INVESTIGATION OF MAXIMUM-POWER TRANSFORMERS FOR MINIMUM REDUCED (TO STEEL) WEIGHT AND ESTIMATED COSTS

L. I. DEL'.

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

Below are presented some results of an investigation of maximum-power transformers for the minimum reduced (to steel) weight of active materials and minimum estimated costs, taking into account the system expenditures for the acquisition, installation, and operation of the transformer, including the capital costs for the additional power required to cover transformer losses during the period of maximum load.

We shall use expressions (2)–(9) from [3], the formula for the nominal power of a transformer (1) [1], and determine the winding height, cm,

\[ h=\frac{4p^{2}S_c}{K_c \varphi(\delta')^{2}u^{2}}, \tag{1} \]

where

\[ u=\varphi\cdot\left[1+\sqrt{1+\frac{4p}{\delta'\varphi}(2b+3\delta_{12})}\right]=\varphi\cdot z, \tag{2} \]

\[ K_c=4.44 f\frac{\pi}{4}k_c \tag{3} \]

We shall find the weights of the active materials.

The weight of the core steel, in kg:

\[ Q_c=\gamma_c l_c q_c, \tag{4} \]

where \(\gamma_c\) is the specific weight of steel, kg/cm\(^3\);

\(l_c\) is the length of the core steel, referred to the cross-sectional area of the limb;

\[ l_c=(n+1)h+(n+1)d+4nb+B, \tag{5} \]

\[ B=8l_i+6\delta_{01}+6\delta_{12}, \tag{6} \]

where \(l_i\) is the insulation distance from the windings to the yoke, cm;

\(n\) is the number of wound limbs.

The weight of the conductor material of the transformer windings, in kg,

\[ Q_m=2n\gamma_m l_m q_m\cdot 10^{-3}. \tag{7} \]

The weight of the active materials reduced to steel (in kg) is

\[ Q_n=Q_c+\beta Q_m, \tag{8} \]

where \(\beta\) is the reduction coefficient, equal to the ratio of the specific costs of the windings and the core.

After substituting the variables by independent ones, equation (7) takes the form:

\[ Q_{\mathrm{p}}=\frac{M_1}{v}+M_2v^3u^3+M_3v^2u^2+\frac{M_4kr(y+\delta)}{P\delta}, \tag{9} \]

where

\[ M_1=\frac{(n+1)\cdot S_c}{K_c}\,a_1, \tag{10} \]

\[ M_2=\frac{(n+1)}{8\rho^3}\,a_1, \tag{11} \]

\[ M_3=\frac{4nb+B}{4p^2}\,a_1, \tag{12} \]

\[ a_1=\frac{k_c\pi\gamma_c}{4}, \tag{13} \]

\[ a_2=2n\beta\gamma_m\tau. \tag{14} \]

An analytical investigation of transformers for a minimum reduced weight in the general form is very complicated; therefore, in the present article, some results are given of a numerical investigation of three-phase two-winding transformers of limiting power in the voltage class 750 kV.

In items 6, 7, 8, 9 of Tables 1 and 2, calculated values are given for the weight of steel, conductor material, and \(Q_{\mathrm{p}}\) for various values of the radial width and height of the copper of an elementary conductor.

Table 1 presents the dependence \(Q_{\mathrm{p}}=f(y)\) for \(x=\mathrm{const}\), and Table 2 the dependence \(Q_{\mathrm{p}}=f(x)\) for \(y=\mathrm{const}\), from which it follows that, as \(x\) and \(y\) increase, \(Q_{\mathrm{p}}\) increases.

In calculating these tables the following initial data were adopted:

\[ S_c=200\ \text{MVA};\quad b=8\ \text{cm};\quad i=0.01\ \text{cm};\quad \delta=0.8\ \text{cm};\quad k_c=0.9; \]

\[ U_p=0.2;\quad \varepsilon=0.3\ \text{W/cm}^2;\quad \rho=2.14\cdot10^{-6}\ \Omega\cdot\text{cm};\quad B=17500\ \text{Gs}; \]

\[ k_R=0.95;\quad f=50\ \text{Hz};\quad k_p=0.78;\quad C=0.92;\quad \delta_{01}=5\ \text{cm};\quad \delta_{12}=18\ \text{cm}; \]

\[ n=3;\quad \beta=3.5;\quad l_i=38;\quad \gamma_c=7.65\cdot10^{-3}\ \text{kg/cm}^3;\quad \gamma_m=8.9\cdot10^{-3}\ \text{kg/cm}^3. \]

In addition, for Table 1 \(x=0.14\ \text{cm}\); for Table 2 \(y=1.8\ \text{cm}\).

Technical-economic investigations of the optimum dimensions of transformers should be carried out by minimizing the total calculated costs, the components of which were mentioned above.

The expression for the calculated costs is represented in the form

\[ Z=(A_1+DB^2)Q_c+(\beta A_2+k_rA^2E)Q_m, \tag{15} \]

where \(A_1\), \(A_2\), \(D\), \(E\) are constants [2], equal to \(A_1=0.2356\); \(A_2=0.185\); \(E=85.7\cdot10^{-7}\); \(D=29.9\cdot10^{-3}\).

Figure 1 shows the dependence \(Z=f(x,y)\) for \(x=0.14\ \text{cm}\) for various values of power per limb.

Figure 2 presents the dependences \(Z=f(x,y)\) for \(y=1.06\ \text{cm}\) for \(b=7\ \text{cm}\); \(y=1.0\ \text{cm}\) for \(b=8\ \text{cm}\); \(y=0.9\ \text{cm}\) for \(b=9\ \text{cm}\).

The values of powers and short-circuit voltage are taken from Table 1 [1] with the corresponding constants.

From the tables and graphs presented, the following conclusions can be drawn:

1. For constructive values of the variables \(x\) and \(y\), the reduced (to steel) weight of the active materials has no minimum.

2. There is a weak dependence of the calculated costs on the value of \(y\) within the limits of its constructive values. The optimum value of the axial height of the coil lies within the range from \(1.5\ \text{cm}\) to \(3.5\ \text{cm}\).

3. The axial dimensions of the elementary conductor, corresponding

Table 1

Fig. 1: 1 — \(S = 70\) MVA,
2 — \(S = 100\) MVA,
3 — \(S = 150\) MVA,
4 — \(S = 200\) MVA,
5 — \(S = 250\) MVA,
6 — \(S = 300\) MVA.

Fig. 2: 1 — \(b = 7\) cm,
2 — \(b = 8\) cm,
3 — \(b = 9\) cm.

the minimum estimated costs, on the one hand, and the maximum power, on the other (all other conditions being equal), differ very little in magnitude (Table 3).

This makes it possible to use \(x^\circ\) (optimal), determined from (17) [1], also for finding the optimal dimensions of transformers corresponding to the minimum cost.

Table 2

Table 3.

REFERENCES

1. I. D. Kutyavin, L. I. Del. “On the limiting power of a transformer.” Izv. TPI, vol. 172.

2. I. D. Kutyavin. “On determining the optimum dimensions of three-phase double-winding transformers.” Izv. TPI, vol. 130.

3. I. D. Kutyavin, L. I. Del. “Determination of the limiting power of transformers with allowance for the short-circuit voltage.”