.5 kV three-phase bank) having a nominal-ratio leakage reactance of 15% on rating (the resistance is ignored for the purposes of this example but not in practical calculations). The transformer is connected to a network with 231 and 21.5 kV base voltages. The transformer, therefore, has a fixed LV nominal-ratio, i.e. t_L = 1. The tap-changer is located on the 231 kV winding hence t_H is variable. We will calculate the transformer reactance and redraw the equivalent circuit of Figure 4.4(b) to correspond to the transformation ratios 207.9 kV/21.5 kV, 231 kV/21.5 kV and 254.1 kV/21.5 kV The three voltage transformation ratios correspond to t_H values of 0.9, 1.0 and 1.1, respectively. Clearly, the transformer impedance value does not change from nominal as illustrated in Figure 4.5(a). It is instructive for the reader to demonstrate that if the tap ratio is placed on the impedance side, then, by making appropriate use of Figure 4.4(c), the reactance values that correspond to t_H = 0.9, 1.0 and 1.1 would become 12.15%, 15% and 18.15%, respectively. This is illustrated in Figure 4.5(b). The reader should also demonstrate that the corresponding circuits of Figure 5.4(a) and (b) are equivalent to each other. This can be done by calculating, for the corresponding equivalent circuits, the short-circuit impedances, i.e. the impedance seen from one side, e.g. H side with the other side, L side, short-circuited and vice versa. The calculated impedances will be identical.

The variation of the tap position of a transformer equipped with an on-load tap-changer results in a value of t_HL, for most transformers used in electric power systems, to be generally within 0.8 and 1.2 pu.

π Equivalent circuit model

The ideal transformer in the general pu equivalent circuit of an off-nominal-ratio transformer shown in Figure 4.4(b) and (c) is, although suitable for use on network analysers, not satisfactory for modern calculations that are almost entirely based on digital computation. Therefore, the ideal transformer must be replaced with a suitable equivalent circuit and this will be achieved by deriving an equivalent π circuit model for Figure 4.4(b). The reader is encouraged to do so for Figure 4.4(c). For convenience, we will drop the explicit pu notation and recall that all quantities are in pu. From Figure 4.4(b), we can write

or

     (4.12a)

also

or

     (4.12b)

or in matrix form

     (4.13a)

The admittance matrix of Figure 4.4(c) can be similarly derived and the result is

     (4.13b)

The equations of the general equivalent π circuit shown in Figure 4.6(a) are

     (4.14a)

     (4.14b)

or in matrix form

     (4.14c)

Equating the admittance terms in Equations (4.13a) and (4.14c), we have

     (4.15a)

and

     (4.15b)

and

     (4.15c)

Similarly, equating the admittance terms in Equations (4.13b) and (4.14c), we have

     (4.16a)

and

     (4.16b)

and

     (4.16c)

The equivalent π circuit models of the off-nominal-ratio transformer of Figure 4.4(b) and (c) are shown in Figure 6.4(b) and (c), respectively. The equivalent K circuit model parameters for the three conditions shown in Figure 4.5(a) are calculated and shown in Figure 4.7. Also shown in Figure 4.7 are the short-circuit impedances calculated from side H with side L short-circuited, and vice versa. It should not come as a surprise to the reader that these impedances are identical to the ones that the reader may have calculated, as recommended previously, for the equivalent circuits shown in Figure 5.4(a) and (b).

As mentioned before, it is generally no longer the case in modern industry practice to neglect the shunt exciting branch representing the iron losses and magnetising losses. The exciting branch admittance can be calculated from the corresponding impedance and inserted in parallel with either Y_B or Y_C or it can also be inserted as a shunt admittance midway through the series impedance. There is also a usual approximation to split it in half and connect each half at either end of the equivalent circuit. The admittance of the exciting or magnetising branch is given by

The general 2 × 2 admittance matrix is given by

     (4.17a)

whose elements, using Equation (4.15), are given by

     (4.17b)

whereas, using Equation (4.16), the matrix elements are given by

     (4.17c)

Winding connections

In three-phase double-wound power transformers, the three-phases of each winding are usually connected as star or delta. There are thus four possible connections of the primary and secondary windings namely star–star, star–delta, delta–star and delta–delta connections. Interconnected star or zig–zag windings are dealt with in Section 4.2.6.

PPS and NPS equivalent circuits

Like all static three-phase power system plant, the impedance such plant presents to the flow of PPS or NPS currents is independent of the phase rotation or sequence of the three-phase applied voltages R, Y, B or R, B, Y. Therefore, the plant PPS and NPS impedances are equal.

It will be shown later in this section that three-phase transformers can introduce a phase shift between the primary and secondary winding currents, and the same phase shift between the primary and secondary voltages, depending on the winding connections. In addition, where the phase shift of PPS currents and voltages is ϕ, it is -ϕ for NPS currents and voltages. The effect of this phase shift is to change the ideal transformer off-nominal turns ratio from a real to a complex number.

Therefore, the general PPS and NPS equivalent circuits of a three-phase two-winding transformer that correspond to the single-phase equivalent circuit of Figure 4.4(b) are shown in Figure 4.8(a) and (b), respectively. Similarly, the general PPS and NPS equivalent circuits of a three-phase two-winding transformer that correspond to the single-phase equivalent circuit of Figure 4.4(c) are shown in Figure 9.4(a) and (b), respectively. As discussed later in this section, the phase shift that may be introduced by a three-phase transformer is not required to be represented initially in short-circuit analysis (or PPS based power flow or stability analysis). Due account of any phase shifts can be made from knowledge of the location of transformers that introduce phase shifts in the system once the sequence currents and voltages are calculated by initially ignoring the phase shifts. Therefore, the transformer PPS and NPS equivalent circuits, ignoring the phase shift, are shown in Figures 4.8(c) and 4.9(c) and the corresponding π equivalent circuit model is the same as that in Figure 6.4(a) and (b). It should be noted that for a three-phase transformer, the transformer nominal turns ratio is equal to the ratio of the phase to phase base voltages on the HV and LV sides of the transformer irrespective of the primary and secondary winding connections. This means that the base voltage ratio and the nominal turns ratio are equal for star–star and delta-delta winding connections but also includes the factor for star–delta winding connection.

ZPS equivalent circuits

The ZPS equivalent circuit of a two-winding transformer is primarily dependent on the method of connection of the primary and secondary windings because the ZPS currents in each phase are equal in magnitude and are in phase. Also, the ZPS equivalent circuit and currents are affected by the winding earthing arrangements. In addition, a practical factor that can influence the magnitude of the ZPS impedances and make them appreciably different from the PPS impedances is the type of construction of the transformer magnetic circuit. The effect of transformer core construction will be dealt with later in this section, but we will now focus on the effect of the winding connection and any neutral earthing that may be present.

Before considering three-phase two-winding transformers of various winding connections, we will restate a basic understanding regarding ZPS currents in simple star-connected or delta-connected three sets of impedances. For ZPS currents to flow in a star-connected set of impedances, the neutral point of the star should be earthed either directly or indirectly, e.g. through an impedance. The ZPS current that flows to earth through this path is the sum of the ZPS currents in the three phases of the star impedances or three times the phase R ZPS current. For a delta-connected three-phase set of impedances, no ZPS currents will flow in the output terminals and hence inside the delta connection. However, ZPS currents can circulate inside the delta by mutual coupling but no ZPS current can emerge out into the output terminals of the delta. Figure 4.10(a) shows the impedance connections and ZPS current flows for the above three cases and Figure 4.10(b) shows the three-phase circuits and their corresponding ZPS equivalent circuits. It should be noted that because the voltage drop across the earthing impedance Z_E is 3I^ZZ_E, the effective earthing impedance appearing in the ZPS equivalent circuit is 3Z_E. The reader should easily demonstrate this.

We will now derive approximate ZPS equivalent circuits for two-winding transformers of various winding connection arrangements. We emphasise that these are approximate because for now we ignore the effect of transformer core or magnetic circuit construction on the magnetising impedance. This is an important practical aspect that should not be ignored in practical short-circuit analysis. This is discussed in detail in Section 4.2.8.

The basic and fundamental principle that underpins the derivation of the approximate ZPS equivalent circuits is based on Equation (4.1b). This states that there is always a magnetic circuit MMF or ampere-turn balance for the transformer primary and secondary windings. That is, no current can flow in one winding unless a corresponding current, allowing for the winding turns ratio, flows in the other winding. In deriving the ZPS equivalent circuits of two-winding transformers of various winding connections, the winding terminal is connected to the external circuit if ZPS current can flow into and out of the winding. If ZPS current can circulate inside the winding without flowing in the external circuit, the winding terminal is connected directly to the reference zero voltage node. Figure 4.11 shows ZPS equivalent circuits for an off-nominal ratio transformer that correspond to Figures 4.8(c) and 4.9(c). These take account of any HV and LV winding connection arrangements, and any neutral impedance earthing that may be used.

By making use of Figure 11.4(a) and (b), and noting that an arrow on a winding signifies the presence of a tap-changer on this winding, Figure 4.12 can be derived. This shows approximate ZPS equivalent circuits for the most common three-phase two-winding transformers used in power systems. It should be noted that Z_E is in per unit on the voltage base of the winding to which it is connected.

Returning to the example shown in Figure 4.5, and assuming the transformer is a three-phase two-winding star (HV winding)-delta (LV winding) transformer, the ZPS equivalent circuits that correspond to each tap ratio condition are shown in Figure 4.13. The reader should find it instructive to obtain the three equivalent circuits shown in Figure 4.13 from the corresponding equivalent π circuits shown in Figure 4.7 by applying a short-circuit at end L and calculating the equivalent impedance seen from the H side.

The elements of the admittance matrix of Equation (4.17a), for the transformer winding connections and their equivalent circuits shown in Figure 4.12, can be easily derived using the following equations:

The results are shown in Table 4.1. The reader is encouraged to derive these results.

Table 4.1 Elements of ZPS admittance matrix of two-winding three-phase transformers

Figure	Two-winding three-phase transformer winding connections	Elements of ZPS admittance matrix of Equation (4.19b)
4.12(a)		Y11 =Y12 =Y21 =Y22 = 0
4.12(b)		Y11 =Y12 =Y21 =Y22 =0
4.12(c)
4.12(d)
4.12(e)		Y11 =Y12 =Y21 =Y22 =0
4.12(f)		Y11 =Y12 =Y21 =Y22 =0
4.12(g)
4.12(h)

Effect of winding connection phase shifts on sequence voltages and currents

The effect of three-phase transformer phase shifts on the sequence currents and voltages will now be considered. The presence of a phase shift between the transformer primary and secondary voltages, and currents, depends on the transformer primary and secondary winding connections. For transformers with a star-star or a delta-delta winding connections, the primary and secondary currents, and voltages, in each of the three phases are either in phase or out of phase, i.e. the windings are connected so that the phase shifts are either 0° or ± 180°. The former case is illustrated in Figure 14.4(a) and (b). British and IEC practices use ‘vector group reference’ number and symbol. In the symbol Yd1, the capital and small letters Y and d indicate HV winding star and LV winding delta connections, respectively, and the digit 1 indicates a phase shift of −30° using a 12 × 30° clock reference. For example, 0° indicates 12 o’clock, 180° indicates 6 o’clock, −30° indicates 1 o’clock and +30° indicates 11 o’clock.

In Figure 4.14, the 0° phase shift is achieved by ensuring that the parallel windings, i.e. same phase windings, are linked by the same magnetic flux. Figure 4.14 also shows that the absence of phase shifts in the phase currents and voltages also translates into the PPS, and NPS, currents and voltages. Consequently, the presence of such transformers in the three-phase network requires no special treatment in the formed PPS and NPS networks under balanced or unbalanced conditions. It should be noted that for the delta winding, although a physical neutral point does not exist, a voltage from each phase terminal to neutral does still exist because the network to which the delta winding is connected would in practice contain a neutral point.

In the case of transformers with windings connected in star-delta (or delta-star), voltages and currents on the star winding side will be phase shifted by a ± 30° angle with respect to those on the delta side (or vice versa depending on the chosen reference). According to British practice, Yd11 results in the PPS phase-to-neutral voltages on the star side lagging by 30° the corresponding ones on the delta side. Also, Yd1 results in the PPS phase-to-neutral voltages on the star side leading by 30° the corresponding ones on the delta side. The example vector diagrams shown in Figure 4.15 for Yd1 and Yd1 1 illustrate this effect.

For RB Y/rby NPS phase sequence or rotation, Figure 4.15 also shows the effect of the Yd1 and Yd11 on the NPS voltage phase shifts and show that these are now reversed. These phase shifts also apply to the PPS and NPS currents in these windings because the phase angles of the currents with respect to their associated voltages are determined by the balanced load impedances only. In summary, where the PPS voltages and currents are shifted by +30°, the corresponding NPS voltages and currents are shifted by −30° and vice versa depending on the specified connection and phase shift, i.e. Yd1 or Yd11. Mathematically, this is derived for the Yd1 transformer shown in Figure 4.15 where n is the turns ratio as follows. The red phase current in amps flowing out of the d winding r phase is equal to I_r = n(I_R – I_B). Using Equation (2.9a) from Chapter 2 for phase currents and noting that because the in-phase ZPS currents cannot exit the d winding, we can write

or

where

     (4.18a)

or

     (4.18b)

or in per unit, where

     (4.18c)

Similarly, from Figure 4.15, the phase-to-neutral voltage in volts on the star winding R phase is

and using Equation (2.9b) for phase r and y voltages, we have

or

where

     (4.19a)

or

     (4.19b)

or in per unit, where

     (4.19c)

The reader is encouraged to derive the equations for the Yd11 transformer.

The American standard for designating winding terminals on star–delta transformers requires that the PPS (NPS) phase-to-neutral voltages on the high voltage winding to lead (lag) the corresponding PPS (NPS) phase-to-neutral voltages on the low voltage winding. This is so regardless of whether the star or the delta winding is on the high voltage side. In terms of sequence analysis, this means that when stepping up from the low voltage to the high voltage side of a star–delta or delta–star transformer, the PPS voltages and currents should be advanced by 30° whereas the NPS voltages and currents should be retarded by 30°. It is interesting to note the following observation on the British and American standards. In American practice, when the star winding in a star–delta transformer is the high voltage winding, this would correspond, in terms of phase shifts, to the Yd1 in British practice. However, when, in American practice, the delta winding in a star–delta transformer is the high voltage winding, this would correspond, in terms of phase shifts, to the Yd11 in British practice.

In terms of fault analysis in power system networks using the PPS and NPS networks, it is common practice to initially ‘ignore’ the phase shifts introduced by all star–delta transformers by assuming them as equivalent star–star transformers and to calculate the sequence voltages and currents on this basis. Then, having noted the locations in the network of such star–delta transformers, the appropriate phase shifts can be easily applied using the above equations as appropriate for the specified Yd transformer.

Winding connections

There are several winding connections for three-phase three-winding transformers which may be used in power systems. Some examples are YNdd, Ydd, YNynyn, YNynd, YNyd and Yyd.

PPS and NPS equivalent circuits

In a three-winding transformer, the three windings are generally mutually coupled although the degree of coupling between the two LV windings of a double secondary transformer can be chosen by design so that the LV windings are either closely or loosely coupled. Figure 4.16(a) shows a single-phase representation of a three-winding three-phase transformer ignoring, for now, the shunt exciting impedance.

From Figure 4.16(a), the following equations in actual physical units can be written

     (4.20a)

     (4.20b)

     (4.20c)

     (4.20d)

Neglecting the no-load current, I₀ = 0, and dividing Equation (4.20d) by N₃, we obtain

     (4.21a)

where

     (4.21b)

Also

     (4.21c)

Using Equations (4.21b), (4.21c) and (4.20c), Equations (4.20a) and (4.20b) become

     (4.22a)

     (4.22b)

In most power system steady state analysis, we are more interested in deriving the three-winding transformer equivalent circuit in pu rather than in actual physical units. The following base and pu quantities are defined

     (4.23a)

     (4.23b)

     (4.23c)

     (4.24a)

     (4.24b)

     (4.24c)

Using Equations (4.23) and (4.24), Equations (4.22a) and (4.22b) become

     (4.25a)

     (4.25b)

Equations (4.25a) and (4.25b) can be rewritten as follows:

     (4.26a)

     (4.26b)

where, the following pu tap ratios are defined

     (4.27a)

     (4.27b)

     (4.27c)

Now, as in the case of two-winding transformers, we define

     (4.28a)

     (4.28b)

     (4.28c)

Substituting Equation (4.28) into Equation (4.26), we obtain

     (4.29a)

     (4.29b)

Using Equations (4.24c) and (4.27) in Equation (4.21a), we obtain

     (4.29c)

Equations (4.29) are represented by the equivalent circuit shown in Figure 4.16(b) where we have replaced the three-winding transformer with three two-winding transformers connected as three branches in a star configuration. Each branch corresponds to a winding and has its own general off-nominal tap ratio to represent a nominal, fixed off-nominal ratio or an on-load tap-changer, as required. The impedance of the magnetising branch Y₀ can be inserted at the fictitious point P in the star equivalent circuit.

For improved applications in practice, Figure 4.16(b) can be further simplified to include only two off-nominal tap ratios instead of three as follows. Let

     (4.30a)

     (4.30b)

     (4.31a)

     (4.31b)

and

     (4.31c)

Substituting Equations (4.30) and (4.31) into Equation (4.26), we obtain

     (4.32a)

     (4.32b)

also, from Equation (4.29c), we have

     (4.32c)

Equations (4.32) are represented by the equivalent circuit shown in Figure 4.16(c) where, again, the three-winding transformer is replaced by three two-winding transformers connected as three branches in a star configuration. However, branch 3 now has an effective tap ratio of unity, t_33(pu) = t_3(pu)/t_3(pu) = 1 as implied from Equations (4.30a) and (4.30b), although t_3(pu) itself does not necessarily have to be equal to unity. The tap ratios included in the remaining two branches include the effect of branch 3 off-nominal tap ratio. This model would need to be used with care since, for example, the presence of a variable tap on one winding, say winding 3, must result in a coordinated and consistent change in the turns ratios between windings 1 and 3, and between windings 2 and 3.

Figure 4.16(b) or (c) represent the three-winding transformer PPS and NPS equivalent circuit if we ignore the windings phase shift. Otherwise, the only difference between the PPS and NPS equivalent circuits would be introduced by the windings phase shift. This was described in Section 4.2.3 and shown to result in a complex transformation ratio instead of a real one. In practice, the impedances of the three-winding transformer required in the equivalent circuit of Figure 4.16 are calculated from short-circuit test data supplied by the manufacturer. This will be covered in detail in Section 4.2.9.

The PPS/NPS star equivalent circuit of the three-winding transformer shown in Figure 4.16(b) can be converted into a delta equivalent circuit. We encourage the reader to do so.

As before, we will convert Figure 4.16(c) into π equivalents suitable for digital computer computation. We obviously obtain three π equivalents, one for each transformer branch as shown in Figure 4.17. It should be remembered that the π equivalents were derived with the convention that the currents are injected into them at both ends. Therefore, the currents flowing into the point P from the three branches would need to be reversed in sign.

ZPS equivalent circuits

The approximate ZPS equivalent circuits of three-winding transformers using various windings arrangements will be derived with the aid of a generic ZPS equivalent circuit as already presented in the case of two-winding transformers. Remembering that in Figure 4.16(c), we have converted the three-winding transformer into three two-winding transformers connected in star. Therefore, the generic ZPS equivalent circuit, ignoring the shunt exciting branch for now is shown in Figure 4.18 and is based on the following assumptions:

The equivalent ZPS circuits for three-winding transformers having YNdd, Ydd, YNyd, YNynd, YNynyn and Yyd winding arrangements are shown in Figure 4.19.

The elements of the admittance matrix of Equation (4.17a) can be derived for each equivalent ZPS circuit shown in Figure 4.19. This is similar to the derivation steps that resulted in Table 4.1. We will leave this exercise for the motivated reader!

Winding connections

Three-phase autotransformers are most often star–star connected as depicted in Figure 4.20(a). The common neutral point of the star–star connection require the two networks they interconnect to have the same earthing arrangement, e.g. solid earthing in the UK although impedance earthing may infrequently be used. Most autotransformers have a low MVA rating tertiary winding connected in delta.

PPS and NPS equivalent circuits

Autotransformers that interconnect extra high voltage transmission systems are not generally equipped with tap-changers due to high costs. However, those that interconnect the transmission and subtransmission or distribution networks are usually equipped with on-load tap-changers in order to control or improve the quality of their LV output voltage under heavy or light load system conditions. Although some tap-changers are connected on the HV winding, most tend to be connected on the LV winding. Most of these are connected at the LV winding line-end and only a few are connected at the winding neutral-end.

A single-phase representation of the general case of an autotransformer with a tertiary winding is shown in Figure 4.20(b). Using S, C and T to denote the series, common and tertiary windings, we can write in actual physical units

     (4.33a)

     (4.33b)

     (4.33c)

Neglecting the no-load current, the MMF balance is expressed as

or

     (4.34a)

where

     (4.34b)

Also

     (4.34c)

Using Equations (4.34b), (4.34c) and (4.33a), Equations (4.33b) and (4.33c) can be written as

     (4.35a)

     (4.35b)

Equation (4.35) can be represented by the star equivalent circuit shown in Figure 4.21(a) containing two ideal transformers as for a three-winding transformer.

The measurement of the autotransformer PPS and ZPS impedances using short-circuit tests between two winding terminals is dealt with later in this section. However, it is instructive to use Equation (4.35) to demonstrate the results that can be obtained from such tests. Using Equations (4.34a) and (4.35), the PPS impedance measured from the H terminals with the L terminals short-circuited and T terminals open-circuited is

hence

     (4.36a)

Also, the impedance measured from the H terminals with the T terminals short-circuited and L terminals open-circuited is

hence

     (4.36b)

Similarly, the impedance measured from the L terminals with the T terminals short-circuited and H terminals open-circuited is

hence

     (4.36c)

To calculate the impedance of each branch of the T equivalent circuit in ohms with all impedances referred to the H side voltage base, let us define the measured impedances as follows:

     (4.36d)

     (4.36e)

     (4.36f)

where the prime indicates quantities referred to the H side.

Solving Equations (4.36d), (4.36e) and (4.36f) for each branch impedance, we obtain

     (4.37a)

     (4.37b)

     (4.37c)

Now, substituting Equations (4.36a), (4.36b) and (4.36c) into Equations (4.37a), (4.37b) and (4.37c), we obtain

     (4.38a)

     (4.38b)

     (4.38c)

Figure 4.21(b) shows the autotransformer PPS T equivalent circuit with all impedances in ohms referred to the H terminals voltage base. In the absence of a tertiary winding, Figure 4.21c shows the equivalent circuit of the autotransformer. Using Equations (4.38) in Equations (4.35), we obtain

     (4.39a)

     (4.39b)

Now, we will convert Equations (4.39) from actual units to pu values. To do so, we define the following pu quantities

     (4.40a)

     (4.40b)

     (4.40c)

     (4.40d)

Using Equations (4.40) in Equations (4.39a) and (4.39b), we obtain

     (4.41a)

     (4.41b)

Equations (4.41a) and (4.41b) can be rewritten as

     (4.42a)

     (4.42b)

where the following pu tap ratios are defined

     (4.43a)

     (4.43b)

Equations (4.42) are represented by the pu equivalent circuit shown in Figure 4.21(d) which represents the autotransformer PPS/NPS equivalent circuit ignoring the delta tertiary phase shift. The autotransformer is clearly represented as three two-winding transformers that are star or T connected. Two of these transformers have off-nominal tap ratios that can represent any off-nominal tap ratios on any winding or a combination of tap-ratios. In some cases, the two variable ratios must be consistent and coordinated where an active tap-changer on only one winding can in effect change the effective turns ratio on another. For example, for a 400 kV/132 kV/13 kV autotransformer having a tap-changer acting on the neutral end of the common winding, the variation of t_LH(pu) caused by HV to LV turns ratio changes will also cause corresponding changes in the HV to TV turns ratio and hence in t_TH(pu). Therefore, t_TH(pu) is a function of t_LH(pu) which varies as a result of controlling the LV (132 kV) terminal voltage to a specified target value around a deadband.

Where the autotransformer does not have a tertiary winding or where the tertiary winding is unloaded, the T terminal in Figure 4.21(d) would be unconnected to the power system network and its branch impedance has no effect on the network currents and voltages. Thus, this branch can be disregarded and the effective autotransformer impedance would then be the sum of the H and L branch impedances given by Z_HL(pu) = Z_H(pu) + Z_L(pu). In this case, the PPS/NPS equivalent circuit of an autotransformer is similar to that already derived for a two-winding transformer and shown in Figures 4.8(c) or 4.9(c). These can be used to represent an autotransformer with ‘series’ winding tap-changer or ‘common’ winding tap-changer, respectively. The latter represents British practice irrespective of whether the tap-changer is connected to the line-end or neutral-end of the ‘common’ winding.

The impedances of the autotransformer required in the equivalent circuit of Figure 4.21(d) are calculated from short-circuit test data supplied by the manufacturer. This is covered in detail in Section 4.2.9.

ZPS equivalent circuits

In addition to the primary, secondary and, where present, tertiary winding connections, the presence of an impedance in the neutral point of the star-connected ‘common’ winding must be taken into account in deriving the autotransformers ZPS equivalent circuit. We will ignore, as before, the shunt exciting impedance, and consider the general case of a star–star autotransformer with a tertiary winding with the common star neutral point earthed through an impedance Z_E. The ZPS equivalent circuit, taking into account the presence of Z_E in the neutral, can be derived in a similar way to the PPS/NPS circuit. However, we will leave this as a minor challenge for the interested reader but we have drawn Figure 4.22 to help the reader in such derivation.

For us, it suffices to say that in Equations (4.35a) and (4.35b), Z_C should be replaced by Z_C + 3Z_E. Therefore, the ZPS equations are given by

     (4.44a)

     (4.44b)

Similar to the derivation of the PPS impedances, the measured ZPS impedances between two terminals and the corresponding T equivalent branch impedances are

     (4.45a)

     (4.45b)

     (4.45c)

     (4.45d)

     (4.45e)

     (4.45f)

Substituting Equations (4.45d), (4.45e) and (4.45f) into Equations (4.44a) and (4.44b), we obtain

     (4.46a)

     (4.46b)

To convert Equations (4.46) to pu, we will use the base quantities defined in Equations (4.40) and define Z_E(pu) = Z_E/Z_L(B), i.e. we choose to use the L terminal as the base voltage for converting Z_E to pu. It should be noted that the choice of the voltage base for Z_E is arbitrary. Therefore, Equations (4.46) become

     (4.46c)

     (4.46d)

where V_T(pu) = 0. Equations (4.46c) and (4.46d) are represented by the ZPS T equivalent circuit of Figure 4.23 for a star-star autotransformer with a delta tertiary winding and with the common neutral point earthed through an impedance Z_E. It is informative to observe how the neutral earthing impedance appears in the three branches connected to the H, L and T terminals

     (4.47a)

     (4.47b)

     (4.47c)

where Z_E(pu) is in pu on L terminal voltage base.

In the absence of a tertiary winding, the equivalent ZPS circuit cannot be obtained from Figure 4.23 by removing the branch that corresponds to the tertiary winding. A shunt branch to the zero voltage node may indeed still be required depending on the transformer core construction. This is covered in detail in Section 4.2.8.

The case of a solidly earthed neutral is represented in Figure 4.23 by setting Z_E = 0. However, the case of an isolated neutral, i.e. Z_E → ∞ cannot be represented by Figure 4.23 because the branch impedances of the equivalent circuit become infinite. This indicates no apparent paths for ZPS currents between the windings although a physical circuit does exist. A mathematical solution is to convert the star circuit of Figure 4.23 to delta or π and then setting ZE → ∞. However, we prefer the physical engineering approach to derive the equivalent ZPS circuit of such a transformer. Therefore, using the actual circuit of an unearthed autotransformer with a delta tertiary winding, shown in Figure 4.24(a), we can write

     (4.48a)

     (4.48b)

     (4.48c)

     (4.48d)

Using Equations (4.48b), (4.48c) and (4.48d) in Equation (4.48a), we obtain

     (4.48e)

The measured ZPS leakage impedance in ohms

is given by

     (4.48f)

The ZPS leakage impedance measured from the H side or L side is the leakage impedance between the series and tertiary winding that is the sum of the series winding leakage impedance and the tertiary winding leakage impedance referred to the series side base turns. Figure 4.24(b) shows a single-phase representation of the series and tertiary winding impedances and turns ratios. Figure 4.24(c) shows the tertiary winding impedance referred to the series winding side and Figure 4.24(d) represents the ZPS equivalent circuit and total ZPS equivalent impedance in actual physical units. The physical explanation for this result is as follows. The ZPS currents flow on the H side through the series windings only and, without transformation, flow out of the L side. This is possible because the delta tertiary winding circulates equal ampere-turns to balance that of the series winding but no ZPS currents flow in the common windings of the autotransformer.

To convert Equation (4.48e) to pu, we first calculate a pu value for ZS–T and note that this is the same whether referred to the series winding or tertiary winding because the base voltages in the windings are directly proportional to the number of turns in the windings. Using Equation (4.40), we have

Dividing by V_H(B) and using

we obtain

     (4.49a)

Equation (4.49a) is represented by the pu equivalent ZPS circuit shown in Figure 4.24(e) from which the ZPS leakage impedance measured from the H side with L side short-circuited is given by

     (4.49b)

Similarly, the ZPS leakage impedance measured from the L side with H side short-circuited is given by

     (4.49c)

The delta or π equivalent circuit can be derived using Figure 4.6(c). The result is shown in Figure 4.24(f) using shunt impedances instead of admittances.

Another similar test may be made but with the delta tertiary open to obtain the shunt ZPS magnetising impedance which may be low enough to be included depending on the core construction. This is dealt with in detail in Section 4.2.8.

Three-phase transformers made up of three single-phase banks

Figure 4.27 illustrates two typical single-phase transformer cores. In both cases, the ZPS flux set up by ZPS voltage excitation can flow within the core in a similar way to PPS flux. Consequently, the ZPS magnetising impedance will be very large and the ZPS leakage impedance of such transformers will be substantially equal to the PPS leakage impedance.

Three-phase transformers of 5-limb core construction and shell-type construction

Figures 4.28 and 4.29 illustrate a 5-limb core-type and a shell-type three-phase transformers, respectively. The 5-limb design is widely used in the UK and Europe whereas the shell-type tends to be widely used in North America. In both designs, the ZPS flux set up by ZPS excitation can flow within the core and return in the outer limbs. Consequently, as in the case of three-banks of single-phase transformers, the ZPS magnetising impedance will be very large and the ZPS leakage impedance of such transformers will be substantially equal to the PPS leakage impedance. The measurement of sequence impedances will be dealt within the next section. However, it is appropriate to explain now that PPS leakage impedances are normally measured at nearly rated current whereas ZPS impedances are measured, where this is done, at quite low current values. Therefore, under actual earth fault conditions giving rise to sufficiently high ZPS voltages on nearby transformers with currents similar to those of the PPS tests, the transformer outer limbs may approach saturation and this lowers the ZPS magnetising impedance. Consequently, the measured ZPS impedance value may be somewhat higher than the actual value for core-type design and substantially higher for shell-type design that exhibits much more variable core saturation than the limb-type core. Nonetheless, these magnetising impedances remain relatively large in comparison with the leakage impedances and hence, it is usually assumed that the PPS and ZPS leakage impedances of such transformers are equal.

Three-phase transformers of 3-limb core construction

Figure 4.30 illustrates a three-phase transformer of 3-limb core-type construction which is extensively used in the UK and the rest of the world. Under ZPS voltage excitation, the ZPS flux must exit the core and its return path must be completed through the air with the dominant part being the tank then the oil and perhaps the core support framework. This ZPS flux will induce large ZPS currents through the central belt of the transformer tank and the overall effect of this very high reluctance path is to significantly lower the ZPS magnetising impedance. This ZPS magnetising impedance will be comparable to other plant values and may be 60–120% on rating for two and three-winding transformers and perhaps 150–250% for autotransformers. These figures can be compared with 5000–20 000% for the corresponding PPS values (these correspond to 2–0.5% no-load currents). Therefore, the effect of the tank for two-winding and three-winding transformers with star earthed neutral windings, and for autotransformers without tertiary windings, can be treated as if the transformer has a virtual magnetic delta-connected winding. We will provide practical examples of this in the next section.

Similar to 5-limb and shell-type transformers, ZPS leakage impedance measurements may be made at low current values and hence may give slightly higher impedances than the values inferred from PPS measurements made at rated current value. In addition, the effect of the tank in the 3-limb construction is non-linear with the reluctance increasing with increasing current that is the magnetising reactance decreasing with increasing current. Therefore, the actual ZPS leakage impedance of three-limb transformers may be slightly lower than the measured values.

PPS and ZPS impedance tests on two-winding transformers

The PPS impedance test is HV–LV//N, i.e. the HV winding is supplied with the LV winding phase terminals joined together, i.e. short-circuited and connected to neutral. The same test may be done for the ZPS impedance if the secondary winding is delta connected. However, for a star earthed-star earthed transformer, the ZPS equivalent circuit that correctly represents a 3-limb core transformer, is a star or T equivalent circuit since we represent the tank ZPS contribution as a magnetic delta tertiary winding. Therefore, at least three measurements are required. It is the author’s practice that four ZPS tests are specified to enable a better estimate of the impedance to neutral branch that represents the tank contribution. The ZPS tests are HV–N with LV phase terminals open-circuited, LV–N with HV phase terminals open-circuited, HV–LV//N with LV phase terminals short-circuited and LV–HV//N with HV phase terminals short-circuited.

PPS and ZPS impedance tests on three-winding transformers

In order to derive the star or T equivalent PPS circuit for such transformers from measurements, let the three windings be denoted as 1, 2 and 3, where:

In three-winding transformers, at least one winding will have a different MVA rating but all the pu impedances must be expressed on the same MVA base. From the above tests, and with the impedances in ohms referred to the same voltage base, and copper losses in kW, we have

     (4.52a)

     (4.52b)

     (4.52c)

Solving Equations (4.52), we obtain

     (4.53a)

     (4.53b)

     (4.53c)

The winding copper loss measurements of Equation (4.52) can be used to calculate the corresponding winding resistances in ohms or in pu on a common MVA base. Then individual resistances are calculated from Equation (4.53). Alternatively, individual winding copper losses could be calculated using Equation (4.53) and used to calculate the corresponding individual winding resistances. When converted into pu, all impedances must be based on one common MVA base.

Regarding the ZPS impedances, the ZPS equivalent circuit that correctly represents a 3-limb core three-winding transformer even when all windings are star-connected, is a star or T equivalent circuit since we represent the tank ZPS contribution as a magnetic delta tertiary winding. Therefore, as in the case of the two-winding transformer, the ZPS tests are HV–N with LV phase terminals open-circuited, LV–N with HV phase terminals open-circuited, HV–LV//N with LV phase terminals short-circuited and LV–HV//N with HV phase terminals short-circuited.

Autotransformers

Generally, two cases are of most practical interest for autotransformers. The first is an autotransformer with a delta-connected tertiary winding and the second is without a tertiary winding. Without a tertiary winding, the PPS test is similar to that of a two-winding transformer described above. With a tertiary winding, the PPS tests are similar to those of a three-winding transformer. However, the ZPS equivalent circuit that correctly represents a 3-limb core autotransformer, with or without a tertiary winding, is a star or T equivalent circuit since we represent the tank ZPS contribution as a magnetic delta tertiary winding. Therefore, the ZPS tests are:

With delta tertiary	Without tertiary	Comments
HV–Tertiary//N	HV–N	LV phase terminals open-circuited
LV–Tertiary//N	LV–N	HV phase terminals open-circuited
HV–LV//Tertiary//N	HV–LV//N	LV phase terminals short-circuited
LV–HV//Tertiary//N	LV–HV//N	HV phase terminals short-circuited

PPS equivalent circuit model

The derivation of QB and PS detailed equivalent circuits needs to take into account both the shunt and series transformers, their winding connections, tap-changer operation, complex turns ratio and leakage impedance variations with tap position. However, we will instead present a simplified treatment but one that is still sufficient for use in practice in PPS power flow, stability and short-circuit analysis. The analysis below is based on the vector diagrams of Figures 4.35(c) and 4.36(c). These allow us to represent the QB or PS as an ideal transformer with a complex turns ratio in series with an appropriate impedance. This ratio is in series with a single effective leakage impedance representing both the series and shunt transformers’ leakage impedances including the effect of the tap-changer on the shunt transformer’s impedance. This representation is shown in Figure 4.37.

For both a QB and a PS, let t = δ V_i(R)/V_i(R) be the magnitude of the injected quadrature voltage in pu of the input voltage. From the QB vector diagram shown in Figure 4.35(c), the phase R open-circuit output voltage phasor is given by

     (4.54a)

Let be defined as the complex turns ratio, at no load, thus,

     (4.54b)

where

     (4.54c)

is the phase angle shift that has a maximum or rated value for practical QBs used in power networks of typically less than or equal to ± 15°. Similar equations apply for the Y and B phases.

Similarly, from the PS vector diagram shown in Figure 4.36(c), the phase R open-circuit output voltage is given by

     (4.55a)

where V_iS(R) is the voltage at the mid-point of the series winding.

The complex turns ratio of the PS, at no load, is given by

     (4.55b)

where

     (4.55c)

is the phase angle shift that has a maximum or rated value for practical PSs used in power networks of typically less than or equal to ±60°.

Let Y_e = 1/Z_e where Z_e is the effective QB or PS leakage impedance appearing in series with the line or the network where the device is connected. The QB and PS are both represented by the same PPS equivalent circuit shown in Figure 4.37. The PPS admittance matrix for this circuit can be easily derived as in the case of an off-nominal-ratio transformer but with an important difference now that the turns ratio is a complex number. From Figure 4.37, we can write

From these equations, the following admittance matrix is obtained

     (4.56a)

which, using Equations (4.54b) and (4.54c), can be written as

     (4.56b)

or, using Equation (4.55b), can be written as

     (4.56c)

The above two admittance matrices for a QB and a PS cannot be represented by a physical π equivalent circuit because the complex turns ratio has resulted in matrices that are non-symmetric, i.e. the off-diagonal transfer terms are not equal. However, recognising that an asymmetrical non-physical π equivalent circuit is simply a mathematical tool, such a circuit is shown in Figure 4.37(b) for a PS. A similar π equivalent for a QB can be derived and the reader is encouraged to derive this equivalent.

NPS equivalent circuit model

Since the QB and PS are static devices, the NPS impedance should be the same as the PPS impedance. However, we have shown that the PPS model includes a complex turns ratio which is a mathematical operation that introduces a phase angle shift in the output voltage (and current) with respect to the input voltage (and current). The mathematical derivation of the phase shift in the QB or PS NPS model requires a detailed representation of all windings of the shunt and series transformers, which, as outlined above, is not presented here. However, we will use vector diagrams to show that the phase shift in the NPS circuit is equal to that in the PPS circuit but reversed in sign. Figures 4.38(i) show the PPS open-circuit vector diagrams of a QB and a PS and the resultant phase angle shift φ in each case. Figures 4.38(ii) show the NPS vector diagrams of a QB and a PS and the resultant phase angle shift in each case.

It is clear that with the reversed NPS phase rotation RBY, the injected series quadrature voltage vector is also reversed, for both QB and PS. Therefore, if the PPS phase angle shift is φ, then the NPS phase angle shift is –φ. Figure 4.39 shows the QB or PS NPS equivalent circuit model where the complex ratio is given by

     (4.57a)

Using the PPS admittance matrices fora QB and a PS, the NPS admittance matrices are easily obtained by replacing φ with -φ in Equations (4.56b) and (4.56c).

ZPS equivalent circuit model

We have already stated that the basic principle of operation of the QB or PS is to inject a voltage across, say phase R, series transformer series winding that is in quadrature with the phase R input voltage. In the PPS and NPS circuits, this is achieved by deriving the injected voltage so as to be in phase or out of phase with the voltage difference between phases Y and B as shown in Figure 38.4(a) and (b), respectively. However, since the three ZPS voltages are in phase with each other, a quadrature voltage component cannot be produced. Therefore, the phase shift between the input and open-circuit output voltages (and currents) in the ZPS circuit is zero. Therefore, the complex turns ratio in the ZPS equivalent circuit for both a QB and a PS becomes unity.

The QB or PS practical ZPS equivalent circuit, however, is different from the PPS and NPS equivalent circuits. The ZPS equivalent circuit is dependent on the winding connections of the series and shunt transformers, their core construction and whether the shunt transformer has any delta-connected tertiary winding. In a similar way to a three-winding transformer, or a 3-limb core autotransformer with or without a tertiary winding, the general ZPS equivalent circuit of a QB or a PS is a star or T equivalent as shown in Figure 4.39(b). The shunt branch in this equivalent represents either the impedance of a tertiary winding or the stray air path ZPS flux through the tank/oil. In addition, in some QBs and PSs, both series and shunt transformers are in the same tank so that the ZPS flux can link both windings either directly or through the tank. In practice, the QB or PS ZPS impedances are derived from a series of manufacturer works-tests and this is described in the next section. The ZPS admittance matrix model of the QB or PS is given by

     (4.57b)

where

     (4.57c)

Types of series capacitor schemes

There are several three-phase series capacitor scheme designs in use in power systems, four of which are depicted in Figure 4.44.

General modelling aspects of series capacitors

The modelling of series capacitor schemes in power flow analysis is straightforward because the magnitude of current flowing through the capacitor would be within its appropriate rated design value. These rated values include not only the highest continuous current, but also the highest short-term overload value permitted and the highest permitted value under power oscillation conditions. The capacitors are therefore represented as three identical series reactances and the ohmic value of the reactance is known from the rated data. The resistive part of the series capacitor is negligibly small. Therefore, the sequence and phase admittance matrix of a three-phase series capacitor is given by

     (4.63)

where

The PPS, NPS and ZPS admittances are equal.

However, for short-circuit analysis, the modelling of series capacitors is not straightforward and requires knowledge of the capacitor protection scheme and capacitor design ratings. It is worth noting that IEC 60909–0:2001 short-circuit analysis standard states that the effect of series capacitors can be neglected in the calculation of short-circuit currents if they are equipped with voltage-limiting devices in parallel acting if a short-circuit occurs. However, as we will see later, this approach is generally inappropriate for varistor protected series capacitors and could result in significant underestimates in the magnitude of short-circuit currents.

Modelling of series capacitors for short-circuit analysis

Old series capacitor schemes were protected by spark gaps against a short-circuit fault that can cause a large capacitor current to flow and a substantial increase in the voltage across the capacitor. The flashover across the spark gap bypasses the series capacitor and the bypass circuit-breaker is immediately closed to extinguish the gap.

In modern series capacitor installations, non-linear resistors are used and these are called metal (usually zinc) oxide varistors (MOVs) known as ZnO. MOVs are used to provide the overvoltage protection mainly against external short-circuit faults, i.e. external to the series capacitor and its line zone. For internal faults, within the series capacitor protected line zone, a forced triggered spark gap can operate typically within 1 ms to limit the energy absorption duty on the varistors. This is then followed by closure of the circuit-breaker to extinguish the gap and hence the capacitor is completely bypassed and removed from the circuit.

The modelling and analysis of varistor protected series capacitors under external or remote short-circuit fault conditions is quite complex. However, we will briefly and qualitatively describe the operation of the modern varistor protected capacitor schemes under external through-fault conditions that result in varistor operation only. As the instantaneous short-circuit current flowing through the capacitor increases, so will the voltage across the capacitor or varistor until this voltage reaches a preset threshold where the varistor starts conducting current in order to keep the voltage across the capacitor constant. This applies under both positive and negative polarity current conditions so that for the first part of each half cycle, the current flows through the capacitor only but for the other part of the half cycle, the current switches to the varistor. The behaviour of the varistor and series capacitor combination is repeated every half cycle during the fault period whenever the voltage across the capacitor/varistor exceeds the preset capacitor protection level. Using Figure 4.45(a), the behaviour of the series capacitor/varistor parallel combination under a through fault condition is analysed using an electromagnetic transient analysis program. The voltage across the capacitor/varistor and the currents through the capacitor and varistor are shown in Figure 4.45(b)–(d), respectively. The fault current is shown in Figure 4.45(e) curve (ii).

Curve (i) of Figure 4.43(e) shows the fault current that would flow through the capacitor if there were no varistor to protect it. Clearly, the conduction of the MOV causes an effective reduction in the fault current magnitude. The extent of reduction is dependent on the electrical proximity of the fault location to the capacitor/varistor location. In addition, curve (iii) of Figure 4.45(e) shows the fault current if the capacitor is assumed to be permanently bypassed. This generic study illustrates that in general, both ignoring the presence of the varistor or assuming the capacitor is permanently bypassed during the fault period can result in a large error in the fault current magnitude. The former can result in an overestimate whereas the latter can result in an underestimate.

The modelling and analysis of short-circuit faults in large-scale power systems is based on the assumption that all power system plant are linear or can be approximated as piece-wise linear. The varistor is, however, a highly non-linear resistor that cannot be directly included in standard quasi-steady state short-circuit modelling and analysis simulations. Figure 4.46 shows a typical voltage/current characteristic of a varistor which presents an almost infinite resistance below the threshold conduction voltage then a very low resistance above it.

The impedance presented by the combined capacitor reactance and varistor resistance over a half cycle is dependent on the total current flowing through their parallel combination. The capacitor current or voltage protection threshold is known because the capacitor reactance is a known design parameter. Therefore, below this threshold, the equivalent impedance of the parallel capacitor/varistor combination is the capacitor reactance in parallel with a very high resistance or alternatively, a very low resistance (practically zero) in series with the capacitor reactance. When a current higher than the threshold flows through the capacitor, the varistor will conduct current for most of the half cycle so that the combination will appear as a low resistance in parallel with the capacitor reactance or, alternatively, a higher resistance than zero in series with a much reduced capacitor reactance. The higher the current is, the more resistive the equivalent impedance becomes reflecting the increased short-circuiting of the capacitor by the varistor. Therefore, the capacitor/varistor parallel combination can be approximated as a resistance and a capacitance in series with both being functions of the total current flowing through them. Figure 4.47(a) shows a three-phase representation of a series capacitor protected by a varistor. Figure 4.47(b) shows the corresponding equivalent circuit where the equivalent series impedance is a function of the total current flowing through the parallel capacitor/varistor combination.

Denoting the capacitor current protective threshold as I_thr, the equivalent series impedance for each phase is given by

     (4.64)

where X_C is the capacitor reactance when all current flows through it, i.e. its nominal design value, I is the total current flowing through the capacitor/varistor parallel combination and I_thr is a known design threshold parameter. The mathematical functions that describe R_Eq(I) and X_Eq(I) as functions of I when I ≥ I_thr can be calculated from the known design data or parameters of the series capacitor and varistor. Figure 4.48 illustrates typical equivalent impedance characteristics.

The series phase admittance matrix of the three-phase series capacitor/varistor device is given by

     (4.65)

Prior to the short-circuit fault or when the current is less than I_thr, the equivalent phase admittances are balanced and equal. This will also be the case under balanced three-phase short-circuit faults. However, under unbalanced short-circuit faults, they are generally unequal depending on the fault type. Therefore, the sequence admittance matrix of Equation (4.65) is given by

where

     (4.66b)

Equation (4.66a) shows the presence of unequal intersequence mutual coupling which should be taken into account if significant errors are to be avoided.

Because fixed impedance short-circuit analysis techniques are linear, Equation (4.64) may be implemented in an iterative linear calculation process. The fault currents flowing in the network and the series capacitors are initially calculated without varistor action, the capacitor current is then used to calculate a new value for the series capacitor equivalent impedance using Equation (4.64). The new impedance is then used to calculate new fault currents in the network and the series capacitor. The final solution would be arrived at when there is a little change in the last two equivalent impedances or capacitor currents calculated.

In power system networks with modern varistor protected series capacitor installations, the operation of the varistor when the capacitor current exceeds I_thr cannot be ignored if very large and unacceptable errors (overestimates) in the calculated short-circuit currents in the network are to be avoided. Also, if these series capacitors are not included in the network model at all, i.e. assumed to be completely bypassed under short-circuit faults, as per IEC 60909, then this could result in significant underestimates in the magnitude of short-circuit currents.

General

The sequence modelling of single-phase and three-phase transformers of various winding connections was extensively covered in Section 4.2. In three-phase steady state analysis in the phase frame of reference, three-phase models are required. For the purpose of steady state e.g. short-circuit analysis, it is sufficient to model the transformer as a set of mutually coupled windings with a linear magnetising reactance. We will not provide a similar extensive coverage to that in Section 4.2 but will instead present the basic modelling technique of a single-phase two-winding transformer, three-phase transformer banks and a three-limb core three-phase transformer. For the three-phase transformer, only a star-delta winding connection is considered. The principles provide the reader with the information and methodology required to develop a three-phase model for a transformer of any winding connection.

Single-phase two-winding transformers

Figure 4.51(a) illustrates a single-phase two-winding transformer represented as a set of two mutually coupled windings where: Z₁₁(Y₁₁) is winding 1 self-impedance (admittance) and Y₁₁ = 1/Z₁₁; Z₂₂(Y₂₂) is winding 2 self-impedance (admittance) and Y₂₂ = 1/Z₂₂; Z₁₂ = Z₂₁(Y₁₂ = Y₂₁) is mutual impedance (admittance) between windings 1 and 2 and Y₁₂ = 1/Z₁₂; Z_HL(Y_HL) is leakage impedance (admittance) of windings 1 and 2 and Y_HL = 1/Z_HL.

From Figure 4.51(a), we can write

     (4.71)

From Equation (4.71), the self-impedance Z₁₁ of winding 1 and the mutual impedance Z₁₂ between the two windings are obtained from the results of an open-circuit test with winding 1 supplied and winding 2 open-circuited and are given by

     (4.72a)

From Equation (4.72a), the mutual impedance can be expressed as

     (4.72b)

where V₁/V₂ is the transformer no-load voltage ratio.

Similarly, the self-impedance Z₂₂ of winding 2 and the mutual impedance Z₁₂ between the two windings can be obtained from the results of an open-circuit test with winding 2 supplied and winding 1 open-circuited. Thus

     (4.73a)

From Equation (4.73a), the mutual impedance can be expressed as

     (4.73b)

From Equations (4.72b) and (4.73b), we have

     (4.74)

The leakage impedance Z_HL between the two windings is obtained from the results of a short-circuit test with winding 1 supplied and winding 2 short-circuited. Thus

     (4.75a)

Using Equation (4.75a) in Equation (4.71), the leakage impedance ZHL can be expressed in terms of the windings self and mutual impedances as follows:

     (4.75b)

Having obtained all four elements of the impedance matrix of Equation (4.71), it is more convenient to use the corresponding admittance matrix given by

     (4.76a)

where

     (4.76b)

Using Equations (4.73b), (4.74) and (4.75b) in Equations (4.76b), we obtain

     (4.77a)

and

     (4.77b)

In the derivation of the above impedance and admittance matrices, we note that, in practice, the only data most likely available are the measured short-circuit leakage impedance and no-load current. The derivation is illustrated using an example.

Three-phase banks two-winding transformers (no interphase mutual couplings)

Consider a large two-winding star–delta transformer constructed as three single-phase banks as shown in Figure 4.52.

In such an arrangement, there is negligible or no interphase coupling between the three phases. Using the admittance matrix given in Equation (4.76a) for each phase and Figure 4.52, the admittance matrix equation that relates the winding currents and voltages for the three-phases is given by

     (4.78a)

or

     (4.78b)

Equation (4.78b) applies to a two set of three-phase windings irrespective of the type of winding connection. Thus, for our star–delta transformer, the three-phase nodal admittance matrix that relates the nodal currents and voltages can be derived from the relationship between the winding currents and voltages, and the nodal currents and voltages. With all nodal voltages being with respect to the reference earth, the relationship between the winding and nodal voltages is found by inspection from Figure 4.52 as follows:

     (4.79a)

where C is defined as the connection matrix. Similarly, the relationship between the nodal and winding currents is found by inspection from Figure 4.52 and is given by

     (4.79b)

Using Equations (4.79a) and (4.79b) in Equation (4.78b), we obtain

     (4.80a)

where

     (4.80b)

Denoting the star-connected winding as H and the delta-connected winding as L, Equation (4.80a) can be written in partitioned form as follows:

     (4.81a)

Using Equation (4.77a), it can be shown that the self and mutual admittance matrices of Equation (4.81a) are given by

     (4.81b)

where Y_HL14 = 1/Z_HL14 and so on.

In the per-unit system with both the star and delta voltages equal to one per unit, the effective turns ratio on the delta side is . If the three phases of the transformer are considered identical, i.e. having identical leakage admittances that is ,then using Equation (4.81), Equation (4.80b) becomes

     (4.82)

To allow for the effect of off-nominal tap ratio, e.g. on the star side, then according to Equation (4.13a), the terms of the self admittance Y_HH should be divided by whereas the terms of the mutual admittances Y_HL and should be divided by t_HL.

The reader is encouraged to show that the nodal admittance matrix of a two-winding star–star transformer with both neutrals solidly earthed and no interphase mutual couplings has the form as the winding admittance given in Equation (4.78a). Using these principles, the reader can easily obtain the nodal admittance matrix for any winding connections of a three-phase banks of transformers.

Three-phase common-core two-winding transformers (with interphase mutual couplings)

For three-phase 3-limb, 5-limb or shell-type core transformers, interphase mutual couplings exist and these are generally asymmetrical, as can be expected from Figures 4.28-4.30, between the outer-limb phases and the phases connected to adjacent limbs. However, in practice, little or no information is available on these asymmetrical coupling affects. Standard transformer test certificates include the normally measured parameters, i.e. the PPS and ZPS impedances and the magnetising current from which the magnetising impedance is calculated. Since the PPS and ZPS impedances are generally different, this implies that equal interphase mutual couplings exist in the three-phase admittance matrix. The asymmetrical interphase coupling effects may also be indirectly averaged out by their short-circuit measurement procedures as described in Section 4.2.10. Figure 4.53 illustrates a common-core star–delta transformer where each phase winding is coupled to all other phase windings and where windings 1 and 4, 2 and 5, and 3 and 6, are wound on the same limb. With the assumption of full flux symmetry, we obtain a 6 × 6 winding admittance matrix that consists of four 3 × 3 symmetrical submatrices. The corresponding nodal admittance matrix can be obtained using Equation (4.80b).

However, since in nearly all practical situations only sequence and magnetising impedances are available, we will present a different technique for the formulation of the transformer three-phase model. The technique is based on using the known PPS and ZPS leakage impedances, ignoring the magnetising impedances for now. To illustrate the technique, we consider the star-delta transformer of Figure 4.53 with a tap-changer on the star side. In Figure 4.53, phase R of the star winding is assumed symmetrically coupled with phases Y and B of the star winding on different limbs, with phase ‘ry’ on the delta winding on the same limb, and with phases ‘rb’ and ‘by’ on the delta winding on different limbs. Similar symmetrical couplings are assumed for other phases. The PPS and NPS equivalent circuits are shown in Figure 8.4(a) and (b). The ZPS equivalent circuit is shown in Figure 4.12(g) with Z_E = 0. Equation (4.13a) corresponds to Figure 4.8(c) where the transformer winding phase shift of 30° for Yd1 and −30° for Yd11 connections was neglected. In the derivation to follow, we denote the star side as H and the delta side as L. Therefore, denoting the phase shift as φ and letting all impedances be expressed in pu, it can be shown that the PPS and NPS nodal admittance matrices of Figures 4.8(a) and (b) are given by

     (4.83a)

     (4.83b)

and the ZPS nodal admittance matrix of Figure 4.12(g) with Z_E = 0 is given by

     (4.83c)

Equations (4.83a), (4.83b) and (4.83c) can be assembled to obtain the transformer sequence admittance matrix, including sequence current and voltage vectors, as follows:

     (4.84a)

or

     (4.84b)

It should be noted that . The phase admittance matrix is obtained using the sequence to phase transformation matrix H noting that in pu, the effective delta winding turns ratio is . Thus

     (4.85a)

where

     (4.85b)

     (4.85c)

Therefore, using Equations (4.84) in Equations (4.85b) and (4.85c), the self (H and L windings) and mutual (H–L and L–H windings) phase admittance matrices are given by

     (4.86a)

     (4.86b)

     (4.87a)

and

     (4.87b)

It should be noted that for star–star and delta–delta connected two-winding transformers, φ = 0 or φ = 180°. However, for the Yd1 transformer of Figure (4.53), φ = 30° and the mutual phase admittance matrices become

     (4.87c)

The transformer PPS magnetising impedance is generally available from the open-circuit test. The ZPS magnetising impedance may also be available from a direct test for some transformers. Alternatively, where it is not available, it can be estimated from the measured PPS and ZPS leakage impedances as discussed in Section 4.2.9. Where no measured ZPS leakage impedance is available, the magnetising impedance and the ZPS leakage impedance may be estimated as discussed in Section 4.2.9. The PPS magnetising impedance is so high in comparison with the PPS leakage impedance that it may be connected on the high voltage side of the transformer with practically negligible error. The ZPS magnetising impedance is also better and more correctly connected nearer the high voltage side of the transformer because the LV winding is wound nearer the core but the HV winding is wound on the outside over the LV winding. Therefore, the effect of this connection is to modify the sequence admittance matrixand the corresponding phase admittance matrixto

     (4.88a)

     (4.88b)

where

     (4.88c)

and and are the PPS and ZPS magnetising impedances, respectively.

Three-phase models for other transformer winding connections including three-winding transformers that result in a 9 × 9 nodal phase admittance matrix can be derived using a similar technique.