Study of the Impact of Fault Conditions and Protection on the Reliability of Asynchronous Electric Motors

STUDY OF THE EFFECT OF EMERGENCY OPERATING MODES AND PROTECTION ON THE RELIABILITY OF INDUCTION ELECTRIC MOTORS

B. A. Itkin, E. K. Strelbitskii

(Recommended by the scientific seminar of the departments of electrical machines and general electrical engineering)

The importance of the problem of increasing the reliability of induction motors is well known. A significant share of failures (10–25%) of these motors occurs as a result of overload and emergency operating modes because their protection system operates with delay or fails to operate. Therefore, one of the primary tasks in the problem of increasing the operational reliability of induction motors is to carry out an analysis and study of the effect of emergency operating modes and the protection system on motor reliability. These questions are considered in the present article.

For induction electric motors, the most common and dangerous emergency modes are considered to be: a) a two-phase short circuit (the motor remaining on 2 phases), b) a three-phase short circuit (full line voltage is applied to the motor with the rotor stationary).

According to our statistical data, based on observation of the fleet of electric motors at the NZTSG plant, for mechanical shops the flows of 2- and 3-phase short circuits per fleet of \(n=1000\) motors per year are, respectively, \(\Phi_2=150\div250\); \(\Phi_3=70\div150\), i.e., the probability of occurrence of the flow of accidents for one motor per year is \(p_2=(0.15\div0.25)\cdot10^{-2}\); \(p_3=(0.07\div0.15)\cdot10^{-2}\); and the mathematical expectation of the number of the flow of accidents per one motor per year

\[ \lambda = np \tag{1} \]

will be \(\lambda_2=1.5\div2.5\); \(\lambda_3=0.7\div1.5\).

It is known [1] that if the probability of occurrence of an event in a small time interval \(\Delta t\) is proportional to \(\Delta t\), and the events occur independently of one another, then the number of occurrences of the event during the given time interval is distributed according to the Poisson law. Usually the Poisson law is applicable to phenomena where the probability \(p\) of the event under consideration is small, while \(n\) is large and \(\lambda \le 15\) [2]. All this fully corresponds to the character of emergency modes. Therefore, the probability distribution for the occurrence of two- and three-phase short circuits is adopted according to the Poisson law

\[ P_{m,n} = \frac{\lambda^m}{m!} e^{-\lambda}. \tag{2} \]

Emergency operating modes lead to wear of the insulation of the stator winding or to complete failure of the electric motor. The magnitude of insulation wear (reduction of the service life) of the motor depends both on the nature

and on the magnitude of the emergency mode, as well as on the characteristic of its protection system. By the characteristic of the protection system is meant the aggregate characteristic of motor disconnection both as a result of operation of the protection according to the generalized current-time characteristic of its protective devices and as a result of the operator’s actions.

Table 1 gives the main protection systems of induction motors and their share in the machine-tool inventory of mechanical shops (according to the results of inspection of three shops of the NZTSG plant).

Table 1.

To take into account the influence of emergency modes, it is necessary to know the probability-density law of protection operation under two- and three-phase short circuits. The corresponding checks of protections showed that in most cases this law is normal,

\[ f(t)_{z}=\frac{1}{\sigma_{n}\sqrt{2\pi}}\exp\left\{-\frac{(t-\bar{t})^{2}}{2\sigma_{n}^{2}}\right\}. \tag{3} \]

Appendix 1 gives the fitting of the statistical distribution of the operation of the protection of 53 motors of type AO41-4 under a two-phase short circuit to the normal law. The obtained probability \(p := 0.891\) by the \(\chi^{2}\) criterion is acceptable.

Disconnection of the emergency mode may also be performed by the operator (machine operator). Data on the law of reaction of the machine operator under emergency motor conditions are lacking. Therefore, to determine the probability density of motor disconnection by the operator, \(f(t)_{\mathrm{op}}\), an experiment was set up: for 60 induction motors the protective apparatus was shunted, and 2-phase short circuits were artificially produced without warning the machine operator, while the time of motor disconnection was recorded. The results of the experiments in the form of a statistical series are given in Table 2. The calculated values \(np_i\) are also given there.

Table 2

The agreement of the obtained statistical distribution with the theoretical Rayleigh law was checked,

\[ f(t)_{\mathrm{op}}=\frac{t}{\sigma_{p}^{2}}\exp\left(-\frac{t^{2}}{2\sigma_{p}^{2}}\right). \tag{4} \]

according to the \(\chi^2\) criterion showed that this hypothesis may be regarded as plausible. In general form, the formula for determining \(p_i\), the probability of falling into categories [3], is

\[ p_i=\int_{t_i}^{t_{i+1}} f(t)\,dt . \tag{5} \]

After substituting expression (4) into (5) and making a number of transformations, we obtain

\[ p_i=\exp\left[-\frac{1}{2}\frac{t_i^2}{\sigma_p^2}\right] -\exp\left[-\frac{1}{2}\frac{t_{i+1}^2}{\sigma_p^2}\right]. \tag{6} \]

The value of the measure of discrepancy, determined according to [3], will be

\[ \chi^2=\sum_{i=1}^{8}\frac{(m_i-np_i)^2}{np_i}=1.84, \tag{7} \]

where

\[ \begin{aligned} m_i&\text{ is the number of observations in the category;}\\ n&\text{ is the total number of trials }(n=60). \end{aligned} \]

With the number of degrees of freedom \(r=6\) and \(\chi^2=1.84\), according to table [3] \(p=0.93\)—the probability that, owing solely to random causes, the measure \(p\) of discrepancy between the theoretical and statistical distributions will be no less than that actually observed in the experiment.

For the indicated laws \(f(t)_z\) and \(f(t)_{\mathrm{op}}\), the probability density of operation of the protection system under emergency conditions is determined [4] by

\[ f(t)=e^{-\frac{t^2}{2\sigma_p^2}} \left[ \frac{1}{\sigma_{\mathrm{n}}\sqrt{2\pi}}\, e^{-\frac{(t-\bar t)^2}{2\sigma_p^2}} +\frac{t}{\sigma_p^2}\, F\left(\frac{\bar t-t}{\sigma_{\mathrm{n}}}\right) \right]. \tag{8} \]

To calculate motor wear due to short circuits, in addition to data on emergency currents and \(f(t)\), data are needed on the reliability of operation of the protection. It is also advisable to take separate account—without including them in the law \(f(t)_z\)—of protections for which the non-operation time is of such an order that it leads to failure of the motor in a single emergency condition.

Two calculation methods are possible. One of them is the application of the method of statistical tests, when the process is modeled on an electronic digital computer, the motors subjected to emergency currents are determined, and for them the consumption of service life is counted [5]. According to the other method, the calculation is performed analytically with the aid of tables calculated in advance [4].

It should be noted that the literature lacks special formulas for calculating wear of motor insulation due to emergency conditions. The available formulas take into account only the factor of thermal aging and were derived mainly for class “A” insulation. To eliminate this gap, we are conducting tests for wear due to short circuits on motors of the new AO2-32-4 series with class “E” insulation, using the statistical method of planning extreme experiments [6]. This method makes it possible to obtain not only an interpolation formula, but also a mathematical model of the process. In this case, insulation wear, expressed by the number of short circuits—\(N\), is determined by the expression

\[ \ln N=b_0-b_1\vartheta_M-b_2\vartheta_{\mathrm{n}}-b_3T_{v0} -b_{12}\vartheta_M\vartheta_{\mathrm{n}}-b_{13}\vartheta_MT_{v0} - \]

\[ -b_{23}\vartheta_{\mathrm{n}}T_{v0}-b_{11}\vartheta_M^2 -b_{22}\vartheta_{\mathrm{n}}^2-b_{33}T_{v0}^{\,2}, \tag{9} \]

where

\[ b_0,\ b_1,\ \ldots,\ b_{33} \]

are the regression coefficients of the quadratic equation (9);

\[ \vartheta_{\mathrm{M}} \]

is the maximum temperature of the stator winding under a short circuit;

\[ \vartheta_{\mathrm{n}} \]

is the initial temperature of the winding;

\[ T_{v0} \]

is the initial rate of temperature rise in the stator winding under a short circuit.

The obtained distribution laws for flows of emergency modes, the trip-density of the protection system, and formula (9) for insulation wear make it possible to investigate the influence of short circuits on the reliability of induction motors. Determination of the actual level of insulation wear due to emergency modes is also necessary for the correct choice of protection, taking into account the full utilization of the motors in terms of power.

Appendix 1

The data from checking the operation of protection using an RT1 No. 28 (or No. 29) thermal relay and PR 2-15 fuses with a 10 A link in the circuit of 53 AO41-4 induction motors under a two-phase short circuit are given in Table 3 in the form of a statistical series.

Table 3

Let us check the agreement of the obtained statistical distribution with the theoretical distribution according to the normal law. Using the formulas in [3], we determine the mathematical expectation \(\bar t^x = 44.3\) and the standard deviation \(\sigma^x = 14.7\) of the statistical series. We adopt a theoretical normal distribution law with parameters \(\bar t = 44.3\), \(\sigma = 14.7\). We find the probabilities \(p_i\) of falling into the classes [3]:

\[ P_i = \frac{1}{2} \left[ \hat{\Phi}\left(\frac{t_{i+1}-\bar t}{\sigma\sqrt{2}}\right) - \hat{\Phi}\left(\frac{t_i-\bar t}{\sigma\sqrt{2}}\right) \right]. \tag{10} \]

From (7) we determine the value of the measure of discrepancy, \(\chi^2 = 1.676\). We determine the number of degrees of freedom as the number of classes minus the number of imposed constraints [3], \(r = 8 - 5 = 3\).

From the table in [3] we find the required probability \(p = 0.891\). The obtained probability is not small; therefore, the hypothesis that the trip density of the protection under a short circuit is distributed according to the normal law may be considered plausible.

References

1. Ya. B. Shor. Statistical Methods of Analysis and Control of Quality and Reliability. “Sovetskoe radio,” 1962.

2. L. N. Bolshev, N. V. Smirnov. Tables of Mathematical Statistics. “Nauka,” 1965.

3. E. S. Venttsel. Probability Theory. Fizmatgiz, 1962.

4. A. S. Gitman, B. A. Itkin, E. K. Strelbitskii. Mathematical model of the influence of protection and emergency modes on the reliability of induction motors. Izv. TPI, vol. 172, 1967.

5. B. A. Itkin, E. K. Strelbitskii. On the influence of protection on the operational reliability of induction motors. Izv. TPI, vol. 160, 1966.

6. V. V. Nalimov, N. A. Chernova. Statistical Methods of Planning Extreme Experiments. “Nauka,” 1965.