Technical and Economic Studies of Optimal Sizes of Power Transformers

G. V. Del, V. P. Krasnov

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

§ 1. Basic Provisions

At the Department of Electric Power Stations of the Tomsk Polytechnic Institute, under the direction of Professor, Doctor I. D. Kutyavin, a method has been developed for technical and economic studies of transformers. At present, studies of the optimal sizes of transformers are being carried out on the basis of this method.

The essence of the method reduces to determining such structural dimensions of the core and windings of a transformer as correspond to the minimum of the calculated costs, which have the expression

\[ З = (A + ДB_1^2 + 1{,}5a_p)Q_c + (\beta A + E\Delta_1^2)Q_{m1} + (\beta A + E\Delta_2^2)Q_{m2}. \tag{1} \]

Here and below, all designations of physical and geometrical dimensions are adopted in accordance with [1].

Using physical and geometrical relationships, we find the expression of \(Q_c\) and \(Q_m\) through the dimensions of transformers. In [1] a derivation is given of equations describing the operation of transformers. These equations may be written in the form of the following system for the case of a two-winding transformer with continuous or helical windings:

\[ \left. \begin{aligned} S &= KB\kappa_1\Delta_1 x_1 y_1 \frac{d^2 h}{y_1+\delta_1},\\[4pt] u_p &= K_p \frac{\kappa_1 x_1 y_1 \Delta_1}{Bd^2 (y_1+\delta_1)} (d+2x_2+l)(x_1+x_2+3\delta_{12}),\\[4pt] 2\sigma_1 k_{y1}(\kappa_{p1}x_1+y_1) &= \rho \kappa_1 x_1 y_1 \Delta_1^2,\\[4pt] 2\sigma_2 k_{y2}(\kappa_{p2}x_2+y_2) &= \rho \kappa_2 x_2 y_2 \Delta_2^2,\\[4pt] \frac{\kappa_1 x_1 y_1 \Delta_1}{y_1+\delta_1} &= \frac{\kappa_2 x_2 y_2 \Delta_2}{y_2+\delta_2}. \end{aligned} \right\} \tag{2} \]

In the system of equations (2) it is assumed that:

\[ \left. \begin{aligned} K &= 1{,}11\, m k_c \pi f \cdot 10^{-11},\\[4pt] K_p &= \frac{8\pi^2 k_r}{33{,}3K_c},\\[4pt] l &= 2\delta_{02}+\delta_{12},\\[4pt] \kappa_{p1} &= \kappa_{x1}:\kappa_{y1};\quad \kappa_{p2} = \kappa_{x2}:\kappa_{y2}, \end{aligned} \right\} \tag{3} \]

\(m\) is the number of active limbs of the transformer.

In system (2) we have 8 variables: \(x_1,\ x_2,\ y_1,\ y_2,\ \Delta_1,\ \Delta_2,\ d,\ h\). The remaining quantities are constants, taken from the technical specification or from GOST (\(S,\ U_1,\ U_2,\ U_p,\ m,\ f,\ \sigma_1,\ \sigma_2,\) main insulation), or else adopted at the very beginning of the calculation (\(B,\ k_c,\ k_r,\) etc.). No-load and short-circuit losses are taken as variables, but it is assumed that their magnitude must not exceed the recommendations of GOST 401—41.

A shortcoming of system (2) is that the copper filling factors (\(\kappa_1\) and \(\kappa_2\)) are taken as constants. The problem of reducing them to variable quantities is solved as follows.

The number of turns in a coil is equal to (Fig. 1)

\[ w_k=\frac{\kappa x y \Delta}{I_n}, \tag{4} \]

where \(I_n\) is the rated phase current.

Fig. 1.

The cross-section of pure conductor material in a coil is

\[ \kappa x y = (x - m_{1(2)} i w_k)y. \tag{5} \]

The number of parallel branches of the winding \(m_{1(2)}\) may be taken as constant, determined by the magnitude of the rated current of the winding \((I_n)\) and the current of one parallel branch \((I_b)\).

Substituting (4) into (5), we obtain, respectively for the HV and LV windings:

\[ \kappa_1=\frac{1}{1+\lambda_1\Delta_1 y_1}; \]

\[ \kappa_2=\frac{1}{1+\lambda_2\Delta_2 y_2}, \tag{6} \]

where

\[ \lambda_1=\frac{i_1 m_1}{I_{n1}}=\frac{i_1}{I_{b1}} \tag{7} \]

and

\[ \lambda_2=\frac{i_2 m_2}{I_{n2}}=\frac{i_2}{I_{b2}}. \]

Taking (6) into account, system (2) is free of the shortcoming indicated above.

For power transformers, two combinations of windings are possible in principle:

1. Continuous (HV)—continuous (LV).
2. Continuous (HV)—helical (LV).

Mathematically, both combinations are described by the same equations. It is only necessary to take into account that the number of turns in the conditional coil of a helical winding is inversely proportional to the number of starts \((n_z)\), which may be regarded as a known quantity,

\[ w_{k2}=\frac{1}{n_z}. \tag{8} \]

The latter circumstance makes it necessary to consider the indicated winding combinations separately.

This article presents the results of a study of transformers with a voltage of 110 kV: three-phase transformers (winding combination C—C) and single-phase transformers (winding combination C—H). No preliminary mechanical restrictions were imposed on the transformers.

§ 2. Three-phase transformers

For the case of three-phase transformers, the system of equations (2), with allowance for (6), gives three independent variables. Studying the costs (1) simultaneously as a function of three independent variables is a very laborious task. Therefore the study was carried out in two stages. In the first stage it was assumed that \(\Delta_1=\Delta_2\), and \(Z=f(x_1,y_1)\) was studied (ratings of the transformers considered: 5.6; 7.5; 10; 15 MVA). It was found that the calculated costs depend only very weakly on the axial dimension of the coil \((y_1)\) (Table 1). The optimum value of \(y_1\) for each of the transformers studied proved to be approximately the same and equal to 1.1 cm. All this made it possible subsequently (second stage) to take \(y_1=\mathrm{const}\) for \(\Delta_1\ne\Delta_2\).

Table 1

Calculated costs (%) as a function of \(y_1\) at \(x_{\mathrm{opt}}\) for various \(S\)

In the second stage, the study of costs was carried out as a function of the independent variables \(x_2, y_2\). In this case, the expressions for the variables through \(x_2, y_2\) from the system of equations (2), with allowance for (6), have the form:

\[ \Delta_2= \frac{\lambda_2+\sqrt{\lambda_2^2+4L_2\,\dfrac{z_2}{y_2}}} {2L_2z_2}, \]

where

\[ z_2=\frac{x_2}{\kappa_{n2}x_2+y_2}; \tag{9} \]

\[ \Delta_1= \frac{ \left(\kappa_{n1}\lambda_1+\frac{y_1X}{y_1+\delta_1}\right) +\sqrt{ \left(\kappa_{n1}\lambda_1+\frac{y_1X}{y_1+\delta_1}\right)^2 +4L_1\frac{\kappa_{n1}}{y_1} } }{2L_1}, \tag{10} \]

where

\[ X=\frac{(y_2+\delta_2)(1+\lambda_2y_2\Delta_2)}{x_2y_2\Delta_2}, \tag{11} \]

\[ L_1=\frac{\rho}{2\sigma_1k_{y1}},\quad L_2=\frac{\rho}{2\sigma_2k_{y2}}; \]

\[ x_1=\frac{(y_1+\delta_1)(1+\lambda_1y_1\Delta_1)}{Xy_1\Delta_1}; \tag{12} \]

\[ d=\frac{V_1+\sqrt{V_1^2+4V_1V_2}}{2}, \tag{13} \]

where

\[ V_1=\frac{K_p x_2 y_2\Delta_2(x_1+x_2+3\delta_{12})} {u_p(y_2+\delta_2)(1+\lambda_2y_2\Delta_2)}, \]

\[ V_2=2x_2+l, \]

\[ h=\frac{S}{K}\cdot\frac{X}{d^2}. \tag{14} \]

The calculated costs according to (1) in the second case take the following form:

\[ З=\frac{\pi\gamma_c k_c}{4}\left(A+ДB^2+1.5a_p\right)\varphi(x_2,y_2), \tag{15} \]

where

\[ \varphi(x_2,y_2)= \frac{mS}{K}10^{-3}X +d^2\left[2ml_u+n_a k_a(x_1+x_2+l_r)+n_dk_d d\right]\cdot10^{-3} + \]

\[ +\frac{M_3}{d^2} \left[ (\beta A+E\Delta_1^2)\frac{d_{\mathrm{avg}.1}}{\Delta_1} + (\beta A+E\Delta_2^2)\frac{d_{\mathrm{avg}.2}}{\Delta_2} \right], \tag{16} \]

\[ M_3=\frac{4\tau_n S\cdot10^{-3}} {K\gamma_c k_c\left(A+ДB^2+1.5a_p\right)}, \]

\[ l_r=\delta_{02}+\delta_{12}+0.5\delta_{11}, \]

\[ d_{\mathrm{avg}.2}=d+x_2+2\delta_{02},\quad d_{\mathrm{avg}.1}=d+2x_2+x_1+2(\delta_{02}+\delta_{12}). \]

For a three-phase three-limb transformer \(n_a=8\), \(n_d=5.6\), \(m=3\). The corresponding expressions for the variables in the first case \((\Delta_1=\Delta_2)\) are analogous in degree of complexity to (9)—(16).

The results of the study of the calculated costs as a function of \(x_2\), \(y_2\) (second case) for a power \(S=10\) MVA are given in Table 2.

It should be noted that the minimum of the function for \(\Delta_1\ne\Delta_2\) proved to be greater than the minimum of the function for \(\Delta_1=\Delta_2\) by 1.4%. Thus, in calculations from the economic point of view, the current densities in the windings should, where possible, be taken close to one another. Unfortunately, in our case this cannot be done, since the values of \(y_2\) po-

For power transformers, in principle two combinations of windings are possible:

1. Continuous (HV)—continuous (LV).
2. Continuous (HV)—helical (LV).

Mathematically, both combinations are described by the same equations. It is only necessary to take into account that the number of turns in the conditional coil of a helical winding is inversely proportional to the number of starts \((n_z)\), which may be regarded as a known quantity,

\[ w_{k2}=\frac{1}{n_z}. \tag{8} \]

The latter circumstance makes it necessary to consider the indicated combinations of windings separately.

The present article gives the results of a study of transformers rated at 110 kV: three-phase transformers (winding combination Hr—Hr) and single-phase transformers (winding combination Hr—Vt). No preliminary mechanical restrictions were imposed on the transformers.

§ 2. Three-phase transformers

For the case of three-phase transformers, the system of equations (2), with (6) taken into account, gives three independent variables. Investigation of the costs (1) simultaneously as a function of three independent variables is a very laborious problem. Therefore the investigation was carried out in two stages. At the first stage it was assumed that \(\Delta_1=\Delta_2\), and \(Z=f(x_1,y_1)\) was investigated (the ratings of the transformers considered were: 5.6; 7.5; 10; 15 MVA). It turned out that the calculated costs depend only very weakly on the axial dimension of the coil \((y_1)\) (Table 1). The optimum value of \(y_1\) for each of the transformers studied proved to be approximately the same and equal to 1.1 cm. All this made it possible subsequently to assume (second stage) \(y_1=\mathrm{const}\) for \(\Delta_1\ne\Delta_2\).

Table 1

Calculated costs (%) as a function of \(y_1\) at \(x_{1\mathrm{opt}}\) for various \(S\)

At the second stage, the study of costs was carried out as a function of the independent variables \(x_2,\ y_2\). In this case, the expressions for the variables through \(x_2,\ y_2\) from the system of equations (2), with (6) taken into account, have the form:

\[ \Delta_2= \frac{ \lambda_2+\sqrt{\lambda_2^2+4L_2\frac{z_2}{y_2}} }{ 2L_2z_2 }, \]

and in the other case we have, although a flat one, a minimum (the studies were carried out for steel E330; 0.35; GOST 802-58).

Fig. 3.

Table 3 gives comparative data for transformers of domestic and foreign manufacture and for transformers calculated by the proposed method.

Table 3

§ 3. Single-Phase Transformers

For single-phase transformers with ratings of 10.5; 20; 40; and 60 MVA, the results of investigations of costs (1) are given for \(\Delta_1 \ne \Delta_2\). In this case the independent variables are \(\Delta_1\) and \(y_2\). Transformers with shell-type and core-type magnetic cores, with windings made of aluminum and copper conductor, are considered. The winding combination adopted is LV--HV. The calculations were carried out on a high-speed electronic digital computer.

For the studies, equations (1--5) and (8) were used with the following initial data:

\[ A = 0,1295;\quad k_T = 0,7 \quad \text{core material E-320, GOST 802-58.} \]

Table 2

are obtained for some transformers (with $\Delta_1=\Delta_2$) as nonconstructive, because of the large difference between the constants $\lambda_1$ and $\lambda_2$.

The investigations indicated above were carried out with the following initial data: $U_1/U_2=110/11$ kV; winding connection $\mathrm{Y}/\Delta$; $u_p=0.105$; $B=16500$ G; $\rho=2.14\cdot10^{-6}$ ohm·cm; $k_{\mathrm{ya}}=1.05$; $\sigma_1=\sigma_2=0.15$ W/cm$^2$; $\gamma_m=8.9$ g/cm$^3$; $\gamma_c=7.65$ g/cm$^3$; $m_1=1$; $i_1=0.14$; $i_2=0.055$; $\delta_1=0.94$; $\delta_2=0.66$; $\delta_{02}=1.5$; $\delta_{12}=5$; $\delta_{11}=6$; $k_{x1}=k_{x2}=0.7$; $k_{y1}=k_{y2}=0.91$; $k_c=0.82$; $k_v=0.95$ (all dimensions in cm); $k_t=0.96$ for $S=5.6$ and $7.5$ MVA; $k_t=0.9$ for 10 and 15 MVA; $\beta=3.2$; $A=0.1775$; $D=45.8\cdot10^{-11}$; $E=47\cdot10^{-7}$; $\alpha_p=19.3$ W/kg.

Winding material—copper. Values of $m_2$ for $S=5.6$; 7.5 MVA—2; 10—3; 15—4.

The revealed weak dependence of the calculated costs on $y$ made it possible in further investigations to take $y_1$ and $y_2=\mathrm{const}$, thereby

Fig. 2.

reducing the number of independent variables to one ($x_2$). Figure 2 gives the dependence of the calculated costs (15) on the specific thermal loading $\sigma$. Figure 3 gives the dependence of $Z$ on the induction $B$, averaged for the entire power range considered. In both cases,

transformers with a shell-type core construction with copper (1–5, 11–15) and aluminum (6–10, 16–20) windings.

From Fig. 4 it is evident that the optimal value of the induction lies within the range of 14,000–16,000 G, and its effect on costs is slight. It should be noted here that the optimal value of the induction may increase as the quality of the core material improves.

Fig. 4.

Fig. 5 shows the dependence of costs \((Z \cdot 10^3)\) on the magnitude of the specific thermal loading of the windings \((s)\), calculated for a 40 MVA transformer at different values of \(B\).

Curves (1, 4, 7, 10) correspond to an induction of 15,000 G; (2, 5, 8, 11) to 16,000 G; (3, 6, 9, 12) to 17,000 G, for transformers with a core-type magnetic circuit (1–3, 8–9) and a shell-type magnetic circuit (4–6, 10–12), with windings made of copper conductor (1–6) and aluminum conductor (8–12).

For comparison, Table 4 gives the principal data for transformers calculated by the above method and data for manufactured transformers [2]. The following data were adopted for the calculation:

\[ B = 16500\ \text{G}; \quad s = 0.15\ \text{W}/\text{cm}^2; \quad z_p = 0.046\ \frac{\text{kW}}{\text{kg}}; \quad \text{steel E-330} \]

GOST 802—54.

The other necessary constants are taken according to § 2.

The calculation equations have the form

\[ y_1 = \frac{ \left[ \frac{\kappa_{p1}}{\Delta_1} + \delta_1\left( \frac{\kappa_{p1} m_1 i_1}{I_{h1}} - L_1 \Delta_1 \right) \right] + \sqrt{ \left[ \frac{\kappa_{p1}}{\Delta_1} + \delta_1\left( \frac{\kappa_{p1} i_1 m_1}{I_{h1}} - L_1 \Delta_1 \right) \right]^2 + \frac{4\kappa_{p1}\delta_1}{\Delta_1} \left[ -\left( \frac{\kappa_{p1} i_1 m_1}{I_{h1}} - L_1 \Delta_1 \right) - \frac{y_2+\delta_2}{I_{h2} w_{k2}} \right] } }{ 2\left[ -\left( \frac{\kappa_{p1} i_1 m_1}{I_{h1}} - L_1 \Delta_1 \right) - \frac{y_2+\delta_2}{I_{h2} w_{k2}} \right] } - \frac{ \frac{4\kappa_{p1}\delta_1}{\Delta_1} \left[ -\left( \frac{\kappa_{p1} i_1 m_1}{I_{h1}} - L_1 \Delta_1 \right) - \frac{y_2+\delta_2}{I_{h2} w_{k2}} \right] }{ 2\left[ -\left( \frac{\kappa_{p1} i_1 m_1}{I_{h1}} - L_1 \Delta_1 \right) - \frac{y_2+\delta_2}{I_{h2} w_{k2}} \right] }, \tag{17} \]

\[ x_1 = \frac{L_1 I_{h2} w_{k2}\Delta_1 (y_1+\delta_1)} {(y_2+\delta_2)\kappa_{p1}} - \frac{y_1}{\kappa_{p1}}, \tag{18} \]

\[ \Delta_2 = \frac{ V_4 + \sqrt{V_4^2 + \dfrac{4L_2 I_{h2}^2 w_{k2}^2}{\kappa_{p2}y_2}} }{ \dfrac{2L_2 I_{h2} w_{k2}}{\kappa_{p2}} }; \]

where

\[ V_4 = \frac{y_2}{\kappa_{p2}} + m_2 i_2 w_{k2} \tag{19} \]

\[ x_2 = (L_2 I_{h2} w_{k2}\Delta_2 - y_2)\frac{1}{\kappa_{p2}}. \tag{20} \]

The remaining expressions are analogous to (13–16).

Figure 4 shows the dependence of costs \((3 \cdot 10^3)\) on the magnitude of the induction in the magnetic core. Here curves (1–10) are for transformers with a core of the core-type construction, and curves (11–20) are for

From Table 4 it is evident that transformers with windings made of aluminum wire have a lower weight of active materials, but somewhat increased short-circuit losses.

The mechanical stresses arising in the transformer during a short circuit ($\sigma_p$), for transformers rated at 20 and 40 MVA with an aluminum winding and a shell-type magnetic core, were found to be greater than the permissible value [3]. In this connection it may be noted that the use of an aluminum winding is more expedient for transformers with a core-type magnetic circuit, despite the increase in losses; however, the possibility of manufacturing transformers with a shell-type magnetic circuit is not excluded.

Fig. 6.

Figure 6 shows the dependence of costs ($3 \cdot 10^3$) on the value of the dens—

the current density in the primary winding at constant values of \(y_2\), respectively, for each power rating (curves 1–12), and the dependence of costs on \(y_2\) at constant values of \(\Delta_1\) for each power rating (curves 13–24).

Curves (7–12) and (19–24) are for transformers with windings made of copper wire; (1–6) and (13–18), of aluminum wire; curves (1–3), (7–9), (13–15), and (19–21) are for transformers with a core-type magnetic circuit, and (4–6), (10–12), (16–18), (22–24) are for shell-type transformers, respectively, of power ratings 10.5 (1, 4, 7…21), 20 (2, 5, 8…23), and 40 (3, 6, 9…14) MVA.

It is evident from Fig. 6 that costs have a weak dependence on the current density \(\Delta_1\), but as the transformer power increases this dependence grows; this can be readily traced in transformers with an aluminum winding and a core-type core (1–3). The dependence of costs on \(y_2\) is sharper, which makes it possible to determine more accurately the value of the optimal \(y_2\).

Conclusions

1. The proposed techno-economic method for studying the optimal dimensions of transformers makes it possible to carry out comprehensive investigations of both three-phase and single-phase transformers. The method makes it possible to find the absolute minimum of costs corresponding to the most advantageous transformer design.

2. The proposed method presupposes the use of electronic computers in its application.

3. The possibility of replacing copper winding wire with aluminum wire is, undoubtedly, of great interest. The proposed procedure makes it possible to study transformers with aluminum windings and to determine the economically advantageous limits (in terms of power rating) for replacing copper, which is a scarce material.

4. A drawback of the method should be considered the fact that, in the proposed form, it does not take additional losses into account, although in principle it is quite possible to take these losses into account.

References

1. I. D. Kutyavin, G. V. Del, V. P. Krasnov. On the techno-economic determination of the optimal dimensions of high-power substation three-phase two-winding transformers. Izvestiya TPI, vol. 130, 1964.

2. G. N. Petrov. Electrical Machines. Part 1, Gosenergoizdat, 1956.

3. P. M. Tikhomirov. Transformer Calculation. GEI, 1962.