METHODology for Calculating Open-Phase Conditions in Long High-Voltage Transmission Lines

R. I. Borisov

(Presented by Professor Doctor V. K. Shcherbakov)

Phase-by-phase control of transmission lines in existing systems has become widespread, since this measure significantly increases the reliability of power supply to consumers without substantial additional capital expenditures.

The use of single-phase automatic reclosing, instead of three-phase reclosing, makes it possible, in the event of ground faults, to increase the time during which line phases remain disconnected without impairing its stability when transmitting the power of the normal operating condition. The limiting value of the operating time with the damaged phases disconnected during SPAR, as is known, must not exceed the deionization time of the air gap at the fault location.

Experimental verification carried out recently [1] has once again confirmed the possibility of applying SPAR on lines with voltages up to 220 kV and lengths up to 500 km, on which the arc is extinguished no later than 250 μsec after disconnection of the damaged phase. As a rule, for 220 kV lines the limiting time of the disconnected state, according to the conditions of dynamic stability, proves to be considerably longer. On 400 kV lines longer than 250 km, the arc at the point of the ground fault does not extinguish at all even after disconnection of the damaged phase, as a result of which successful automatic reclosing proves impossible. In this case, maintenance of the current in the arc is effected by the voltage induced from the healthy phases.

For lines at a voltage of 400 kV, a section length of 250 km is limiting: at this length the arc at the point of damage will be extinguished and it will be possible to carry out phase-by-phase control of the line.

On a long line equipped with phase-by-phase control, for a whole number of reasons prolonged open-phase conditions may arise.

Thus, in the case of a sustained single-phase short circuit on some section, the damaged phase will be disconnected from both sides; then, if a second circuit is present, the installation will operate in

on three phases, but five conductors. In the damaged phase the equivalent resistance will increase somewhat compared with operation under normal conditions, but such an unbalanced condition will be lighter than the conditions that arise when one phase is disconnected in a bundled configuration. This will mean the loss of one phase and a reduction in the power transmitted along the line. The resulting unbalance of currents and voltages will be more pronounced.

Thus, the most frequent single-phase faults, if they prove to be stable, will lead to single unbalances.

With separate control of the line phases, it becomes possible to expand considerably the range of work on inspection and repair of lines and equipment performed without complete disconnection of the transmission line and carried out by the method of phase-by-phase repair.

When such work is performed on one phase, with the others energized, unbalanced conditions may be superimposed in the event of an emergency disconnection of some phase and an accident at another point of the line. The probability of such coincidences increases with increasing line length and number of circuits. For these reasons, we consider the following schemes of open-phase conditions as the most probable in the practice of operating long lines: 1) interruption of a phase in one section, 2) interruption of unlike phases in one section, 3) interruption of two unlike phases in different sections.

Prolonged operation of a given transmission line under such conditions is associated with a number of features and adverse consequences that may limit their application. Therefore, when deciding whether to equip a line with phase-by-phase control, it is necessary in each specific case to prove not only the expediency of this, but also the possibility of prolonged operation of the lines under the unbalanced conditions that will undoubtedly arise during operation of such a line.

As was established earlier, applying a general analysis method for any case of unbalance is disadvantageous because of the laboriousness of the calculations [2].

The use of a general method may be appropriate only when calculating unbalanced conditions in an asymmetrically constructed circuit. Therefore, many authors who analyze unbalanced conditions and schemes use the method of symmetrical components, but seek to find computational techniques that would facilitate carrying out the calculations for the particular conditions under consideration.

Usually, by such techniques the calculation of the actual circuit of an unbalanced condition can be reduced to the calculation of a symmetrical circuit, but with different e.m.f.’s acting in the phases, or to the calculation of a symmetrical circuit with certain additional parameters at the point of unbalance. This greatly facilitates the calculation, since the principle of independence of action of symmetrical components can be used.

One such technique, which can significantly facilitate the calculation of open-phase conditions of long lines, is the use of the theory of matrix-vector parameters of electrical circuits. In this approach, all elements of the electrical system in the equivalent circuits of each sequence are replaced by matrix parameters if they are passive, or by matrix-vector parameters if the elements are active.

Since the disconnection of a phase or of different phases of a line will occur not along the entire length, but on remote sections no longer than 250 km,

which can be represented by lumped parameters, the switching operations at the ends of the section can be taken into account by a phase interruption at one point.

The proposed calculation method is as follows.

The equivalent circuit of the system elements is represented in the form of three-terminal networks connected to one another in accordance with the actual conditions. As the input terminals for the circuit of each sequence, the point of interruption is taken for the section where the special phase coincides with the calculated one, and as the secondary terminals—the point of interruption where the special phase differs from the calculated one. Then the circuits of each sequence are interconnected in accordance with the equations of the boundary conditions for the given type of fault. The connection for the interruption section where the special phase coincides with the calculated one must be electrical; for the section where the special and calculated phases do not coincide it must be electromagnetic, implemented through ideal transformers. The interconnection of the circuits of the separate sequences through ideal transformers with complex transformation ratios corresponds to transformations of the special phase into the calculated one for those points of unbalance where these phases do not coincide.

By transforming the circuits of each sequence to the resulting circuit and using the boundary conditions, the parameters are found of an equivalent three-terminal network composed of the negative- and zero-sequence circuits, which account for the effect of unbalance in the positive-sequence circuit. In the case of a connected circuit and double interruptions, the resulting circuits of each sequence reduce to complete four-terminal networks, to which the usual equations relating the currents and voltages of the terminal pairs through the circuit parameters cannot be applied.

To overcome this computational difficulty when determining the equivalent three-terminal network, a transition is made from the complete matrix of active and passive elements to a simplified one. Substituting the found value of the equivalent three-terminal network at the point of unbalance into the positive-sequence circuit makes it possible to calculate the open-phase condition.

Let us consider the application of this method to a long compensated double-circuit 400 kV transmission line connecting a remote station and a receiving system and divided by switching stations into four sections of equal length (Fig. 1).

Fig. 1.

Assume that when phase \(a\) is opened on section \(L\), phase \(b\) is also opened on section \(M\). We neglect the effect of grounding when the conductors are opened.

The switching stations make it possible to carry out phase-by-phase control of line sections whose length is no more than 250 km. In normal operation, each circuit transmits power equal to the natural—

…which makes it possible to determine the e.m.f.’s of the equivalent generators of the transmitting station and the receiving system.

In accordance with what has been set forth above, the transmission elements in the circuit of each sequence are replaced by three-terminal networks.

In the capacity of the terminal clamps of the three-terminal networks for the circuits of the different sequences, we take the points of the breaks; then the circuits on section \(L\), where the special and the calculated phases coincide, are connected electrically in accordance with the relation of the boundary conditions for this section

\[ \dot I_{AL_1}+\dot I_{AL_2}+\dot I_{AL_0}=0. \]

Fig. 2.

For section \(M\), the connection is made, according to its own relations of the boundary conditions, through ideal transformers with a complex transformation ratio in such a way that the current in the connecting part is equal to the current in the damaged phase

\[ \dot I_{BM_1}+\dot I_{BM_2}+\dot I_{BM_0}=0 \]

or, in the coordinates of the calculated phase \(A\),

\[ a^2\dot I_{AM_1}+a\dot I_{AM_2}+\dot I_{AM_0}=0, \]

\[ [\bar A_{TL}];\ [\bar A_{\Pi L}] \]
— matrices of the lines on section \(L\) of the circuit of the corresponding sequence,

\[ [\bar A^V] \]
— matrix of the passive elements between the open-circuit sections,

\[ [\bar A'];\ [\bar A''] \]
— matrices of the active transmission elements in the positive-sequence circuit; in the negative- and zero-sequence circuits they represent passive elements.

The parameters of each of the three-terminal networks of the equivalent circuit of any sequence relate the currents and voltages with respect to the neutral conductor isolated within its own circuit, i.e., having no connections with the neutral conductors of the other circuits. In this case the problem of obtaining an equivalent passive circuit becomes somewhat more complicated.

Let us consider the circuits of each sequence separately, and, since they have the same configuration, we shall omit the indices in what follows.

After adding the three-terminal networks \([\bar A']\) with \([A_{\Pi L}]\), and also \([\bar A_{1M}]\), \([\bar A_{\Pi M}]\) and \([\bar A'']\), and transforming the resulting three-terminal networks into equivalent T-shaped circuits, we obtain for each sequence the equivalent circuit shown in Fig. 3, a.

Reducing it to the resultant with respect to the input and output terminals and transforming the parameters, we obtain the circuit in Fig. 3, b.

As R. A. Voronov and G. E. Pukhov have shown in their works [3, 4], the usual equations of a three-terminal network cannot determine the electrical state of such circuits in which the equality of the currents in the branches of the input and output terminals is violated, or, what is the same thing, if one of the sides is not superconducting.

Fig. 3.

Since the circuit of Fig. 3, b does not satisfy the conditions under which the simplified formulas for a four-terminal network can be applied, we shall use the relations for the four-terminal network in their complete form, in the form \(Y\).

For the adopted positive directions of current and voltage, shown in Fig. 3, b, we shall have:

\[ \dot I_1=\overline{Y}_{11}\dot U_1-\overline{Y}_{12}\dot U_2+\overline{Y}_{1x}\dot U_x, \]

\[ \dot I_2=\overline{Y}_{21}\dot U_1-\overline{Y}_{22}\dot U_2+\overline{Y}_{2x}\dot U_x, \]

\[ \dot I_x=\overline{Y}_{x1}\dot U_1-\overline{Y}_{x2}\dot U_2+\overline{Y}_{xx}\dot U_x, \]

where

\[ \dot I_1=\frac{1}{2}(\dot i_{a_1}+\dot i_{b_1}), \]

\[ \dot I_2=\frac{1}{2}(\dot i_{a_2}+\dot i_{b_2}), \]

\[ \dot I_x=\dot i_{a_1}-\dot i_{b_1}=\dot i_{a_2}-\dot i_{b_2}, \]

or, in matrix form,

\[ \begin{bmatrix} \dot I_1\\ \dot I_2\\ \dot I_x \end{bmatrix} = \begin{bmatrix} \overline{Y}_{11} & \overline{Y}_{12} & \overline{Y}_{1x}\\ \overline{Y}_{21} & \overline{Y}_{22} & \overline{Y}_{2x}\\ \overline{Y}_{x1} & \overline{Y}_{x2} & \overline{Y}_{xx} \end{bmatrix} \begin{bmatrix} \dot U_1\\ -\dot U_2\\ \dot U_x \end{bmatrix}. \tag{1} \]

The determination of the individual components of the conductance matrix is carried out from equations (1) for different cases of short circuits of the circuit in Fig. 3, б [3].

Thus, by short-circuiting terminals \(a_2\) and \(b_2\), and connecting them to the midpoint between terminals \(a_1\) and \(b_1\), we shall have:

\[ U_2 = 0,\quad U_x = 0; \]

in this case

\[ Y_{11}=\frac{\dot I_1}{\dot U_1};\qquad Y_{21}=\frac{\dot I_2}{\dot U_1};\qquad Y_{x1}=\frac{\dot I_x}{\dot U_1}; \tag{2} \]

\[ \dot I_{a_1}=\dot U_1\left[\frac{1}{z_1+z_2}+\frac{1}{2\left(z_1+z_3+z_4\right)}\right]; \]

\[ \dot I_{b_1}=\dot U_1\left[\frac{1}{z_1+z_2}+\frac{1}{2\left(z_2+z_3+z_5\right)}\right]; \]

\[ \dot I_{a_2}=\frac{1}{2}\dot U_1\left[\frac{1}{z_1+z_3+z_4}\right]; \]

\[ \dot I_{b_2}=\frac{1}{2}\dot U_1\left[\frac{1}{z_2+z_3+z_5}\right]. \]

\[ \dot I_1=\frac{1}{2}\dot U_1\left[ \frac{2}{z_1+z_2} +\frac{1}{2\left(z_1+z_3+z_4\right)} +\frac{1}{2\left(z_2+z_3+z_5\right)} \right]. \]

\[ \dot I_2=\frac{1}{4}\dot U_1\left[ \frac{1}{z_1+z_3+z_4} -\frac{1}{z_2+z_3+z_5} \right]. \]

\[ \dot I_x=\frac{1}{2}\dot U_1\left[ \frac{1}{z_1+z_3+z_4} -\frac{1}{z_2+z_3+z_5} \right]. \]

Substituting \(\dot I_1\), \(\dot I_2\), and \(\dot I_x\) into equations (2), we find

\[ Y_{11}=\frac{1}{2}\left[ \frac{2}{z_1+z_2} +\frac{1}{2\left(z_1+z_3+z_4\right)} +\frac{1}{2\left(z_2+z_3+z_5\right)} \right], \]

\[ Y_{21}=\frac{1}{4}\left[ \frac{1}{z_1+z_3+z_4} -\frac{1}{z_2+z_3+z_5} \right]. \]

\[ Y_{x1}=\frac{1}{2}\left[ \frac{1}{z_1+z_3+z_4} -\frac{1}{z_2+z_3+z_5} \right]. \]

From the reverse short-circuit experiment we shall have:

\[ U_1=0,\quad U_x=0. \]

Then

\[ \bar{Y}_{12}=-\frac{\dot I_1}{U_2} =\frac{1}{4}\left[ -\frac{1}{z_1+z_3+z_4} +\frac{1}{z_2+z_3+z_5} \right], \]

\[ \bar{Y}_{22}=-\frac{\dot I_2}{U_2} =\frac{1}{2}\left[ \frac{2}{z_1+z_5} +\frac{1}{2\left(z_1+z_3+z_4\right)} +\frac{1}{2\left(z_2+z_3+z_5\right)} \right], \]

\[ \bar{Y}_{2x}=-\frac{\dot I_x}{U_2} =-\frac{1}{2}\left[ \frac{1}{z_2+z_3+z_5} -\frac{1}{z_1+z_3+z_4} \right]. \]

Finally, assuming the voltages \(U_1\) and \(U_2\) to be equal to zero, i.e., short-circuiting the input and output terminals, we obtain

\[ U_1=0,\qquad U_2=0. \]

Then

\[ Y_{1x}=\frac{\dot I_1}{U_x} =\frac{1}{2}\left[ \frac{1}{z_1+z_3+z_4} -\frac{1}{z_2+z_3+z_5} \right], \]

\[ Y_{2x}=-\frac{\dot I_2}{U_x} =\frac{1}{2}\left[ \frac{1}{z_1+z_3+z_4} -\frac{1}{z_2+z_3+z_5} \right], \]

\[ \bar{Y}_{xx}=\frac{\dot I_x}{U_x} =\frac{1}{z_1+z_3+z_4} +\frac{1}{z_2+z_3+z_5}. \]

Let us denote

\[ \frac{1}{z_1+z_3+z_4}=\gamma,\qquad \frac{1}{z_2+z_3+z_5}=\mu. \]

The complete admittance matrix of the four-terminal network is written as follows:

\[ [Y]= \left[ \begin{array}{ccc} \displaystyle \frac{1}{2}\left( \frac{2}{z_1+z_3} +\frac{1}{2}\gamma+\frac{1}{2}\mu \right) & \displaystyle \frac{1}{4}(\gamma+\mu) & \displaystyle \frac{1}{2}(\gamma-\mu) \\[1.2em] \displaystyle \frac{1}{4}(\gamma+\mu) & \displaystyle \frac{1}{2}\left( \frac{2}{z_1+z_5} +\frac{1}{2}\gamma+\frac{1}{2}\mu \right) & \displaystyle \frac{1}{2}(\gamma-\mu) \\[1.2em] \displaystyle \frac{1}{2}(\gamma-\mu) & \displaystyle \frac{1}{2}(\gamma-\mu) & \displaystyle \gamma+\mu \end{array} \right]. \]

Having determined the complete admittance matrix, one can, by equations (1), pass to the simplified admittance matrix, valid in the absence of a longitudinal equalizing current \(I_x\).

For this it is necessary to create a superconducting side by short-circuiting terminals \(b_1\) and \(b_2\) of the circuit in Fig. 3, \(b\). Since this cannot be done directly, we insert an additional resistance \(z\) between terminals \(b_1\) and \(b_2\), find for it the complete admittance matrix, and add it to the matrix of the four-terminal network. Then, by equations (1), we determine the parameters of the simplified matrix for re-

resulting for \(I_x=0\), and assign to \(\bar z\) a value equal to zero.

The complete admittance matrix \(\left[\dfrac{1}{z}\right]\) is found analogously to the preceding one:

\[ \left[\frac{1}{z}\right]= \begin{bmatrix} \dfrac{1}{4\bar z} & \dfrac{1}{4\bar z} & -\dfrac{1}{2\bar z}\\[6pt] \dfrac{1}{4\bar z} & \dfrac{1}{4\bar z} & -\dfrac{1}{2\bar z}\\[6pt] -\dfrac{1}{2\bar z} & -\dfrac{1}{2\bar z} & \dfrac{1}{\bar z} \end{bmatrix}. \]

Adding the additional-conductance matrix to the admittance matrix of the four-terminal network, we obtain the general longitudinal-admittance matrix

\[ |\bar Y|+\left[\frac{1}{\bar z}\right]= \begin{bmatrix} \dfrac{1}{2}\left(\dfrac{2}{z_1+z_2}+\dfrac{1}{2}\bar\gamma+\dfrac{1}{2}\bar\mu\right)+\dfrac{1}{4\bar z} & \dfrac{1}{4}(\bar\gamma+\bar\mu)+\dfrac{1}{4\bar z} & \dfrac{1}{2}\left(\bar\gamma+\bar\mu\right)+\dfrac{1}{4\bar z} \\[8pt] \dfrac{1}{4}(\bar\gamma+\bar\mu)+\dfrac{1}{4\bar z} & \dfrac{1}{2}\left(\dfrac{1}{z_4+z_5}+\dfrac{1}{2}\bar\gamma+\dfrac{1}{2}\bar\mu\right)+\dfrac{1}{4\bar z} & \dfrac{1}{2}(\bar\gamma-\bar\mu) \\[8pt] \dfrac{1}{2}(\bar\gamma-\bar\mu)-\dfrac{1}{2\bar z} & \dfrac{1}{2}(\bar\gamma-\bar\mu)-\dfrac{1}{2\bar z} & \bar\gamma+\bar\mu+\dfrac{1}{\bar z} \end{bmatrix}. \]

The simplified admittance matrix can be obtained from the general one from the formulas by setting \(I_x=0\) and substituting the found value of \(U_x\) into the two other equations. Then we shall have:

\[ \bar Y^{1}_{11}=Y_{11}-\frac{Y^2_{1x}}{Y_{xx}}; \qquad \bar Y^{1}_{12}=Y_{12}-\frac{Y_{2x}\bar Y_{1x}}{Y_{xx}}; \qquad \bar Y^{1}_{22}=Y_{22}-\frac{\bar Y^2_{2x}}{Y_{xx}}. \]

Substituting the admittance values of the general matrix, after transformations and evaluation of the indeterminacy by L’Hôpital’s rule as \(z\to 0\), we obtain the values of the terms of the simplified matrix

\[ |\bar Y^{1}|= \begin{bmatrix} \dfrac{1}{z_1+z_2}+\dfrac{1}{2}\bar\mu & -\dfrac{\bar\gamma\bar\mu}{\bar\gamma+\bar\mu} \\[8pt] -\dfrac{\bar\gamma\bar\mu}{\bar\gamma+\bar\mu} & \dfrac{1}{z_4+z_5}+\dfrac{1}{2}\bar\mu \end{bmatrix} = \begin{bmatrix} Y^{1}_{11} & Y^{1}_{12}\\ Y^{1}_{21} & Y^{1}_{22} \end{bmatrix}. \]

Using the known formulas, let us pass from the form \([\bar{Y}]\) to the matrix of the form \([\bar{A}]\)

\[ [\bar{A}^{1}] = \begin{bmatrix} -\dfrac{\bar{Y}^{1}_{22}}{\bar{Y}^{1}_{21}} ; & \dfrac{1}{\bar{Y}^{1}_{21}} \\[6pt] -\dfrac{|\bar{Y}^{1}|}{\bar{Y}^{1}_{21}} ; & \dfrac{\bar{Y}^{1}_{11}}{\bar{Y}^{1}_{21}} \end{bmatrix} = \begin{bmatrix} \bar{A}^{1}; & \bar{B}^{1} \\ \bar{C}^{1}; & \bar{D}^{1} \end{bmatrix}. \]

Such transformations should be carried out with the inverse- and zero-sequence circuits, after which the matrix \([\bar{A}]\) of the equivalent passive four-terminal network can be found:

\[ [\bar{A}] = \begin{bmatrix} \dfrac{\bar{A}_{2}\bar{B}_{0}+\bar{B}_{2}\bar{A}_{0}}{\bar{B}_{2}+a\bar{B}_{0}} ; & -\dfrac{\bar{B}_{0}\bar{B}_{2}}{\bar{B}_{2}+a\bar{B}_{0}} \\[10pt] a^{2}\bar{C}_{2}+\bar{C}_{0} + \dfrac{(\bar{A}_{2}-a\bar{A}_{0})(a\bar{D}_{0}-\bar{D}_{2})}{\bar{B}_{2}+a\bar{B}_{0}} ; & \dfrac{\bar{B}_{2}\bar{D}_{0}+\bar{B}_{0}\bar{D}_{2}}{\bar{B}_{2}+a\bar{B}_{0}} \end{bmatrix}. \]

In an analogous manner, the parameters of the equivalent four-terminal network that takes account of the influence of the break can also be found for other cases of open-phase conditions. Without presenting the derivations for determining the parameters of the matrices \([\bar{A}]\), we record the results in Table 1.

Table 1

Calculations of open-phase-condition circuits performed for the parameters of the Kuibyshev–Moscow power transmission line showed [5] that, when operating in the block scheme, disconnection of a phase on one section leads to a current unbalance in the generator phases equal to 65%, and to a voltage unbalance on the buses of the receiving system of 1.0%. Operation of the installation in such a mode without special measures for reducing the unbalance is impermissible.

When operating in the interconnected scheme, calculations showed that, when a phase is disconnected on one of the sections, the resulting current unbalance in the generator phases is no more than 8%, with voltage unbalance.

equal to 1%, and in the case of the interruption of two unlike phases on different sections the current unbalance will be 14% when the voltage unbalance on the buses of the receiving system is 2%. Operation of generators under such conditions may be permissible. The calculations of open-phase conditions were carried out with values of the e.m.f. of the equivalent generator of the sending station and of the receiving system, as well as with the phase angle between the vectors of these e.m.f.’s, equal to the corresponding values of the normal symmetrical condition.

REFERENCES

1. The question of the stability of Swedish transmission lines. Novosti zarubezhnoi tekhniki, 1956.

2. Gershengorn A. I., Melnikov N. A. Calculation of unbalance in an electrical system with an increased length of the transposition cycle on long lines. Proceedings of the All-Union Correspondence Power Engineering Institute, Gosenergoizdat, 1954.

3. Voronov R. A. General Theory of Four-Terminal and Multi-Terminal Networks. Gosenergoizdat, 1951.

4. Pukhov G. E. Fundamentals of the new theory of the four-terminal network. Scientific Notes of LPI, Lviv, 1949.

5. Borisov R. I. Steady-state unbalanced conditions in open-phase operation of long transmission lines. Candidate dissertation. Tomsk, 1955.