Comparative analysis of the reliability of electric locomotives based on semi-Markov models of restorable systems

Based on the general theory of semi-Markov recovery processes and their application to Markov recovery processes, the applicability of stationary reliability indicators to assessing the reliability of electric locomotives with an asynchronous motor, the functioning of which is considered as a superposition of independent alternating semi-Markov processes, is shown. By calculating the main indicators of system reliability in a steady state, a comparative analysis of the operational reliability of electric locomotives with an asynchronous DC and AC motors, used respectively on the Azerbaijan and Kazakhstan railways, was carried out.


Introduction
The foundations of the theory of semi-Markov processes (SMP) and their applied possibilities are described in detail in the monograph [1], in which classes of semi-Markov processes were considered that simulate the behavior of recoverable systems with fully accessible (unrestricted) recovery and are implemented as a superposition of independent semi-Markov processes with a finite number of states, in particular, a superposition of alternating renewal processes with constraints such as the exponentiality of the original distributions.
In [2], super-positions of both independent and dependent SMPs with a common phase space are introduced as SMPs that simulate the functioning of restored systems with various types of redundancy, a limited and unlimited number of recovery devices (RD). In principle, in this class one can study any of the systems given, for example, in the reference book [3] without additional restrictions such as the exponentiality of the original distributions. As in the well-known work [4], it is noted that the applied capabilities of the SMP are limited for systems with a finite number of states over a long time interval.
An effective way of specifying the SMP is Markov renewal processes (MRPs) -two-dimensional Markov chains, the first component of which (in turn, being a Markov chain) describes the state of the system, and the second fixes the sojourn times of this system in states. The semi-Markov property of the system states is that the residence time in the state depends only on the given (and possibly the next) state and does not depend on the previous evolution of the system. Generally speaking, the physical state of a real system does not possess the semi-Markov property. However, if in a certain way "expand" the physical state { 1 , … , } (the number of possible states of the system at each moment of time), adding to it some vector = { 1 , … , }, 1 ≥ 0, then in many cases it is possible to IOP Publishing doi:10.1088/1757-899X/1021/1/012003 2 achieve that the new extended state { 1 , … , , 1 , … , } will already have a semi-Markov property. This allows us to describe the functioning of a number of restored systems using PMB [5]. The problem of modeling complex technical recoverable systems using semi-Markov processes remains relevant at the present time, being the subject of research by scientists from the union of independent states (UIS) countries [6][7][8] and far abroad [9][10][11][12][13].
In this work, to analyze the reliability of operation of electric locomotives with an asynchronous motor, an approach based on semi-Markov processes and Markov recovery processes is used, the mathematical apparatus for studying which was developed in [1] and was further developed in [2,14,15]. By calculating the main reliability characteristics of restored systems in a steady state (stationary availability factor, mean time between failures and mean time to recovery), a comparative analysis of the operational reliability of electric locomotives with DC and AC induction motors, currently used at the Azerbaijan and Kazakhstan Railways, was carried out.

Processes of Markov recovery
A jump-like homogeneous Markov process can be constructively given by a stochastic kernel that determines the transition probabilities of the Markov chain, which specifies changes in the states of the process, and a non-negative function on states that specifies the parameters of exponentially distributed random variables that determine the residence times in the states of the process between neighboring jumps.
The concept of semi-Markov processes is a generalization of this construction of jump-like processes. The change in the states of these processes is also controlled by a discrete Markov chain called an embedded Markov chain (EMC), and the sojourn times in states between adjacent jumps have arbitrary distribution functions that depend on the state of the process. Thus, for the constructive specification of semi-Markov processes, the initial material is the sequence { , Θ , ≥ 0}, the first component of which { , ≥ 0} forms a homogeneous Markov chain that fixes the state of the process at the nth step (after the nth jump), and the second component Θ ≥ 0 records the time the system is in the state . The two-component sequence { , Θ , ≥ 0} is a homogeneous Markov chain, the transition probabilities of which do not depend on the values of the second component (the residence time in the previous state does not affect the residence time in this state and the change in this state). Such a Markov chain { , Θ , ≥ 0} is called the Markov renewal process. The times Θ turn out to be conditionally independent random variables for a fixed trajectory sojourn of the embedded Markov chain { , ≥ 0} [11].
According to Kolmogorov's extension theorem (see, for example, [16]), a random process with a fixed phase space is uniquely (in a probabilistic sense) determined by specifying all its finitedimensional distributions.
In the case of Markov chains with discrete time and phase space (Y, ), where (Y, ) is an arbitrary measurable space, all finite-dimensional distributions are calculated using the initial distribution 0 ( ),

Semi-Markov processes
When modeling the evolution of refurbished technical systems (RTS), it is of interest to calculate the characteristics not only at the moments of state change, but also at any current moments of time t. With every MRP { , Θ , ≥ 0}, which is given by the right-hand sides (1) and (2), is connected with (at least one) random process with continuous time ( ) with the same as for { , ≥ 0} phase space. In order to associate a single random process with continuous time with the indicated PMW, we define on the trajectory 0 , 1 , 2 , … nested in the given MRP Markov chain { , ≥ 0} family of conditionally independent random variables (RV) , +1 with conditional distribution functions (DF) [1]: ( ). Through RV , +1 the so-called counting process is introduced:

Superposition of independent semi-Markov processes
Consider a system of n independently operating elements (subsystems), the functioning of each of which is described by the SMP ( ) with arbitrary phase space (Ζ ( ) , ( ) ), = 1, ̅̅̅̅̅ (processes ( )) collectively independent, ( ) --algebra of subsets Ζ ( ) . Foe every Ζ ( ) there is a partition The set of values of the vector d for which the system is operational is denoted by 1 , and the set of values at which the system is inoperable is denoted by 0 , i.е. 1 = { : ( ) = 1}, 0 = { : ( ) = 0}. By assumption = 1 ∪ 0 , 1 ∩ 0 = ∅. Let us investigate the reliability characteristics of such a system, in particular, the average stationary (steady-state) residence times of the process ( ) in sets 1 и 0 (mean time between failures and mean time to recover the system, respectively) and the limit value at → ∞ the likelihood of catching the process ( ) in the set 1 (stationary availability factor). The availability factor, mean time between failures and mean recovery time in the stationary mode of the PMP ξ (t) can be determined, respectively, by the following formulas [2]: (1) According to the data on the Azerbaijan Railways (shortly AR) we have (taking into account 365x12x24 = 8760 working hours per year), denoting through 0, ( ) and 1, ( )respectively, the average VR and the average FOT of the -th element of the -th electric locomotive. Since for the considered system with disconnection of elements 1, 1, = 0, ⁄ = 1, , This means that for the considered system with disconnection of elements, the average stationary reliability characteristics of the system are equal to the corresponding characteristics of the elements.
Comparing the average indicators (10) and (11), we come to the following conclusion: although the mean time between failures ̅ 1 for DC electric locomotives of AR is more than for AC electric locomotives of KR, by about 140 hours, their average availability factor ̅ Г is less 2% and the average recovery time is 13 hours longer, which indicates the greater reliability of AC electric locomotives in comparison with DC electric locomotives.

Conclusion
The operation of electric locomotives with an induction motor based on data on unscheduled repairs associated with violations of individual elements of an electric locomotive can be represented as a recoverable system with disconnection after failure of one element (or several) of all remaining elements. Under the conditions 1 and 2 for the obtained semi-Markov process of restoration, the formulas for calculating the reliability indicators in the steady state (stationary) mode are applicable. To calculate these indicators, simplified formulas are obtained in the work, which can be used for a comparative analysis of the reliability levels of various types of complex systems with shutdown in case of failures, having non exponential distribution functions of the residence times in states.