Use of Terminal Devices in Tuning Lines to a Half-Wavelength

V. T. Agafonov

(Presented by Professor, Doctor of Technical Sciences I. D. Kutyavin)

One of the ways of ensuring the static stability of extra-long alternating-current transmissions is tuning to a half-wavelength. The possibility of using this method for extra-long electric power transmissions becomes increasingly evident for line lengths greater than 1500 km. The use of compensation in the form in which it is now carried out (series connection of capacitors and shunt connection of reactors) encounters serious difficulties on such lines [1, 2, 3], since: a) the required rating of the compensating devices is large; b) the short-circuit currents and the required equipment rating increase considerably; c) phenomena of self-excitation, self-buildup, and subharmonic oscillations arise.

Power transmission with intermediate synchronous compensators, whose reactance is compensated by series capacitance, constitutes a second solution to this problem. But in order to determine definitively the feasibility of applying such power transmission to extra-long AC lines, a great deal of research work is required. There are very many questions that have been little studied or not studied at all. Such questions as parametric resonance, dynamic overvoltages, the most appropriate degree and scheme of series compensation, self-excitation, self-buildup during parallel operation of several compensators, and instability of two machines connected by capacitive reactance require solution. The transient processes, protection behavior, and other operating conditions of the entire system have likewise not been investigated. It is possible that, in order to eliminate all these phenomena, the degree of compensation of the compensators’ reactance will have to be substantially reduced, and this may greatly increase their required rating [4, 5, 6]. It should be noted that the very use of intermediate synchronous compensators improves the dynamic stability of power transmission only slightly, and special measures are needed to improve it [7].

Tuning lines to a half-wavelength also has its drawbacks, one of which is the comparatively large rating of the compensating devices, especially for a line length of 1500 km and when it is necessary to tune to a mode greater than a half-wavelength \((Q_p > 2,\ Q_{sk} > 2)\).

The rating of the compensating devices can be reduced by using the generators and transformers at the sending and receiving ends of the power transmission as compensating devices, as was first proposed in [8]. But in that work only a simple case was considered, namely the possibility of tuning by means of inductive resistances of constant magnitude, to which, without doubt, only transformers and reactors can be assigned. The question of whether a generator can be used as a compensating device,

was not considered. For example, it is unknown what the generator reactance should be in this case; the magnitude of the generator reactance that must be introduced into the equivalent circuit of the power transmission for various cases of voltage regulation is unknown. Since the phase-shift angle between the generator \(E\) and \(U_g\) depends on the load, the effectiveness of the generator as a compensating device also depends on the load and, consequently, on the operating conditions of the line itself with terminal compensating devices. As a result, the possibility of using generators as a compensating device must be treated as a complex problem.

Further, the case of tuning only with the aid of inductive devices at the line ends is characterized by a small bandwidth and poor utilization of the line insulation, and therefore has little practical value. This makes it necessary to apply, in addition to longitudinal-inductive compensation, transverse-capacitive compensation.

As was indicated above, the power of static capacitors or synchronous compensators, when the line is fully tuned to the half-wave regime both in current and in voltage, will be rather large. However, by applying incomplete tuning in current, the required power of the static capacitors can be reduced by approximately half. The placement of the elements of the transverse-capacitive compensating device along the line must be clarified from the standpoint of the acceptability of the voltage regime and of maximum efficiency. Without a detailed consideration of all these questions it is impossible to speak of using a synchronous generator as a compensating device. Before proceeding to the presentation of the tuning method with the idea of using a synchronous generator as a compensating device (what is meant is the equivalent generator of the station), it is necessary to clarify the question of what will be taken as the generator reactance.

It is known that the synchronous reactance is the ratio of the no-load voltage to the current under a three-phase short circuit at the same ampere-turns of excitation. If the machine is unsaturated, this ratio will be constant under all steady-state operating conditions of the generator. Therefore, for the investigation of steady-state regimes the synchronous reactance may be taken. The purpose of tuning, however, is to ensure stable operation of the power transmission and, chiefly, static stability. If the power transmission were tuned to the exact half-wave regime (the phase shift between \(E\) and \(U_n\) were exactly \(180^\circ\)), then \(E\) would be constant for any transmitted power and for any small change in it. The voltage at the generator terminals (as an intermediate point of the power transmission tuned to a half-wave) would vary within certain limits. In the analysis of static stability the generator in this case is replaced by the synchronous reactance. But such tuning to the exact half-wave regime with varying voltage on the station buses has no practical significance. The operation of a tuned power transmission must proceed with an angle between \(E\) and \(U_n\) greater than \(180^\circ\), for reasons of dynamic stability and phenomena associated with wave detuning (change of frequency, line parameters). It is also necessary to strive to ensure constancy of the voltage on the station buses. Under conditions of tuning greater than \(180^\circ\) (approximately \(-\pi + \frac{\pi}{6}\)), this requires regulation of \(E\) when the transmitted power changes. Depending on the methods of such regulation, as is known [5], in calculations of the static stability of a power transmission the synchronous reactance is introduced if regulation is carried out very slowly (manual regulation); the transient reactance is introduced if electronic voltage regulators or compounding circuits are used; or, finally, the generator reactance is altogether omitted when

regulation by derivatives of the regulated parameters (strong regulation).

The last two methods of regulation are generated chiefly by static stability and have the aim of increasing the limit of the transmitted power of the electric-power transmission and, consequently, the margin of static stability. In electric transmissions tuned to a regime somewhat greater than a half-wave, the limit of transmitted power is so high [9] that the carrying capacity is determined not by static stability but by the voltage regime in the line. Therefore these methods of regulation should not be applied in electric transmissions tuned to a half-wave. The requirements of voltage regulation under stationary operating conditions of the system (meaning voltage regulation at definite points of the electric transmission when the transmitted power changes, which may be carried out with some delay) can be satisfied even by manual regulation. This gives the right to introduce a synchronous reactance into the electric-transmission circuit, which will be done below. The phase shift between \(E\) and \(U_2\) depends on the magnitude of the synchronous reactance, i.e., the greater \(x_d\), the greater \(\delta_2\). Therefore the first question that must be considered will be the possible limit of increasing the generator reactance.

Limit of increasing \(x_d\) of the generator

In essence, this question would have to be examined in detail from three sides: a) from the standpoint of the design possibilities for making generators with increased reactive resistance; b) from the standpoint of the acceptability of the operating regimes of generators: stability, short circuits, voltage regulation, self-excitation, etc.; c) and also from the standpoint of economics. Not being able to consider the first and third aspects, we may note only that in the literature [8, 10] there are statements that the reactance of transformers can be brought comparatively simply to \(e_k\,[\%]=30\%\), and the \(x_d\) of synchronous generators can be significantly increased; moreover, the cost of such generators will even be lower than that of ordinary ones, not to mention generators with reduced reactive resistance. In practice there are synchronous machines made with \(x_d=2\div 2.5\). The increase in reactance occurs mainly by reducing the air gap.

It is necessary to dwell somewhat more closely on the second aspect of the question. It is known that the stability of the generator itself depends on \(x_d\). At a very large value of \(x_d\), the overload capacity of a synchronous generator may prove to be very small, and the generator itself, as an element of the electric transmission, will be unstable. Therefore it is desirable to determine that magnitude of the synchronous reactance beyond which one should not go. The electromagnetic power of the generator (salient-pole), neglecting the active resistance of the stator winding, is equal to the electrical power and is expressed by the formula

\[ P_\psi=P_c=m\left[ \frac{u_n^2\cos\theta\sin\theta}{x_q} + \frac{u_n e_0\sin\theta}{(x_d-x_q)} - \frac{u_n^2}{x_d}\cos\theta\sin\theta - \frac{u_n e_0\sin\theta\cdot x_q}{x_d(x_d-x_q)} \right], \tag{1} \]

where \(m\) is the number of phases;
\(u_n\) is the rated voltage.

Thus, the electrical power of the generator

\[ P_c=f(u_n,\theta,x_d,x_q,e_0) \tag{2} \]

is a function of many quantities. This function can be investigated for an extremum by the method of Lagrange multipliers.

In the given case, knowing the additional relationships between the variables \(x_d = 1.5 x_q\), and also specifying an acceptable ratio \(\dfrac{e_0}{u_k}\) and considering it unchanged, this can be done more simply by taking partial derivatives with respect to the various parameters. It turns out that if one specifies an allowable overload of \(20\%\), then the synchronous reactance of the generator may be increased to \(x_d = 2\) (Fig. 1).

Of concern are the sudden load-shedding condition of such a generator and the operating condition with a capacitive load. Under sudden load shedding, the increase of voltage at its terminals, defined as

\[ \Delta u[\%] = \frac{e_0 - u}{u}\,100, \]

will be significant. Therefore, fast-acting field suppression must be provided on the generators. With existing line lengths, a capacitive load is obtained in the no-load condition of the transmission line or in the operating condition with \(P < P_{\text{nat}}\). In transmissions tuned to a half-wave, this need not be feared. In fact, the equivalent constants of a transmission tuned to a half-wave will be:

\[ A = D = -1,\quad B = C = 0. \]

The surge impedance

\[ \left(z_c = \sqrt{\frac{L}{C}} = \sqrt{\frac{0}{0}}\right) \]

is equal to an indeterminate quantity, i.e., any power will be natural, which ensures coincidence in phase of the voltage and current vectors at the ends of the transmission. Taking losses into account, the input resistance will be active. In the case of incomplete tuning by current, when the phase shift of the currents at the ends is less than \(180^\circ\), the input resistance will be a complex quantity with an inductive component. This practically excludes operation of the generator on a capacitive load. The most favorable condition for such a generator is the short-circuit condition. The steady-state short-circuit current

\[ I_{k.u.*} = \frac{E_{0*}}{x'_d}, \tag{3} \]

i.e., with an increase of \(x'_d\), will decrease, although not inversely proportionally to \(x'_d\), since \(E_{0*}\) increases.

Fig. 1.

Thus, according to the operating conditions of the generator, the synchronous reactance may be taken equal to \(x_d = 2\). We take the transformer reactance equal to \(e_k[\%] = 30\%\).

Selection of the transmission-line circuit

We shall assume that at the beginning of the line the role of the compensating device will be performed only by the generators and transformers. At the end of the line (if the receiving system is regarded as buses of infinite power) there will be only transformers. The magnitude of their reactance will be insufficient for tuning. Therefore, it is necessary to connect additional reactance at the end of the line. The magnitude of this reactance cannot be determined without knowing the point of connection and the magnitude of the elements of shunt capacitive compensation. As already indicated, the point of connection and the magnitude of the elements of shunt capacitive compensation are deter-

is determined not only by static stability, but also by the conditions for ensuring acceptable operating conditions of the power transmission (voltage conditions, p.f., generator power factor, etc.). Consideration of the operating conditions of a power transmission with transverse-compensation elements placed at various points of the line shows that the insertion of transverse-compensation elements at intermediate points of lines 1,500–2,000 km long is undesirable, mainly from the standpoint of the voltage conditions in the line, the p.f., etc. Therefore we shall assume that the transverse-compensation elements are connected only at the ends of the line. If the line were tuned to the half-wave condition without taking into account the generator reactance, then it would be necessary to insert capacitive conductance at the beginning of the line. But when the generator reactance is used as the element of longitudinal inductive compensation, the insertion of capacitive conductance at the beginning of the line can be dispensed with.

Fig. 2.

Under these conditions the main problem is to choose such a connection scheme for the compensating device at the end of the line, and to determine the magnitudes of its elements (longitudinal reactance and transverse capacitance), that acceptable operating conditions of the power transmission are ensured. In considering the possible variants of the terminal compensating-device circuit and the resulting operating conditions of the power transmission, the circuit shown in Fig. 2 was selected. As the transverse capacitive element \(b_{k1}\), static capacitors connected on the high-voltage side may be used; and as \(b_{k2}\), both static capacitors and synchronous compensators may be used. Synchronous compensators of the receiving system may also be used successfully as \(b_{k2}\), providing a small leading \(\cos \varphi\) on the power-transmission buses.

The magnitudes of the individual elements of the terminal compensating device can be determined on the basis of the following conditions: a) \(\cos \varphi = 1\) at the end of the line when transmitting the natural power of the line; b) the voltage at the end of the line in this case would be equal in magnitude to the voltage of the receiving buses, i.e. \(|u_l| = |u_n|\); c) the displacement of the voltage vectors in the compensating device would be equal to any value required by the tuning conditions, for example, \(\theta_{ku}\). Since it is not possible to compute \(\theta_{ku}\) analytically, it must simply be selected. Let us choose it, for example, as follows. When transmitting the natural power of the line, \(\theta_c + \theta_m\) from the sending end is approximately \(70^\circ\), and in the line \(\theta_l = 90^\circ\) \((l = 1500\ \text{km})\). With construction at an angle \(\theta_c = 210^\circ\), about \(50^\circ\) falls on the terminal compensating device, i.e. \(\theta_{ku} = 50^\circ\). Neglecting losses and solving the equations relating the voltage and current at the input and output of the compensating device with respect to \(x_k\), \(b_{k1}\), and \(b_{k2}\), we obtain:

\[ x_k = \frac{u_n \sin \theta_{ku}}{I_n \cos \varphi_n}; \tag{4} \]

\[ b_{k1} = \frac{I_n \cos \varphi_n \operatorname{tg} \dfrac{\theta_{ku}}{2}} {\cos \dfrac{\theta_{ku}}{2}\, u_n}; \tag{5} \]

\[ b_{k2} = \frac{I_n \left(\sin \varphi_n + \cos \varphi_n \operatorname{tg} \dfrac{\theta_{ku}}{2}\right)} {u_n}. \tag{6} \]

These dependences are presented in Fig. 3. It is not difficult to see that the required value of the elements of the terminal compensating device depends on the transmitted power, if one tries, under all loads, to satisfy the conditions: \(|u_l|=|u_n|\), \(\cos\varphi_l=1\), and \(u_l^{v}u_n=\theta_{ku}\). But there is no necessity for this. On the contrary, if the voltage at the end of the line is kept constant when the load changes, then the voltage at the beginning of the line, i.e., at the generator terminals, will vary, and a mode of generator operation will be obtained with varying voltage and with a fixed angle \(\theta_g\). This mode of generator operation is analogous to the operating mode with regulation of the generator excitation along the longitudinal and transverse axes of the rotor [11]. The efficiency of the generator from the standpoint of static stability can be considerably increased here. The negative aspect of such an operating mode is that operation takes place with varying voltage on the station buses. In order for the voltage on the station buses to be constant, it is necessary to provide conditions for changing the voltage at the end of the line (for \(l=1500\text{--}2000\ \mathrm{km}\)). In such a mode, as the transmitted power decreases, the angle \(\theta_g\) decreases. To compensate for this decrease, the line must be loaded with leading reactive power at the end. For this purpose, the values of the transverse elements of the terminal compensating device at small loads must be taken larger than is determined by dependences (4)--(6). We assume that \(x_k\) and \(b_{k1}\) are not regulated, while \(b_{k2}\) is regulated as a function of the transmitted power. Taking their values equal to the values required under conditions of transmitting the natural power of the line, we obtain the following power-transmission scheme (Fig. 4).

Fig. 3.

Fig. 4.

Calculations have been carried out for the normal operating mode of the power transmission both in the case of a constant value of the reactive power supplied by the synchronous compensators and in the case of a value varying as a function of the transmitted power, as given by dependence (6). The operating characteristics of the terminal compensating device, the voltage-distribution mode, and the characteristics of the sending end of the power transmission for these two cases are shown, respectively, in Figs. 5--10.

The operating mode of the power transmission in the case of a constant value of the reactive power supplied by the synchronous compensators differs little from the operating mode of a power transmission with a short line. The voltage on the station buses varies very little. The small rise in voltage at \(P<0.4P_{\text{nat}}\) can be eliminated by slight regulation carried out at the end of the line by the synchronous compensators. The voltage-distribution mode along the line is also favorable: when transmitting power \(P=P_{\text{nat}}\), at no point of the line does it exceed \(5\%\) above nominal. The total phase shift of the e.m.f. of the sending station and the voltage at the receiver buses, equal to \(183^\circ\) at no load, increases with the growth of transmitted power to \(204^\circ\).

Operation of the transmitting station in the second case takes place under conditions of varying voltage. The power transmission does not possess static stability when transmitting power below \(0.5P_{\text{nat}}\), which requires a greater power of synchronous compensators than is given by

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

dependence (6). Thus, from the standpoint of ensuring static stability, adjustment of the transverse element \(b_{k2}\) of the terminal compensating device is not required. But, on the other hand, its value is also determined by \(\cos \varphi_k\), which may require adjustment under conditions of varying \(\cos \varphi_k\).

Conclusions

1. Circuits for complete tuning of lines to the half-wave regime require considerable power of compensating devices (for \(l = 1500\ \text{km}\), \(Q_p = 2;\ Q_{sk} = 2\)).

1. The power rating of compensating devices can be substantially reduced by using generators and transformers as compensating devices, and also by applying the case of incomplete tuning.

2. The synchronous reactance of the generator \((x_d)\) may be increased to \(x_d = 2\), and the reactance of the transformers to \(e_k[\%] = 30\%\).

3. The selected power-transmission scheme, tuned to a regime somewhat greater than a half-wave, is statically stable and is the most economical in comparison with other possible tuning schemes. For a line \(1500\ \text{km}\) long, the required power of the compensating devices will be: \(Q_p = 0.5;\ Q_{ck} = 1\) (at \(\cos \varphi_k = 1\)).

4. The transmission capacity of the power transmission is determined not by static stability, but by the voltage regime in the line. With a \(50\%\) increase in voltage above nominal, the transmitted power is equal to the natural power of the line.

5. Allowing for a rise in voltage (by reinforcing the insulation at certain points of the line), the transmission capacity of the power transmission can be substantially increased.

6. The efficiency of the proposed power transmission will be higher than that of power transmissions containing fully tuned lines, since when the power transmission is loaded with a power close to the natural power of the line, there is no transmission of reactive power along the line. In addition, by using generators and transformers as compensating devices together with incomplete tuning, the magnitude of the total power of the compensating devices and the magnitude of the energy lost in them are considerably reduced.

Literature

1. Pospelov G. E., Basic Technical and Economic Questions of the Application of Series Static Compensation in Long-Distance Power-Transmission Systems, Academy of Sciences of the Uzbek SSR, Tashkent, 1950.

2. Longitudinal Capacitive Compensation of Power-Transmission Lines (review), Elektrichestvo, 1952, No. 8.

3. Venikov V. A., Transmission of Electric Power by Alternating Current over Long Distances, Elektrichestvo, 1954, No. 5.

4. Azariev D. I., Sokolov N. N., Venikov V. A., Long-Distance Power Transmissions with Intermediate Synchronous Compensators, Elektrichestvo, 1955, No. 4.

5. Zhdanov P. S., Stability of Electrical Systems, Gosenergoizdat, 1948.

6. Gruzdev I. A. and Levinshtein M. L., Operating Regimes of Long-Distance Power Transmissions with Intermediate Synchronous Compensators, Interuniversity Scientific-Technical Conference on Long-Distance Power Transmissions, Abstracts of Reports, Ministry of Higher Education of the USSR, LPI, Leningrad, 1956.

7. Rozovskii Yu. A. and Andreiuk V. A., Support Compensated Synchronous Compensators for Long-Distance Power Transmissions, Interuniversity Scientific-Technical Conference on Long-Distance Power Transmissions, Abstracts of Reports, Ministry of Higher Education of the USSR, LPI, Leningrad, 1956.

8. Chernysheva E., Transmission Capacity of a Transmission Line with Transformers at the Ends and Tuning of Lines to Half-Waves, Elektrichestvo, 1933, Nos. 6–7.

9. Wulf A. A., The Problem of Transmitting Energy over Ultra-Long Distances by Compensated Lines, Gosenergoizdat, 1941.

10. Berger A. Ya., Synchronous Machines, GONTI, 1938.

11. Botvinnik M. M., Regulation of the Excitation of a Synchronous Machine along the Longitudinal and Transverse Axes of the Rotor, Interuniversity Scientific-Technical Conference on Long-Distance Power Transmissions. Abstracts of Reports, Ministry of Higher Education of the USSR, LPI, Leningrad, 1956.