On the Influence of Protection on the Operational Reliability of Induction Motors

B. A. Itkin, E. K. Strelbitskii

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

Preliminary studies and a number of statistical data [1] show that failures of induction motors caused by the protection against the action of short circuits and overload conditions amount to 15–25 percent of their total accident rate. The influence of individual characteristics of protection and of accident streams on the operational reliability of a motor—\(H(t)_{\text{op}}\)—is expediently studied by means of a mathematical model.

The initial data for constructing the model are the density of the distribution of the protection operating time—\(f(t)\), the parameters of the Poisson streams of 3- and 2-phase short circuits—\(\Phi_3\), \(\Phi_2\), the overload stream \(\Phi_1\), and certain parameters of the motors. The values of the streams are taken from statistical data on the operation of electric drives of the motor population being analyzed. The function \(f(t)\) is determined as a result of processing tests of protections.

We shall solve the following two main problems:

1. Determination of the probability of motor failure due to the protection—\(Q_3\)—from streams of accidents of one type.
2. Determination of \(Q_3\) under the action of several types of emergency conditions.

Let us consider the model for the first problem. Given a known density of distribution of wear of motor insulation—\(f(V)\)—the accident rate due to the protection over the calculated time will be expressed by the formula

\[ Q_3=\int_{1}^{\infty} f(V)\,dV. \tag{1} \]

As a result of the action of one emergency condition, wear of the stator-winding insulation occurs—\(V(t)\), or, correspondingly, expenditure of resource—\(\Delta r\). The quantity \(V(t)\) is determined by the formula

\[ V(t)= \frac{2.94\cdot 10^{-6}T_0\exp(-b\vartheta_4)} {b\Theta_m} \left\{ 4\exp\left(\frac{b\Theta_m}{2}\right)+\exp(b\Theta_m) \right\}, \tag{2} \]

where \(T_0\) is the time constant of the stator winding or of the whole machine,

\(b\) is the coefficient for class-A insulation, equal to 0.0866,

\(\vartheta_4\) is the permissible temperature rise of the stator-winding insulation,

\(\Theta_m\) is the maximum temperature rise as a result of an emergency condition of duration \(t_i\).

It may be assumed that the operating time of the protection system is distrib-

distributed according to the normal law \((n, \bar t, \sigma_t)\). For a fixed value of \(t_i\) and an initial resource \(r_{0i}\), the motor can withstand \(n_i\) emergency conditions

\[ n_i=\frac{r_{0i}}{\Delta r_i}. \tag{3} \]

On the basis of service-life tests, the distribution of the initial resources of a fleet of induction motors may be regarded as normal \((n, \bar r_0, \sigma_{r0})\).

Analytical determination of the probability density \(f(n_i)\) is practically impossible because of the complex form of function (2). It is therefore expedient to determine \(f(n_i)\) by the method of statistical tests. An approximate form of \(f(n_i)\) is shown in Fig. 1.

Fig. 1.

The curve \(f(n_i)\) may be represented as consisting of three parts. Each part bounds an area respectively \(P'_1\), \(P'_2\), \(P'_3\). The area \(P'_1\) is equal to the fraction of machines having in reserve a resource of no more than one emergency condition. When an emergency condition occurs in these machines, failure occurs with probability equal to unity. The probability of the occurrence of an emergency condition is equal to

\[ P''_1=1-e^{-\lambda}, \tag{4} \]

where \(\lambda=\Phi T\) is the mean value of the flow of emergencies per motor over \(T\) years of operation.

The fraction of failed machines in the interval \(n_i \le 1\) is

\[ P_1=P'_1 \cdot P''_1=P'_1(1-e^{-\lambda}). \tag{5} \]

With an increase in the parameter of the flow of emergencies per year \(\Phi\) or in the operating time \(T\), \(P_1 \to P'_1\), i.e., this interval is rapidly “burned out.” In practice, for \(\lambda=3 \leftarrow \to P_1=0.95P'_1\). With a further increase in \(\lambda\), the fraction of failed machines increases insignificantly.

The probability of failure of machines with reserve \(n_i>16\) is practically zero. As calculations have shown, the probability of failure of motors in the interval \(1<n_i<16\) for \(\lambda<1.5\) may be neglected. For \(\lambda>1.5\), it is necessary to take into account the probability of the arrival at a motor of two — \(P''_{n=2}\); three — \(P''_{n=3}\), etc., flows of emergencies per motor.

Taking the distributions of the flow of emergencies according to the Poisson law,

\[ P''_{n=i}=\frac{\lambda^{n_i}e^{-\lambda}}{n_i!}+\frac{\lambda^{n_i+1}e^{-\lambda}}{(n_i+1)!}+\ldots, \tag{6} \]

we find the fraction of failed machines in the interval \(n_i>1\)

\[ P_2=P'_{n=2}\cdot P''_{n=2}+P'_{n=3}\cdot P''_{n=3}+\ldots \tag{7} \]

where \(P'_{n=2};\ P'_{n=3}\) are, respectively, the proportion of machines having in reserve a service life for two and three emergency conditions. The values \(P'_1,\ P'_{n=2}\) are determined by the formula

\[ P'_1=\frac{N_1}{m}; \qquad P'_{n=2}=\frac{N_2}{m}, \tag{8} \]

where \(N_2,\ N_2\) are the number of realizations of the given multiplicity of expenditure of service life out of \(m\) statistical trials.

The process of modeling problems No. 1 and No. 2 on the “Minsk-1” electronic digital computer is given in the flow chart of Fig. 2. The protection tripping time \(t_i\) and the service life \(r_i\) in each trial, or for each motor \(K_i\), are determined by the formulas

\[ t_i=\bar{t}\pm \xi_t\sigma_t, \tag{9} \]

\[ r_i=\bar{r}\pm \xi_r\sigma_r, \tag{10} \]

where \(\xi\) is a random variable distributed according to the law

\[ (n,\ 0,\ 1). \]

In computation on the electronic digital computer, \(\xi\) is formed by a random-number generator [2].

Let us consider the model for the second problem. We take a motor fleet \(K>500\). On the basis of the Poisson distribution of emergency conditions, we determine the motors on which faults occur. For each of these motors, by (10) we find the service life and by (9) the protection tripping time or the duration of the fault (overload). For these motors we calculate the wear according to (2), and then, for each motor, sum the wear from all emergency flows. We determine the ratio of the motor service life to the wear it has accumulated,

\[ n_i=r_i:eV(t_i). \]

We count the number of motors, \(K_v\), for which \(n_i\leq 1\). We determine the probability of motor failure in the given fleet due to the protection as

\[ Q_3=\frac{K_v}{K}. \tag{11} \]

This model is constructed under the assumption that the operation of the given motor-protection specimen is uncorrelated for different types of faults. If such relationships are present, this can be taken into account by introducing the corresponding correlation coefficients into the model.

Table 1 gives, as an example, the results of calculating on the “Minsk-1” the probability of failure due to protection in a hypothetical fleet of induction motors from three-phase short circuits for different protection parameters, fault flows, their service life, and operating time. The initial rate of temperature rise with a locked rotor was taken as \(5.8^\circ\text{C}/\text{s}\).

Analysis of the data in Table 1 shows that \(Q_3\) depends most strongly on the standard deviation of protection operation, to a lesser extent on its mathematical expectation, while the scatter of the initial motor service life has little effect on \(Q_3\).

The mathematical models obtained for determining the influence of protection on \(H(t)_{\mathrm{ek}}\) make it possible to study the actual state of this question, which is necessary for choosing the optimum protection–motor system.

Literature

1. VNIIEM, Technical Information No. OAB. 143.834, 1955.
2. N. P. Buslenko. Method of Statistical Tests, FM, 1962.

No. 1

• Input of initial data
• Obtaining $\xi_t,\ \xi_z$
• Calculation of $t_i,\ z_i,\ V(t_i),\ n_i$
• $n_i \le 1$
• yes → Counter $n_i \le 1$
• no → $1 < n_i \le 16$
  • yes → Counter $1 < n_i \le 16$
  • no → Counter $n_i > 16$
• $m = 500$
• no
• yes → Calculation of $P_1,\ P_2,\ Q_3$ → Stop

No. 2

• Input of initial data
• Obtaining $0 \le N_{L3} \le 1$
• $P(\Phi_3) \ge N_{L3}$
• no
• yes → Calculation of $\xi_{ti3};\ t_{i3};\ V(t_i)_3$
• Obtaining $0 \le N_{L2} \le 1$
• $P(\Phi_2) \ge N_{L3}$
• yes → Calculation of $\xi_{ti2};\ t_{i2};\ V(t_i)_2$
• Obtaining $0 \le N_{L1} \le 1$
• $P(\Phi_1) \ge N_{L3}$
• no
• yes → Calculation of $\xi_{ti1};\ t_{i1};\ V(t_i)_1$
• Calculation of $\sum V(t_i)$
• Calculation of $\xi_{zi};\ z_i;\ n_i$
• $n_i \le 1$
• yes → Counter $n_i \le 1$
• no → Counter $n_i > 1$
• $k = 500$
• no
• yes → Calculation of $Q_3$ → Stop

Fig. 2. Flowcharts for modeling problems No. 1 and No. 2 on an electronic digital computer.

Table 1