Simplification of Checking Conductors and Equipment for Thermal Stability
Abstract Generated abstract
This paper proposes an approximate procedure for simplifying thermal stability checks of conductors and electrical equipment under short-circuit conditions. Instead of calculating both three-phase and two-pole short-circuit currents, it relates their thermal effects through a correction coefficient K, defined as the ratio of energy released by the two-pole current to that released by the three-phase current, with values depending on calculated reactance, fault duration, and generator type. The authors identify a critical reactance at which the two effects are equal and show when the three-phase current alone is decisive. For cases where two-pole faults govern, they provide formulas converting standard three-phase thermal stability quantities, including minimum conductor cross section and stability currents, by multiplying with factors involving K.
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Simplification of Checking Conductors and Equipment for Thermal Stability
I. D. Kutyavin and A. S. Sharygina
The guidelines for calculating short-circuit currents, 1944 edition, recommend (see § 60) determining the thermal action of the short-circuit current on conductors and equipment either from the current of a three-phase short circuit or from the current of a two-pole short circuit, depending on which of them produces the greater thermal effect.
For inexperienced designers it is difficult to decide in advance which current causes the greater heating; therefore they have to calculate the short-circuit current for both two- and three-phase short circuits and check the equipment and conductors for both currents.
It appears possible to simplify these calculations by an approximate estimate of the thermal action of the two-pole short-circuit current from the thermal action of the three-phase current, which eliminates the need to determine the two-pole short-circuit current.
The relation between the currents of a two-pole and a three-phase short circuit and their thermal action depends on the calculated reactance \(x_p\) and the actual duration of flow of the short-circuit current \(t_d\).
To solve the problem in general form, the quantity of energy released by the three-phase current in the conductor, relative to current, can be found from the expression:
\[ A_3 = I_{\sim *}^{(3)\,2} t_{\phi}^{(3)} = f_1(x_p, t_d), \tag{1} \]
where \(I_{\sim *}^{(3)}\) is the steady-state three-phase short-circuit current in relative units, determined for the calculated reactance \(x_p\);
\(t_{\phi}^{(3)}\) is the fictitious time;
\(x_p\) is the calculated reactance.
The quantity of energy released by a two-pole short-circuit current, relative to current, will have the following expression:
\[ A_2 = 3 I_{\sim *}^{(2)\,2} t_{\phi}^{(2)} = f_2(2x_p, t_d), \tag{2} \]
where \(I_{\sim *}^{(2)}\) is the steady-state direct-sequence current for a two-pole short circuit in relative units, determined from the nomogram for the reactance \(2x_p\).
An approximate accounting of the heating of conductors by the two-pole short-circuit current can be carried out with the aid of a coefficient \(K\), equal to the ratio:
\[ K = \frac{A_2}{A_3} = f_3(x_p, t_d). \tag{3} \]
In Figs. 1, 2, 3, and 4 the values of this coefficient are plotted for various standard types of generators. The determination of \(A_2\), \(A_3\), and \(t_{\phi}\) was carried out from the nomograms given in the guidelines for calculating short-circuit currents.
Fig. 1. Correction curves (for a turbine generator with a voltage regulator)
Fig. 2. Correction curves (for a turbine generator without a voltage regulator)
On the basis of these figures the following conclusions may be drawn:
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For all types of generators there exists a certain critical reactance \(x_{rk}\), at which the heating of conductors by the current of a two- and three-pole short circuit is the same and, consequently, \(K=1\). The magnitude of this critical reactance \(x_{rk}\) depends on the calculated time \(t_o\), the type of generator, and whether the latter has an automatic voltage regulator.
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In the case when the calculated reactance \(x_r > x_{rk}\), the current of a three-pole short circuit has a greater thermal effect than the current of a two-pole short circuit and, consequently, the check for thermal stability in this case should be made only for the current of a three-pole short circuit.
Fig. 3. Correction curves (for a hydrogenerator with a voltage regulator with a damper winding)
Fig. 4. Correction curves (for a hydrogenerator with a voltage regulator without a damper winding)
- For \(x_r < x_{rk}\), the current of a two-pole short circuit is decisive with respect to thermal stability. However, there is no need in this case to determine this current, since the amount of energy released by the current of a two-pole short circuit, \(A_2\), can be determined from \(A_3\) by multiplying by \(K\):
\[ A_2 = K A_3 = K I_{\sim *}^{(3)\,2} t_{\phi}^{(3)}. \tag{4} \]
The value \(K\) is determined as a function of \(x_r\) and the type of generator from one of the figures given.
For checking conductors and apparatus for thermal stability, various formulas are widely used. Below is given the transition from calculated quantities, determined by these formulas for a three-pole short circuit, to those for a two-pole short circuit by means of the coefficient \(K\).
- The minimum required cross section of a conductor from the condition of heating by a two-pole short-circuit current is
\[ q_m^{(2)} = I_b I_{\sim *}^{(3)} \sqrt{\frac{K t_{\phi}^{(3)}}{A_{\theta m}-A_{\theta n}}} = \sqrt{K}\, q_m^{(3)}, \tag{5} \]
where \(I_b\) is the base current,
$A_{\theta m}$ and $A_{\theta n}$ are quantities depending on the maximum permissible and initial temperature of the conductor.
2. Five-second thermal-stability current for a two-pole short circuit
\[ I_{5}^{(2)}=I_{b} I_{\sim *}^{(3)} \sqrt{\frac{K t_{\phi}^{(3)}}{5}}=I_{5}^{(3)}\sqrt{K}. \tag{6} \]
3. Thermal stability of a reactor under a two-pole short circuit:
\[ \left(I_{tn}\sqrt{t_n}\right)^{(2)} = I_{b} I_{\sim *}^{(3)} \sqrt{K t_{\phi}^{(3)}} = \sqrt{K}\left(I_{tn}\sqrt{t_n}\right)^{(3)}, \tag{7} \]
where $I_{tn}$ is the thermal-stability current,
$t_n$ is the time in seconds to which, according to factory data, the thermal-stability current of the reactor is referred.
4. Ratio of the one-second thermal-stability current under a two-pole short circuit:
\[ n_{1}^{(2)} = I_{b} I_{\sim *}^{(3)} * \frac{\sqrt{K t_{\phi}^{(3)}}}{I_n} = \sqrt{K}\, n_{1}^{(3)}, \tag{8} \]
where $I_n$ is the rated current of the apparatus.
When taking into account the individual attenuation of separate components of the current, the check of conductors and apparatus for thermal stability should be made with respect to the three-pole short-circuit current if the reactances of the branches are greater than the critical value. In the case where the reactance of one of the branches is less than the critical value, an approximate allowance for the heating of conductors by two-pole short-circuit currents could be made according to expression (4), determining the coefficient $K$ for this smallest reactance. In doing so, it must be borne in mind that expression (4) will give an overestimated result. If, under such a check, the conductor proves to be thermally stable, the question may be considered settled; but if the conductor is unstable, then, if necessary, its stability under the two-pole short-circuit current may be checked by the usual method.
The method recommended above for allowing for the heating of conductors and apparatus by the two-pole short-circuit current will make it possible to considerably simplify calculations connected with the determination of short-circuit currents and the selection of electrical equipment for switchgear installations.