Degree of Compensation of Long-Line Parameters

O. B. TOLPYGO

(Presented by the scientific seminar of the Power Engineering Faculty)

Compensation of the inductive resistance and capacitive conductance of a long line is carried out by inserting reactive elements into the line: capacitance, inductance, or both simultaneously. Since the operating mode of electric power transmission is determined by the operation of all the elements entering into it, the inclusion of compensating devices will introduce a change into the operating mode of the electric power transmission, will change its properties as a whole, and, first of all, will change the properties of the line itself, i.e., its constants \(A, B, C, D\). The reflection of the new properties of the line acquired under compensation will be expressed in a change of the coefficients. The degree of change of the coefficients will depend on the power of the compensating devices and on the place where they are inserted in the line.

The degree of change of the line parameters, i.e., chiefly its reactive resistance and reactive conductance, is usually characterized by the degree of compensation. The concept “degree of compensation” must be clarified, since this concept is often interpreted, in our opinion, incorrectly. G. E. Posselev [1] asserts that “the measure of series compensation is the degree of compensation \(k_c\), defined as the ratio of the capacitive resistance inserted in series into the line to the inductive resistance of the line \(x_\ell\)

\[ k_c = \frac{x_c}{x_\ell} \tag{1} \]

There also the degree of shunt compensation is defined as the ratio of the conductance of the reactor inserted into the line to the capacitive conductance of the line

\[ k_L = \frac{b_l}{b_c}. \tag{2} \]

The following objections arise against such a definition of the degree of compensation:

a) the degree of compensation depends on the location of the compensating device on the line, which formulas (1, 2) do not take into account;

b) the determination of the compensation coefficient for circuits having distributed parameters (lines) and lumped inductances and capacitances (compensating devices), according to the above formulas, is in principle incorrect, since they do not reflect the specific features of the combination of distributed and lumped parameters of a compensated line. The compensated line may be represented as a chain of four-terminal networks. The generalized parameters of the compensated line will be found as

\[ \left| \begin{array}{cc} A_e & B_e \\ C_e & D_e \end{array} \right| = \left| \begin{array}{cc} A_1 & B_1 \\ C_1 & D_1 \end{array} \right| \left| \begin{array}{cc} A_k & B_k \\ C_k & D_k \end{array} \right| \left| \begin{array}{cc} A_2 & B_2 \\ C_2 & D_2 \end{array} \right|. \tag{3} \]

Since \(B_e\) represents the reactance of the compensated line, and \(C_e\) its reactive conductance, the degree of compensation should be understood as the relative change in the corresponding parameter of the compensated line in comparison with the corresponding parameter of the uncompensated line:

degree of longitudinal compensation

\[ k_c=\frac{B-B_e}{B}, \tag{4} \]

degree of transverse compensation

\[ k_L=\frac{C-C_e}{C}. \]

Fig. 1. Equivalent circuit of a long line.

Since the coefficients \(C_e, B_e\) depend on the type and location of the compensating devices connected to the line, one should expect the magnitude of the degree of compensation to vary for the reasons mentioned above.

With single longitudinal compensation, the degree of compensation as a function of the point at which the capacitance is connected to the line is determined by the formula (for \(r_p=0,\ g_0=0\))

\[ k_c=\frac{x_c}{(\operatorname{tg}\lambda_1+\operatorname{tg}\lambda_2)Z_v} \quad [2] \]

or, in relative units,

\[ k_c=\frac{x_c}{\operatorname{tg}\lambda_1+\operatorname{tg}\lambda_2}. \]

It is easy to prove \(\left(\frac{\partial k_c}{\partial \lambda_1}=0\right)\) that the greatest degree of compensation for

\[ \dot{x}_c=\text{const} \]

is obtained when the capacitance is placed at the middle of the line.

Taking \(x_c\) to be constant, we obtain the following dependence of the degree of longitudinal compensation on the position of the capacitive CU on the line (line length \(l=1000\) km, \(x_c=-j\,0.433\)).

As is seen from Fig. 2, depending on the location of the CU on the line, the degree of longitudinal compensation is smallest when the capacitive CU is placed at the ends of the line and greatest when it is connected at the middle of the line.

Incidentally, we note that if one adheres to the method of determining the degree of compensation by the formula \(k_c=\frac{x_c}{x_l}\), then \(k_c=40\%\) for any position of the CU on the line.

Figure 3 shows the changes in the reactance of the capacitive CU when the degree of longitudinal compensation is kept constant for any position of the CU on the line.

It follows from the curve that, in order to maintain one and the same degree of series compensation, for different positions of the capacitive compensating device on the line, the greatest capacity is required when the compensating device is located at the end.

Fig. 2. Curve of the dependence of the degree of series compensation on the point of connection of the capacitive compensating device.

For shunt compensation, the expression for the degree of shunt compensation can be determined from formula 3, bearing in mind that

\[ A_k=D_k=1,\qquad B_k=0,\qquad C_k=-jy_k, \]

where \(-jy_k\) is the conductance of the reactor.

Therefore

\[ \left| \begin{matrix} A_e & B_e\\ C_e & D_e \end{matrix} \right| = \left| \begin{matrix} A_1 & B_1\\ C_1 & D_1 \end{matrix} \right| \left| \begin{matrix} 1 & 0\\ -jy_k & 1 \end{matrix} \right| \left| \begin{matrix} A_2 & B_2\\ C_2 & D_2 \end{matrix} \right| = \]

\[ = \left| \begin{matrix} A_1A_2-jA_2B_1y_k+B_1C_2 & A_1B_2-jB_1B_2y_k+B_1D_2\\ A_2C_1-jA_2D_1y_k+C_2D_1 & C_1B_2-jB_2D_1y_k+D_1D_2 \end{matrix} \right|. \tag{6} \]

Whence

\[ k_L=\frac{C-C_e}{C}=\frac{A_2D_1y_k}{A_2C_1+C_2D_1}. \tag{7} \]

But

\[ A_2=\cos\lambda_2,\qquad C_2=j\frac{\sin\lambda_2}{Z_b}. \tag{8} \]

\[ C_1=j\frac{\sin\lambda_1}{Z_b},\qquad D_1=\cos\lambda_1 \]

and

\[ k_L=\frac{y_k Z_b}{\tg\lambda_1+\tg\lambda_2}, \tag{9} \]

or, in relative units \((Z_b=1)\),

\[ k_L=\frac{y_k}{\tg\lambda_1+\tg\lambda_2}. \tag{10} \]

Consequently, the degree of shunt compensation is determined by an expression analogous to the expression for the degree of series compensation, and thus the change in the degree of shunt compensation with the position of the reactor on the line can be represented by the same graph (Fig. 3).

Thus, the degree of compensation is the ratio of the change in the coefficient \(B_e\) of the compensated line \(K\) to the coefficient \(B\) of the uncompensated—

[[unclear: beginning of word]] line for series compensation and the corresponding ratio of the coefficients \(C\) for shunt compensation.

Fig. 3. Curves of the variation of the reactance of a capacitive compensating device depending on its position on the line.

But the connection of only a series CU causes a change not only in the coefficient \(B\), reflecting the inductive resistance of the line, but also in the coefficient \(C\), reflecting the capacitive conductance of the line.

In fact, for series compensation, from (6) it follows that

\[ \dot C_e = C - j x_c \dot C_1 \dot C_2 \]

or

\[ \dot C_e = j(C + x_c C_1 C_2). \tag{11} \]

From the latter expression one may conclude that series capacitive compensation, by reducing the coefficient \(B\), i.e., the inductive resistance of the line, simultaneously increases the coefficient \(C\), in other words its capacitive conductance; therefore the no-load current of a line with series compensation will be greater than the no-load current of an uncompensated line,

\[ I_{10e} = U_2 C_e > I_{10} = U_2 C,\quad \text{since } C_e > C. \tag{12} \]

The conclusion suggests itself that series capacitive compensation produces negative compensation of the parameter \(C\), i.e., an increase in capacitive conductance; moreover, this degree of compensation (we shall denote it by \(k'_L\)) will be equal to

\[ k'_L = \left|\frac{\dot C - \dot C_e}{\dot C}\right| = \left|\frac{x_c C_1 C_2}{C}\right|. \tag{13} \]

Transforming expression (13), we obtain

\[ k'_L = \frac{x_c}{Z_b\left(\operatorname{ctg}\lambda_1+\operatorname{ctg}\lambda_2\right)} \tag{14} \]

or, in relative units,

\[ k'_L = \frac{x_c}{\operatorname{ctg}\lambda_1+\operatorname{ctg}\lambda_2}. \tag{15} \]

Analysis of expression (15) permits the conclusion that \(k'_L\) will be maximal when the capacitance is located in the middle of the line \(\left(\lambda_1=\lambda_2=\frac{\lambda}{2}\right)\),

\[ k'_L = \frac{x_c}{2\operatorname{ctg}\frac{\lambda}{2}}, \tag{16} \]

the smallest value of \(k'_L\)—when the capacitive CU is placed at the ends of the line \((\lambda_1=0,\lambda_2=\lambda\) or \(\lambda_1=\lambda,\lambda_2=0)\), since in this case

\[ k'_L=\frac{x_c}{\operatorname{ctg}0^\circ+\operatorname{ctg}\lambda}=0. \tag{17} \]

Applying analogous reasoning only to shunt compensation, we find that the coefficient \(B\) in such a compensated line is greater than the coefficient \(B\) of an uncompensated line. From expression (6) it follows that

\[ \dot B_e=\dot B-j\dot B_1\dot B_2 y_k \]

or

\[ \dot B_e=j(B+B_1B_2y_k), \tag{18} \]

whence it is necessary to conclude that the use of shunt compensation increases the parameter \(B\) of the compensated line. This negative degree of compensation of the parameter \(B\) (denote it by \(k'_c\)) will be equal to

\[ k'_c=\left|\frac{B-B_e}{\dot B}\right| =\left|\frac{y_k Z_B}{\operatorname{ctg}\lambda_1+\operatorname{ctg}\lambda_2}\right| \tag{19} \]

or, in relative units,

\[ k'_c=\frac{y_k}{\operatorname{ctg}\lambda_1+\operatorname{ctg}\lambda_2}. \tag{20} \]

Fig. 4. Curve of the dependences of the negative degrees of compensation on the positions of the compensating devices on the line.

In Fig. 4 are given the plot and the values of the coefficients \(k'_L\) and \(k'_c\) for a line \(l=1000\) km. With constant reactance \(x_c\) for the case of series compensation, and with constant susceptance for the case of shunt compensation, depending on the point at which the CU is connected to the line. From expressions (15) and (20) it follows that an increase in the reactance of a capacitive CU or an increase in the susceptance of a shunt CU is accompanied by a corresponding increase in the capacitive susceptance or inductive reactance of the line. When series and shunt (single) CUs are used jointly, the degree of compensation depends not only on the magnitude of the reactance or susceptance of the CU and on their point of connection into the line, but also on the mutual position of the series and shunt CUs, as is seen from Figs. 5 and 6, where the dependences of the degrees of compensation are presented for a constant location of the capacitive CU and various positions of the reactor on the line.

It is easy to see that the nature of the variation of the degrees of compensation differs from circuits with only one CU, series or shunt. For example, for the circuits (Fig. 5) the degree of series compensation changes not so significantly when the reactor is moved toward the end of the line. For circuits with a series CU at the end of the line (Fig. 6), the degrees of compensation \(k_l\) and \(k_c\) with displa-

with the displacement of the reactor along the line, change in mutually opposite directions. Moreover, it is characteristic that moving the reactor beyond the lumped capacitive resistance is accompanied by a sharp change in the degree of compensation: \(k_c\) decreases by a factor of 1.75, while \(k_L\) increases by the same amount. Another interesting question is how the degree of compensation will change when the compensating device is subdivided, under the condition that their total resistance or conductance remains constant. Figs. 7 and 8 give the answer to this question.

Fig. 5

Fig. 6

Figs. 5, 6. Curve of the dependence of the degrees of compensation on the mutual arrangement of the compensating devices.

The change in the degree of compensation when one CU is displaced, with the CU of the other type subdivided, is not as significant as in the case of single CUs; and it must be assumed that, with a large number of CUs, i.e., when they are placed at small intervals, the degree of compensation will remain constant.

Calculations of the degree of compensation, carried out for a large number of circuits, make it possible to draw the following conclusions:

1. Determining the degree of compensation by a simple ratio of the magnitude of the resistance (conductance) of the CU to the resistance (conductance) of the line is incorrect, since this method does not take into account the combination of distributed (line) and lumped (compensating devices) constants, the place at which the CU is connected on the line, and their mutual arrangement.

1. The degree of compensation must be determined by comparing the corresponding parameters of the compensated and uncompensated lines

\[ k_C=\frac{B-B_e}{B}, \qquad k_L=\frac{C-C_e}{C}. \]

1. With the same impedance (admittance) of the compensating device, the degree of compensation depends on the point at which the compensating device is connected to the line and on the relative arrangement of the compensating devices.

Fig. 7

Fig. 8.

Figs. 7, 8. Curves of the dependence of the degrees of compensation on the position of the compensating devices on the line.

1. The use of only longitudinal compensating devices causes an increase in the capacitive conductance of the line, while the use of only transverse compensating devices causes an increase in the inductive impedance of the line.

2. Splitting a compensating device of one type, with a different placement of compensating devices of another type, causes an insignificant change in the degree of compensation.

3. When split compensating devices are used, in order to maintain constancy of the degree of compensation as a function of the position of the compensating devices on the line, a smaller compensating-device rating is required than when single compensating devices are used.

REFERENCES

1. Pospelov G. E. On the limiting compensation of electric-transmission parameters. Elektrichestvo, 2, 1952.

2. Atabekov G. I. Problems of relay protection of power-transmission lines with longitudinal capacitive compensation. Elektrichestvo, 8, 1953.