TECHNICAL AND ECONOMIC DETERMINATION OF THE OPTIMAL DIMENSIONS OF MEDIUM- AND HIGH-POWER SMOOTHING CHOKES WITH A GAP EQUAL TO THE WINDING HEIGHT

L. I. Piletskii

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

Unlike [1], which gives an approximate formula for the rated power, the present article gives a more accurate expression for the power of a choke having the same construction as in [1].

The induction inside the winding and across its width is distributed, according to [2], by the trapezoidal law (Fig. 1).

The induction inside the winding is \(B_z\).

\[ B_z=\frac{0.4\pi I W}{H}, \tag{1} \]

where \(I\) is the dc component of the current flowing through the choke winding, A;

Fig. 1

\(W\) — number of turns of the winding;
\(H\) — length of the air gap, cm.
Induction across the width of the winding \(B_x\):

\[ B_x=\frac{B_zD}{2b}\left(1-\frac{2x}{D}\right). \tag{2} \]

The magnetic flux created by the winding and passing across the width of the winding links unequally with the turns of the winding. The magnetic flux of an elementary tube of radius \(x\) is equal to

\[ d\Phi_x=B_x2\pi x\,dx. \tag{3} \]

This flux links with all turns located to the right (outward) from the tube, i.e., with the number of turns \(\dfrac{W}{b}\left(\dfrac{D}{2}-x\right)\).

The total flux linkage across the width of the winding will be

\[ \psi_b=\int\limits_{\frac{D-2b}{2}}^{\frac{D}{2}} \pi B_z\frac{DW}{b^2}\left(1-\frac{2x}{D}\right)x\left(\frac{D}{2}-x\right)\,dx = \frac{\pi B_z W b}{6}(2D-3b). \tag{4} \]

Flux linkage inside the winding

\[ \psi=\frac{\pi B_z W}{4}(D-2b)^2. \tag{5} \]

Total flux linkage \(\psi_p\):

\[ \psi_p=\pi B_zW\left[\frac{(D-2b)^2}{4}+\frac{b}{6}(2D-3b)\right]. \tag{6} \]

The rating capacity of the choke \(S\):

\[ S=\omega LI^2, \tag{7} \]

where \(L\) is the inductance of the choke, H;
\(\omega\) is the industrial-network frequency.

\[ L=\frac{\psi_p}{I}. \tag{8} \]

According to [1], the cross section of the conducting material of the winding \(q_m\), taking into account axial cooling ducts, is

\[ q_m=\frac{k_0bhxy}{(x+i)(y+\delta)} \tag{9} \]

and the current density \(\Delta\), expressed through the principal geometric dimensions, is

\[ \Delta=\sqrt{\frac{aP(x+i)}{b_1xy}}, \tag{10} \]

where

\[ P=k_pb_1+y. \tag{11} \]

Then the power of the choke in kVA, expressed through the geometric dimensions, will take the form

\[ S=K\frac{k_0^2b^2h^2xyP}{b_1H(x+i)(y+\delta)^2} \left[ \frac{(D-2b)^2}{4}+\frac{b}{6}(2D-3b) \right], \tag{12} \]

where

\[ K=0.4\pi^2\omega a\cdot10^{-11}; \]

\(S\) is a function of five variables: \(b\), \(h\), \(x\), \(y\), and \(D\). With an increase in either \(h\), or \(x\), or \(D\), the power increases without bound. With an increase in \(y\) or \(b\), the power first increases and then decreases.

Using

\[ \frac{\partial S}{\partial b}=0, \tag{13} \]

\[ \frac{\partial S}{\partial y}=0, \tag{14} \]

one can obtain those optimal values \(b_0\) and \(y_0\) at which \(S\) reaches a maximum.

(13) and (14) give, respectively,

\[ b_0=0.5D; \tag{15} \]

\[ y_0=\frac{k_{\mathrm{p}}b_1}{k_{\mathrm{p}}b_1-2\delta}. \tag{16} \]

Since the construction of a shell-type core must provide, in the upper plane, an opening for the outgoing high-voltage insulator and, in the lower plane, an opening for connecting or disconnecting the winding lead to the insulator, the optimal value \(b_0\) must be taken not as \(0.5D\), but somewhat smaller, i.e.,

\[ b_0=k_1D, \tag{15′} \]

where

\[ k_1<0.5. \]

A change of \(y\) toward a decrease by 30 percent, or an increase by 45 percent, from \(y_0\) gives a decrease in power of only 2.2 percent from its maximum value.

By selecting the radial size of the conductor \(x\) from the existing standard sizes of winding wire, adopting \(b_0\) and \(y_0\), and proceeding either from the railway clearance dimensions, or from the minimum consumption of active materials, or from the minimum calculated costs, one can obtain the remaining independent variables characterizing the choke.

To determine the weight of the choke at a given constant induction in the steel over its entire length \(B_{\mathrm{c}}\), it is necessary to determine the flux produced by the winding. According to Fig. 1, the flux produced by an elementary tube over the section

\[ \frac{D}{2}-\frac{D-2b}{2} \]

is

\[ d\Phi_x=B_x2\pi xdx. \tag{17} \]

The flux produced by all the ampere-turns over the section \(\frac{D}{2}-\frac{D-2b}{2}\) is equal to

\[ \Phi_b=\int_{\frac{D-2b}{2}}^{\frac{D}{2}} \pi B_{\mathrm{z}}\frac{D}{b}\left(1-\frac{2x}{D}\right)x\,dx =\frac{\pi B_{\mathrm{z}}b}{6}(3D-4b). \tag{18} \]

The flux inside the winding is

\[ \Phi=\pi B_{\mathrm{z}}\frac{(D-2b)^2}{4}. \tag{19} \]

Adding (18) and (19), we obtain the total flux produced by the choke winding \(\Phi_{\mathrm{p}}\):

\[ \Phi_{\mathrm{p}}=\pi B_{\mathrm{z}}\left[\frac{(D-2b)^2}{4}+\frac{b}{6}(3D-4b)\right]. \tag{20} \]

Proceeding from the assumption that the flux produced by the winding closes only through the steel, i.e.,

\[ \Phi_{\mathrm{p}}=\Phi_{\mathrm{s}}, \tag{21} \]

one can determine the steel cross section \(q_{\mathrm{s}}\):

\[ q_{\mathrm{s}}=k_{\mathrm{s}}\cdot n\cdot r\cdot \tau, \tag{22} \]

where \(k_{\mathrm{s}}\) is the coefficient of filling the stack with steel;
\(n\) is the number of stacks of the split core \((n=6\div 12)\);
\(r\) is the radial dimension of the core stack, cm;
\(\tau\) is the tangential width of the stack, cm;

\[ \tau = D\cdot \sin \frac{180^\circ}{n}. \tag{23} \]

On the other hand,

\[ \Phi_{\mathrm{s}}=B_{\mathrm{s}}\cdot \pi\cdot D\cdot r \tag{24} \]

or

\[ r=\frac{\Phi_{\mathrm{p}}}{\pi B_{\mathrm{s}}D}. \tag{25} \]

Using the condition of maximum power (15′), we write expression (25) as

\[ r=\frac{B_{\mathrm{z}}}{B_{\mathrm{s}}}D \left[ \frac{(1-2k_1)^2}{4}+\frac{k_1}{6}(3-4k_1) \right]. \tag{26} \]

The weight of the steel of the yokes \(Q_{\mathrm{sy}}\) is

\[ Q_{\mathrm{sy}}=2\pi b(D-b)k_{\mathrm{s}}\cdot r\cdot \gamma_{\mathrm{s}}, \tag{27} \]

where \(\gamma_{\mathrm{s}}\) is the specific weight of steel.

The weight of the steel of the legs \(Q_{\mathrm{ss}}\) is

\[ Q_{\mathrm{ss}}=n\cdot r\cdot \tau\cdot k_{\mathrm{s}}(H+2r+2\delta_0)\cdot \gamma_{\mathrm{s}}, \tag{28} \]

where \(\delta_0\) is the insulation distance from the winding to the steel stack, cm.

The total weight of steel \(Q_{\mathrm{s}}\) is

\[ Q_{\mathrm{s}}=Q_{\mathrm{sy}}+Q_{\mathrm{ss}}. \tag{29} \]

The weight of copper \(Q_{\mathrm{m}}\) is

\[ Q_{\mathrm{m}}=q_{\mathrm{m}}l_{\mathrm{m}}\cdot \gamma_{\mathrm{m}}, \tag{30} \]

where \(l_{\mathrm{m}}\) is the length of the mean turn of the winding, cm,

\[ l_{\mathrm{m}}=\pi(D-b); \tag{31} \]

\(\gamma_{\mathrm{m}}\) is the specific weight of the conducting material.

Then, taking (31) and (9) into account, the weight of copper can be represented in the following form:

\[ Q_{\mathrm{m}}=\pi\gamma_{\mathrm{m}} \frac{k_0bhxy(D-b)}{(x+1)(y+\delta)}. \tag{32} \]

For a techno-economic investigation it is necessary to reduce the weight of the conducting material to the weight of steel, or conversely. Then the total reduced weight of the choke \(Q_{\mathrm{p}}\) is

\[ Q_{\mathrm{p}}=Q_{\mathrm{s}}+\beta Q_{\mathrm{m}} \tag{33} \]

in (33),

\(\beta\) is the reduction coefficient, characterizing how many times greater the specific cost of the winding material is than the specific cost of the steel in the product.

It is of interest to investigate the influence of the geometrical dimensions of the choke, proceeding from the condition of minimum specific consumption of active materials, i.e.,

\[ \frac{Q_{\mathrm{p}}}{S}=\frac{Q_{\mathrm{s}}}{S}+\beta\frac{Q_{\mathrm{m}}}{S}. \tag{34} \]

Expression (34) is a function of five variables.

To determine the minimum value of \(\dfrac{Q_p}{S}\), it is necessary to reduce the number of variables. For this we shall use the condition for maximum power (15′) and the power expression (12). In (15′) we replace \(b\) by \(D\), i.e., \(b=k_1D\), and express \(D\) in terms of the power \(S\).

Consequently,

\[ D=\sqrt[4]{\frac{12Sb_1H(x+i)(y+\delta)^2}{Kk_o^2k_1^2h^2xyP(3-8k_1+6k_1^2)}} . \tag{35} \]

Then (34) will be a function of only three variables \(x\), \(y\), and \(h\):

\[ \begin{aligned} \frac{Q_p}{S}={}& a_1\beta\gamma_M(1-k_1) \sqrt{\frac{Hxy}{\alpha SP(x+i)T}} + a_2\frac{(1-k_1)(y+\delta)M}{hT} \sqrt{\frac{x+i}{\alpha Pyx}} \\ &+ a_3M\sqrt[4]{\frac{h^2(x+i)(y+\delta)^2}{S\alpha k_1^2HxyPT}} + 2a_4\delta_0M\sqrt[4]{\frac{(x+i)(y+\delta)^2}{S\alpha Hh^2k_1^2xyPT^3}} \\ &+ a_5\frac{M^2}{T}\sqrt[4]{\frac{S(x+i)(y+\delta)^2}{\alpha k_1^2H^3h^2xyPT}} , \end{aligned} \tag{36} \]

where

\[ \begin{gathered} a_1=\pi\sqrt{\frac{12b_1}{A}};\qquad a_2=\frac{0.8\pi^2\gamma_c\sqrt{b_1}}{AB_ck_o};\qquad a_3=\frac{0.4\pi^2\gamma_cn\sin\frac{180^\circ}{n}}{B_c} \sqrt[4]{\frac{b_1}{12A^3k_o^2}}, \\ a_4=\frac{0.8\pi\gamma_cn\sin\frac{180^\circ}{n}}{B_c} \sqrt[4]{\frac{b_1}{12A^3k_o^2}};\qquad a_5=\frac{0.32\pi^2\gamma_cn\sin\frac{180^\circ}{n}}{AB_c^2k_c^2} \sqrt[4]{\frac{b_1}{12A^3k_o^2}}; \\ A=0.4\pi^2\omega\cdot10^{-11};\qquad M=(3-6k_1+4k_1^2);\qquad T=(3-8k_1+6k_1^2). \end{gathered} \tag{37} \]

Investigations for the minimum specific consumption have shown that, as the power increases, the weight of the reduced materials per unit power decreases. The optimum values of the winding height \(h_0\) are obtained as small in relation to its diameter. The elementary dimensions of the conductor \(x_0\) and \(y_0\) are obtained as nonconstructive, i.e., very small; thus, for a power \(S=10^3\) kVA, \(y<0.1\) cm and \(x<0.1\) cm. Variation of \(x\) at \(y_0\) and \(h_c\) gives a smaller change in \(\dfrac{Q_p}{S}\) than variation of \(y\) at \(x_0\) and \(h_0\).

Changing \(h\), with \(x_0\) and \(y_0\), in the direction of decrease by 5 percent gives a change in \(\dfrac{Q_p}{S}\) of 0.7 percent from the minimum value, while increasing it by 5 percent gives a change in \(\dfrac{Q_p}{S}\) of 3 percent from the minimum value.

Of great practical interest is the investigation of the choke for a minimum of calculated costs. The investigation was carried out according to the method of [3].

Starting from the choke power, the adopted induction in the steel, and the permissible specific density of the heat flow from the winding surface, it is possible to determine the optimum values of the main dimensions of the choke by minimizing the calculated costs, consisting of the costs of manufacturing and operating the choke, as well as the capital investments for the additional capacity of the system with the fuel base associated with operation of the choke.

As applied to smoothing chokes, the total calculated costs \(Z\) for the acquisition and operation of the choke are expressed through the weights of active materials in the following form:

\[ З=(A_1+A_2)Q_c+(\beta A_1+E\cdot \lambda_2)Q_m, \tag{38} \]

where \(A_1,\ A_2\), and \(E\) are constants.

Using (15′) and (35) to eliminate variables, one can obtain the calculated costs as a function of three variables: \(x,\ y\), and \(h\).

\[ \begin{aligned} З={}&(A_1+A_2)a_2\,\frac{(1-k_1)S(y+\delta)M}{hT} \sqrt{\frac{x+i}{aPxy}}+ \\[4pt] &+a_3(A_1+A_2)M \sqrt[4]{\frac{S^3h^2(x+i)(y+\delta)^2}{ak_1^2HxyPT^3}}+ \\[4pt] &+2a_4(A_1+A_2)\delta_0M \sqrt[4]{\frac{S^3(x+i)(y+\delta)^2}{aHh^2k_1^2xyPT^3}}+ \\[4pt] &+a_5(A_1+A_2)\frac{SM^2}{T} \sqrt[4]{\frac{S(x+i)(y+\delta)^2}{ak_1^2H^3h^2xyPT}}+ \\[4pt] &+\beta A_1a_1\gamma_m \sqrt{\frac{SHxy(1-k_1)^2}{aP(x+i)T}} +Ea_1\gamma_m\frac{a}{b_1} \sqrt{\frac{SHP(x+i)(1-k_1)^2}{xyT}} . \end{aligned} \tag{39} \]

Investigations of the minimum calculated costs once again confirmed that the optimum value of the winding width \(b_0\) is equal to half its diameter, and the dependence \(З=f(k_1)\) (Fig. 2) in the interval \(k_1=0.35\div0.5\) is almost linear. The optimum value of the axial dimension of the conductor \(y_0\), on the contrary, in comparison with the condition of maximum power, changed upward and reached such dimensions as are not produced by industry, i.e. \(y_0 \cong (2.7\div3.3)y_{\max}\),

Fig. 2

where \(y_{\max}\) is the maximum axial dimension of the conductor produced by industry.

If not \(y_0\), but \(y_{\max}\), is used, then the calculated costs increase. Figure 3 shows the dependence \(З=f(y)\) for power \(S=10^3\) kVA, where the calculated costs corresponding to the optimum value \(y_0\) are taken as 100 percent.

If, nevertheless, one aims at a choke with minimum calculated costs, then it is necessary to use doubled disk coils. The optimum value of the height \(h_0\) tends to yield a choke design with a small air gap and a large value of \(D\), which confirms the necessity of introducing axial cooling ducts.

With an increase in the ratio \(\dfrac{b}{D}=k_1\), \(h_0\) decreases linearly (Fig. 4).

Fig. 3

Fig. 4

The studies were carried out for chokes with powers \(S=10^3;\ 10^4;\ 10^5\) and \(5\cdot 10^5\) kVA with the following data: induction in the steel \(B_c=1.6\) T; value of the radial cooling duct \(\delta=1.4\) cm [4]; specific heat-flux density from the winding \(\varepsilon=0.16\) W/cm\(^2\); width of the “elementary” coil \(b_1=9\) cm; \(k_p=0.78;\ k_1=0.5;\ 0.45;\ 0.2\). The insulation distance between the winding and the choke steel is \(\delta_0=20\) cm; specific weight of steel \(\gamma_c=7.65\cdot 10^{-3}\) kg/cm\(^3\); specific weight of copper \(\gamma_m=8.9\cdot 10^{-3}\) kg/cm\(^3\); \(\beta=3.5;\ \alpha=13.5\cdot 10^4\). The coefficients \(A_1,\ A_2\), and \(E\) were calculated by the method set forth in [3], taking [5] into account.

\[ A_1=0.084;\quad A_2=0.204;\quad E=8.1\cdot 10^{-6}. \]

Determination of the optimal values of the variables \(x,\ y\), and \(h\) for expressions (36) and (39) was carried out on the electronic digital computer (EDC) of the Computing Center of Tomsk Polytechnic Institute.*

* The work on minimizing the calculated costs on the EDC was performed by L. M. Valkova, staff member of the Department of Electric Power Stations of TPI.

References

1. I. D. Kutyavin, L. I. Piletskii. On the limiting power of smoothing chokes with a linear characteristic. Izv. TPI, vol. 179, 1969.

2. G. N. Petrov. Electrical Machines, part 1, GEI, 1956.

3. I. D. Kutyavin. On determining the optimal dimensions of three-phase two-winding transformers. Izv. TPI, vol. 130, 1964.

4. S. A. Farberman, Yu. A. Bun. Repair and Modernization of Transformers, “Energiya,” 1966.

5. I. A. Morel. Remarks and proposals on a new methodology for techno-economic calculations in power engineering. Elektricheskie stantsii, no. 12, 1967.