Influence of Line Capacitance and the Inductance of Smoothing Reactors on the Magnitude and Nature of Short-Circuit Current in a DC Line
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Influence of Line Capacitance and the Inductance of Smoothing Reactors on the Magnitude and Character of the Short-Circuit Current in a DC Line
V. K. Shcherbakov and A. I. Vasiliev
1. The magnitude of the current in a lossless transmission line is determined by the equation:
\[ i=\frac{U}{Z}, \tag{1} \]
where \(Z=\sqrt{\frac{L}{C}}\) is the surge impedance of the line. According to this formula, it appears possible to approximately determine the magnitude of the short-circuit current in a high-voltage DC transmission line that was operating at no load under voltage \(U_0\).
It follows from expression (1) that short-circuit currents will be considerable in magnitude in cable lines, which have a large capacitance and a small inductance.
For high-voltage DC cable lines, when one half of the electric transmission is in operation, the surge impedance is \(Z \approx 30\), while for overhead lines it is more than 170.
When both halves of the electric transmission are in operation, the surge impedances correspondingly increase by a factor of 2.
2. In order to determine more accurately the magnitude of the short-circuit current, as well as the law of its variation with time, a more general formula was obtained. If the short-circuited part of the line is represented in the form of a chain equivalent circuit consisting of series-connected four-terminal networks, each of which represents a T-shaped equivalent circuit of an elementary section of the line with parameters \(R\), \(L\), and \(C\), then for the short-circuit current in the line the expression obtained is
\[ i_{k.z}=\frac{2U_0}{nL}e^{-\frac{R}{2L}t}\sum_{m=1}^{m=n}\frac{\sin \beta_m t}{\beta_m}, \tag{2} \]
where
\[ \beta_m=\sqrt{\frac{2}{CL}\left[1-\cos\frac{2m-1}{2n}\pi\right]-\frac{R^2}{4L^2}}. \]
According to formula (2), the variation of the short-circuit current with time has a damped oscillatory character. The rate of damping of the oscillations, determined by the value \(\frac{R}{2L}\), does not depend on the location of the short circuit and is a constant quantity for the given electric transmission line.
The period of the oscillations depends on the location of the short circuit. As the length of the short-circuited section of the line increases, the period increases, and vice versa.
For a transmission line at a voltage of \(2 \times 400\) kV, having a cable of considerable cross-section, the surge current of a single-pole short circuit ahead of the smoothing reactor of the inverter substation, caused only by the discharge of the line capacitance, reaches a value 12.5 times as large.
In Fig. 1 a short-circuit-current curve is plotted for such a transmission line when its equivalent circuit is replaced by a chain circuit with \(n = 4\).
In Fig. 2 an oscillogram of the short-circuit current in the same line is presented for \(n = 4\).
Fig. 1
The amplitude value of the short-circuit current in this transmission line decreases to the value of the rated current after \(0.05\) sec.
Calculation of the fault current by formula (2) for \(n = 24\) gives a short-circuit-current value very close to the actual one and sufficiently accurate for practical calculations.
Fig. 2
3. The influence of the inductance of the smoothing reactor on the magnitude and character of variation of the short-circuit current in the line can be determined with sufficient certainty if the line is represented by a T-shaped equivalent circuit, bearing in mind that the inductance of the line is small in comparison with the inductance of the reactor.
In this case the current in the line is
\[ i_{kz}=\frac{U_0}{\sqrt{\dfrac{L}{C_n}-\dfrac{R^2}{4}}}\, e^{-\frac{R}{2L}t}\, \sin \sqrt{\frac{1}{LC_n}-\frac{R^2}{4L^2}}\;t. \]
Here
\[ R=\frac{R_l}{2}+R_{op}, \]
\[ L=\frac{L_l}{2}+L_{op}. \]
Calculations of short-circuit currents beyond the smoothing reactor of the receiving end, carried out by this formula for an unloaded transmission line, under the assumption of an unchanged value of the reactor-coil inductance \((L_{op})\), show that the maximum short-circuit current exceeds the rated current by a factor of 2--3 in the case of cable lines of various lengths, and does not reach the value of the operating current in the case of overhead transmission lines.
At the same time it should be noted that the values of the short-circuit currents at the end of transmission lines, with \(L_{op}=\mathrm{const}\), do not depend on the type of short circuit, since in a two-pole short circuit the voltage, inductance, and resistance increase, while the line capacitance decreases by a factor of two in comparison with the case of a single-pole short circuit.
The frequency of oscillation of the emergency current in this short-circuit process is considerably reduced in comparison with the preceding case and, for various cable transmission lines, lies within the range 8--16 cycles per second.
One of the principal parameters determining the value of the s.c. current in the given emergency condition is the inductance of the smoothing reactor, which in the preceding discussion was assumed to remain unchanged during the short-circuit process.
Fig. 3
In reality, the sharp increase of current at the instant of short circuit leads to strong saturation of the reactor iron and to a considerable reduction of the inductance. This circumstance makes it necessary, in calculations of emergency currents in the line, to take into account the change in the inductance of the reactors.
The change in inductance of one constructed smoothing reactor, as a function of the magnitude of the current flowing through it, is shown in Fig. 3.
The established dependence \(L_{\partial p}=f\left(\dfrac{i}{I_n}\right)\) shows that at a 3-fold current the inductance decreases by a factor of 1.85, and at a 6-fold current—by a factor of 5.6.
The change in the inductance of reactor coils in short-circuit calculations according to formula (3) can be taken into account by the method of successive approximations.
Fig. 4
Such accounting showed that the multiplicity of the surge short-circuit current at the end of cable lines increases by 2–4 times in comparison with the case of constant reactor inductance.
Together with the change in \(L_{\partial p}\) during the short-circuit process, the period and frequency of oscillations of the fault current also change.
Fig. 5
Figure 4 gives the calculated short-circuit curves beyond the smoothing reactor for a powerful transmission line:
\[ 1\text{—curve for } L_{\partial p}=\mathrm{const}, \]
\[ 2\text{—curve for } L_{\partial p}=\mathrm{var}. \]
Figure 5 presents an oscillogram of the short-circuit beyond the reactor for the same transmission line.
Conclusions
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Short circuits in DC cable lines upstream of the smoothing reactor are accompanied by the occurrence of currents of considerable magnitude due to the energy stored in the line capacitance. This makes it necessary to take account of the capacitance of transmission lines in short-circuit calculations.
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The short-circuit current in faults in the line upstream of the smoothing reactor at the receiving end of the transmission decreases to the value of the rated current over a time interval of the order of 0.05 sec.
In short circuits beyond the reactor, the damping process of the short-circuit current is significantly prolonged, and the decrease to the value of the rated current occurs only over a time interval on the order of 0.5 sec.
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In the case of a short circuit beyond the reactor at the receiving end of the transmission line, it is necessary in short-circuit calculations to take into account the change in the reactor inductance due to saturation.
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In order to limit the short-circuit current at the end of the transmission line, it is desirable to design the smoothing device in such a way that saturation of the reactor iron does not occur, or is minimal, when the emergency current flows through its winding.