COMPUTATION OF LIMITING OPERATING CONDITIONS OF LONG-DISTANCE AC TRANSMISSION LINES WITH DIFFERENT METHODS OF REPRESENTING AN INTERMEDIATE CONNECTION*

R. I. BORISOV, Yu. V. KHRUSHCHEV

(Presented by the Department of Electric Systems and Networks)

Sufficient attention has been devoted to the question of the influence of load on the limit of transmittable power of long-distance AC transmission lines. Thus, studies have been carried out on the nature and magnitude of such influence by analyzing the free term of the characteristic equation for different values of the regulating effects of the complex load connected to the buses of the receiving system [1]. Studies of a similar kind [2] contain results for a load connected at the sending end of the circuit.

In our opinion, a separate and independent interest is presented by posing the question of the influence of different methods of representing a complex load connected at intermediate points of long-distance transmission lines on the limit of transmittable power under static stability conditions, owing to the established tendency to use long-distance AC transmission lines both as trunk lines and as lines for intersystem ties [3]. Accordingly, for the intended scheme (Fig. 1) the load is represented in the form of static characteristics with different values of the regulating effects (Fig. 2):

\[ \begin{aligned} &1)\ \frac{dP_{\mathrm{н}}}{dU}=2;0;-2; \quad \text{in this case } \frac{dQ_{\mathrm{н}}}{dU}=0,\\ &2)\ \frac{dQ_{\mathrm{н}}}{dU}=2;0;-2; \quad \text{in this case } \frac{dP_{\mathrm{н}}}{dU}=0. \end{aligned} \tag{1} \]

The calculations were carried out by the method set forth in [4], for determining the actual limit of transmittable power. For a lossless line of length 1000 km, \(U_{\mathrm{н}}=500\) kV, with an intermediate connection at the center, a no-voltage-drop operating condition is established. The power of the connection is equal to the natural power of the line and is balanced at the nominal voltage. Positive or negative values of the regulating effects of the connection indicate an increase in the surplus or deficit of power, respectively, when the voltage is lowered. With zero values of the regulating effects, the intermediate connection has no influence on the lim—

* Note. Student V. V. Kupchikov of the EEF took part in carrying out the calculations.

Fig. 1

Fig. 2. Connection characteristics:

a—1. \(P_{\mathrm{n}}(U)\) for \(\dfrac{\partial P_{\mathrm{n}}}{\partial U}=2\); 2. \(P_{\mathrm{n}}(U)\) for \(\dfrac{\partial P_{\mathrm{n}}}{\partial U}=-2\).

b—1. \(Q_{\mathrm{n}}(U)\) for \(\dfrac{\partial Q_{\mathrm{n}}}{\partial U}=2\); 2. \(Q_{\mathrm{n}}(U)\) for \(\dfrac{\partial Q_{\mathrm{n}}}{\partial U}=-2\).

the limit of the line’s transmitted power. This corresponds to the limit of transmitted power \(P_{m1}=1.15\) at the limiting value of the angle \(\delta_{12\mathrm{pr}}=90^\circ\) (Fig. 5, curve 2).

Calculations show that the positive regulating effect of the active load of the connection changes the limiting values of the transmitted power at the beginning and end of the line.

\[ P_{m1}=1.03 \text{ at } \delta_{12}=70^\circ \quad \text{and} \quad P_{m2}=1.3 \text{ at } \delta_{12}=86^\circ . \]

The negative regulating effect of the active load

\[ \frac{dP_{\mathrm{n}}}{dU}=-2 \]

changes the limiting powers at the beginning and end of the line to the values

\[ \begin{aligned} P_{m1}&=1.32 \text{ at } \delta_{12}=91^\circ,\\ P_{m2}&=1.01 \text{ at } \delta_{12}=75^\circ . \end{aligned} \]

To determine the limiting angle with respect to static stability, let us consider the circuit as a two-machine system.

For the adopted directions of power (Fig. 1), we can write:

\[ M_1 \frac{d^2\delta_1}{dt^2}=\Delta P_1;\qquad M_2 \frac{d^2\delta_2}{dt^2}=-\Delta P_2, \tag{2} \]

where \(\delta_1;\ \delta_2\) are the angles of divergence of the vectors of the beginning and end of the line relative to the connection voltage;

\(M_1,\ M_2\) are the moments of inertia of the machines of electrical systems 1 and 2.

Having written the equations of small oscillations and carried out the corresponding transformations, we obtain

\[ a_{12}=\frac{1}{M_1}\frac{dP_1}{d\delta_{12}}+\frac{1}{M_2}\frac{dP_2}{d\delta_{12}}, \tag{3} \]

where \(a_{12}\) is the specific relative acceleration of the rotors of the machines of the second and first electrical systems.

We carry out a graphical determination of the angle \(\delta_{12}\) that is limiting with respect to static stability. Differentiating \(P_1\) and \(P_2\) with respect to the angle and multiplying the derivatives by \(\frac{1}{M_1}\) and \(\frac{1}{M_2}\), respectively, we construct dependences 3, 4 (Figs. 3, 4) and their sum (dependence 5, Figs. 3, 4)*. The points of intersection of dependence 5 with the abscissa axis determine the limiting angles and the corresponding powers.

\[ \delta_{\text{lim};1}=80^\circ;\qquad P_1=1.0;\qquad P_2=1.29. \]

\[ \delta_{\text{lim};2}=82^\circ;\qquad P_1=1.3;\qquad P_2=1.0. \]

Thus, a change in the sign of the regulating effect of the active load does not increase the limiting values of the transmitted power, but merely redistributes the limits among the sections of the line.

The regulating effects of the reactive load influence the limiting values of power and angle unambiguously (Fig. 5), i.e., a negative regulating effect reduces the limit of the transmitted power \((P_m=1.03;\ \delta_{12\,\text{lim}}=67^\circ)\), while a positive one increases it \((P_m=1.3;\ \delta_{12\,\text{lim}}=100^\circ)\). Consequently, the limit of the transmitted power increases—

Fig. 3. Angular power characteristics for

\[ \frac{\partial Q_{\mathrm{н}}}{\partial U}=0,\qquad \frac{\partial P_{\mathrm{н}}}{\partial U}=2: \]

\[ 1.\ P_1(\delta_{12});\quad 1.\ P_2(\delta_{12});\quad 3.\ \frac{1}{M_1}\frac{\partial P_1}{\partial\delta_{12}};\quad 4.\ \frac{1}{M_2}\frac{\partial P_2}{\partial\delta_{12}};\quad 5.\ \frac{1}{M_1}\frac{\partial P_1}{\partial\delta_{12}}+ \frac{1}{M_2}\frac{\partial P_2}{\partial\delta_{12}}. \]

* In Figs. 3, 4, curves 3, 4, 5 are constructed under the condition \(M_{*1}=M_{*2}=1\).

Fig. 4. Angular power characteristics for

\[ \frac{\partial Q_{\mathrm{н}}}{\partial U}=0,\qquad \frac{\partial P_{\mathrm{н}}}{\partial U}=-2: \]

\[ 1.\ P_1(\delta_{12});\qquad 2.\ P_2(\delta_{12});\qquad 3.\ \frac{1}{M_1}\frac{\partial P_1}{\partial \delta_{12}};\qquad 4.\ \frac{1}{M_2}\frac{\partial P_2}{\partial \delta_{12}} \]

\[ 5.\ \frac{1}{M_1}\frac{\partial P_1}{\partial \delta_{12}} +\frac{1}{M_2}\frac{\partial P_2}{\partial \delta_{12}}. \]

Fig. 5. Angular power characteristics for

\[ \frac{\partial P_{\mathrm{н}}}{\partial U}=0 \]

and for

\[ 1.\ \frac{\partial Q_{\mathrm{н}}}{\partial U}=2;\qquad 2.\ \frac{\partial Q_{\mathrm{н}}}{\partial U}=0;\qquad 3.\ \frac{\partial Q_{\mathrm{н}}}{\partial U}=-2. \]

is manifested only in the case where, as the voltage is lowered, the surplus of reactive power at the connection increases.

The influence of the regulating effects with respect to the active and reactive powers of the intermediate connection has a significant effect on the limit of the power transmitted along the line.

REFERENCES

1. N. A. Kachanova, L. V. Tsukernik. Analysis of static stability with allowance for the characteristics of a composite load. Collection of Proceedings of the Institute of Electrical Engineering, Academy of Sciences of the Ukrainian SSR, issue 17, Kiev, Publishing House of the Academy of Sciences of the Ukrainian SSR, 1959.

2. R. S. Zainullina. The influence of load on the static stability of a system. Proceedings of the Siberian Scientific-Research Institute of Power Engineering, issue 4, Publishing House “Nauka,” Siberian Branch, Novosibirsk, 1966.

3. R. I. Borisov. The influence of generalized static characteristics of intermediate systems on the limit of power transmitted over long-distance AC transmission lines. (This collection).

4. P. S. Zhdanov. Stability of electrical systems. Gosenergoizdat, 1948.