On application of semi-Markov processes superposition to energy systems modeling

One of the important tasks of the theory of reliability and efficiency of energy systems is the task of creating information systems for managing energy systems and the transition to intelligent management and engineering. The solution to this problem is possible based on the construction of mathematical models relating to various aspects of the structure and functioning of these systems. The paper, using the example of a superposition of independent semi-Markov processes constructed in the works of V.S. Korolyuk, A.F. Turbin, examines the possibilities of using semi-Markov processes with a common phase state space for modeling energy systems; an illustrative example of the application of this approach is given.


Introduction
Semi-Markov processes (SMP) are widely used to build models and analyze systems for various purposes: technical, industrial, energy, information, economic, biological, etc.
The leading world publishing houses Elsevier, Springer, World Scientific have published a number of monographs on the subject in recent years. [1][2][3][4][5].
Semi-Markov processes can be effectively used to build models relating to various aspects of the structure and functioning of energy systems: reliability, efficiency, monitoring, diagnostics, maintenance and forecasting.
In this paper, by the example of a superposition of independent semi-Markov processes constructed in the works of V.S. Korolyuk, A.F. Turbin and their apprentices [6,7], the possibility of using semi-Markov processes with a common phase state space for energy systems modeling is considered.

Description of independent semi-Markov processes superposition
Let us consider the semi-Markov model of independent SMPs superposition, following to [6,7].
The system consists of N independent elements (subsystems), functioning of each of which is described by Vector process (t) characterize the system's states at the instant t by its element's (subsystem's) up or down states. The set of possible (t) process values is the set of binary vectors In [6,7] a superposition of independent processes i =1,N was constructed, and it was shown that the average stationary time to failure T1, the average stationary renewal time T0 and the stationary availability factor Ka can be determined by the following formulas:  The paper [7] shows that the formulae (1) -(3) can be used for immediate calculation of reliability indicators of various typical structures and also, using these formulas and step-by step structure merging algorithm [7], for reliability characteristics estimating for the systems with high number of elements.
Using the formulae (1) -(3) one can also take into account the presence of unloaded reserve, inspection, maintenance, availability of time reserve, etc.
To do this one should find the appropriate expressions for T1, T0 of subsystems elements, e.g. from [8 -11] and use them in the formulae (1) -(3).
The paper [7] represents approximate formulae for T1, T0, Ka, valid under the condition of system's elements (subsystems) rapid recovery. Using the semi-Markov superposition model built in [6,7] and the embedded Markov chain stationary distribution found there one can also find other characteristics of the system including that determine the effectiveness of system's operation.

Example of electric energy system modeling
As an example to illustrate the application of the formulae (1) -(3) to model the electric energy systems let us determine the reliability characteristics of the electric energy system with the structural diagram represented in the Figure 1 and the elements reliability characteristics represented in the Table 1.   Finding the characteristics of the system considered with the aid help of the formulae (1) -(3) is carried out according to the following scheme: 1. Parallel elements 3, 4, and also 6, 7 are merged to one element.
2. Sequential elements 2, 8 and the result of the merging of 3 and 4, as well as 5, 9 and the result of 6 and 7 merging are merged to one element.
The same result for of the system considered is obtained with the help of the structural functions method [10,11].

Conclusion
The paper shows the possibilities of semi-Markov processes with common phase space of states application to model construction and operation analysis of energy systems using an example of independent SMP superposition. This approach lets to avoid of assumption on exponential distribution law and also take into account inspection, maintenance, etc. in the model of system. Unlike some other methods for finding the reliability and efficiency characteristics of energy systems, it allows to find not only probabilistic characteristics of the system, but also temporary ones. To solve the dimension problem one can use the algorithms of asymptotic and stationary phase merging of the systems [4 -7].