MATHEMATICAL MODEL

OF THE INFLUENCE OF PROTECTION AND EMERGENCY MODES

ON THE RELIABILITY OF INDUCTION MOTORS

A. S. Gitman, B. A. Itkin, E. K. Strelbitskii

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

The share of failures of induction electric motors caused by protection amounts to 15–25%. It is advisable to investigate the quantitative assessment and the influence of individual characteristics of the protection of emergency currents on the operational reliability of electric motors by means of mathematical models [1].

Depending on the problem posed and the available initial data, various models may be used. The proposed model makes it possible to evaluate the influence of short circuits on the wear of the machine insulation and its reliability.

In constructing the model, it is assumed that the wear of the insulation \(\Delta r_i\) under emergency mode \(i\) does not depend on the influence of previous short circuits and overloads. Failure of the \(k\)-th motor occurs after exhaustion of its insulation resource \(R_k\),

\[ \sum_i \Delta r_i > R_k \qquad (i = 1, 2,\ldots;\quad k = 1, 2,\ldots). \tag{1} \]

Let us introduce into consideration the limiting temperature in the stator winding of the motor under emergency mode \(\vartheta\) (over the time \(t_{\mathrm{pr}}\)), after reaching which the machine fails. The accident rate of the motor fleet due to protection, when the operating time does not exceed \(t_{\mathrm{pr}}\), will be denoted by \(A'\). The accident rate when operation occurs over a time exceeding \(t_{\mathrm{pr}}\) (failure of the protection to operate, or absence of protection, corresponds to \(t_{\mathrm{pr}}=\infty\)) will be denoted by \(A''\). The total accident rate, according to the theorem of addition of probabilities, is

\[ A = A' + A'' - A'A''. \tag{2} \]

The stream of accidents has an unequal effect on motors with different insulation resources. On the basis of service-life tests, the distribution of the initial resources of a fleet of induction motors may be considered normal. In integral form, the distribution law of the resource has the form:

\[ P(r,\bar r,\sigma_r)= \frac{1}{\sqrt{2\pi}\,\sigma_r} \int_{0}^{r} e^{-\frac{(r-\bar r)^2}{2\sigma_r^2}}\,dr = F\left(\frac{r-\bar r}{\sigma_r}\right), \tag{3} \]

where \(F(x)\) is the integral Laplace function.

Suppose that an estimate of the average wear is available for the given protection parameters. Then the insulation life of the motors can be calculated by the number of faults that they can withstand:

\[ n_i=\frac{r_i}{\Delta r}. \tag{4} \]

Expression (3) then takes the form

\[ P(n,\overline{n},\sigma_n)= \frac{1}{\sqrt{2\pi}\,\sigma_n} \int_{0}^{n} e^{-\frac{(n-\overline{n})^2}{2\sigma_n^2}}\,dn = F\left(\frac{n-\overline{n}}{\sigma_n}\right), \tag{5} \]

where

\[ \overline{n}=\frac{\overline{r}}{\Delta r};\qquad \sigma_n=\frac{\sigma_r}{\Delta r}. \]

And the probability that a motor of the fleet under consideration has withstood not fewer than \(n-1\), but not more than \(n\), emergency modes will be

\[ \left.F\right|_{n-1}^{n} = F\left(\frac{n-\overline{n}}{\sigma_n}\right) - F\left(\frac{n-1-\overline{n}}{\sigma_n}\right); \qquad (n=1,2\ldots). \tag{6} \]

It may be assumed that the quantity \(\left.F\right|_{n-1}^{n}\) determines the share of the motor fleet for which \(n\) emergency modes will lead to failure.

Under the condition of the probability that the occurrence of exactly \(n\) emergency modes will lead to motor failure on the interval \(SS_{n-1,n}\), it is determined by the formula

\[ P_{\lambda,n\,(m\le n)} = \frac{\dfrac{e^{-\lambda}\lambda^n}{n!}} {\displaystyle\sum_{m=0}^{n}\frac{\lambda^m e^{-\lambda}}{m!}} = \frac{\dfrac{\lambda^n}{n!}} {\displaystyle\sum_{m=0}^{n}\frac{\lambda^m}{m!}}. \tag{7} \]

The fraction of motors that have failed through the fault of the protection on the segment \(S_{n-1,n}\) will be equal to

\[ P(\lambda)= \frac{\dfrac{\lambda^n}{n!}} {\displaystyle\sum_{m=0}^{n}\frac{\lambda^m}{m!}} \left.F\right|_{n-1}^{n}, \tag{9} \]

where \(\lambda\) is the parameter of the Poisson stream of emergency modes.

Summing all segments, we obtain the overall failure rate when the protection operates in a time less than \(t_{\mathrm{pr}}\):

\[ A' = P(\lambda)= \sum_{n=1}^{\infty} \frac{\dfrac{\lambda^n}{n!}} {\displaystyle\sum_{m=0}^{n}\frac{\lambda^m}{m!}} \left.F\right|_{n-1}^{n}. \tag{10} \]

We have calculated the values of function (10) for various values of \(\lambda\), \(n\), and \(\sigma_n\).

Let us determine the failure rate of motors through the fault of the protection when it operates in a time exceeding \(t_{\mathrm{pr}}\). Suppose that the probability of the event is known:

\[ p\,(t\ge t_{\mathrm{pr}})=p''. \tag{11} \]

for the given type of protection. Then the accident rate \(A''\), for the mean value of the accident flow \(\lambda\), is determined by the formula [2]

\[ A''=1-(1-p'')^\lambda . \tag{12} \]

To estimate insulation wear in the emergency mode it is necessary to know the distribution density of the protection operating time. Protection must be understood in the broad sense of the word, i.e., taking into account not only the characteristics of the protective apparatus, but also the actions of the operator. On the basis of works [3, 4], the distribution of the time for disconnecting the motor by the operator in the emergency mode obeys the Rayleigh law

\[ F(t)=1-e^{-\frac{x^2}{2\sigma_p}} . \tag{13} \]

The distribution of the time for disconnecting the motor by protective apparatus for a given type of short circuit obeys a truncated normal law with parameters \(\bar t\) and \(\sigma_{\mathrm н}\) [4]. The normalizing factor in most cases may be neglected,

\[ F(t)=\frac{1}{\sqrt{2\pi}\sigma_{\mathrm н}}\int_0^t e^{-\frac{(t-\bar t)^2}{2\sigma_{\mathrm н}^2}}\,dt = F\!\left(\frac{t-\bar t}{\sigma_{\mathrm н}}\right). \tag{14} \]

Since disconnection of the motor by the operator or by protective apparatus are independent events, the probability of disconnecting the motor during time \(t\) is determined from the expression

\[ P(t)=1-\left[1-F\!\left(\frac{t-\bar t}{\sigma_{\mathrm н}}\right)\right] \left[1-\left(1-e^{-\frac{t^2}{2\sigma_p^2}}\right)\right] = 1-F\!\left(\frac{\bar t-t}{\sigma_{\mathrm н}}\right)e^{-\frac{t^2}{2\sigma_p^2}} . \tag{15} \]

From (15) we determine the probability of the event \(p(t\ge t_{\mathrm{pr}})\)

\[ P(t\ge t_{\mathrm{pr}})=1-P(t<t_{\mathrm{pr}}) = F\!\left(\frac{\bar t-t_{\mathrm{pr}}}{\sigma_{\mathrm н}}\right) e^{-\frac{t^2}{2\sigma_p^2}} . \tag{16} \]

Taking the derivative of function (15) with respect to \(t\), we find the distribution density of the time of motor disconnection by protection in the emergency mode

\[ f(t)=p'(t)= e^{-\frac{t^2}{2\sigma_p^2}} \left[ \frac{1}{\sqrt{2\pi}\sigma_{\mathrm н}} e^{-\frac{(t-\bar t)^2}{2\sigma_p^2}} + \frac{t}{\sigma_p^2} F\!\left(\frac{\bar t-t}{\sigma_{\mathrm н}}\right) \right]. \tag{17} \]

For a quantitative estimate of the accident rate due to the fault of the protection, it is necessary to know the distribution density of insulation wear \(\varphi(V)\). Determining \(\varphi(V)\) analytically presents considerable difficulties. A less exact, but simpler, path consists in finding the initial moments \(v_k\) and establishing, with their aid, a suitable law of the wear distribution. First let us derive the function of insulation wear during heating. According to [7], insulation wear over the time interval \(t\) is determined by the formula

\[ V(\vartheta)=\int_0^t \exp\!\left[b(\vartheta-\vartheta_{\mathrm{dop}})\right]\,dt, \tag{18} \]

where \(b\) is a coefficient depending on the insulation class;
\(\vartheta\) is the insulation temperature;
\(\vartheta_{\mathrm{dop}}\) is the permissible insulation temperature for the given insulation class.

The value of \(\vartheta\) for the short-circuit case under consideration is found from formula [6]

\[ \vartheta_{\mathrm{kz}}=(T_v-at)t+\vartheta_{\mathrm{nach}}, \tag{19} \]

where \(T_v\) is the initial rate of temperature rise in the stator winding during a short circuit;

\(a\) is a constant depending on the motor type;

\(\vartheta_{\mathrm{nach}}\) is the initial temperature of the machine.

Substituting (19) into (18), after a number of transformations we obtain

\[ V_1(\vartheta)= e^{-b\left(\vartheta_{\mathrm{nach}}-\vartheta_{\mathrm{dop}}+\dfrac{T_v^2}{4a}\right)} \sqrt{\frac{\pi}{ab}} \left[ F(\beta t-\alpha)-F(-\alpha) \right], \tag{20} \]

where

\[ \beta=\sqrt{2a\cdot b};\quad \alpha=\frac{\sqrt{b}\,T_v}{\sqrt{2a}}. \]

The insulation-wear function during cooling \(V_2(\vartheta)\), in accordance with [8], is

\[ V_2(\vartheta)= \frac{T}{b\Delta\Theta_{\mathrm{kz}}} \exp b(\vartheta_{\mathrm{nach}}-\vartheta_{\mathrm{dop}}) \left[ \exp b\Delta\Theta_{\mathrm{kz}} + 4\exp\frac{b\Delta\Theta_{\mathrm{kz}}}{2} - 5 \right], \tag{21} \]

where \(T\) is the time constant of the winding during cooling,

\(\Delta\Theta_{\mathrm{kz}}\) is the excess of the short-circuit temperature over the initial temperature.

Substituting (19) into (21), we finally obtain

\[ V_2(\vartheta)= \frac{T}{b(T_vt-at^2)} \exp b(\vartheta_{\mathrm{nach}}-\vartheta_{\mathrm{dop}}) \left[ \exp b(T_vt-at^2) + 4\exp\frac{b(T_vt-at^2)}{2} - 5 \right]. \tag{22} \]

The initial moments of the wear functions during heating and cooling can be calculated from the formulas

\[ \nu_k\left[V_1(\vartheta)\right] = \int_{0}^{t_{\mathrm{pr}}} \left[ V_1(\vartheta_2) \right]^k f(t)\,dt \tag{23} \]

\[ \nu_k\left[V_2(\vartheta)\right] = \int_{0}^{t_{\mathrm{pr}}} \left[ V_2(\vartheta_2) \right]^k f(t)\,dt, \tag{24} \]

where the function \(f(t)\) is determined by (17).

When calculating the total wear, it is necessary to take into account the presence of a correlation between the wear during heating and during cooling. Denoting by \(V(\vartheta)\) the total wear, for the first initial moment we obtain

\[ \nu\left[V(\vartheta)\right] = \nu_1\left[V_1(\vartheta)\right] + \nu_2\left[V_2(\vartheta)\right], \tag{25} \]

as the mathematical expectation of the sum of random variables. The second initial moment for the function \(V(\vartheta)\) is found from the expression \(\nu_2[V(\vartheta)]=M[V(\vartheta)^2]\), and finally

\[ \nu_2\left[V(\vartheta)\right] = \nu_2\left[V_1(\vartheta)\right] + 2\nu_{12}\left[V(\vartheta)\right] + \nu_2\left[V_2(\vartheta)\right], \tag{26} \]

where \(\nu_{12}\) is the mixed initial moment

\[ \nu_{12}\left[V(\vartheta)\right] = M\left[V_1(\vartheta)V_2(\vartheta)\right] = \int_{0}^{t_{\mathrm{pr}}} V_1(\vartheta)V_2(\vartheta)f(t)\,dt. \tag{27} \]

The calculations we carried out for the first four initial and mixed moments of the wear functions during heating and cooling, for the most common four-pole motor and for various protection parameters, showed that the insulation wear, both during heating and during cooling, obeys the Weibull law.

\[ F(x)=1-e^{-\frac{x^m}{x_0}}. \tag{28} \]

An example calculation is given in the Appendix.

When using the mathematical model of the influence of protection on the operational reliability of induction motors, for practical calculations some simplifications may be made:

1. Take into account insulation wear only during cooling.
2. Compute only the first two initial moments.

The mathematical model obtained makes it possible to analyze the influence of the motor protection system on its operational reliability for various protection characteristics and emergency modes.

Appendix

Example of calculating the influence of coarse protection of induction motors on their reliability with the following parameters: \(\sigma_p=16\) sec.; \(t_n=50\) sec.; \(\sigma_n=25\) sec.; \(\lambda=4\); \(\vartheta_{\text{init}}=40^\circ\mathrm{C}\); \(t_{\text{pr}}=59\) sec.

1. From the previously calculated tables we find the initial and mixed moments of insulation wear:

\[ \nu_1[V_1(\vartheta)]=15,58\ \text{h.} \qquad \nu_1[V_2(\vartheta)]=56,93\ \text{h.} \]

\[ \nu_2[V_1(\vartheta)]=0,8136\cdot 10^5\ \text{h.}^2 \qquad \nu_2[V_2(\vartheta)]=0,8786\cdot 10^6\ \text{h.}^2 \]

\[ \nu_3[V_1(\vartheta)]=0,8334\cdot 10^9\ \text{h.}^3 \qquad \nu_3[V_2(\vartheta)]=0,2785\cdot 10^{11}\ \text{h.}^3 \]

\[ \nu_4[V_1(\vartheta)]=0,10441\cdot 10^{14}\ \text{h.}^4 \qquad \nu_4[V_2(\vartheta)]=0,1088\cdot 10^{16}\ \text{h.}^4 \]

\[ \nu_{12}[V_1(\vartheta)V_2(\vartheta)]=0,2671\cdot 10^6\ \text{h.}^2 \]

1. We find the parameters \(m\) and \(x_0\) of the Weibull law for wear during heating and cooling:

a. During heating,

\[ \nu_2:\nu_1^2=0,8136\cdot 10^5:15,58=336, \]

\[ m=0,1975 \]

Determine the parameter \(x_0\):

\[ x_0= \left[ \frac{\nu_1}{\Gamma\left(\frac{1}{m}+1\right)} \right]^m = \left[ \frac{15,58}{\Gamma\left(\frac{1}{0,192}+1\right)} \right]^{0,192} = 0,63. \]

The 3rd and 4th initial moments:

\[ \nu_3=x_0^{\frac{3}{m}}\Gamma\left(\frac{3}{m}+1\right) = 0,63^{\frac{3}{0,192}} \Gamma\left(\frac{3}{0,192}+1\right) = 0,589\cdot 10^{10}, \]

\[ \nu_4=x_0^{\frac{4}{m}}\Gamma\left(\frac{4}{m}+1\right) = 0,63^{\frac{4}{0,192}} \Gamma\left(\frac{4}{0,192}+1\right) = 1,83\cdot 10^{14}, \]

The agreement of the 3rd and 4th moments is sufficiently good.

b. During cooling,

\[ \frac{\nu_2}{\nu_1^2} = \frac{0,8786\cdot 10^6}{56,53} = 274. \]

\[ m=0,1975 \]

\[ x_0=\left[\frac{56{,}53}{\Gamma\left(\frac{1}{0{,}1975}+1\right)}\right]^{0{,}1975}=0{,}845;\quad \nu_3=0{,}845^{\frac{3}{0{,}1975}}\Gamma\left(\frac{3}{0{,}1975}+1\right)= \]

\[ =0{,}164\cdot 10^{11}, \]

\[ \nu_4=0{,}845^{\frac{4}{0{,}1975}}\cdot\Gamma\left(\frac{4}{0{,}1975}+1\right)=0{,}159\cdot 10^{16}. \]

The agreement of the 3rd and 4th moments is also good. Let us find the upper confidence limits of wear with confidence probability 0.995. In the case of wear only during cooling,

\[ x=(5{,}3\cdot x_0)^{\frac{1}{m}}=(5{,}3\cdot 0{,}845)^{\frac{1}{0{,}1975}}=2130\ \text{hours}. \]

Taking into account joint wear according to (25) and (26),

\[ \nu_1[V(\vartheta)]=56{,}53+15{,}58=72{,}11\ \text{hours}. \]

\[ \nu_2[V(\vartheta)]=0{,}8786\cdot 10^6+0{,}8136\cdot 10^5+20{,}2671\cdot 10^6=1{,}4991\cdot 10^6 \]

\[ \nu_1:\nu_2=287,\quad m=0{,}196 \]

\[ x_0=\left[\frac{72{,}11}{\frac{1}{0{,}196}+1}\right]^{0{,}196}=0{,}873; \]

\[ \Delta r=x=(5{,}3\cdot x_0)^{\frac{1}{m}}=(5{,}3\cdot 0{,}873)^{\frac{1}{0{,}196}}=2190. \]

In this case the correction for wear during heating is insignificant. Let us recalculate the service-life parameters

\[ \bar n=\frac{\bar r}{\Delta r}=\frac{13000}{2190}=5{,}94;\qquad \sigma_n=\frac{\sigma}{\Delta r}=\frac{2500}{2190}=1{,}14. \]

For rounded values \(n=6\) and \(\sigma_n=1\). \(A'=0{,}1001\). Let us determine the probability of protection operation over the time

\[ t\geq t_{\mathrm{pr}}\ \text{by (16)}\quad p''=F\left(\frac{50-59}{25}\right)e^{-\frac{59^2}{2\cdot 16^2}}=0{,}402\cdot 10^{-3}. \]

and the accident rate \(A''\) by (12)

\[ A''=1-(1-0{,}402\cdot 10^{-3})^4\approx 0{,}161\cdot 10^{-2}. \]

The total accident rate by (2)

\[ A=0{,}1001+0{,}00161-0{,}1001\cdot 0{,}00161=10{,}16\%. \]

LITERATURE

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

2. E. S. Venttsel. Introduction to Operations Research, Sovetskoe Radio Publishing House, 1964.

3. Yu. V. Chuev et al. Fundamentals of Operations Research in Military Engineering, Sovetskoe Radio Publishing House, 1965.

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

5. Ya. B. Shor. Statistical Methods of Analysis and Quality and Reliability Control, Sovetskoe Radio Publishing House, 1962.

6. B. A. Itkin, E. K. Strelbitskii. On the rate of temperature rise under emergency modes and the cooling laws of motors of the AO2 series, Izv. TPI, vol. 162, 1967.

7. G. Goter. Heating and Cooling of Electrical Machines, GEI, 1961.

8. I. A. Syromyatnikov. Operating Modes of Induction and Synchronous Electric Motors, GEI, 1963.