On the Calculation of Dry-Type Transformers

I. D. Kutyavin, V. N. Kopeev

(Presented by the scientific seminar of the Department of Electric Power Stations)

The question of the optimal calculation and design of dry-type transformers of the II–III sizes is not at present receiving due attention in the literature. Meanwhile, dry-type transformers of these sizes have a number of advantages over oil-filled ones: explosion safety, ease of operation, and, in economic terms, the cost of dry-type transformers, even if it exceeds the cost of oil-filled ones, does so only very slightly.

The calculation and optimization of dry-type transformers have distinctive features in comparison with oil-filled transformers. This is connected, first of all, with problems of heat exchange. Very few works in the Soviet and foreign literature are devoted directly to the question of calculating and studying thermal processes in dry-type transformers.

The most widespread method at present for calculating dry-type transformers is the method of L. M. Shnitser [1]. L. M. Shnitser considers the thermal process in dry-type transformers, taking into account heat exchange between surfaces and cooling in channels. Between the temperature rise and the specific thermal loads, the following dependence is recommended:

\[ \Theta = 0.2 q^{0.89}, \]

where \(\Theta\) is the average temperature rise, deg;
\(q\) is the heat-flux density, W/m\(^2\).

For internal surfaces (in cooling channels), an efficiency coefficient less than unity is introduced, taking into account the geometry of the channel. The proposed dependences are empirical in character and are distinguished by comparatively low accuracy. A number of works that appeared later may be regarded as a further development of the method of thermal calculation proposed by L. M. Shnitser. These are the works of Sh. I. Lapidus, G. B. Fridman, and N. Yu. Rustamov.

Thermal calculation is an independent problem when it serves to evaluate an already existing design. However, another, opposite approach is also possible, when, proceeding from an assessment of the thermal regime, the geometry and specific loads of the transformer are selected so as to obtain a more satisfactory design. Obviously, in the optimal design of dry-type transformers, the second point of view proves to be more acceptable.

To determine the heat flux from the open surface of the winding and from the surface of the ventilation channel, we used the dependences recommended in [2]:

\[ q = 9.8 \Theta^{1.1}; \]

\[ q = 4.9 \left[ 1 - 0.11 \left( \frac{\delta}{h} \right)^{-0.32} \right] \cdot \Theta^{1.1}, \]

where \(\delta\) is the channel width, mm;
\(h\) is the channel height, mm.

Any design, including a transformer, must meet certain technical requirements. When designing a transformer, we shall regard the following quantities as specified:

1) transformer power \(S\);
2) voltage of the lower-voltage side \(U_1\);
3) voltage of the higher-voltage side \(U_2\);
4) winding connection scheme;
5) permissible temperature rise for the given insulation class;
6) insulation clearances and dimensions of insulating structures;
7) copper losses \(P_{\text{k.z.}}\) and steel losses \(P_{\text{x.x.}}\), which, for the optimal variant, must not exceed the values corresponding to the GOST standard.

In the course of the calculation it is necessary to determine the optimal values of the conductor dimensions \(x_1, x_2, y_1, y_2\),
the current densities \(\Delta_1\) and \(\Delta_2\),
the core diameter \(d\),
the winding height \(h\),
the induction \(B\),
the winding temperature rise \(\Theta\),
the short-circuit voltage \(U_k\),
and to select the material and the structural design of the winding.

Since the problem of designing a transformer, like any optimization problem, has more unknowns than the number of constraint equations, some of them are specified before the calculation begins, while some are taken as independent variables.

On the basis of transformer-design practice, at the first stage of the calculation one may exclude from the number of unknown quantities the induction \(B\) and the structural design of the winding, in particular, for a multilayer winding—the number of parallel conductors in the winding, the number of layers, and the number of cooling channels. If necessary, the possibility remains of varying these quantities discretely.

As independent variables we take the core diameter \(d\), the design temperature rise \(\Theta\), and the short-circuit voltage \(U_k\).

In formulating the constraint equations and the calculation algorithm, the following assumptions are introduced:

1. Neglect of the magnetizing current, \(U_p = U_k\).
2. In the calculation, permissible mechanical stresses are not taken into account. This assumption is sufficiently well substantiated in the literature, for example, in [3].
3. Additional losses are accounted for by constant coefficients, rather than by variables, because of the substantial and unjustified increase in the complexity of the calculation.
4. Heat exchange between the windings and between the windings and the magnetic core is not taken into account.
5. The discreteness of the dimensions and cross sections of the conductors is not taken into account.

The optimality criterion chosen is the minimum of the objective function of calculated costs \(Z_{\mathrm{p}}=f(d,\Theta,U_{\mathrm{k}})\). According to [4], the calculated-cost function has the form

\[ Z_{\mathrm{p}}=Z_{\mathrm{t}}+Z_{\mathrm{s}}+Z_{\mathrm{p}}+Z_{\mathrm{n}}, \]

where

\(Z_{\mathrm{t}}\) is the calculated price of the transformer with depreciation charges;

\(Z_{\mathrm{s}}\) is the calculated cost of the additional power required to cover losses in the transformer during the period of maximum load in the system;

\(Z_{\mathrm{p}}\) is the calculated cost of electrical-energy losses in the transformer;

\(Z_{\mathrm{n}}\) is the cost of reactive-power compensation.

The determination of the optimality criterion is based on solving the equation

\[ h=f(d,\Theta,U_{\mathrm{k}}). \]

For \(d_{0}\), \(\Theta_{0}\), and \(U_{\mathrm{k}0}\), \(h_{0}\) is found. When \(d\), \(\Theta\), \(U_{\mathrm{k}}\), and \(h\) are known, all the remaining parameters and characteristics of the transformer are determined quite simply: the numbers of turns, the dimensions and cross sections of the winding conductors, the current densities, the copper and steel losses, the weights of copper and steel, and the values of the objective functions for losses, weights of active materials, and calculated costs. All the above-mentioned characteristics and parameters are printed out. Next, by discrete variation of the possible values of the diameter \(d\), the temperature rise \(\Theta\), and the short-circuit voltage \(U_{\mathrm{k}}\),

\[ d_{0}\le d\le d_{\max}; \]

\[ \Theta_{0}\le \Theta\le \Theta_{\max}; \]

\[ U_{\mathrm{k}0}\le U_{\mathrm{k}}\le U_{\mathrm{k}\max} \]

the region of possible structural designs of the transformer is determined. Selection of the optimal variant is carried out by the designer by graphically or tabularly finding the minimum of the objective function of calculated costs.

The proposed calculation algorithm is convenient at the preliminary-design stage, but is somewhat cumbersome in the processing of the machine calculation.

Using this method, dry-type transformers of the III size rating with capacities of 1000 and 1600 kVA were calculated. Since the conclusions from the research results for transformers of both capacities practically coincide, for convenience all graphs below will be given only for the 1000 kVA transformer. In the course of calculation and investigation, two most competitive variants were considered for a transformer of each capacity:

1) the low-voltage winding is made of bare busbar;

2) the low-voltage winding is made of metal strip, with the strip width equal to the winding height; hereinafter this winding will be called a foil winding.

The high-voltage winding for both variants is multilayer, made of rectangular wire.

As already indicated above, the optimality criterion chosen is the minimum of the objective function of calculated costs. Figure 1 presents graphs of calculated costs as a function of temperature, \(Z_{\mathrm{p}}=f(\Theta)\), at optimal values of the core-leg diameter \(d=d_{\mathrm{opt}}\) and at ...

Fig. 1. Influence of economic coefficients on the position and magnitude of the minimum of the objective function \(Z_p=f(\theta)\) for \(d=d_{\mathrm{opt}}\) and \(U_k=5.5\%\).

Fig. 2. Change in the objective function of calculated costs for different winding materials at \(U_k=5.5\%\) and \(d=d_{\mathrm{opt}}\).

at a constant short-circuit voltage value \(U_k = 5.5\%\) and at different economic coefficients \(\beta\) and \(K_t\)*. The low-voltage winding was made according to variant 1; the winding material was copper. It turned out that, with this formulation of the problem, it is very important to know the actual value of these coefficients. It follows from the graphs that the economic coefficients affect not only the absolute value, but, most importantly, the position of the minimum of the objective function of calculated costs. When \(\beta\) changes from 2 to 3.5 and \(K_t\) from 0.7 to 1.1, the range of optimal operating overheating temperatures changes from 60 to \(110^\circ\). As the economic coefficients increase, the minimum of the objective function shifts into the region of higher temperatures. Consequently, when the cost of the transformer is higher, it is advisable to increase the operating temperature (current densities); when the cost of the active materials is low, it is necessary to choose an operating regime with low current densities, since in this case the cost of electrical-energy losses begins to prevail over the cost of the design.

From the standpoint of the economics of a dry-type transformer, the question of what to use as the winding material—copper, aluminum, or a combination of the two, i.e., making the low-voltage winding of copper and the high-voltage winding of aluminum—is also of interest. A number of literary sources report that aluminum as a winding material is quite competitive with copper. We have obtained no fundamentally new conclusions. As follows from Fig. 2, for a transformer rated at 1000 kVA, the design with aluminum windings proved more advantageous, followed by the design with combined windings and, finally, the design with copper windings. The results do not change for different structural embodiments of the windings. It is quite likely that this conclusion will also prove valid for the entire series of dry-type transformers of this size.

When comparing two winding types according to variant 1 and variant 2, the second winding type proved preferable both in calculated costs, which are about 5% lower, and in thermal conditions (strip material eliminates the possibility of local overheating).

Conclusions

1. The proposed method can be used at the preliminary-design stage for calculating dry-type transformers of new series.

2. In designing technically rational and economical transformers, it is necessary to know the actual values of the economic coefficients \(\beta\) and \(K_t\) in the objective function of calculated costs and the most probable transformer load.

3. Aluminum as a winding material for transformers of this size is preferable to copper, except in specific and special cases.

4. A new type of winding made of strip copper (aluminum) is very promising and deserves more careful consideration.

* \(\beta\)—the ratio of the specific calculated cost of copper to the cost of steel in the product;

\(K_t\)—the calculated specific price of the transformer in rubles/kg.

References

1. L. M. Shnitser. The thermal process in a dry-type transformer. Elektrichestvo, 1948, No. 4.

2. Z. N. Kyazim-Zade, N. Yu. Rustamov. A method for calculating the heating of a dry-type transformer under steady-state conditions. Elektrotekhnika, 1969, No. 3.

3. K. K. Balashov. Transformers with aluminum windings. Vestnik elektrotekhnicheskoy promyshlennosti, 1958, No. 5.

4. I. D. Kutyavin. On determining the optimal dimensions of three-phase two-winding transformers. Proceedings of TPI, vol. 130, Tomsk, TTU Publishing House, 1964.