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QUANTITATIVE ESTIMATION OF INDIVIDUAL RELIABILITY OF THE EQUIPMENT AND DEVICES OF THE POWER SUPPLY SYSTEM
Farhadzade E.M., Muradaliev A.Z., Farzaliev Y.Z.
Azerbaijan Scientific-Research and Design-Prospecting Institute of Energetic
AZ1012, Ave. H.Zardabi-94 •
e-mail :fem1939@rambler. ru
ABSTRACT
The basic stages of a design procedure of parameters of individual reliability the equipment and devices of electro power systems are considered. The recommended method illustrated on an example of parameters of reliability calculated as average arithmetic random variables. The method based on imitating modeling of random variables and the theory of check of statistical hypotheses.
1. Statement of a problem and some definitions
The objective estimation of parameters of reliability (PR) the equipment and devices of electro power systems (EPS) always was and remains to one of priority problems which decision directed on decrease in expenses at designing and operation of electro installations [1]. In addition, despite of urgency of this problem, calculations PR, traditionally, spent for assumptions rather far from the validity. The basic assumption is the opportunity of representation of statistical data of operation by representative sample of general set of these data, i.e. these data represented homogeneous. Calculated PR thus carries, naturally, average character. At the same time dependence PR on those or other factors as a class of a voltage, type of the equipment, duration and conditions of operation, the system of service and so forth, that already contradicts this assumption is marked.
Set of the statistical data describing reliability of the equipment, actually represents so-called final set of multivariate data (MD) [2]. MD essentially differs from sample of general set. First, MD are set not only set of the random variables describing reliability of objects of research, but also set of versions of attributes (VA), describing each random variable. Practically, these data formed in the so-called empirical table which lines allocate objects, and columns: a serial number of objects, the attributes describing object, realizations of random variables and casual events. The set of objects is limited to frameworks of a solved problem and shown in set VA. But distinction not only in it.
It is known, that about reduction of number of random variables of sample of general set accuracy of estimations PR decreases (width of a confidential interval increases) [2]. At classification MD on set significant VA, decrease in number of realizations of a random variable in sample accompanied by decrease in disorder of possible values of a random variable, i.e. accuracy of estimations PR increases.
As an example, consider duration of restoration of deterioration at emergency repair (iem) witches of a power supply system. It is known, that with increase in a class of a voltage of switches average value of a random variable iem also increases. Thus, if values iem for air switches in final set of multivariate data change in an interval (6-105) hr., for air switches with nominal voltage (15-20)kV this interval appears essentially less and is equal (6-25) hr.
The account of these features allows passing from calculation of average values PR to calculation of parameters of individual reliability, i.e. PR for set VA. It is necessary to have in view of, that parameters of individual reliability, in fact, also are average. However, averaging here spent on "rustling" VA. Difficulty of an estimation of parameters of individual reliability in many respects caused by necessity of ranging set VA on their importance.
The problem of an estimation of parameters of individual reliability of the equipment from the methodical point of view is a special case of a problem of classification of park of objects on groups for which calculated PR differs not casually.
Significant interest at the decision of some operational problems caused with laws of change of parameters of individual reliability in function VA. The scale of these attributes can be not only in a quantitative kind (for example, service life, an interval of time after scheduled repair, etc.), but in serial (for example, a class of a voltage, capacity, etc.) and in nominal (for example, units of object, its importance, etc.). Difficulty of the decision of noted problems increases also because at the constant approach algorithms of estimation various PR are various.
2. Algorithm of comparison statistical functions of distribution final set MD and not casual sample of these MD.
Classification of statistical data on set VA, first, assumes an opportunity of an estimation of its expediency. One of ways of the characteristic of expediency of classification of data is the estimation of character of a divergence of statistical functions of distribution (s.f.d.) final set MD and sample of these MD on set VA. The approach to such comparison we shall consider on example PR, calculated as an average arithmetic X. Let us specify initial data:
— in the empirical table some final set MD of a random variable of X. Numerical value X is set depends from «n» considered attributes. Each of «n» attributes is presented to one of ri VA with i=1,n.
F*(X)- s.f.d., and M£(X) - average value of final set MD (index £ carries parameters and characteristics of reliability to final set MD);
— certain combination VA sets object, PR that in the form of estimation M*( X) should estimated. Directly to estimate M*( X) it is impossible, since data about X at this object practically are
absent. And without taking into account these VA, about any individuality to speak it is not necessary;
— not casual sample of values of random variable X, as result of classification MD on one VA is set. S.f.d. this sample we shall designate, as F*(X). To compare F£*(X) and F*(X), we spend following sequence of calculations:
2.1. We count the greatest empirical deviation A.
For this purpose:
— for each value Хj from set {X}V samples it is defined absolute size of a deviation s.f.d. F£*(X) from s.f.d. F*( X) under the formula
A (X j) = |fs* (X j) — Ff* (X j )| (1)
with j=1,m, where m- number of realizations of random variable X in sample;
— define the greatest value among m realizations A(X) under the formula
A £ = max {A (Xx); A (X 2);...; A (X} );...A (Xm)} (2)
As distributions F£*(X) and F*(X) constructed on statistically given operation, size AE there is the greatest empirical deviation.
It is necessary to note, as statistics of criterion of a divergence can be chosen not only size A, but also average value of the greatest deviation Aav, average quadratic value Aaq., average geometrical
value and a number of others. However, as shown in [4], statistics A has at the fixed value of a error of I type, the greatest capacity of criterion.
2.2. Modeling of distribution F * [A(H 1)]
S.f.d. F* [A(H 1)] = P[A < A(H 1)] - distribution of realization of absolute size of the greatest deviation of modeled realizations s.f.d. F*( X) from s.f.d. F*(X) for assumption H1 (divergences of realizations s.f.d. Fs*(X) also F*(X) has casual character) Modeling F * [A(H1)] spent in following sequence:
— on distribution, F*(X) it is modeled m random variables X. According to [3] calculation of realization of random variable X it is carried out under the formula
X = Xt + (X i+1 — Xt )[(n + 1)f — i ] (3)
where proggrammatic a modeled pseudo-random variable with uniform distribution in an interval [0,1]. Let's designate this set of values X as {X }*
— according to {X }* is under construction s.f.d. FV**( X);
— transformation of final set MD is spent. For this purpose:
— from set of values X of final set MD are withdrawn m the values describing {X}v ;
— instead of {X}v values {X}* are entered;
— pays off s.f.d. on transformed final set MD. Designate it is Fs** (X) ;
— for realizations of random variables Xj with j=1,m samples are calculated absolute deviations s.f.d. F*( Xj ) and s.f.d. Ff**( Xj ) under the formula:
A( X j) = |fs** ( Xj ) — F"( xJ )|; (4)
— the greatest deviation s.f.d. defined FS**(X) from s.f.d. F*( X) under the formula
A( H j) = max{A(Xx); A (X2);...; A(Xj );...A(Xm )}; (5)
— it is modeled N realizations of a random variable A(H1);
— N realizations A(H1) placed in ascending order. Further to each value A(H1) the probability f * [Ai (h 1)] = yn, where i-serial number of realizations of set of values is compared A(H1).
Calculations F * [A(Hx)] come to the end with that.
2.3. Modeling of distribution F * [A(H 2)].
S.f.d. F * [A(H2)] = P[A<A(H2)] - distribution of realizations of absolute size of deviations s.f.d. F*( X) from s.f.d. Fs*( X) for assumption H2 (the divergence Fs*( X) and F*( X) is not casual). The algorithm of modeling F* [A(H2)] is similar to algorithm of modeling of distribution F * [A( H J] with that essential difference, that modeling of sample from m values of random variable X is spent not on s.f.d. final set MD Fs*(X) , and on s.f.d. F*(X).
2.4. Decision-making
To make a decision on character of a divergence Fs*(X) and F*(X), i.e. to choose one of two
assumptions (H1 or H2) and by that to estimate expediency of classification of statistical data to the set attribute, it is necessary:
1. To define average value N of realizations A(H1) under the formula
N / m*[A(hi)] = £A_(hi) n
i=i /
2. To define average value N of realizations A(H2) under the formula
N /
m *[A( h 2)] = i ( h 2) n
3. To construct s.f.d., describing error of I a*(A) and the II P*(A) types
3.1. If M * [A(H 1 )] < M * [A(H2 )], that
a * [ A( H1 )] = 1 - F * [ A( H1 )] p\ A( H 2 )] = F * [ A(H 2 )]
3.2. If M * [A(H 1)] > M * [A(H 2)], that
a[A(H2)] = 1 - F*[A(H2)] /[A^)] = FXHJ]
4. On s.f.d. a*(A) and P*(A) to define critical values of absolute size of the greatest deviation Acr.. Size Acr. in practice are calculated for the set significance values acr. and pcr., usually accepted equal acr=pcr=0.05 (0.1). As actually distributions a*(A) and P*(A) have discrete character, and among discrete values s.f.d., as a rule, there are no probabilities acr and pcr, equal 0,05 or 0,1, recommended to accept as an admissible error of I type the nearest to acr smaller value among set of discrete values s.f.d. a*(A), and as an admissible error of II type - the nearest to pcr, smaller value among set of discrete values s.f.d. P*(A). The valid boundary values of these errors at M * [A(H1)] < M * [A(H2)] designate accordingly: for a error of I type - through sh1[A(H1)], and for a error of II type - through sh2[A(H2)]. Corresponding sh1[A(H1)] critical value of the greatest deviation will be Acr[sh1(H0], and for sh2[A(H2)] - will be Acr [sh2(H2)]. If M * [A(H1)] > M * [A( H 2)] boundary values of these errors accordingly will be sh1[A(H2)] and sh2[A(H1)]. Corresponding mistakes of the first and second sort sh1[A(H2)] and sh2 [A(H1)] critical values of the greatest deviation will be Acr[sh1(H2)] and Acr[sh2(H1)]
5. To compare with an empirical deviation AE with critical values of mistakes of the first and second sort. Thus
5.1. If M * [A( H1)] < M * [A(H 2)] and AE>Acr[sh1(H0], with a significance value sh1[A(H0] assumption H2 is accepted. If M * [A ( H1)] > M * [A(H 2)] and AE>ACT[sh2(H2)], with a significance value sh1[A(H2)] assumption H1 is accepted
5.2. If M * [A(H 1 )] < M * [A(H 2)], and AE<ACTsh1(H1)] and AE<ACT[sh2(H2)], with a significance value sh2[A(H2)] assumption H1 is accepted. If M * [A(H 1)] > M *[A(H 2)], and AE<Acr[sh1(H2)] and
AE<Acr[sh2 (H1)], with a significance value sh2[A(H1)] assumption H2 is accepted
The total risk of the erroneous decision pays off under the formula:
Ri ( A ) = A • F [A ( H1) ]+ B • F [A ( H 2) ]= Ri [ A ( H1)] + Ri [ A ( H 2)] where A and B - factors of the importance of errors of I and II types; A+B=1. If the information on consequences of possible errors of I and II types is absent, is accepted A=B=0.5, and Ri(A) calculated as an average arithmetic errors of I and II types.
Choice of one of two assumptions is spent on following conditions: If Ri[Ae (H1)] >> Ri[Ae (H2)], that H=Hi
If Ri[Ae (H2)] >> Ri[AE (H1)], that H=H f" (6)
If Ri[AE(H 1)]s Ri[Ae(H2)], that H=Hi
As an example in table 1 initial data are cited: final set MD (X}s and sample of these MD {X}V, s.f.d. F* [X ] and F* [X ] and results of calculation AE.
In table 2 results of calculations of critical values of the greatest deviation and risk of the erroneous decision are resulted. From this table it is evidently visible, that errors of I and II types should be calculated not proceeding from corresponding assumptions (H1 and H2), and proceeding from a parity of average values of a random variable of sets {X}s and {X}V. Not the account this parity leads to essential decrease in significance values (errors of I and II types). Data of tables 1 and 2 testify to inexpediency of classification MD, i.e. H^H1.
In table 3 results of calculation for a case, when H^H2
Table 1
Illustration of calculation of the greatest empirical deviation
N { X }e F*( x ) { X }v x ) A( X )
1 105.8 0.059
2 109.6 0.118
3 109.9 0.176
4 110.3 0.235
5 111.3 0.294
6 111.7 0.353 111.7 0.333 0.02
7 112.7 0.412
8 113.7 0.471 113.7 0.667 0.196
9 113.9 0.529
10 114.7 0.588
11 115.2 0.647
12 115.5 0.706
13 117.2 0.765
14 117.4 0.824
15 117.7 0.882 117.7 1.000 0.118
16 119.2 0.941
17 119.6 1.0
Note: Ae=0,196
3. Algorithm of a choice of the most significant VA
It would seem algorithm of a choice it is simple enough:
— it is necessary to receive not casual sample of data of final set MD on everyone VA;
— to compare F*(X) and F*( X);
— to define risk of the erroneous decision;
— to define VA with the minimal risk of the erroneous decision
However simplicity of algorithm is deceptive, since it is required to lead generally n! calculations, that on time exceeds comprehensible opportunities of computer facilities. The problem consists in comparison not casual выборок data on everyone VA. It is necessary to allocate sample, s.f.d. This to the greatest degree would differ from s.f.d. final set MD. The preference is given sample, numerical characteristics s.f.d. this to the greatest degree differed from numerical characteristics s.f.d. F*( X) in comparison with the others s.f.d. In particular considered:
Table 2
Illustration of calculation of critical values of the greatest deviation and risk of the erroneous decision
N A( H !) F * [A( Hj] A ( H 2) F * [A(H2)] {1 - F * [A( H 2)]} Ri* (A) Results of calculation
1 0.000 0.000 0.000 0.000 1.000 0.500 M*[A(H0] = 0,271
2 3 4 5 6 7 8 0.020 0.039 0.059 0.078 0.098 0.118 0.137 0.003 0.006 0.012 0.031 0.045 0.097 0.140 0.118 0.137 0.150 0.206 0.850 0.794 0.474 0.467 M*[A(H2)] = 0,245 Sh1[A(H2)] = 0,023 ACr[sh1(H2)] = 0,471 Sh2[A(H0] = 0,045 ACr[sh2 (HO] = 0,098 Ri *(A ) = 0,433
9 0.157 0.214 0.157 0.332 0.668 0.441 H^Hj
10 0.176 0.256 0.176 0.390 0.610 0.433
11 0.196 0.355 0.196 0.475 0.525 0.440
12 0.216 0.412
13 0.235 0.460 0.235 0.568 0.432 0.446
14 0.255 0.545 0.255 0.641 0.359 0.452
15 0.275 0.622
16 0.294 0.679 0.294 0.759 0.241 0.460
17 0.314 0.728 0.314 0.798 0.202 0.465
18 0.353 0.772 0.353 0.838 0.162 0.467
19 0.373 0.806
20 0.412 0.847 0.412 0.911 0.089 0.468
21 0.431 0.876
22 0.471 0.941 0.471 0.977 0.023 0.482
23 0.490 0.966
24 0.529 0.995 0.529 1.000 0.000 0.498
25 0.549 1.000 0.000 0.500
Table 3
Results of calculations of expediency of classification final set MD
N (X}z Nv {X} v Results of calculations
1 100 1 100 M*[A(H1)] = 0,188
2 103 2 104,6 M*[A(H2)] = 0,269
3 104,6 3 105,9 Aэ=0,265
4 105,9 4 113,5 Sh1[A(H0] = 0,029
5 107,3 5 116 ACr[sh1(H0] = 0,265
6 7 8 108,3 109,7 113,5 6 139,6 Sh2[A(H2)] = 0,030
ACr[sh2(H2)] = 0,147
9 114 Ri* (A) = 0,245
10 116 H^H2
11 116,7
12 118,9
13 122,1
14 132
15 134,7
16 139,6
17 140,5
— the greatest value of absolute size of distinction of average values of random variables of final set MD and samples of this set on each of set VA, calculated under the formula:
A[M*(X)] = max{A1[M*(X)];A2[M*(X)];...;A, [M*(X)];...;A„[M*(X)]} (7)
where:
A[M •( X)] = m *( X) — m (X);
i=1,n;
m •( x ) = £ x l;
m r, ( x ) = £X,
j=1
the greatest value of absolute size of distinction of average quadratic deviations of random variables of final set MD and samples of this set on each of set VA, calculated under the formula:
A[G *( X)] = max{Ai[G *(X)]; A 2[G *(X)];...;A. [G*(X)];...;A „ [G*(X)]} (8)
where: a.[G(X)] = G(X) — g "vi (X)|; i=1,n;
G *( X) =
£ Im V( X)—Xj ]2
L — 1
G V,.( X) =
± IM V,.( X) — Xj ]2
L - Number of random variables of final set MD; mi - number of random variables of i-th sample
— the greatest value of absolute size of distinction of estimations of factors of a variation of final set MD and samples of this set on each of set VA, calculated under the formula:
A[ K *( X)] = max{A[ K*(X)]; A[K2*(X)];...; A. [K*(X)];...; A[ K*(X)]} (9)
where: Afc (X)] = |k*(X) — K*, (X)|; i=1,n;
K *(X) = G*(X ^
K Vi (X) =
G V,i (X).
m V(X) V/MV, (X)
the least relative value of disorder of random variables samples, calculated under the formula:
S(X) = minXX);........,Sn (X)}
(10)
m. — 1
where: i=1,n; S, (X) = (Xmax,V,, " Xmin, V,i )/(Xmax,S _ Xmin,S ) ;
X max,v ,i = maxlXv,,,i; xv,,,2;.....Xv,,,m};
Xmin, V,i = min{Xv,i,1; XV,i,2 ;.....XV,,,m };
Xmax,S = max{XS,i; XS,2;.....XS,L };
Xmin,S = min{XS,i; XS,2;.....XS,L } .
Thus it was supposed, that if s.f.d. samples with extreme values of distinction of numerical characteristics casually differs from s.f.d. FS*(X), casually differs from FS*(X) and s.f.d. all others samples. How much this assumption is true? Experiences of calculations have allowed drawing following conclusions:
_ by comparison F* (X) and FV* (X) in a kind of small number of realizations of sample, there is an uncertainty of the decision caused by classification that testifies to inexpediency of classification. I.e. the preference given H1. Here finds the reflection influence not so much numbers of random variables, how many sizes of their disorder concerning disorder of final set MD. It is established, that the importance VA above, the concerning the average value random variables of sample more concentrate on axes of set of values of final set MD. This conclusion explains cases of erroneous decisions on a condition (7) _ the average quadratic deviation characterizes disorder of random variables concerning their average value. Consequently, it should seem to carry out a choice of the most significant VA more authentically. But it has appeared that the decision depends on average value of sample
and number of random variables. Than it is less M** (X) and more M*, (X), and the number
of random variables of sample is more, the reliability of the decision of a choice between H1 and H2 is more;
_ the condition (9) in which basis there is a comparison of change of factors of a variation, eliminates dependence of estimations of an average quadratic deviation on average value of sample of random variables. Reliability of the decision in comparison with a condition (8) has increased. The condition (9) has eliminated the errors caused by influence of average values
M*( X ) and M*, (X) with i=1,n, but has kept their dependence on number of random
variables that is shown already at number of attributes i>3
_ the condition (10), reflecting physical essence of importance VA, has appeared the most
sensitive and authentic. It precisely proves to be true graphically by comparison s.f.d. F*( X)
and FV*, (X) with i=1,n.
Thus, algorithm of definition of the most significant VA and consequently working sample, at each stage of classification MD it reduced to following sequence of calculations: _ formation (n+1-i) samples from final set MD of realizations of random variables X for set VA,
where i- number simultaneously considered VA, i=1,n is spent; _ the interval of change of random variables X for final set MD and (n+1-i) samples is defined;
_ under the formula (10) relative values of an interval of change of random variables in (n+1-i) samples are calculated and defined sample with the minimal value of relative value of an interval is 5(X);
_ constructions under s.f.d. this sample F*( X) and final set MD Fs*( X)
According to the algorithm stated in p.2 comparison s.f.d. is spent. F*(X) and F*(X).
4. The integrated algorithm of definition of parameters of individual reliability
The essence of algorithm reduced to following sequence of calculations:
_ at the first stage most significant of VA defined. The methodology of the decision of this problem considered us in section 3. Designate the sample corresponding most significant VA as {X (i, j)} V, where i—1,n. j—1,ri, and a serial number of it VA - (i, j);
_ check of assumptions of character of a divergence F*(X) is spent and F* P (X). If assumption
H2 of difference F*(X) also F* P(X) is rejected (theoretically it probably) and data do not
contradict assumption Hi of casual character of difference significant VA are absent, classification of final set MD is inexpedient, and PR are calculated on final set MD. If with the set significance value assumption H1 is rejected and data do not contradict assumption H2 we pass to the second stage of calculations; _ at the second stage calculations similar to calculations at the first stage with that essential difference carried out, that as final set MD sample {X(i, j)}V undertakes. From this final set
samples on all set VA except for VA with a serial number (i,j) undertake. Among (n-1) samples there is a sample relative size of an interval of which changes of random variables the least. S.f.d. the sample it compared with s.f.d. final set MD.
Thus, naturally, there is a question on to what distribution to compare s.f.d. samples on two
VA - with initial s.f.d. final set MD Fs*( X) or with s.f.d. the final set MD received to the most
significant attribute F*( X) = F*1 (X) ?
At the decision of this problem, it is necessary to start with following three axiomatic positions:
P 1. If Fs*(X) and FV*1(X) differ not casually ^
and if F*1(X) and FV** 2(X) differ not casually L^ (11)
that F*( X) and FV* 2( X) also differ not casually P 2. IfF*(X) and FV* 1(X) differ casually :
and if Fy1(X) and FV*2(X) differ not casually (12)
that F*( X) and FV* 2( X) also differ not casually In other words, the neglect casual character of a divergence Fz*( X) and FV* 1( X) conducts
to artificial distortion of size and understating of accuracy of an estimation of parameters of individual reliability, decrease in number of stages of classification of data, to the erroneous list significant VA
P 3. If Fs (X) and FV*,1 (X) differ not casually
and if Fy1(X) and FV*2(X) differ casually v (13)
that F*(X) and FV 2(X) also differ not casually
J
Positions (P1-P3) testify that at each i- th stage of classification distributions FV*(i_1)(X) should be compared and F*t (X). At all subsequent stages of classification MD the calculations similar to the above-stated are spent, and come to the end provided that distinction s.f.d. FV*( i_1)( X) and F*i (X) becomes casual
CONCLUSIONS
As a result of the lead researches methodical bases are developed:
— quantitative estimation of parameters of individual reliability of the equipment and devices of power supply systems;
— classifications set VA on significant and insignificant;
— ranging of significant attributes in ascending order the importance;
— transition of the decision of operational problems on the basis of ranging reliability of the equipment and devices at an intuitive level, to the decision on the basis of comparison of quantitative estimations of parameters of their individual reliability
LITERATURE
1. Voropay N.I., Kovalyov G.F. About substantive provisions of the Concept of maintenance of reliability in electric power industry. The power policy, 2010, №3, 7-10 p.
2. Anderson T. Introduction in the multivariate statistical analysis. M. Phismathpub, 1963, 500 p.
3. Ryabinin I.A. Basis of the theory and calculation of reliability of ship electro power systems. 2 pub., "Shipbuilding", 1971, 453 p.
4. 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.
5. Farhadzadeh E.M., Muradaliyev A.Z., Farzaliyev Y.Z. Method and algorithm of comparison of empirical characteristics of relative duration of non-working conditions of the equipment of power supply systems. M.: Electricity.№6, 2010, 10-15 p.
