RESONANCE PHENOMENA DURING STEADY-STATE SHORT CIRCUITS IN HALF-WAVE TUNED POWER TRANSMISSION LINES

V. T. Agafonov

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

The occurrence of significant overvoltages in the line and in other elements of the system during short circuits is a major drawback of power transmission lines tuned to a half-wave. Even in the case of exact tuning of the line itself to the half-wave regime, when the constants of the tuned line take the values: \(A = D = -1,\ B = C = 0\) (without losses), the magnitude of the short-circuit current that occurs near the end of the tuned line is determined only by the parameters of the terminal devices. However, as shown in [1], an external short circuit is not the most severe for a half-wave tuned transmission line. The most dangerous for such a transmission line is a short circuit at a point separated from the equivalent emf of the generators by a zero equivalent resistance, i.e., at the point of resonant short circuit. The multiplicity of the short-circuit current and of the overvoltages, even taking corona into account, reaches 4–5, and without taking corona into account, 15–20.

All this applies to a line fully tuned to a half-wave by means of any compensating devices (hereinafter abbreviated as c. d.) connected into the line.

In [1], only the scheme with central placement of the c. d. is investigated. In reality, however, resonance phenomena will depend to a large extent on the tuning scheme, the length of the line being tuned, and the parameters of the terminal devices. If the reactances of the terminal devices are used as c. d. [2], resonance phenomena must be considered additionally. Of great importance is the study of resonance phenomena not only for three-phase short circuits, which occur very rarely, but also for unsymmetrical short circuits. Let us consider the resonance phenomena of a transmission-line scheme tuned to a half-wave by means of terminal c. d., for which generators and transformers are used. We shall be interested only in the qualitative aspect of the question and shall not take into account losses, which affect mainly its quantitative side.

I. Three-Phase Short Circuit

The calculation scheme is shown in Fig. 1. Let the given power transmission line be characterized by the constants \(A,\ B,\ C,\ D\) (in the general case \(A \ne D\)). In the case of a short circuit at the end and at the beginning of this system, as is evident from the general equations,

\[ \begin{aligned} E_1 &= A U_2 + B I_2,\\ I_1 &= C U_2 + D I_2; \end{aligned} \tag{1} \]

\[ \begin{aligned} U_2 &= D E_1 - B I_1,\\ I_2 &= -C E_1 + A I_1; \end{aligned} \tag{2} \]

the short-circuit impedances, measured respectively from the beginning and from the end of the system, are:

\[ Z_k=\frac{B}{D}, \qquad Z_k=-\frac{B}{A}. \tag{3} \]

The resonance point for the adopted circuit can be found in the following way. Since the generator impedance (meaning the equivalent generator of the station) enters the system, then, proceeding from the beginning of the system, one can find the point at which \(Z_k=\frac{B}{D}\) would be equal to zero or, taking losses into account, would be purely active. The same can also be done with respect to the receiving system. It is easy to see that \(Z_k=0\) if \(B=0\).

Fig. 1.

Fig. 2.

Equating the coefficient \(B\) to zero and solving the resulting equation, we find:

\[ \lambda=\operatorname{arctg}\left(-\frac{x_{\mathrm{r}}+x'_{\mathrm{t}}}{z_c}\right). \tag{4} \]

This dependence is presented in Fig. 2. It is not difficult to see that the location of the resonance short-circuit point depends on the impedance of the terminal device and lies the farther beyond \(\lambda=90^\circ\), the smaller the reactive impedance of the sending end of the transmission line. If the length of the tuned line is not more than \(2000\ \mathrm{km}\), then, for a three-phase short circuit, there will be no resonance within the length of the line. It is of interest here to determine how close the short-circuit regime at any point of the line comes to resonance. This can be learned by considering the change in the input impedance of the transmission line under a short circuit at various points, which will be equal to:

\[ X_{k.z.}=[z_c \operatorname{tg}\lambda+(x_{\mathrm{r}}+x'_{\mathrm{t}})]. \tag{5} \]

It follows from (5) that the short-circuit impedance increases continuously as the short-circuit point moves away from the sending end and becomes equal to \(\infty\) at \(\lambda=90^\circ\). The short-circuit impedance becomes negative for \(\lambda>90^\circ\) and rapidly falls to zero at the resonance short-circuit point, and then smoothly increases to a value equal to the impedance of the sending end of the transmission line \((\lambda=180^\circ)\). With a short circuit in a line whose length is somewhat less or greater than the resonant length (up to \(\lambda=180^\circ\)), a reduced value of the short-circuit impedance occurs, and, consequently, there lies a zone of the most unfavorable short circuits for the transmission line. Naturally, in tuning it is necessary to strive to eliminate this dangerous zone. For this purpose, the most suitable are circuits with terminal short-circuiting devices.

It was shown above that, for a line length of not more than \(2000\ \mathrm{km}\), the resonance point does not fall within the line. Let us see where the resonance point will be located relative to the sending end. We find the equiva-

...the system constants: generator—transformer—line—terminal switching station and the short-circuit resistance, which will be equal to:

\[ X_{k.z} = \frac{ \left\{ \left( \cos \lambda - \frac{x_{\mathrm{g}} + x'_{\mathrm{t}}}{z_c}\sin \lambda \right) + \left[ j z_c \sin \lambda + j(x_{\mathrm{g}} + x'_{\mathrm{t}})\cos \lambda \right]j b_1 \right\} }{ \left( j\frac{\sin \lambda}{z_c} + j\cos \lambda\, b_1 \right) j(x_p + x''_{\mathrm{t}}) + } \times \frac{ j(x_p + x''_{\mathrm{t}}) + j z_c \sin \lambda + j(x_{\mathrm{g}} + x'_{\mathrm{t}})\cos \lambda }{ +\cos \lambda }. \tag{6} \]

Using equation (6), taking \(x_{\mathrm{g}}, x'_{\mathrm{t}}, \lambda, z_c, b_1\) and \(b_2\) as given quantities, one can construct the dependence
\(X_{k.z} = f\bigl((x_p + x''_{\mathrm{t}})\bigr)\) (Fig. 3).

It is seen from Fig. 3 that, with respect to the beginning of the system, resonance will occur for a short circuit within the reactance of the terminal switching station. But since, in the case of single-phase reactor and transformer design, three-phase short circuits are practically impossible, it is evident that resonance phenomena for a three-phase short circuit in such a power transmission system will not occur with respect to its beginning. The most severe case here will be a three-phase short circuit on the receiving buses. In power transmission systems with a central arrangement of the switching station, however, the resonance point would necessarily be located on the line. The resonance point for a three-phase short circuit can be determined with respect to the receiving system (Fig. 4).

Fig. 3.

Fig. 4.

Equating the input resistance of the short circuit to zero and solving with respect to \(\lambda\), we obtain:

\[ \lambda = \operatorname{arctg} \left[ - \frac{x_p + x''_{\mathrm{t}}} {1 - (x_p + x''_{\mathrm{t}})b_1} \right]. \tag{7} \]

Fig. 5.

The dependence \(\lambda = f\bigl((x_p + x''_{\mathrm{t}})\bigr)\) for \(b_1 = \mathrm{const}\) (curve 1) and \(\lambda = f(b_1)\) for \((x_p + x''_{\mathrm{t}}) = \mathrm{const}\) (curve 2) is shown in Fig. 5. As is seen from Fig. 5, the line length under short circuit for which resonance is possible will be greater than 2000 km. If the finite power of the receiving system is taken into account, i.e., if it is replaced by its equivalent emf and reactance, then one can obtain:

\[ \lambda = \operatorname{arctg} \left\{ \frac{ -\left[(x_p + x''_{\mathrm{t}} + x_{nc}) + b_2(x_p + x''_{\mathrm{t}})x_{nc}\right] }{ 1 - (x_p + x''_{\mathrm{t}})b_1 - x_{nc}(b_1 + b_2) + x_{nc}(x_p + x''_{\mathrm{t}})b_1 b_2 } \right\}. \tag{8} \]

The dependence of \(\lambda\) on \(x_{nc}, b_2, (x_p + x''_{\mathrm{t}})\), with the other quantities constant, is shown in Fig. 6:
\(\lambda = f(x_{nc})\) for \((x_p + x''_{\mathrm{t}}); b_2; b_1 = \mathrm{const}\) (curve 1);
\(\lambda = f(x_p + x''_{\mathrm{t}})\) for \(x_{nc}; b_1; b_2 = \mathrm{const}\) (curve 2);
\(\lambda = f(b_2)\) for \(x_{nc}; b_1; (x_p + x''_{\mathrm{t}}) = \mathrm{const}\) (curve 3).

In this case the reactance of the receiving system, as it were, lengthens the line, and the location of the resonance point shifts to the right.

Thus, the most suitable tuning scheme for half-wave operation from the standpoint of three-phase short-circuit conditions is the scheme with terminal short circuits, in the capacity of which generators and transformers are used. If the length of the line being tuned is not more than 2,000 km, then no resonance phenomena arise in it under a three-phase short circuit.

2. Unsymmetrical short circuits

The investigation of resonance phenomena under unsymmetrical short circuits can be carried out most conveniently and simply by using the method of symmetrical components [3, 4, 5]. As is known, the determination of currents and voltages under an unsymmetrical short circuit on the basis of this method is reduced to determining these quantities under a certain fictitious three-phase short circuit. In the analysis, a procedure is used in which the elements of the electric system are represented in the form of four-terminal networks, making it possible to take into account the distribution of constants. The matrix parameters of the elements of the positive- and negative-sequence circuits

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

are determined by known methods. For lines they are obtained as a result of calculating the line by means of hyperbolic functions. The determination of the matrix parameters of circuits with distributed parameters in the zero-sequence circuit is given in [6]. Having determined the propagation coefficient and the surge impedance for zero-sequence currents by means of hyperbolic functions, one can find the matrix parameters of the equivalent four-terminal network, which on the superimposed side will be a three-terminal network. The three-terminal networks of the individual elements are combined with one another in the circuit of each sequence, as a result of which three equivalent three-terminal networks of the positive, negative, and zero sequences are obtained.

By connecting the complex equivalent circuits of the negative and zero sequences in accordance with the equations of the boundary conditions, one can find the parameters of the equivalent three-terminal network which, when introduced into the positive-sequence circuit, would take into account the effect of the unsymmetrical short circuit (calculation diagram of Fig. 7). Let us compose the equivalent circuit for the positive-sequence current (Fig. 8).

The elements of the matrices \([A_1']\) and \([A_1'']\) depend on the distance to the short-circuit point. Similarly, equivalent circuits are constructed for the currents of the zero and negative sequences. We transform the circuits of the negative and zero sequences into the equivalent branch of Fig. 9. From Fig. 9 it is evident that

\[ Z_2' + Z_3' = \frac{D_2' - 1}{C_2'} + \frac{1}{C_2'} = \frac{D_2'}{C_2'} ; \tag{9} \]

\[ Z_1'' + Z_3'' = \frac{A_2'' - 1}{C_2''} + \frac{1}{C_2''} = \frac{A_2''}{C_2''}. \tag{10} \]

Combining them in parallel, we obtain:

\[ (Z_2' + Z_3') \parallel (Z_1'' + Z_3'') = \frac{A_2'' D_2'}{D_2' C_2'' + A_2'' C_2'} . \tag{11} \]

An equation of the same structure is obtained from the zero-sequence circuit:

\[ (Z_2' + Z_3') \parallel (Z_1'' + Z_3'') = \frac{A_0'' D_0'}{D_0' C_0'' + A_0'' C_0'} . \tag{12} \]

For a single-pole short circuit, the resistance of the equivalent branch will be equal to:

\[ Z_{\text{equiv}} = \frac{A_0'' D_0'}{D_0' C_0'' + A_0'' C_0'} + \frac{A_2'' D_2'}{D_2' C_2'' + A_2'' C_2'} . \tag{13} \]

As is known, the equivalent resistance for a two-pole short circuit to ground is obtained by the parallel combination of the resultant resistances of the negative- and zero-sequence circuits, while for a two-pole short circuit it is obtained only from the resistance of the negative sequence.

Fig. 10.

Fig. 11.

Substituting the found values of \(Z_{\text{equiv}}\) into the equivalent circuit of the positive sequence at the fault location, we find the generalized matrix of all elements of the positive sequence \([A]\) and reduce the circuit to the form of Fig. 10. In the circuit:

\[ Z_1 = \frac{A - 1}{C}, \qquad Z_2 = \frac{D - 1}{C}, \qquad Z_3 = \frac{1}{C}. \]

The operating mode of the entire system can be represented as the result of two independent modes. One of them occurs with \(E_g\) and \(E_c = 0\), the second with \(E_c\) and \(E_g = 0\). By superposing the currents and voltages of these two modes, we obtain the actual operating mode of the system.

Since the equivalent circuit of each sequence contains capacitance and inductance elements that depend on the distance to the short-circuit point, it is in principle possible to select such a line length up to the short-circuit point at which the input resistance under short circuit will be equal to zero (neglecting losses). Let us consider a single-pole short circuit. The positive-sequence circuit for all types of short circuits (only \(Z_{\text{equiv}}\) changes) is shown in Fig. 8. To determine the value of \(Z_{\text{equiv}}\) for a single-pole short circuit, it is necessary to construct the equivalent circuits of the negative and zero sequences.

sequences. The negative-sequence circuit is shown in Fig. 11, and the zero-sequence circuit in Fig. 12.

It is first necessary to clarify the question of resonance in the negative- and zero-sequence circuits. The point is that, in the event of a short circuit at some point of a half-wave-tuned transmission system, resonance may occur in the negative- and zero-sequence circuits. This leads to a considerable increase in the currents of the corresponding sequences and to overvoltages.

The circuits of the negative and zero sequences given above (Figs. 11, 12) may be regarded as general for all types of unsymmetrical short circuits, bearing in mind, of course, that for a two-phase short circuit there will be no zero-sequence circuit. Only the voltages applied to these circuits at the point of the short circuit will depend on the type of short circuit. For simplicity we assume that zero-sequence currents flow only between the transformers at the sending and receiving ends of the line. By successive composition of four-terminal networks we find the generalized matrices of the negative- and zero-sequence circuits to the right and to the left of the short-circuit point. Then, equating to zero the values of the input impedances of the short circuit and solving with respect to \(\lambda\) and \(\lambda_0\), we obtain the line length required for resonance, as a function of the parameters of the short circuit and of the terminal devices both to the right and to the left of the short-circuit point:

Fig. 12.

1) for the negative-sequence circuit on the right

\[ \tg \lambda_{(L-l)} = \frac{-(x_k''+x_{c2}-x_k''x_{c2}b_2)} {x_k''x_{c2}b_2b_1-b_1x_{c2}-b_2x_{c2}-x_k''b_1+1}; \tag{14} \]

2) for the negative-sequence circuit on the left

\[ \tg \lambda_l=-(x_{r2}+x_T'); \tag{15} \]

3) for the zero-sequence circuit on the right

\[ \tg \lambda_{0(L-l)} = -\frac{x_{k0}} {z_{c0}-x_{k0}z_{c0}b_1}; \tag{16} \]

4) for the same circuit on the left

\[ \tg \lambda_{0l}=-\frac{x_T'}{z_{c0}}. \tag{17} \]

Analysis of relations (14), (15) shows that resonance in the negative-sequence circuit is possible, in the case of a short circuit in the line, only if its length is greater than 2000 km. The location of the resonance point depends on the magnitudes of the elements of the short circuit and of the terminal devices. Resonance in the zero-sequence circuit, because the wave length of the line for the zero-sequence current is considerably greater than for the positive and negative sequences (approximately by a factor of 1.5), occurs for a line length even less than 1500 km.

Now let us return to the positive-sequence circuit. Substituting the values of the matrix parameters of the generalized matrices of the negative- and zero-sequence circuits to the right and to the left of the short-circuit point into the formula for \(Z_{\mathrm{eqv}}\), we determine its value and, then finding the generalized matrices

of the positive-sequence network to the left and to the right of the short-circuit point, \([A'_1]\) and \([A''_1]\), we find the generalized matrix of the entire positive-sequence network as the product \([A'_1] Z_{\text{eq}} [A''_1]\).

By the superposition principle, the short-circuit impedance in the mode when only the e.m.f. \(E_r\) acts, while \(E_c=0\), will be equal to:

\[ Z'_{\text{s.c.}1} = \frac{ \left(A'_1+B'_1 \frac{1}{Z_{\text{eq}}}\right) B''_1+B'_1D''_1 }{ \left(C'_1+D'_1 \frac{1}{Z_{\text{eq}}}\right) B''_1+D'_1D''_1 }. \tag{18} \]

In the mode when \(E_r=0\) and \(E_c\) acts, one obtains

\[ Z''_{\text{s.c.}1} = \frac{ \left(A'_1+B'_1 \frac{1}{Z_{\text{eq}}}\right) B''_1+B'_1D''_1 }{ \left(A'_1+B'_1 \frac{1}{Z_{\text{eq}}}\right) A''_1+B'_1C''_1 }. \tag{19} \]

Determining, in general form, the location of the resonant points in the positive-sequence network is not possible because of the difference between the line parameters for the positive- and negative-sequence currents and the zero sequence. This difference in parameters detunes the pure resonance, by which is meant equality to zero of the imaginary part of the short-circuit impedance in the positive-sequence network. In this case, resonance should be understood as the minimum value of the short-circuit impedance, and the resonant length as that length, at short circuit, at which this minimum is ensured. Calculations show that the minimum value of the short-circuit impedance in the positive-sequence network occurs at a greater line length than the resonant length for a three-phase short circuit. The location of the point, at short circuit, at which the impedance to positive-sequence currents will be a minimum does not coincide with the resonant points of the negative- and zero-sequence networks. This indicates that overvoltages in power transmissions tuned to a half-wave may arise under single-phase short circuit over a fairly wide range of line length.

For the case of a two-phase short circuit and a two-phase short circuit to ground, the resonance conditions in the negative- and zero-sequence networks have already been considered. As for the positive-sequence networks, only the magnitudes of the equivalent impedances from the negative- and zero-sequence networks change. By determining these impedances for a two-phase short circuit and a two-phase short circuit to ground and inserting them into the positive-sequence network, one can then determine the short-circuit impedances and their minimum. Since the parameters of the zero-sequence network do not enter into the general expression for the short-circuit impedance for the positive-sequence current in a two-phase short circuit, resonance also occurs in the positive-sequence network. The resonance point is located farther away than in a three-phase short circuit. In a two-phase short circuit to ground there will be no pure resonance in the positive-sequence network, but resonance is possible in the zero-sequence network at a line length of less than \(1\,500\) km. For a clearer treatment of this question, in order to clarify its quantitative aspect, and also to study the influence of corona, further investigations are required.

Conclusions

1. All power transmissions tuned to a half-wave are characterized by resonance phenomena both under three-phase short circuits and under unsymmetrical short circuits.

1. Depending on the tuning scheme and the length of the line included in it, resonance phenomena during a short circuit arise either in the line itself or during a short circuit in other elements of the power transmission system.

2. When choosing a tuning scheme, one must seek to eliminate resonance phenomena during a short circuit in the line itself. For this purpose, the most suitable scheme is one with terminal short-circuiting devices, in which generators and transformers are used.

3. If the length of the line included in such a power transmission system is not more than 2,000 km, then there will be no resonance phenomena either in the case of a three-pole short circuit within the length of the line or in the case of unsymmetrical short circuits, except for resonance in the zero-sequence circuit during a single-pole short circuit and a two-pole short circuit to ground.

References

1. A. A. Vul’f and O. V. Shcherbachev, “Short-circuit conditions in power transmission lines tuned to a half-wave,” Elektrichestvo, 1945, No. 4.

2. V. T. Agafonov, “The use of terminal devices in tuning lines to a half-wave” (published in the present collection).

3. K. F. Wagner and R. D. Evans, The Method of Symmetrical Components, ONTI, 1936.

4. P. N. Shchedrin, Short-Circuit Currents in High-Voltage Systems, ONTI, 1935.

5. S. A. Ul’yanov, Short Circuits in Electrical Systems, Gosenergoizdat, 1952.

6. G. I. Atabekov, The Distance Principle of Protection for Long-Distance Power Transmission, Yerevan, 1953.