CC BY
63Journal of Siberian Federal University. Engineering & Technologies, 2019, 12(3), 314-322
yflK 621.315.1:536.755
Application of the Entropy Theory
for Electric Connection Schemes Reliability Analisys
of Traction Substations
Vladislav G. Belov, Alexey V. Bobrov and Vladimir A. Tremyasov*
Siberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia
Received 21.12.2018, received in revised form 18.01.2019, accepted 21.02.2019
Entropy criteria of traction substations of110-220 kV schemes' unreliability are considered in this article in the conditions of indeterminacy of schemes' conditions. It is shown how the application of entropy estimates during scheduling reconstruction of traction substations allows to estimate effectiveness of measuring to increase non-failure operation and substations' schemes electrical connections availability.
Keywords: traction substation, reliability, entropy estimates.
Citation: Belov V.G., Bobrov A.V., Tremyasov V.A. Application of the entropy theory for electric connection schemes reliability analisys of traction substations, J. Sib. Fed. Univ. Eng. technol., 2019, 12(3), 314-322. DOI: 10.17516/1999-494X-0138.
Применение теории энтропии для анализа надежности схем электрических соединений тяговых подстанций
В.Г. Белов, А.В. Бобров, В.А. Тремясов
Сибирский федеральный университет Россия, 660041, Красноярск, пр. Свободный, 79
В статье рассматриваются энтропийные критерии ненадежности схем тяговых подстанций 110-220 кВ в условиях неопределенности состояний схем. Показано, что применение энтропийных оценок при планировании реконструкции тяговых подстанций позволяет оценить эффективность мер по повышению безотказности и готовности схем электрических соединений подстанций.
Ключевые слова: тяговая подстанция, надежность, энтропийные оценки.
© Siberian Federal University. All rights reserved
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). Corresponding author E-mail address: belov_v2012@mail.ru
Introduction
The main consumer of the electric power on railway transport is the alternating current electric train (ACET). Therefore, the functions of the traction substation (TS) are transformation of the electric power from one voltage level to another and power distribution through contact network by means of a feed line, with its subsequent transfer on the ACET current collector. Besides this ACET, traction substations implement the electric power supply of railway stations, locomotive and carriage depots, communication devices and automatic lock-outs, adjacent houses and the industrial enterprises. The violation of power supply and distribution, in turn, involves violation of the schedule and idle time of trains, shutdowns of the adjacent enterprises operation and the electric power supplyofnon-traction consumers, etc ii].
The cemoaratioe anaCysis and abilities sssnssmeni ef tlea existino mothods chows the necessity to devaiopthemeshod sf Tt'c Sunciioeong cebiabtlityandeifettivtnessanaeysis as a complex technical system with a large numberof sentes, besing nnlraditionalmethoOts censlderingonfeaCuresvf construction and functioning of this system.
Thepue^aeef she ircut isthn elevelopmenc ef tna metiiod besed on entropy assetsment and its application reliability assessment of electrical connections traction substations of 110-220 kV scCemet.
Thefottowinptcsksarnsnlnedtn ale nsssent crtitiel
• the research of entropy estimeteeuiing possibilsty forCheanalsasofTP selisbtlity;
• the calculation of entropyinOccetort ofrflibSvlityeorvariousTPl 10-220 kVsoXeme s;
• scheduling the reconstruction of TP 110-220 kV schemes.
Application of an entropy theory fou fhe eleetric insicllati<)ns i-atyel:il^lt:j' anaiys1
The nets —^asv ol vng elements tnnyols, sSnteci depecde ontoial ounsber cS tlns^ ekments
entcrinn thiso— andeheie spiritc gi—Uty in this si Tpecific \neigssjn ey valid ekmsnts utlanyarc defined by the probabllity to receive an element of this type at random sampling of this set.
There is the entropy of Ulisset si elsmentsSyierminedbyexpression )V]seooeslnyo0ernetil^sif^^ the variety measuremen^y! sei:
H(A) = -£p, leg- p, , (1)
i=i
w-ereH(A1 - the enScopy oS the setA \pi s s^t^sget^s^^nent)<ll^itiicl(jf anslement t'ajrannTm sampling
n
ufa set uf A, "Rjt^r^g ismyde wiA all etemeseu ol0 tge acmpieSe sst eeel elements ^tp. =e t.
,=t
The entropy defined on(1) hasthe followingproperties:
• the entropy reaches in zero when emergence of one element of a set is reliable, and others is impossible;
• at the given number of various elements the entropy has a maximum when emergence of these elements has thesameprobability;
• the entropy increases at increasing innumber of elementsinaset;
• the entropy has property of additivity: when sets of independent elements (events) unite in one,
their entropy is summarized, and the sum is an entropy of the integrated set. The entropy can be used as reliability assessment of a system by definition of indeterminacy measuring of system conditionsin the quantitative lookthrough its probabilistic characteristics [3].
Entropien for the difficult sets (for the difficult sets of their entropy) are calculated by means of tables of a highe r traesc endental function r|(p) = -p log2 p,
Undeo union oftwo sets el and B with possible elements, an,..., an and bu... ,b m is understooda set (A, B) whichelements oepoe sent nil pnssible onmbtnationsof tlie a, anO bj elements which number is entail t o nvq
Ddsenating ^^i^oiLijijli/so the emergence protiabffity oil0 the aibi elemnetin a set (A, B). The definitioe il) chncnthyts an eatropy of a bet lei, A):
n m
Hso,Bt = -'Z Iptr3S2/ts.
p=1 (=1
Tlia untted entropy i[ equef to tie sum of items' entropies at uniting of independent events sets (dement), artiees):
HS(A , B) = H (A) + H (B). (2)
We rscdvr tht lacioiaedesi ia ea/e ei uniiing <ie(^nnient sets:
H(A, B) = //(14) + SA(B/A) = H{Bi + H{A/B), (3)
Anrs arm H3AhB), HtB/Ai - ^cmer^jE1^ pogthikmaf eetaof iei o:f oe;ts A eMfi.
Tgp eoneisinnai nneorpy efti cet of B provkidy eAh i n a set of oh she at, rlemenf appeared, is called the valen dnteomijt(r<^ by tbe equatiun:
eo (*/«,) = jp -.i «,.)is
jo
wltere P(/j /a ¡t - tise cqndttid sal oeebatsility oh s mnrgeice qf the bj element a set of B if in a set A the cst doment appealed.
Theaverageconditionalentropy H(B/A) isdefinedasanexpectationoftheconditionalentropy:
H(B/A) = fj p t HlBI om)
i=i
n m
H(B/A) = t t Pin(bj /1). (4)
i=1 j=1
Theinequality iscalledthemain inequality of Shannon:
H(B/A)<H(B). (5)
From(5)and(3)followsthat H(A,B)<H(A)+H(B), H(A,B,C)<H(A)+H(B)+H(C). Thus, the entropy is a measure of the elements variety of any sets. If these elements are casual events or conditions of any physical or technical system, then the entropy is a measure of indeterminacy
of a condition of this system. As shown in [4], the entropy can become also a measure of unreliability ofasystem undersomeconditions.
If the task of all system entropy detesmination in general is set, the information measuring for a set of elements n (everyone from which has two states) can be determined by expression:
e
H = "X (Pi loi2 Pi + l-r 2 q), (6)
wOerc />, o- tilt prabalility of tionsfeil^e^ opeeation; <7, - tho probability of /-element failure of the system.
Expression (6) reflects existence of the general measure of information or the generalized entropy.
Statistical data on behavior of a system are necessary to be used in this formula: the number of failures and restitutions, a response time in operating condition, in the condition of failure, etc.
Any power supply system or traction substation which is exposed various accidents and devices of relay protection and false drops of automatic relaying protection equipment (ARPE) can be presented by the following scheme (Fig. 1). I) is a set of potential damages and accidents; L is a system of couplings and conditions in electric installation (EI); E is a set of conditions of EI; R is a set of possible actions of ARPE.
According to [5] the following condition is satisfied for the specified sets:
H(y)>H{Dl + H{RID)-H(R), (7)
where H(E) - the entropy of the conditions set of EI; H(R), H(D) - the entropie s ofs ets of Rand D; H(R/D) - the average conditional entropy of sets of R at the influence determined by the managing directorfrom a gre at number of D.
Inequality (7) is a formula of a law of requisite variety [5]. According to (7) entropy of conditions of H(E) has amiuimum,andthis minimum wül bnthr leastif-R iethe emgle-vahierl fnncüon from D, i.e.ARyE has fault-freeoperation:
nr(£) = HOPt -ewnui Hrenf-CR/D^O. (8)
It is posstbrefoeraim-hatrneHinimumvaiuu of an enheury ofsteter H(Ef canUeecduocd concerning auentrrjiy 0-indignvticos sitH(D- 0wly Oue to true rame inoreaec una-entropyofauttons of ARPE. Tlit; eftroue of n sef of souditions of EI HiEO is defineU bothits nonnal operaciona!, and the criticalcoaditsons endis cmeasoreof inUotermivacyoacopdátiont of EI. Therefore, (S) f^ the expres-
Fig. 1. Relations between sets of D, Eand R
sio nfor minimum ^i^i^el^uibility inl^ic;hc^iaiibe provided at the given set of accidents D and a set of the chosen iefluences R ARPEin of their non-failure operation.
The expression (7) allows to estimate a minimum of unreliability of the automated and operated EI taking into account a set of damages, a variety of the revolting influences of a variety of actions and seliabHity of rtsARPE,bocaost Che entropy of H(R/D) is a measure of unreliability of actions of ARPE. Probabilities of element's emergences of a set E can be connected with probabilities of EI ele-monesiibding in vaeeeusseetef: normal (admissible) pi and the emergency q. Then the entropy will beteadiness measutement, measureng elie indeterminacy to find EI in operational or non-operational stota ehleselements.
As the availability quotients of electric equipment are close to 1, the value of Hq(E) is close to zero and its application is limited. Having determined probabilities of P and Q for all EI, it is possible to consider/f/^JesenunintersupSad ope retkon measure.
Tie psobabilities ofelemonts'a^earance in set E can connected with probabilities of the failure evesids Ceodeng te nen-operational conditions of Q(t), and non-failure operation of P(t) during the given trme ob e InShis eatethe enCrepy^n^ wiit be non-failure operation measurement, measuring the tnheterminacstoaet nrn-failure op e^i^l^ien or any EI's failures.
The entropy HQ(E) is the function of time. HQ(E) is close to zero due to high probability of non-failure operation by consideration of an infinitesimal interval of time. HQ(E) grows increasing the considered interval of time due to increasing in probabilities of failures and accidents and its maximum comes with a efehueocaofiíUhlses eloseto P
if in EI there are noC —m efement's failures leads to failure of all installation or to accident, then Savlwg a amiecteh eadcopysi ze oolywith two probabilities of Pa and Qa = 1 - Pa, where Pa - the probability of fault-free operation, and Qa - the probability of failure emergence, we will receive a fail-safety measure HQ(E^.
The probability of elements emergence of a set D can be connected with probabilities Q(t) of vari-iE^ele ma eCs fartum i nhsst allation or probability of lack of any failures P(t).
ahecnlroppaiIda(IDlwilt benefineil as an entropy or an indeterminacy measurement of damages and failures, and the entropy of indignations [6].
^ni^r^j^i^i; oi rerdinrss and uninterrupted operation of one element H^EJ = Hq(E) and
Hq(R) = n(r) + n(q), (9)
where p is availability quotient; q - idle time coefficient, here p + q=1.
The equation (10) is in case of one element for assessment of an entropy of non-failure operation Hq(E) and fail-safety of HQ(E„)
Hq (E) = n(P) + n(Q), (10)
where P(t) + Q(t) = 1, P(t) - the probability of non-failure operaion during t time; Q(t) - failure probability during t time.
In case of a Poisson stream of failures:
P(t) = e-a,
where a - theaverage numberof failuresduring t time; in case of the simplest stream a = It.
There are various methods of assessment of reliability of power supply systems. However many of them were developed for a research of reliability of electric installations at design stages [4]. The analysis of the existing methods shows that assessment of system's reliability of traction power supply under operating conditions and also in the course of reconstruction is expedient to use entropy criteria when scheduling versions of schemes of traction power supply.
Example. Schemes of systems of traction power supply of 110-220 kV are provided on Fig. 2.
The first scheme (Fig. 2a) is more prime and cheap. An alternative for this scheme is the scheme with application of more expensive, but also more reliable gas-insulated switches (Fig. 2b) instead of cheap one, but less reliable devices OD-KZ [7].
The scheme "Two Blocks with OD-KZ, and a Non-automatic Crossing Point from Lines" is forbidden by the modern norms of technological projection of PS [8] to application. However, it often meets on traction substations, and still is on stream for a delivery of contact network of the railroads. It is necessary to use alternating-current vacuum switches with the spring drive without oil isolation accordingtothe requirements establishedbytheowner of infrastructure[9],both onanain constructed, andonthe reconstructed ianenrdevices of traction power supply.
Indicators of reliability of settlement elements of schemes are given in Table 1.
Estimating effectiveness of these actions by means of entropy indicators [4, 10]:
• HQ(EJ is an entropy of fail-safety;
• ng/Ot) is antntrooyofannienafft tr;
• Hq(R) is an entropy of actions of ARPE and a control system.
It is necessary to estimate those values to calculate presented indicators:
a - median number of repayments of consumers;
7V]V -averagotime ofrestitutiong^ delioniy;
O,s -tPe vctueohyooflrciant of idrotimt.
te rTSschemet(Fig.O), calcutoteh byt rableanSftgirtltothnfnue r4)atsp = 1 year, the value of the specifiedindexesnre providedsn Tabft 2.
Fig. 2. a) scheme "Two Blocks with Separators and a Non-automatic Crossing Point from Lines"; b) scheme "Two Blocks with Switches and a Non-automatic Crossing Point from Lines"
Table 1. Indexes of reliability of the capital equipment of 110 kV [7].
Equipment X, 1/year Т„ hours Tpl, hours/year
The transformer, damage with short circuit (s.c.) 0,014 70 28
Disconnector, damage with s.c. 0,01 11 8
Separator, damage with s.c. 0,01 3,5 10
Short-circuit-maker, damage with s.c. 0,01 6 6
Gas-insulated switch 110 kV, damage with s.c. 0,2 30 25
Power line 35-220 kV 0,81 8,4 12
Note: X - intensity of equipment failures; Tr - restoring time; fpl - delivery break time because of scheduled repairs.
Table 2. Indexes of reliability of traction substations (Fig. 2)
Scheme by the Fig. 2 a, 1/year Ttav, hours qab
a 0,8340 0,1325 1,21 0,1810-4
b 0,2125 0,0229 5,32 0,1410-4
The TS failure will be considered as the complete short and long periods of lost power. The reservation is carried out in case of refusal only one line of 110 kV. It means that the system will pass to a delivery from other line (reserved) in the absence of tension on a base line of a delivery, in the automatic mode.
For the scheme in Fig. 2b we will consider calculations of entropy indicators in more details (index Q from HQ(E) and other indexes we will lower for record reduction). From Table 2 for the scheme (Fig. 2b) ofa = 0.0229,then by definition
H(Ea) = itfe"0) + Л(1 -e~a) =r|(0,9774) + r|(0,0226) = 0,156,
H(D) = ¿Я(£>0 = £ [ti(e"A-'p) + [TI(1 - e)] =h(e"081) + W - e-0'81)] + i=i i=i
+ We41'014) + [rid " е"'014 XI + M е-0'") + [Л(1 +
+[ц(е ~0-2)+^T1a-"-0^)] =1 ¿602-, The entropy of actions of relay protection of a source
h (R) = n(QSd)+n(i - Qsd), где Qsd - the probability of a source shutdown
k
i=n
where qi - the probability of findidg eleme nit in arepairstate during the year.
Qsd = 0,0041.
H(R) = n(0,0041) + n(1 - 0,0041) = 0,038 .
Table 3. Indexes schemes' reliability
Scheme according to Fig. 2 H(Ea) H(D) H(R)
a 0,541 1,338 0,0398
b 0,156 1,8602 0,038
Entropy indicators were defined by using this technique. Results of similar calculations of these two schemes on the Fig. 2a are presented in Table 3.
Conclusions
It is possible to draw the following conclusions analyzing the results given in Table 3:
1. The scheme in Fig. 2b is the best by entropy estimates in terms of fail-safety. Entropy indicators are more sensing and allow to judge a condition of the traction power supply system, and its structural reliability.
2. It is possible to estimate a share of presence of an entropy of a disabled state in total amount of the obtained information inherent in all structure of a system of traction power supply, comparing indicators. Such indicators are very useful as criterion at problem solving of reconstruction and modernization of traction substations. Thus, the entropy can be used as the size characterizing the level of structural reliability of a system of traction power supply.
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