The planning operation process of electrotechnical equipment using the Markov process theory

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  • ISSN 10683712, Russian Electrical Engineering, 2011, Vol. 82, No. 11, pp. 592595. 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. 2024.



    Operation of electric power energy systems is acontinuous process that needs regular scheduled control that influences an object or constituent elementsthereof and provides operating conditions and troublefree 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 optimum operational regulations. The latter task entailsselection of the servicing system, development of control 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 control of the technical state (TS) of an electrotechnicalsystem, which is a very important problem, it is necessary to plan and correctly organize the entire processof technical maintenance so as to provide effectiverepair processes during the whole period of use [13].


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

    Since electrotechnical equipment generally has afinite (countable) number of possible states in the process of use and the operating process can be represented 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 discretestate Markovprocesses. The corresponding theory has been welldeveloped and allows one to solve a large number ofapplied problems, since it matches allows relative simplicity and easy computer formalization of the problem 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 initial data and mostly do not exceed 35% [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 characteristics. 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 troublefree operationP(t) or fault Q(t), the density distribution of faults f(t),or the failure intensity (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

    AbstractProblems 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



    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 troublefree 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 preventive reconstruction (3).

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

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



    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 corresponding to the ith state of the process. The probabilities Pj and pij are related by the system equations


    Since the instants of EE system transitions fromstate to state are generally unknown and the corresponding transitions occur randomly (for example, thefaults of elements or the terminations of their reconstruction), 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 thesemiMarkov 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 = 1/tk);

    from state 1 to state 4 with an intensity of transitions that is equal to the fault intensity of equipment(14 = );

    from state 2 to state 1 with the intensity that isdetermined by the control duration k(21 = 1/k);

    from state 2 to state 4 of emergency reconstruction (the transition which is caused by detection offailure during periodic control) with the transition frequency 24;

    Kg Pi,i Xp=

    g pijZijPj,j 1=


    i 1=



    Pj pijPi; Pji 1=


    i 1=


    1; piji 1=


    1.= = =

    from state 2 to state 3 with the transition frequency 23;

    from state 3 to state 1, the last one being thetroublefree state, with the intensity which is determined by duration of repair work pv (31 = 1/pv); and

    from state 4 to state 1 with an intensity that isdetermined by duration of repair work ab (41 = 1/ab).

    Parameters tk, k, pv, av, and and, correspondingly, the transition intensities 12, 21, 31, 41, and 14can be obtained using the results of experimental studies or tests or they are given by the rules of technical service and can be taken as the known values. The intensities23 and 14 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


    This system must be supplemented by the normalization condition


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


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

    dP1 t( )dt

    12 14+( )P1 t( ) 41P4 t( ) 31P3 t( );+ +=

    dP2 t( )dt

    21 23 24+ +( )P2 t( ) 12P1 t( );+=

    dP3 t( )dt

    31P3 t( ) 23P2 t( );+=

    dP4 t( )dt

    41P4 t( ) 24P2 t( ) 14P1 t( ).+ +=

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

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


    2 34





    p21(t) p31(t)


    Fig. 1. The structural model of the operation process of electrotechnical equipment corresponding to the TS strategy.

  • 594




    12 14+( )P1 41P4 31P3+ + 0;=

    21 23 24+ +( )P2 12P1+ 0;=

    31P3 23P2+ 0;=

    41P4 24P2 14P1+ + 0.=

    Solving the system taking account (5) and (6), onecan find the expression P1,2 = P1 + P2 that corresponds to the probability of the troublefree state of

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

    (8)P1 2,12 21 23 24+ + +

    21 23 24+ + 1

    1241 31 23+( ) 31 2412 14 21 23 24+ +( )+[ ]+3141 21 23 24+ +( )



    The probability dependence of the troublefreestate on the periodicity tk of the state control at different values of failure intensity , and the value of duration control k 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 conclusions.

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

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

    values tk = 1000 h at = 104 h1 and tk = 5000 h at =106 h1); in addition, the slope of the graph becomessmaller.

    The optimal periodicity of technical service tk isproportional to the increase in its duration k. Forinstance, at = 104 h1 the increase of k 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 considerably decrease the probability of an objects readiness,specifically at large values of failure intensity. Thus, at = 104 h1 and the given duration of control k = 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 technical service (direct optimization problem); and

    providing the maximum possible level of reliability with limited loss and expense (inverse optimization 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








    0.856000500040003000200010000 tk, h


    = 104 h1

    = 106 h1 = 105 h1

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



    [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.


    (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 efficiency indices according to the model of the process ifuse is made of statistical data and experimental investigations 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 optimization 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.


    This work was supported by grant no. MK2773.2011.8 Management of the Technical State of

    Electrotechnical Objects to Increase the Parameters ofTheir Energetic Efficiency.


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