STUDY OF THE GROUNDED INSULATION OF INDUCTION MOTORS WITH RANDOM-WOUND WINDINGS BY THE MONTE CARLO METHOD

E. K. STREL’BITSKII, A. S. GITMAN, Yu. P. POKHOLKOV

In designing induction motors, it becomes necessary to construct single- or multilayer ground insulation capable of withstanding both operating voltages and overvoltages arising during service. For this purpose it is necessary to investigate various combinations of insulating materials, i.e., to solve a synthesis problem. The opposite problem (the analysis problem) arises, for example, in the process of manufacturing motors, when it is necessary to determine the causes affecting the magnitude of rejects due to breakdown of the ground insulation.

The scatter of insulation properties, the random character of the occurrence of damage during manufacture, and also the influence of random environmental effects during operation require the use of statistical methods of investigation. Taking into account also the fact that the breakdown process is described by nonlinear dependences, the most expedient approach for solving the stated problems is the construction of a mathematical model and the application of the Monte Carlo method.

For this purpose we adopt the following assumptions:

1. Insulation breakdown is equally probable within the limits of an “elementary” region determined by the area of the test electrode.

2. The insulation of conductors adjacent to the “elementary” region is a continuous layer with a thickness equal to the one-sided thickness of the conductor insulation.

The adopted assumptions make it possible to represent the ground insulation within the limits of an “elementary” region in the form of a chain of series-connected capacitors, where each capacitor is equivalent to one insulation layer.

The insulation between the plates of each capacitor has its own law of distribution of breakdown voltages. All random effects on the insulation can be taken into account by changing the parameters of the laws. Breakdown of the ground insulation is equivalent to breakdown of all the capacitors of the chain; moreover, the breakdown sequence is independent and the punctured insulation does not recover. To find the statistical parameters, it is necessary to subject to breakdown tests all materials included in the composition of the ground insulation. For this purpose the insulating materials are removed from the slots while taking precautions that exclude additional damage. The law of distribution of the breakdown voltages of the conductor insulation is determined for a conductor with a length equal to the sum of the lengths of the sections adjacent to the “elementary” region.

Since in the overwhelming majority of cases the thickness of the ground insulation is less than 10 percent of the radius of curvature of the slot, the field may be considered plane. In this case the voltage drop across the capacitors is determined by the formula:

\[ U_i=\frac{U d_i \prod_{i=1}^{n}\varepsilon_i} {\varepsilon_i \sum_{j=1}^{n}\frac{d_j}{\varepsilon_j}\prod_{i=1}^{n}\varepsilon_i} \tag{1} \]

\[ (i=1,2,\ldots,n);\quad (j=1,2,\ldots,n), \]

where \(U\) is the applied voltage;
\(d_i\) is the thickness of the insulation of the \(i\)-th layer.
\(\varepsilon_i\) is the dielectric constant of the \(i\)-th layer.

The insulation thickness and dielectric constant entering into (1) are random variables, and their distribution laws must be known.

In the simulation, at each step it is necessary to compute the probability of breakdown of a chain of capacitors. Consider a chain consisting of two capacitors. Let \(A(B)\) denote the event that the first (second) capacitor breaks down after the initial distribution of voltages; let \(A/B\) \((B/A)\) denote the event that the first (second) capacitor breaks down after breakdown of the second (first). Let \(AB\) denote breakdown of the entire chain. We find the probability of breakdown of the chain \(P(AB)\) through the probability of the opposite event

\[ P(AB)=1-P(\overline{AB}) \]

The event \(\overline{AB}\) (the chain is not broken down) is equivalent to the union of incompatible events

\[ \overline{A}\cap\overline{B},\quad A\cap\overline{B}/A \quad \text{and} \quad \overline{A}/B\cap B,\quad \text{i.e.} \]

\[ \overline{AB}=(\overline{A}\cap\overline{B})\cup(A\cap\overline{B}/A)\cup(\overline{A}/B\cap B). \tag{2} \]

Passing to probabilities, we obtain

\[ P(\overline{AB})=P(\overline{A})P(\overline{B})+P(A)P_A(\overline{B})+P(B)P_B(\overline{A})= \]

\[ 1+P(A)P(B)-P(A)P_A(B)-P(B)P_B(A) \tag{3} \]

Hence the probability of breakdown of the chain in accordance with (2) is

\[ P(AB)=P(A)P_A(B)+P(B)P_B(A)-P(A)P(B) \tag{4} \]

For a chain with three capacitors, the probability of breakdown of the chain is

\[ \begin{aligned} P(ABC)=&\ P(AB)P_{AB}(C)+P(AC)P_{AC}(B)+P(BC)P_{BC}(A)\\ &-P(A)P_A(B)P_A(C)-P(B)P_B(A)P_B(C)-P(C)P_C(A)P_C(B)\\ &+P(A)P(B)P(C) \end{aligned} \tag{5} \]

In (5), the probabilities \(P(AC)\) and \(P(BC)\) are computed analogously to the probability \(P(AB)\) according to (4).

For example,

\[ P(BC)=P(B)P_B(C)+P(C)P_C(B)-P(B)P(C). \]

The ground insulation of asynchronous motors with random-wound windings consists of a number of layers (including the layer of conductor insulation) not exceeding four.

For a chain of four capacitors,

\[ \begin{aligned} P(ABCD)=&\ P(ABC)P_{ABC}(D)+P(ABD)P_{ABD}(C)+\\ &+P(ACD)P_{ACD}(B)+P(BCD)P_{BCD}(A)-P(AB)P_{AB}(C)P_{AB}(D)-\\ &-P(AC)P_{AC}(B)P_{AC}(D)-P(AD)P_{AD}(B)P_{AD}(C)-\\ &-P(BC)P_{BC}(A)P_{BC}(D)-P(BD)P_{BD}(A)P_{BD}(C)- \end{aligned} \]

\[ \begin{aligned} &{}-P(CD)P_{CD}(A)P_{CD}(B)+P(A)P_A(B)P_A(D)P_A(C)+\\ &\quad +P(B)P_B(A)P_B(C)P_B(D)+P(C)P_C(A)P_C(B)P_C(D)+\\ &\quad +P(D)P_D(A)P_D(B)P_D(C)-P(A)P(B)P(C)P(D) \end{aligned} \tag{6} \]

In (6), the probabilities \(P(ABD)\), \(P(ACD)\), and \(P(BCD)\) are computed analogously to the probability \(P(ABC)\) by (5).

The modeling process is carried out on an electronic digital computer (EDC). For each realization of breakdown of the groundwall insulation, consisting of \(n\) layers, values \(d_i\) and \(\varepsilon_i\) are formed in accordance with the specified distribution laws. After this, breakdown of the number of layers \(i\) is simulated successively: from \(i=0\) to \(i=m-1\). For a number \(i\) of broken-down layers it is necessary to compute the breakdown probabilities of the remaining intact layers in all possible combinations, whose number is equal to \(C_m^i\). The total number of combinations is

\[ M=\sum_{i=0}^{m-1} C_m^i - 1 = 2^m - 1. \]

Then the breakdown probability of the “elementary” section is determined as a function of the number of layers by formulas 4, 5, or 6. After the required number of realizations has been performed, statistical processing and printing of the results take place.

The transition from the breakdown probability of the “elementary” section to the breakdown probability of the entire groundwall insulation of the motor is carried out in accordance with [1]. Let \(k\) denote the ratio of the entire insulation area of the motor to the area of the “elementary” section, i.e.,

\[ k=\frac{S_{\text{tot}}}{S_{\text{el sec}}}. \]

Then the probability of breakdown of the insulation to ground is

\[ p=1-\left[1-p\left(S_{\text{el sec}}\right)\right]^k \tag{7} \]

Since the quantity \(P(S_{\text{el sec}})\ll 1\), then

\[ p\simeq k p(S_{\text{el sec}}) \tag{8} \]

The estimate of the required number of realizations \(N\) on the EDC is determined by the formula [2]

\[ N=\frac{t_\alpha^2(1-p)}{p d_{\max}^2}. \tag{9} \]

where \(t_\alpha\) are the quantiles of the normal distribution,

\[ \Phi(t_\alpha)=\alpha \]

and \(d_{\max}\) is the maximum relative error of the solution.

REFERENCES

1. A. S. Zingerman. Determination of breakdown voltage from tests of samples. Elektrichestvo, No. 3, 1950.

2. N. P. Buslenko. Mathematical Modeling of Production Processes, Nauka Publishing House, 1964.