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COMPARISON PARAMETERS AVERAGE AND INDIVIDUAL RELIABILITY EQUIPMENT OF ELECTROPOWER SYSTEMS
E.M.Farhadzadeh, Y.Z.Farzaliyev, A.Z.Muradaliyev
•
Azerbaijan Scientific-Research and Design-Prospecting Institute of Energetic AZ1012, Ave. H.Zardabi-94, e-mail:[email protected]
ABSTRACT
Existing comparison criteria average values of random variables of general population cannot be used at comparison of average random variables of multivariate data. The method, algorithm and an example of calculation of critical values recommended statistics offered.
I. INSTRUCTION
One of the most representative group parameters reliability of the equipment and devices of electro power systems (EPS) are the parameters calculated as an average arithmetic random variables. To them at the characteristic of non-failure operation concern a time between failures and between refusals, at the characteristic of maintainability - averages value of duration of non-working conditions (emergency idle time, being in a reserve, emergency repair at automatic switching-off owing to damage and switching-off under the emergency application owing to defect, capital, average and current scheduled repairs), and at the characteristic of a shelf life- an average idle time at restoration of the adjacent equipment.
Assume, that population is known {x}i from ni multivariate continuous random x variables, the average arithmetic which is equal ME (x) . On some version of an attribute (VA) these data (for example type of the equipment, a class of a voltage, service life and so forth) sample is lead {x}v from nv random variables, an average which arithmetic realizations equally MV (x). The
expediency of classification of data is defined by probability distinction Mi (x) and MV(x).
Two methods of the decision of similar problems [1] known. Both of method assumes normal distribution of random variables. In the first method the hypothesis about a casual divergence of average value of random variables is checked x samples MV(x) from a population
mean of general population of Mi(x). The dispersion of general population is unknown. The criterion looks like:
t > t /
where t = Mz (x) - M*v(x)
S =
t
and a significance value a.
£ [xi - MV(X)]2
(nv -1)
1-a^ ;(nv -i) - a random variable distributed under law Student with (n-1) by degrees of freedom
In the second method the hypothesis about equality of average values mVj(tj) and
mv 2(t2) two samPles normally distributed random variables which dispersions are equal is checked, but are unknown. The criterion looks like:
t > t /
1"%;(nv1 + nv2 -2) '
where t = [MVi(xi) - MV?(x 2)]
S . .
[MVi(Ti)-M*v2(x2)]
[Mvi(X)-MV2(X)] (n + n )
(ni - 1)S2 + (n2 - 1)S2
S2 =
(ni + n2 - 2)
^i, - Mvi(Ti)]2 £[T 2,i - M^T 2 )] 2 Sf = --S2 =-
K -1) 2 (n„, -1)
Thus, initial preconditions of these methods are the normal law of distribution of random variables of sample of general population. Population of statistical data of maintenance service, repair equipment and devices of EES concerns to group of multivariate data. They depend on set of attributes and their versions, and their distribution, as a rule, dissymmetrical. Therefore, to apply at classification of statistical data the methods noted above certainly it is possible, but reliability of result definitely will not correspond to the set significance a value. Below the algorithm of the decision of a problem of comparison of parameters of average and individual reliability on the basis imitating modeling and the theory of check of statistical hypotheses is resulted.
S
2. ALGORITHM OF COMPARISON M* (x) and M*V(x)
For an illustration of recommended algorithm of comparison M*,e(x) and MV,e(x) (the
index «e» allocates the estimation MV(x) calculated according to operation) enter following
designations:
F*(x) - statistical function of distribution (s.f.d.) random variables of population of
multivariate data {x}*;
• FV(x) - s.f.d. not casual sample of random variables {x}v;
• Hi and H2 - assumptions, accordingly, about casual and not casual distinction M*,e(x) and MV,e(x);
• F*[MV(x)] - s.f.d. realizations of average value of sample from nv modeled random variables x provided that sample {x}v concerning a data population {x}* it is representative
F [Mv(x)] = 1 - R [Mv(x)]
• F*[M**(x)] - s.f.d. realizations of average values of sample from nv the random variables modeled on s.f.d. Fv*(x)
Functions of distribution R*[MV(x)] and F*[M**(x)] are necessary, first of all, for an
estimation of critical values, accordingly, MVa (x) for a preset value of a error I type a and
MV p (x) for a preset value of a error II type p.
Algorithm of calculation MVa (t) reduced to following sequence of calculations:
V,a v
1. On a data population {t}i pays off M2 (t) ;
2. For set VA from (t}e sample nv random variables is spent t and further pays off MV e(t)
3. It is modeled s.f.d. F*[MV(t)]. For what:
3.1. Method of imitating modeling on s.f.d. Fs(t) it is modeled nv random variables t. Calculations are spent under the formula recommended in [2]:
T = Ti + (tw -Ti)[n(i-1)] (1)
where E, - a random variable with uniform distribution in an interval [0,1];
3.2. The estimation MV(t) pays;
3.3. Items 3.1 and 3.2 repeat N time, where N - number of iterations of modeling {t}v and calculation of realizations MV (t) . The number of iterations N is defined as follows:
- First are modeled N=500 realizations of casual values MV(t) and placed in ascending order;
- Realization corresponding F [MV (t)] = 0,5 is defined;
- Relative deviation is calculated:
MV;0,5(t) - MV;e (T)
5MV(t) =
MV;e(T)
(2)
- If 5MV e(T) > 0,01, the next sample from N=500 random variables {MV(t)}n is modeled and on 2N to realizations MV(t) the next value 5MV (t) is calculated
- Modeling MV(t) comes to the end at 5MV;e(T) < 0,01;
3.4. N realizations mV(t) are placed in ascending order and to each value mV(t) the probability F*[M*v(t)] = yN with i=1, N is appropriated;
3.5. For the fixed value of a error I type a=0,05 on s.f.d. R*[MV(t)] = {-F*[MV(t)]} Critical
[V(t)] = { - F*[MV,
value MVa (t) is defined;
LV,a '
4. If MV 0 05 (t) < MV e (t) , H ^ H2, i.e. sample {t}v it is unpresentable. Process of classification proceeds with that distinction that as population of multivariate data unpresentable sample is accepted. If MV,e (t) < MV;005 (t) , we pass to check of assumption H2;
5. Modeling s.f.d. F*[M7(t)] . For what:
5.1. Method of imitating modeling on s.f.d. FV(t) modeled nv random variables t. Calculations are spent under the formula (1);
5.2. Average value nv realizations pays off t, which we shall designate as MV (t);
5.3. Items 5.1 and 5.2 repeat N time;
5.4. N realizations MV*(t) are placed in ascending order and to each value of some MV*(t) the
probability F*[MV*(t)] = yN with i=1, N is appropriated;
5.5. On s.f.d. F*[MV*(t)] critical value MV*(t) for a error II type is calculated P=0,05.
6. If it will appear, that MV;e(T) < MV* 0 05(t), H ^ H2, i.e. classification is expedient. Otherwise
we pass to comparison of risk of the erroneous decision, accordingly, assumptions H1 and H2, i.e. sizes Ri*(H0 and Ri*(№);
7. Risk of the erroneous decision Ri*(Hj) = R*[M*Ve(x)]; Ri (H2) = F [Mz,e(i)]. Remind, that Ri*(H2) >p, and Ri*(Hj) >a
8. If
Ri*(Ht)-a Ri*(H2)-ß Ri*(H2)-ß Ri*(Hj)-a
> (1 + a), to H ^ Hj
> (1 + ß), to H ^ H2
(3)
3. REALIZATION OF ALGORITHM
In the illustrative purposes practical realization of algorithm of modeling of distribution F*[MV m(X)] consider on an example of pseudo-random numbers £ with uniform distribution in an
interval [0,1]. Model nv pseudo-random numbers, define their average statistical value MV(£) and absolute value of relative change under the formula
8M*V(^) =
|M(Q - M*(i;)| Mß)
1 -
M*v(^)/
= 1 - 2MV(d
Further dependence is required to us
1 -SM'vß)
Mv(5) =■
2
(4)
Calculate N realizations SMV(^), and ranging SM*(^) in ascending order, we build s.f.d. F* [SM*V(^)J. Transition from realizations MV(£) to realizations SMv(£) allows to compare distributions F* [SMV(^)J not only for different nv, but also for various Fe(£), for example, for uniform distribution in an interval [0.5; 1]. In table 1 are resulted quantile s. f.d. F* [SMV(^)J for of some nv and discrete values of probabilities of these distributions. Laws of change s.f.d. R*[SM*(^)J= 1 -F*[SM*(^)J for nv=4; 22 and 150 are resulted on fig.1, and on fig.2. of some critical values quantile distributions R* [SM*(^)J depending on nv. Regression analysis of these dependences has shown-laws of change, that laws of change MVma (Ç) = f(nv), with high accuracy (factor of determination> 0.999) correspond to following dependence:
ÔMV;m;a ß)
-0,5
(5)
Values of factor A depending on a error I type are resulted in table 2
Under the standard program at a<0,2 greatest convergence (R2=0.994) the equations of regress A=f (a) took place for a polynom
A =-18,5a2 - 7,33a + 1,48 (6)
Table 2
Experimental estimations of constant factor the equations of regress (6)
Error I type 0,01 0,05 0,1 0,2
Value of A factor 1,42 1,13 0,95 0,75
Thus, dependence SMVa (£) = f(a,nv) looks like
«M* (É) =
(18.5a2 - 7.33a +1.48
(7)
v
Table 3
Quantile distributions M* [SM*(^)]
2 3 4 5 6 7 11 22 29 40 90 150
0.001 0.001 0.001 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.01 0.006 0.004 0.004 0.003 0.003 0.003 0.002 0.002 0.001 0.001 0.001 0.001
0.05 0.026 0.022 0.018 0.017 0.015 0.014 0.011 0.008 0.007 0.006 0.004 0.003
0.1 0.05 0.044 0.037 0.033 0.031 0.027 0.023 0.015 0.014 0.011 0.008 0.006
0.15 0.077 0.066 0.056 0.049 0.046 0.041 0.034 0.024 0.02 0.017 0.012 0.009
.02 0.105 0.088 0.076 0.066 0.061 0.056 0.046 0.031 0.027 0.023 0.016 0.012
0.25 0.136 0.111 0.095 0.083 0.076 0.071 0.057 0.040 0.035 0.03 0.02 0.015
0.3 0.166 0.134 0.116 0.101 0.092 0.085 0.069 0.048 0.042 0.036 0.024 0.018
0.35 0.196 0.158 0.136 0.119 0.109 0.101 0.08 0.056 0.05 0.042 0.028 0.022
0.4 0.226 0.182 0.158 0.138 0.125 0.116 0.093 0.065 0.057 0.049 0.032 0.025
0.45 0.259 0.207 0.180 0.158 0.142 0.134 0.106 0.074 0.065 0.055 0.037 0.028
0.5 0.293 0.233 0.202 0.178 0.16 0.151 0.12 0.083 0.073 0.062 0.041 0.032
0.55 0.33 0.259 0.225 0.199 0.179 0.168 0.134 0.093 0.081 0.07 0.046 0.036
0.6 0.369 0.288 0.25 0.223 0.20 0.186 0.15 0.103 0.091 0.077 0.052 0.040
0.65 0.409 0.320 0.277 0.246 0.223 0.206 0.165 0.115 0.101 0.086 0.057 0.045
0.7 0.453 0.353 0.307 0.271 0.247 0.228 0.182 0.127 0.112 0.095 0.064 0.049
0.75 0.5 0.393 0.339 0.301 0.274 0.253 0.202 0.142 0.123 0.105 0.07 0.055
0.8 0.554 0.435 0.374 0.334 0.304 0.282 0.225 0.158 0.137 0.117 0.079 0.061
0.85 0.614 0.489 0.417 0.372 0.339 0.317 0.252 0.177 0.154 0.131 0.088 0.068
0.9 0.687 0.555 0.474 0.423 0.385 0.361 0.287 0.202 0.176 0.15 0.101 0.078
0.95 0.776 0.639 0.557 0.498 0.453 0.425 0.341 0.241 0.210 0.179 0.12 0.093
0.99 0.905 0.785 0.71 0.638 0.59 0.545 0.439 0.311 0.276 0.235 0.157 0.121
0.999 0.969 0.901 0.839 0.769 0.71 0.667 0.56 0.386 0.351 0.296 0.198 0.151
1 0.995 0.958 0.917 0.909 0.872 0.816 0.656 0.467 0.455 0.342 0.244 0.139
Fig. 1. Laws of change s.f.d. R* [SM*(^)J for nv=4; 22 and 150
Fig.2. Laws of change of critical values quantile distributions R* [SM*(^)J depending on nv and a
significance value a=0.05; 0.25 and 0.95
4. EXAMPLE OF CALCULATION.
In the present example on the basis of the lead researches analytically we shall confirm
intuitively clear conclusion: casual character of Mi(X) and MV(X) it is possible at an essential
divergence of corresponding functions of distribution Fi(X) and FV (X). It is easy to notice, that the
example corresponds to the approach at the decision of «return problem», when the result known and efficiency of the approach is checked.
Let Fe(X) is a function with uniform distribution in an interval [0,1]. Random variables (y) samples in volume nv are calculated under the formula:
y = a + Fz(y).(b-a) (8)
where Fi(y) - function of distribution of a random variable y.
Fs(y) corresponds to the uniform law of distribution in an interval [a, b], where a=0,5, and b=1. Hence
y = 0.5 [№(y)] (9)
As realizations Fs(y) take advantage to first four (nv=4) of random numbers of table 9.1 [3] with uniform distribution in an interval [0,1.] This: 0,1009; 0,3754; 0,0842 and 0,9901. Having substituted them in the equation (10), receive accordingly 0,55; 0,688; 0,542 and 0,995. Check of hypothesis about casual divergence Fi(X) and FV(Z) lead according to table 5 [4], and a hypothesis
about a casual divergence of Mi(X) and MV(y) according to table 1. Results of calculations are resulted in table 3.
Table 3
The data checks of statistical hypotheses
i Xi Yi Zi FV(Zi) A i
1 2 3 4 0.1009 0.3754 0.0842 0.9901 0.550 0.688 0.542 0.995 0.550 0.542 0.688 0.995 0.25 0.50 0.75 1.0 0.300 0.042 0.062 0.005 Mi(=0,5; Ms(y) =0.75 MV(Z) =0.694; 8MV(Z) =0.075 MV(A)=0.103
Total 2,775 - 0,409 R*[MV(Z)=0.80
R*[MV(A)<0.01
As the hypothesis about casual divergence follows from table 3 Fi(X) and FV (y) does not prove to be true R*[MV (A)] « a, that completely corresponds to valid parity Fi(X) and FV (y). A hypothesis about a casual divergence of Mi(X) and R*[MV(A)] >>a . In other words, sample {y}nv from the point of view of distinction of M(X) and MV(y) proves to be true homogeneous, and classification of data inexpedient.
CONCLUSION.
1. For the parameters of reliability calculated as an average arithmetic multivariate of random variables which distribution differs from the normal law, the method, algorithm and criterion of comparison of estimations of parameters of reliability are developed at classification of multivariate data;
2. Application for these purposes of a method of comparison s.f.d. to multivariate population of initial data and samples of this population leads to unjustified decrease in a significance
value owing to the unreasonable account of characteristics of disorder of random variables.
REFERENCES
1. Ryabinin I.A. Base of the theory and calculation of reliability of ship electro power systems., Shipbuilding. 1971, 454 p.
2. Farhadzadeh E.M., Muradaliyev A.Z., Rafiyeva T.K., Nazirova U.K. Method of statistical modeling of random variables on empirical distributions. Kazan: News of High schools. Problems of Power №9-10, 2008, 112-120 p.
3. Shor Y,B. Kuzmin F.I. Table for the analysis and the control of reliability. M:, publishers «Soviet radio» 1968, 288 p.
