ISSN 1068�3712, Russian Electrical Engineering, 2011, Vol. 82, No. 11, pp. 592–595. © Allerton Press, Inc., 2011.Original Russian Text © A.B. Petrochenkov, S.V. Bochkarev, A.V. Romodin, D.K. Eltyshev, 2011, published in Elektrotekhnika, 2011, No. 11, pp. 20–24.

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INTRODUCTION

Operation of electric power energy systems is acontinuous process that needs regular scheduled con�trol that influences an object or constituent elementsthereof and provides operating conditions and trou�ble�free running. In addition, a high level of reliabilitymust be reached with an adequate profitability.

Operation reliability of electrotechnical objectsand their constituent electrotechnical equipment(EE) can be achieved when two main problems aresolved [1]: (1) ensuring normal working regimes ofindividual elements and the object as a whole and(2) predicting individual resources and assigning opti�mum operational regulations. The latter task entailsselection of the servicing system, development of con�trol and diagnostics systems, processing and analysisthe data for obtaining information on performancequality, carrying out various stages of maintenanceaccording to technical and economy criteria, andincreasing the quality of reconstruction. In thisrespect, to provide a given level of reliability and con�trol of the technical state (TS) of an electrotechnicalsystem, which is a very important problem, it is neces�sary to plan and correctly organize the entire processof technical maintenance so as to provide effectiverepair processes during the whole period of use [1–3].

CONSTRUCTION OF AN OPERATION MODEL FOR ELECTROTECHNICAL EQUIPMENT

In contemporary demands for organizing EE sys�tems, the main attention is devoted to the servicingstrategy of technical conditions, which means that alltypes and terms of repair must be stated in the depen�dence of the TS of an object that are determined dur�ing the period of control of the TS [1, 4].

Since electrotechnical equipment generally has afinite (countable) number of possible states in the pro�cess of use and the operating process can be repre�sented as a plurality of discrete transitions from onestate to another, to solve the problems of modeling andplanning of the operation process, it is expedient toemploy theoretical principles of discrete�state Markovprocesses. The corresponding theory has been welldeveloped and allows one to solve a large number ofapplied problems, since it matches allows relative sim�plicity and easy computer formalization of the prob�lem and is reasonably applicable to the real process[5, 6]. In most practical problems, use of the Markovapproximation allows one to obtain solutions theerrors of which are within the limits of accuracy of ini�tial data and mostly do not exceed 3–5% [1, 6].

With such an approach to the solution of problemsof planning and optimization, the operation process ofEE involves plotting a graph of states, construction ofa reliability model, and determination of the TS char�acteristics. In addition, stable regimes and stationarymodels of reliability are usually considered, as are thefinal values of the probabilities that an object is foundin each state [1, 5].

In constructing operation models with differenttypes of TSs, one can indicate a series of characteristicgroups of flows (transitions from one state to another)[1, 3, 7]: the appearance of the faults and disrepairs,the use of different types of TSs, and the outcome thatflows from the state of the TS.

In the general case, when planning and optimizinga TS, the initial information can be given in the formof the probability function of trouble�free operationP(t) or fault Q(t), the density distribution of faults f(t),or the failure intensity λ(t) [1, 6].

The structural model of the operation process as aMarkov process (Fig. 1) is represented in the form ofan oriented graph G with a finite value of vertices N

The Planning Operation Process of Electrotechnical Equipment Using the Markov Process Theory

A. B. Petrochenkov, S. V. Bochkarev, A. V. Romodin, and D. K. EltyshevReceived October 21, 2011

Abstract—Problems are considered of using the Markov process for managing and planning the operation ofelectrotechnical equipment using a servicing strategy relevant to the technical state. Methods for determiningindices of operation effectiveness are described. The problem of the process optimization is formulatedaccording to the chosen criterion.

Keywords: electrotechnical equipment, Markov process, reliability, maintenance.

DOI: 10.3103/S1068371211110113

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THE PLANNING OPERATION PROCESS OF ELECTROTECHNICAL EQUIPMENT 593

that characterize the space of operation states X and acomplex of transition probabilities Pij(t) from state i tostate j (i, j ∈ X, t ≥ 0).

The space of the model states includes X1, a trou�ble�free state (1); X2, the periodic control of the state(2) aiming to find a prefault state; X3, the state ofemergency reconstruction (4), and X4, the state of pre�ventive reconstruction (3).

As the main indices of effectiveness of technical EEmaintenance, it is convenient to use the average life�time of a system in different states ti, coefficient oftechnical usage Kg, and the average expenses on tech�nical use g [1, 5].

The values of EE effectiveness indices are definedby the following expressions [1, 2, 5]:

(1)

(2)

where Pj is the stationary probability for EE being in theworking state i, pij is the transition probability from statei to state j in an unlimited time, Pj are the stationary(final) probabilities of states, and Zij are the resourceexpenses corresponding to the transition from state i tostate j; Zii are the resource expenses per unit time corre�sponding to the ith state of the process. The probabili�ties Pj and pij are related by the system equations

(3)

Since the instants of EE system transitions fromstate to state are generally unknown and the corre�sponding transitions occur randomly (for example, thefaults of elements or the terminations of their recon�struction), obtaining certain characteristics of suchsystems can be carried out using the properties ofMarkov processes with continuous time that have arelatively wide area of engineering applications. In thiscase, one can use the dependences characterizing thesemi�Markov processes [1, 2, 5, 8].

Taking into account what has been said, in the caseof a model which is depicted on Fig. 1, the followingtransitions are possible:

—from state 1 to state 2 with periodicity tk (theintensity of transition is λ = 1/tk);

—from state 1 to state 4 with an intensity of transi�tions that is equal to the fault intensity of equipment(λ14 = λ);

—from state 2 to state 1 with the intensity that isdetermined by the control duration τk(λ21 = 1/τk);

—from state 2 to state 4 of emergency reconstruc�tion (the transition which is caused by detection offailure during periodic control) with the transition fre�quency λ24;

Kg Pi,i Xp∈

∑=

g pijZijPj,

j 1=

X

∑i 1=

X

∑=

Pj pijPi; Pj

i 1=

X

∑i 1=

X

∑ 1; pij

i 1=

X

∑ 1.= = =

—from state 2 to state 3 with the transition fre�quency λ23;

—from state 3 to state 1, the last one being thetrouble�free state, with the intensity which is deter�mined by duration of repair work τpv (λ31 = 1/τpv); and

—from state 4 to state 1 with an intensity that isdetermined by duration of repair work τab (λ41 = 1/τab).

Parameters tk, τk, τpv, τav, and λ and, correspond�ingly, the transition intensities λ12, λ21, λ31, λ41, and λ14can be obtained using the results of experimental stud�ies or tests or they are given by the rules of technical ser�vice and can be taken as the known values. The intensitiesλ23 and λ14 can be obtained using the statistical data, theresults of experimental investigations, or the properties ofMarkov processes with continuous time [1, 8].

The system of differential equations for an orientedgraph of states (Fig. 1) is given by

(4)

This system must be supplemented by the normal�ization condition

(5)

and the initial conditions. For example, the last onescan be written by

(6)

To find the generalized characteristics of the modelin a stable (stationary) regime (at t ∞) when theprobabilities have their final values, system (4) trans�forms into the system of linear equations of the form

dP1 t( )dt

������������ λ12 λ14+( )P1 t( )– λ41P4 t( ) λ31P3 t( );+ +=

dP2 t( )dt

������������ λ21 λ23 λ24+ +( )P2 t( )– λ12P1 t( );+=

dP3 t( )dt

������������ λ31P3 t( )– λ23P2 t( );+=

dP4 t( )dt

������������ λ41P4 t( )– λ24P2 t( ) λ14P1 t( ).+ +=⎩⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧

P1 t( ) P2 t( ) P3 t( ) P4 t( )+ + + 1=

P1 0( ) 1; P2 0( ) P3 0( ) P4 0( ) 0.= = = =

1

2 34

p41(t)

p14(t)

p12(t)

p24(t)

p21(t) p31(t)

p23(t)

Fig. 1. The structural model of the operation process of elec�trotechnical equipment corresponding to the TS strategy.

594

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PETROCHENKOV et al.

(7)

λ12 λ14+( )P1– λ41P4 λ31P3+ + 0;=

λ21 λ23 λ24+ +( )P2– λ12P1+ 0;=

λ31P3– λ23P2+ 0;=

λ41P4– λ24P2 λ14P1+ + 0.=⎩⎪⎪⎨⎪⎪⎧

Solving the system taking account (5) and (6), onecan find the expression P1,2 = P1 + P2 that corre�sponds to the probability of the trouble�free state of

the EE (in the control regime without shutting down)and determines the readiness coefficient accordingto (1)

(8)P1 2,λ12 λ21 λ23 λ24+ + +

λ21 λ23 λ24+ +��������������������������������������� 1

λ12λ41 λ31 λ23+( ) λ31 λ24λ12 λ14 λ21 λ23 λ24+ +( )+[ ]+λ31λ41 λ21 λ23 λ24+ +( )

���������������������������������������������������������������������������������������������������������+1–

.=

The probability dependence of the trouble�freestate on the periodicity tk of the state control at differ�ent values of failure intensity λ, and the value of dura�tion control τk is represented in Fig. 2. The displayeddependence shows evidence of optimal regulation oftechnical operation and, correspondingly, the limitingmaximum value of the readiness coefficient. Thisallows one to plan and control the terms of necessarywork in the process of EE use according to the specificstatement of the problem.

Using Eq. (8), one can draw the following conclu�sions.

—The greatest impact on the value of the readinesscoefficient has the failure intensity λ and periodicity ofstate control tk.

—The maximum value of the readiness coefficientis displaced to larger values of periodicity of the statecontrol with decreasing λ (which corresponds to the

values tk = 1000 h at λ = 10–4 h–1 and tk = 5000 h at λ =10–6 h–1); in addition, the slope of the graph becomessmaller.

—The optimal periodicity of technical service tk isproportional to the increase in its duration τk. Forinstance, at λ = 10–4 h–1 the increase of τk from 10 to100 h results in an increase of optimal periodicity oftechnical service from 700 to 1800 h and the maximumvalue of readiness coefficient decreases by 6%.

—Deviations in the value of the periodicity oftechnical service from the optimum value consider�ably decrease the probability of an object’s readiness,specifically at large values of failure intensity. Thus, atλ = 10–4 h–1 and the given duration of control τk = 100 h,the decrease in tk from 1800 to 1000 h leads to thedecrease in the value of readiness coefficient byapproximately 2%, whereas the increase in tk to 2500 hresults in the corresponding reduction value of 1%.

Using the probabilities Pj that are obtained fromsystem of equations (7) and the known values ofexpenses Zij, the probabilities pij are computed from(3). This allows one to find the optimal parameters ofEE maintenance (the system of technical service)when solving two tasks [1, 2, 5]:

—providing the necessary level of reliability (i.e.,the value of reliability index, particularly the readinesscoefficient) with minimum loss and expense on tech�nical service (direct optimization problem); and

—providing the maximum possible level of reli�ability with limited loss and expense (inverse optimi�zation problem).

Since electroenergetic objects and EEs are a classof complex and expensive objects and their failuresand decreasing reliability (below the acceptable level)may result in serious consequences, the reliabilityindices that are most important are usually consideredto be limitations, while expenses are less importantthan the goal optimization function [1, 5]. Thus, it iseffective so solve the direct problem of optimization

0.99

0.97

0.95

0.93

0.91

0.89

0.87

0.856000500040003000200010000 tk, h

Kg

γ = 10–4 h–1

γ = 10–6 h–1γ = 10–5 h–1

Fig. 2. Dependence of the probability of the trouble�freestate of electrotechnical equipment on the periodicity ofstate control. The solid line corresponds to τk = 10 h, thedashed lane to τk = 100 h, and the dotted line to τk = 200 h.

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THE PLANNING OPERATION PROCESS OF ELECTROTECHNICAL EQUIPMENT 595

[1, 2, 5]. Taking into account expressions (1) and (2),it is written as

min{g} at Kg ≥ (9)

where is the necessary value of the effectivenessindex.

CONCLUSIONS

(1) To solve the problems of planning relevant tothe operation of electrotechnical equipment, it isworthwhile to use the basics of the Markov theory,which allows one to calculate the corresponding effi�ciency indices according to the model of the process ifuse is made of statistical data and experimental inves�tigations and can be used when finding the optimalparameters for controlling an EE taking into accountthe chosen criteria.

(2) In total, the certain problem statement in opti�mization the process of operation is determined by thetype and complexity of an object, the character of itsfunctions, the number and type of possible states, andstrategy of TS, as well. The difficulty in the solution ofthe optimization problem depends first and foremoston the complexity of the TS system itself and on thenumber of optimizing parameters. In addition, inmost cases, special optimization methods should beused for solving the problem.

ACKNOWLEDGMENTS

This work was supported by grant no. MK�2773.2011.8 “Management of the Technical State of

Electrotechnical Objects to Increase the Parameters ofTheir Energetic Efficiency.”

REFERENCES

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2. Barzilovich, E.Yu., Modeli tekhnicheskogo obsluzhi�vaniya slozhnykh sistem (Service Models for ComplexSystems), Moscow: Vysshaya shkola, 1982.

3. Guk, Yu.B., Analiz nadezhnosti elektroenergeticheskikhustanovok (Reliability Analysis for Electric PowerPlants), Leningrad: Energoatomizdat, 1988.

4. Yashchura, A.I., Sistema tekhnicheskogo obsluzhivaniyai remonta energeticheskogo oborudovaniya: Spravochnik(Service and Repair System for Power Equipment:Handbook), Moscow: NTs ENAS, 2006.

5. Mine, H. and Osaki, S., Markovian Decision Processes,New York: AEPCI, 1970; Moscow: Nauka, 1977.

6. Sugak, E.V., Teoriya sluchainykh protsessov. Osnovnyepolozheniya i inzhenernye prilozheniya: Uchebnoe poso�bie dlya vtuzov (Random Processes Theory. GeneralPrinciples and Engineering Appendixes: Student’sBook for Technical Universities), Krasnoyarsk:KFAGA, 2004.

7. Guk, Yu.B., Dolgov, P.P., Okorokov, V.R., et al.,Kompleksnyi analiz effektivnosti tekhnicheskikh resheniiv energetike (Complex Analysis of Efficient DecisionMaking in Power Engineering), Leningrad: Energoat�omizdat, 1985.

8. Korolyuk, V.S. and Turbin, A.F., Polumarkovskie prot�sessy i ikh primeneniya (Semi–Markov Processes andTheir Application), Kiev: Naukova Dumka, 1976.

Kg*,

Kg*