ON THE ECONOMIC FEASIBILITY OF USING TRANSFORMERS WITH SPLIT WINDINGS¹

G. V. Del

(Presented by the scientific seminar of the Departments of Electric Power Stations and Substations and of Electrical Networks and Systems)

To limit the magnitude of short-circuit currents in distribution networks with transformers rated 20 MVA and higher per unit, current-limiting reactors are currently used most often. In support of this, it is sufficient to cite the major work by L. I. Dvoskin [1], where the principal and most characteristic layouts of high-voltage distribution devices are considered. Unfortunately, in this book the reactorless method of limiting short-circuit currents by using transformers with split windings found no reflection.

At the same time, transformers with split windings may be used in the following cases [2]:

1) to supply two transmission lines instead of two less powerful transformers;

2) to distribute large powers at voltages of 6–10 kV, which limits the short-circuit power on each of the sections;

3) to sectionalize generator-voltage busbars at stations;

4) to connect a station with a system, or as tie transformers between systems.

In the present work an attempt is made to find criteria determining the economic feasibility of replacing a two-winding transformer with a transformer having a split winding.

Let us consider the indicated replacement of transformer types in the indoor switchgear of 6–10 kV step-down substations. The schematic connection diagram of the indicated distribution device for a two-winding transformer is shown in Fig. 1.

It is assumed that, in the case of using transformers with split windings, the schematic connection diagram is transformed into the form shown in Fig. 2.

In the economic comparison of these two alternatives, the following costs must be taken into account [3]:

1) capital expenditures for the purchase and installation of the transformer, including depreciation charges;

2) capital expenditures for the creation of additional system capacity necessary to cover losses of active power in the transformer during the period of maximum load in the system;

3) the cost of active-energy losses in the transformer over a certain calculation period of \(T\) years.

Fig. 1. Connection diagram of 6–10 kV switchgear with one busbar system, with installation of group split reactors in the transformer circuits and with separate operation of the latter on the 6–10 kV side.

Let us take the depreciation charges on capital expenditures, in relative units, as equal to \(a\). Then the depreciation charges over a calculation period of \(T\) years are

\[ Z_a = Z_p \cdot aT, \tag{1} \]

where \(Z_p\) is the calculated cost of the transformer. The total costs associated with the installation and operation of the transformer are equal to

\[ Z_t = Z_p + Z_a = Z_p(1 + aT). \tag{2} \]

Fig. 2. Connection diagram of 6–10 kV switchgear with one busbar system, using transformers with split windings.

The installation of a transformer in the system is associated with the need to create additional capacity in it to cover losses in the transformer during the period of maximum load. According to [3], the magnitude of this additional capacity is expressed as follows:

\[ P_d = p_c + p_m \left( \frac{k_c \cdot S_{m1}}{S} \right)^2, \tag{3} \]

where \(p_c\) and \(p_m\) are the losses in the transformer steel and copper under rated conditions;

\(S_{mn}\) is the maximum load of the annual schedule for the future year, for which the transformer capacity is selected;

\(k_c\) is the coefficient of coincidence with the system maximum;

\(S\) is the rated capacity of the transformer.

The capital costs for creating additional capacity in the system are

\[ Z_d = c_1 \cdot P_d, \tag{4} \]

where \(c_1\) is the cost, in rubles, of one installed kilowatt of additional system capacity.

Creating additional system capacity intended to cover losses in the transformer will require an increase in capital investments in the construction of power stations, the fuel base, and fuel transport. If only the first two components are taken into account, then the capital costs per \(1\ \text{kW}\) of additional capacity will be

\[ c_1 = 1.1\,(c_c + c_t \cdot q_t), \tag{5} \]

where \(1.1\) is a coefficient taking into account the station’s own needs;

\(c_c\) is the average cost of an installed kilowatt, in rubles/kW, for the systems of the Soviet Union;

\(q_t\) is the average annual consumption of standard fuel per \(1\ \text{kW}\) of additional capacity, in tons;

\(c_t\) is the average capital cost of creating the fuel base, referred to 1 ton of annual production, in rubles/ton.

The average cost of a newly installed kilowatt in the systems of the Soviet Union is planned in the coming years to be reduced to \(c_c = 100\ \dfrac{\text{rub}}{\text{kW}}\).

The annual fuel consumption per \(1\ \text{kW}\) of additional capacity will be equal to the product of the specific consumption of standard fuel per \(1\ \text{kWh}\) and the annual duration of use of this capacity. If the average specific fuel consumption is taken as \(400\ \text{g/kWh}\), and the duration of use of the additional capacity as \(T_m = 5000\) hours, then the annual fuel consumption by the additional capacity will be

\[ q_t = 0.4 \cdot 10^{-3} \cdot 5000 = 2.0\ t. \]

The average capital costs for creating the fuel base, referred to a ton of annual production, may be taken as \(c_t = 20\ \text{rub/t}\). Then

\[ c_1 = 1.1\,(100 + 2.0 \cdot 20) = 154\ \text{rub/kW}. \tag{6} \]

The average annual power losses in the transformer [3] are

\[ p_{\mathrm{avg}} = p_c + p_m \cdot \frac{\tau}{t} \cdot \left(\frac{S_{mn}}{S}\right)^2, \tag{7} \]

where \(\tau\) is the annual time of maximum losses;

\(t\) is the operating time of the transformer during the year.

Then the cost of electric-energy losses in the transformer over the calculated period of \(T\) years will be

\[ Z_p = c_2 \left[ p_c T + p_m \cdot T_p \frac{\tau}{t} \left(\frac{S_{mn}}{S}\right)^2 \right], \tag{8} \]

where \(c_2\) is the cost, in rubles, of one kilowatt-year of losses.

\(T_p\) is the reduced time of copper losses, in years.

For substation transformers, \(T_{\mathrm{p}}<T\) and can be calculated from the following expression:

\[ T_{\mathrm{p}}= \left(\frac{S_{\mathrm{m}1}}{S_{\mathrm{mp}}}\right)^2+ \left(\frac{S_{\mathrm{m}2}}{S_{\mathrm{mp}}}\right)^2+\ldots+ \left(\frac{S_{\mathrm{m}T}}{S_{\mathrm{mp}}}\right)^2, \tag{9} \]

where \(S_{\mathrm{m}1}, S_{\mathrm{m}2}, \ldots, S_{\mathrm{m}T}\) are the maximum loads of the first, second, ..., \(T\)-th year. Taking into account (2), (3), (4), and (8), the total costs for the transformer will be

\[ Z_1=Z_{\mathrm{t}}+Z_{\mathrm{d}}+Z_{\mathrm{p}} =Z_{\mathrm{r}}(1+aT)+p_{\mathrm{s}}(c_1+c_2T)+ \]

\[ +p_{\mathrm{m}}\left(\frac{S_{\mathrm{mp}}}{S}\right)^2 \left(c_1 k_{\mathrm{s}}^2+c_2\cdot\frac{\tau}{t}\cdot T_{\mathrm{p}}\right). \tag{10} \]

The costs for the reactor will obviously have the same components as those for the transformer, i.e.:

1) capital costs for the purchase and installation of the reactor, with depreciation deductions;

2) the cost of active electric-energy losses in the reactor over the calculation period;

3) capital costs for creating additional system capacity during the period of maximum load.

Analogously to (10), for the reactor we have

\[ Z_2=Z_{\mathrm{rr}}(1+aT)+p_{\mathrm{mr}} \left(\frac{I_{\mathrm{t}}}{2I_{\mathrm{rn}}}\right)^2 \left(c_1 k_{\mathrm{s}}^2+c_2\cdot\frac{\tau}{t}\cdot T_{\mathrm{p}}\right). \tag{11} \]

Here \(Z_{\mathrm{rr}}\) is the calculated cost of the reactor;

\(p_{\mathrm{mr}}\) is the copper losses in the reactor (rated);

\(I_{\mathrm{t}}\) is the current in the secondary winding of the two-winding transformer;

\(I_{\mathrm{rn}}\) is the rated current of one branch of the reactor.

In (11) it is assumed that: a) the operating mode of the reactor completely coincides with the operating mode of the transformer; b) the iron losses of the reactor are zero; c) the percentage of depreciation deductions for the transformer and the reactor is the same.

Thus, the total costs for the transformer and reactor for the switchgear circuit according to Fig. 1 will be expressed, according to (9) and (10), as follows:

\[ \sum Z=Z_1+Z_2. \tag{12} \]

In the proposed switchgear circuit (Fig. 2), the costs for the reactor disappear, and only the costs for the transformer with a split winding remain.

The use of a split winding will probably cause some increase in the cost of the transformer because of the larger number of leads and the somewhat greater labor intensity of its manufacture, compared with a transformer of the same power but with a non-split winding. The exact percentage increase in cost, naturally, can be determined only after a detailed calculation of the transformer with a split winding. There is no doubt that this percentage will be very small [4].

As for the no-load and short-circuit losses of a transformer with a split winding, they will probably be equal to the losses of an equivalent two-winding transformer, provided the electromagnetic loads are the same.

In the general case, the costs for a transformer with a split winding will be expressed as

\[ \sum Z'=Z_1'=Z_{\mathrm{r}}'(1+aT)+p_{\mathrm{s}}'(c_1+c_2T)+p_{\mathrm{m}}' \left(\frac{S_{\mathrm{mp}}}{S}\right)^2 \left(c_1k_{\mathrm{s}}^2+c_2\cdot\frac{\tau}{t}\cdot T_{\mathrm{p}}\right). \tag{13} \]

The economic benefit from using a transformer with a split winding according to the scheme of Fig. 2, taking into account expressions (10), (12), (13), will be expressed as

\[ Z = Z_1 + Z_2 - Z_1' = \]
\[ = Z_2 - \left[(1 + aT)(Z_p' - Z_p) + (p_c' - p_c)(c_1 + c_2 T) + \right. \]
\[ \left. + (p_m' - p_m)\left(\frac{S_{m}}{S}\right)^2 (c_1 k_c^2 + c_2 \cdot T_p)\right]. \tag{14} \]

If the cost and losses of the transformer with a split winding are equal to the cost and losses of a two-winding transformer, then the economic effect from replacing the transformers will be expressed by the cost of the total expenditures for the reactor \(Z_2\).

If, however, the difference \((Z_1 - Z_1') > Z\), then the economic feasibility of using a transformer with a split winding disappears.

Example. Let us calculate the economic benefit from using transformers with a split winding, each with a capacity of \(31.5\ \mathrm{MVA}\), voltage \(110/6\ \mathrm{kV}\), for a \(6\ \mathrm{kV}\) switchgear with one bus system.

The calculated cost of a two-winding transformer of \(31500\ \mathrm{kVA}\) without an on-load tap changer, with a high voltage of \(110\ \mathrm{kV}\), according to Table II-17 [1], is equal to 37,500 rubles in new prices.

We take (according to Table II-25 [1]) \(a = 0.08\). According to (6), \(c_1 = 154\ \mathrm{rubles/kW}\). From [3], \(c_2 = 70\ \mathrm{rubles/(kW\cdot year)}\) (with the average cost for the Soviet Union of about \(0.8\ \mathrm{kopecks/kWh}\)); \(k_c = 0.8\); \(\dfrac{S_m}{S} = 0.8\); \(t = 8760\) hours; \(\tau = 4000\) hours; \(T_p = 5\) years.

According to the data of the Zaporizhzhia Transformer Plant, the losses of the indicated transformer are: \(p_m = 201.2\ \mathrm{kW}\); \(p_c = 81.3\ \mathrm{kW}\). The rated current of the secondary winding of the transformer is \(2700\ \mathrm{A}\).

According to [1], the minimum current of each branch of the split reactor must be not less than 0.675 of the rated current of the transformer feeding the two sections, i.e., it must be equal to \(1860\ \mathrm{A}\).

To make it possible to limit the short-circuit power in the \(6\ \mathrm{kV}\) network to \(200\ \mathrm{MVA}\), according to preliminary calculations and taking the above into account, one may take a reactor of \(2000\ \mathrm{A}\), \(6\ \mathrm{kV}\), \(8\%\).

The losses in each phase of the reactor, when a current of \(2000\ \mathrm{A}\) flows in each half-phase, are equal to \(9200\ \mathrm{W}\).

The factory price of the reactor is 1600 rubles per phase (Appendix II-1 [1]).

According to [5], the calculated cost of a transformer amounts to 1.8 of the selling price. Taking the same coefficient for the reactor, we find that \(Z_{rr} = 9000\) rubles.

Let us assume that the transformer with a split winding is more expensive than the two-winding transformer by a factor of \(K_z\), i.e.,

\[ Z_p' = K_z \cdot Z_p . \tag{15} \]

We shall also consider that the losses of the first transformer are greater than the losses of the second by a factor of \(K_p\), i.e.,

\[ \sum P' = K_p \cdot \sum P . \tag{16} \]

In this case, for simplicity we assume that

\[ p_c' = K_p \cdot p_c, \]

\[ p_m' = K_p \cdot p_m . \tag{17} \]

Then the total costs for the reactor on the basis of (13) will be equal to

\[ Z_2 = 9000(1 + 0.06 \cdot 8) + 27.6 \left(\frac{2200}{4000}\right)^2 \cdot \left(154 \cdot 0.8^2 + \]

\[ + 70.5 \cdot \frac{400}{8760}\right) = 15460 \text{ rubles}, \]

\[ I_t = \frac{0.8 \cdot 31500}{0.6} = 2200 \, a . \]

The economic benefit from the use of a transformer with a split winding, according to (14), taking into account (15), (16), and (17), will be equal to (per transformer)

\[ Z = 15460 - \left[1.48 \cdot (K_z - 1) \cdot 37500 + (K_p - 1) \cdot 81.3 \cdot (154 + 70 \cdot 8) + \right. \]

\[ \left. (K_p - 1) \cdot 201.2 \cdot 0.6^2 \cdot \left(154 \cdot 0.8^2 + \]

\[ + 70 \cdot 5 \frac{4000}{8760}\right)\right] = 162260 - 53500 \cdot K_z - 91300 \cdot K_p . \]

Figure 3 gives the dependences \(Z = f(K_z, K_p)\) for the entire substation. In the case of equality of the costs and losses of a two-winding transformer and a transformer with a split winding, an economic effect of about 31,000 rubles is obtained.

Fig. 3. Dependence \(Z = f(k_z, k_p)\).

If, however, it is assumed that the cost of the transformer with a split winding is greater than the cost of the two-winding transformer by 10%, and the losses of the former are likewise greater than the losses of the latter by 10%, then the economic effect, even in this unlikely case, will amount to 1,500 rubles (in new prices).

Conclusions

1. The use of transformers with split windings at electrical substations with voltages of 6–10 kV and capacities of 40 MVA and above is economically expedient. The magnitude of this benefit may be estimated, approximately, at 10–20 thousand rubles for each substation over an estimated period of 8 years.

1. By eliminating one of the intermediate links (the reactor), the operational reliability of a substation circuit with split-winding transformers increases.

References

1. L. I. Dvoskin. Layout and Designs of High-Voltage Switchgear. Gosenergoizdat, 1960.

2. D. D. Chase, A. N. Garin. Transformers with split windings. El. Eng., No. 6, 1934; translation—Energy Review, No. 1, 1935.

3. I. D. Kutyavin. On the economic selection of transformer capacity. Izvestiya Vuzov. Energetics, No. 1, 1962.

4. L. N. Bantidanov and V. N. Tarasov. Electrical Equipment of Electric Stations and Substations. Gosenergoizdat, 1960.

5. K. K. Balashov. On the question of the technical-economic foundations of transformer design. VEP, No. 1, 1960.

1. The work was carried out under the supervision of Professor I. D. Kutyavin, Doctor of Technical Sciences. ↩