CC BY
15
ESTIMATION LIKELIHOOD OF SPEED OF CHANGE OF DIAGNOSTIC
PARAMETERS OF TRANSFORMERS •
Farhadzadeh E.M., Muradaliyev A.Z., Ismayilova S.M.
Azerbaijan Scientific-Research and Design-Prospecting Institute of Energetic AZ1012, Ave. H.Zardabi-94 E-mail :fem1939@rambler. ru
ABSTRACT
Methods of classification retrospective data on independent groups of homogeneous data and estimations of reliability the assumption of constant speed of deterioration during normative service life are developed. Keywords. The transformer, diagnostics, criteria, speed of the change, the guaranteed estimations
I. INSTRUCTION
Increase of efficiency of the control of conformity of a technical condition power transformers and autotransformers (further: TR) to shown requirements represents the important and difficult problem. Its importance caused by the high cost TR, expenses increasing in process of ageing TR for diagnostics, restoration of deterioration, and growth of influence of the human factor. Difficulty of the decision this problem connected with an insufficient computerization of process the analysis of retrospective data, including results of measurement diagnostic parameters (DP). Stochastic character of change DP, influence on DP numerous factors, deterministic the approach in methodology of the analysis of the technical condition TR, not considering these features, is a principal cause of observable discrepancy of results the analysis to real process.
Application of modern methods to research of technical condition TR demands automation of calculations. Considering, that number DP TR is estimated in tens, and number of versions of attributes of distinction TR - hundreds, application of computer technologies allows to solve not only challenges, but also extremely bulky.
"Tool" of practical realization of these technologies are the intellectual automated information systems (IAIS) with that difference from known AIS, that alongside with formalization and storage of retrospective data in special "database", ordering and a press of the information necessary for the analysis, they carry out this analysis and represent recommendations on maintenance service and repair TR.
As bright example of such approach, recommendations [1] serve at chromatographic analysis of the dissolved gases in oil TR. The essential contribution to perfection of system of the analysis of results of measurement DP brought with the researches [2] focused on use of expert systems. Authors of clause spend the researches for more severe constraints - when number experts is limited by units, and IAIS provides with their necessary information and the recommendations, allowing to prove made decisions with the set size of risk of the erroneous decision.
At the analysis of data of measurement DP, along with comparison DP with maximum permissible values, the great value has also the analysis of speed of change DP. This parameter calculated under the formula:
3{n,(t2 -t,)}= 3[n,t] = n^); (1)
where: n(t2 ) and n(tx) - accordingly current and preceded values DP (n) during the moments of time 12 and tx. Its local character, which does not allow comparing with speed of change various DP (owing to distinction of dimensions) concerns to, lacks of this parameter.
The lack it is deprived speed of change of the relative values DP, calculated under the formula:
mn, (t2 - ®=4m t )]>(n ■ (2 >-ffn ■)] (2)
where: Iz(n, t) - relative size DP (n), a describing degree of deterioration of property of a material of units TR during the moment t. In conformity with the developed practice, the size Iz(n, t) in abbreviated form named by "deterioration" during the moment t and calculated under the formula:
Iz(n, t )= n (t)-no (3)
n0 - n
0 o
where: nd and no - accordingly, maximum permissible and initial values DP. Having substituted (3) in (2), we shall receive:
*Hn, (t2 - ^ )]}=3[Mn, t )]= ,[n M-n (t1 )], = \nAzt)] v V2 1/1 v ! (t2-11 )(n-no) - -
n - n
(4)
So that to pass to relative values of speed of change DP, it is necessary to divide absolute value of speed of change DP on (nd - no ). In some cases, the size no ignored. It is inadmissible, if in process of deterioration size
DP decreases. If in process of deterioration size DP increases, the error depends on a paritynd nd . This parity is more
the error of calculations is more. The estimation of size S\_Iz(n, t )] is not end in itself. According to [3] S\_Iz(n, t )] it compared to precede value.
So, according to [1] change of speed for concrete DP more, than on 10 % a month testifies to presence of quickly developing defect in TR. In other words:
S\(n,t) = №, (t3 -;2 )}-\in; (t2 -11 )}j < 0,1 (5)
V ' [ \in, (t2 - t1 )} J If (t3 -12 ) = (t2 -11 ) the formula (5) becomes simpler and looks like:
\n, t ) = }< 1,1
V ' [[n(t2 )-n(t1 )]J
(6)
On fig. 1 law of change DP according to four measurements, accordingly, during the moments t0, t1, t2 , t3
where t, - the moment of measurement n„ is resulted.
Fig.1. Graphic illustration of change DP before restoration of deterioration
As follows from fig.1, the given measurements during the moment t3 testify to that that n (t 3 ) < n d . However, speed of change DP does not satisfy to a condition (6). Speed on a site (t2 ^ t3 ) essentially is more, than on a site (tx ^ t2 ), ô\ (n, t) > 1,1. If to extrapolate line cd, that is to assume, that at t > t3 speed of change DP
remains constant it will appear, that n (t ) will be equal nd during the moment of time t4 .
A seeming simplicity of these calculations is deceptive. Process of deterioration TR far not always corresponds a broken curve abcdke. The analysis of features of real process of deterioration, the account of these features is an indispensable condition of objectivity of the automated calculations.
If deterioration of the transformer connected with growth DP, in process of increase in service life, TR observed not only natural continuous increase in numerical value DP, but also its discrete reduction at use of those or other forms of restoration of deterioration or discrete increase at influence of operational factors. For example [1], at decontaminations of oil, addition of the decontaminated oil and of some other ways of improvement quality of oil TR, concentration of the gases dissolved in oil decreases. Moreover, at refusal of system of cooling, influence of through currents of short circuit, concentration of the gases dissolved in oil sharply increases and at absence of defects TR during one - two months decreases. Dependence of many DP from temperature of oil known.
Let us consider algorithm of ordering of data of speed of change DP. Let in empirical table ET ( n ) databases the sequence of results of tests of park TR of a power supply system is placed.
1. We spend sample of measurement set DP;
2. From this sample systematized given measurements DP for concrete TR on time. These data include: 2.1. Initial data (result of measurement DP at input of unit TR in work during the moment t0
For separate units the moment of implementation coincides with the moment of implementation TR. Chances
when these moments are various).
2.2. Results of measurement DP in process of increase in service life tj > t0 ; j = p,Mi ; Mt - number of measurements DP for i -th TR.
3. Under the formula (2) speed of change of relative values DP during the moment tj_ and tj is calculated with
j = 1, Mt ;
4. The negative values \[lz(n, t )] corresponding this or that form of restoration of deterioration are excluded from consideration.
Results of calculation are brought in empirical table ET ( n ) together with data about service life ( tcn j = (tj._1 + tj )/ 2 ), design features TR and conditions of operation.
Enter concept of rated speed of change DP and designate as \H (n). Further assume, that
\H (n) = (nd _ no ) / Tsl, where Tsl - normative service life TR. Hypothetical law of change DP thus corresponds to
a line af fig.1. It is obvious, that t4 < Tsl. And that it has not occurred, necessary to restore deterioration TR in an
interval t3 ^ t4 . The parity speed of change DP on sites (t0 ^ t1 ) and (t1 ^ t2 ) does not satisfy to a condition (6), but
equality n(t) and nd occurring the moment t5 >> Tsl. In other words, a condition (6) and n(t) < nd are often inconsistent.
As fuller characteristic of conformity of technical condition TR shown requirements are served with a condition not excess of size of relative change \ [n(t2 _ t1 )] of unit. Calculations spent under the formula:
\n, t) = \(n, t)/\(n) (7)
If now in the formula (7) to substitute values \(n, t ) and \H (n ) and to consider (3), receive:
ô\[lz(n, t )] = ô\(n, t ) =
r Tsl J r n (t 2 )_ n (ti )1
_(t2 _ t1 )_ L (n _ no ) J
= Tsl\[n,(t2 _ti)] < 1 (8)
What as much as possible admissible value DP should be after restoration of deterioration during the moment t3 to provide non-failure operation of work TR till the moment of time Tsl at speed of deterioration on an interval (tsl ^ 13 ) no more \ [n, (t3 _ 12 )]. Designate a size as well as n * (t3 ) and calculate it under the formula:
n*(t3)={n(t3Mt„ _t3)[n((3)_n)2)] } (9)
V3 _ 12)
Thus, shown, that:
□ Dnot excess current value DP of maximum permissible size DP does not testify yet to absence of defect TP. The reasons of such discrepancy are or the overestimated (underestimated) value □ nd, or the underestimated
(overestimated) value of admissible change of speed □ \(n, t ). This conclusion based on known in the theory of reliability process of deterioration of materials («a curve life») when after the normal period (speed of deterioration is constant) there comes the period of ageing and catastrophic deterioration (speed of deterioration nonlinear increases);
□ □ Excess of speed of change DP more than on 10% in comparison with preceded value is not necessarily connect with occurrence of local defect. It speaks casual character of changed \(n) □ and essential influence of some factors (design features and conditions of operation);
□ □ Essential growth speed of change DP and not excess of predicted value of residual service life of normative size is a significant attribute of presence of the defect demanding restoration;
□ □ Relative value of speed of change DP DS\ [lz(n, t)] in view of reference value DP allows to compare with speeds of change various DP.
Noted above a ratio have been received in the assumption of constant speed of deterioration on an interval of
service life TR ( Tsl ) and not excess DP of maximum permissible value ( nd ). In real conditions of operation, TR can
appear that this assumption is erroneous. A principal cause to that is heterogeneity of set results of measurement DP and noted above discrepancy of limiting values DP and speeds of change DP to real process of deterioration.
So that to raise accuracy of the forecast of a residual operating time to excess DP of maximum permissible value it is necessary to consider first of all stochastic character of deterioration TR and to develop:
1. The method of classification of retrospective data of speed of change DP on groups of variety of attributes (VP).
2. The method of an estimation of reliability of the assumption of constant speed of deterioration on an interval of
time Tsi ;
As a matter fact, the first method provides an opportunity of application of the second method. According to the terminology accepted in mathematical statistics, agree to name set of data of calculation relative speed of deterioration park TR of a power supply system a final data set (FDS), and a data set, chosen of FDS on the set version of one or of some attributes - sample.
Agree that data of relative values of speed of change DP TR collected and placed in the empirical table (ET).
In columns ET the serial number of measurement, numerical values S\ [lz(n, t)], the name of distinctive attributes are consistently registered. To distinctive attributes concern not only nameplate data TR, i.e. its design features, but also attributes of conditions of operation TR, such as: service life, an operating time after major overhaul, the name of the enterprise and substation, etc.
Designate number of considered distinctive attributes through n , and number of variety of attributes (VP) -
through ri with i = 1, n . Set of results of calculation \[lz(n)] concrete DP, forming FDS, we shall designate as [iz (nv j )]}E , where v = 1, k ; k - number DP; j = 1, Mv ; Mv - number of realizations for j -th DP, and set of realizations \[iz(n)] of sample with set VP - as {<\ [iz(nv j )]}B with v = 1, k and j = 1,Mv B ; Mv B -number of realizations \[iz(n)] for v -th VP in sample ( B ).
II. QUALITY MONITORING OF IMPOSING APPEARANCE OF SAMPLE OF REALIZATIONS
{53 [iz (nvj)]}.
The method is based on a following axiom: if sample of realizations 53[iz(n,t)] for some from set VP, having the greatest absolute value of the maximal divergence of statistical function of distribution (s.f.d.) from s.f.d. FDS, it is representative, other samples of set of versions of considered attributes are representative also all.
Under representative, we shall understand sample, the maximal divergence s.f.d. Which from s.f.d. FDS satisfies to a condition:
ak < {i Fm W* 5)J}>> F*m Iaf" (5)J (1°)
where: 5v] = 53 [iz {pv], t)]- symbolic notation; ak = 1 - R; ak - mistake of the first sort; AF* (5) - the
greatest divergence between s.f.d. FDS (designate it as Fz*(5)) and s.f.d. Samples (designate its FB*(5)). Calculated under the formula:
AF; (5) = max[AF * (5y)] (11)
where: j = 1, Mv, B
Let's agree size AF* (5) to name the greatest deviation empirical distributions AFZ (5 j) and AFB (8j).
AF * (5j )= |FE* (5j)-FB (5 j )| (12)
F^ (5) = i / Mv (13)
(5) = i/Mvb (14)
F*m [aAFJ* (5)]- s.f.d. The greatest divergence between s.f.d. FDS Fz*(5) and modeled on Fz (5) s.f.d. Samples
F*m (5).
AFm (5) = max[AFm (5)] (15)
where: j = 1,Mv , and
AFJ (5j )= Km (5j )-Flm (5 j )| (16)
Fii IaC (5)]= i / N (17)
where: N - number of modeled realizations of the greatest divergence between Fz (5) and FB m (5).
F*m [AFJ**(5)] - s.f.d. The greatest divergence between s.f.d. FDS Fz (5) and modeled on F*B (5) s.f.d. Samples
5).
AFT (5) = max[AF*' (5 )J (18)
where: j = 1,Mv , and
AF'm' fa )= IC, (5)-F**m (5j ) (19)
F*., [aF*;,. (5)J= i / N (20)
To check of a condition (10) precede:
1. Formation of sample of measurements for each variety of considered attributes;
2. Under formulas (1R14) the greatest empirical divergence of all versions i -th an attribute is calculated with i = 1, n ;
3. The greatest divergence among AF3 i (5) Is calculated with i = 1, n . Designate its AF* m (5).
4. The hypothesis about imposing appearance of the sample corresponding size AF*m (5). Is checked. if sample is
representative, according to an axiom other samples for set VP are representative also all. In other words, FDS it is homogeneous, and it is non-uniform - otherwise.
III. METHOD OF CHECK OF THE ASSUMPTION OF CONSTANT SPEED OF DETERIORATION
Casual character of speed of change DP essential influence on this size of operational factors cause difficulties of recognition on retrospective data of law of change in time. Construction of confidential area with the set factor of
trust does not allow to solving a task in view since so as speed of deterioration on an interval Tsl is constant, appear
assumptions of nonlinear laws of its change are fair.
Below the method of check of the assumption of constant speed of change DP, based, as well as a method of classification of data, on statistical modeling units and theories of check statistical hypotheses is resulted. The integrated block diagram of algorithm promoting representation about a method is resulted on fig.2. We shall consider some features of program realization of algorithm.
Block 1. FDS is formed of realizations 53 [lz(n, t)] ET (5). Designate this FDS as {5,9 [lz(n, t)]}E = 5s; Block 2. S.f.d. Fz* (S) Pays off under the formula:
Fs"\(5) = i M (21)
where: M z - number of lines ET (S).
Block 3. Sample (SB ) is formed from (Ss ), which realizations satisfy to a condition:
0,75 Tsi < tj < Tsl (22)
with j = 1,MB, where: MB - number of sample units (B ). Block 4. S.f.d. FB* (S) Pays off under the formula:
FBii (S) = i/M* (23)
Block 5. The greatest empirical divergence s. f.d. Fz* (5) also F* (5) calculated under the formula:
AF3 (5) = max {Fz* (5,) - FB (5 j) |} (24)
Block 6. Average values of speed of change DP FDS and samples under formulas pay off:
m ; (5)=||>j №,t)]}/Ms (25)
MB (5) = { MB 9j [lz(n, t)]!/M B (26)
j=i
Further management is transferred to modeling (m ) s.f.d. The greatest divergence N of realizations FS m (5)
and F*m (5) (block 7). It is originally modeled s.f.d. F* m (5) A method of "inverse functions" on the basis of M B
random numbers with uniform distribution in an interval [0,1] and s.f.d. Fz (5), accommodations of sample of MB realizations of speed of change DP in ascending order and calculation of probability of display of these realizations
under the formula (23). It is formed FDS and s.f.d. Fz m (s), under the formula (16) the greatest divergence AFm (s.) is calculated. These calculations repeat N time, is under construction s.f.d. Fm*[AFm*(ö)J= R(S) and at last is calculated s.f.d. a(s) = 1 - R(s) (block 8).
1 7
10
Formation
FDS
2 1
Calculation s.f.d.
F;{S)
3
1
Formation
of sample
4 1
Calculation s.f.d.
F'b (?)
5 г
Estimation of the max.
divergence AF*(S)
6 г
Estimation
Ml(S) и M* (£)
Modeling s.f.d.
Modeling s.f.d
-4
7.1
Modeling s.f.d.
KM
7.2
Formation FDS and Flm{S)
7.3
Estimation of the max. divergence AF~, (ô )
7.4
i>N
no
7.5
yes
Formation
k№(4
10.1
Estimation of the max. divergence
10.4
no
i>N
10.5
yes
Formation
14
Speed of change DP during Tsi decreases
13
yes
Ml(S)>M*B(ô)
no
15
no
a[AF;(ô)]>ak
yes
Speed of change DP during Tsi is constant
Speed of change DP during Tsi increases
Fig.2 Integrated block diagram of algorithm of the statistical analysis of speed of change DP in current TsL
If it will appear, that a[aF3 (j)] < ak , where ak - critical value of a significance value it means, that on an interval [0,75 Tsl ; Tsl ] relative speed of change DP not casually differs from the average characteristic for FDS.
8
9
Moreover, the parity corresponds to decreasing speed of change DP (block 14), and a parity - to increasing speed of change DP (block 15).
If had data do not contradict a hypothesis Ho of constant speed of deterioration, i.e. a[AF3* (5)J> ak, where ak - critical size of a significance value management transferred the block 10 which is similar to the block 7 with
that difference, that modeling of sample is carried out on distribution F B* (5 ). In other words, the hypothesis H2 about constant speed of deterioration on an interval normalized service life is checked.
If a mistake of the second sort for an empirical greatest divergence s.f.d. Fz* (5 ) Also F B* (5 ) it appears it is less, than a mistake of the first sort it is possible to accept a hypothesis H1 with the certain degree of confidence. If the
return parity (the hypothesis H2 is fair) takes place, management is transferred to the block 11 for an estimation of character of change of speed of deterioration.
CONCLUSION.
1. Operating experience, literary data testify to necessity to show care at use of criterion not excess of speed of deterioration of preceded value, and a diagnostic parameter - maximum permissible size. A principal cause to that is not the account of stochastic character of process of deterioration TR.
2. Methods of classification of retrospective data on independent groups of homogeneous data and estimations of reliability of the assumption of constant speed of deterioration during normative service life are developed. Consistency of data about constant speed of deterioration allows raise objectivity of the forecast of technical condition TR on the basis the guaranteed estimations.
REFERENCES
1. RD 153.-34.0-46.302-00. Methodical instructions on diagnostics of developing defects of the transformer equipment by results of chromatographic analysis of the gases dissolved in oil. M., 2001, p.31.
2. Davidenko I.V. «System engineering of a multidimensional estimation of a technical condition and service highvoltage oil the filled electric equipment». The author's abstract of the dissertation on competition of a scientific degree of Dr.Sci.Tech. Ural state technical university - Yekaterinburg - 2009.
3. RD 34.45-51.300-97. Volume and norms test of the electric equipment. 6 publ. - M.: NC ENAC, 1998.256 p.
