Mechanics, Materials Science & Engineering, December 2016
Process Modeli
ISSN 2412-5954
ystem with a Help of 10
Markov Discrete Chain
Victor Kravets1,a, Vladimir Kravets2, Olexiy Burov3 1
National Mining University, Dnipro, Ukraine
2
Dnipropetrovsk National University of Railway Transport, Dnipro, Ukraine
3
Jack Baskin School of Engineering, University of California-Santa Cruz, CA, USA
a
prof.w.kravets@gmail.com DOI 10.13140/RG.2.2.34948.32643
Keywords: smart house, Markov discrete chains, possible states, transition probabilities matrix, transition costs matrix, mathematical expectations of transitions costs, cost of Markov random process.
ABSTRACT. Method for evaluating economic efficiency of technical systems using discrete Markov chains modelling illustrated by the system of "Smart house", consisting, for example, of the three independently functioning elements. Dynamic model of a random power consumption process in the form of a symmetrical state graph of heterogeneous discrete Markov chain is built. The corresponding mathematical model of a random Markov process of power consumption in the "smart house" system in recurrent matrix form is being developed. Technique of statistical determination of probability of random transition elements of the system and the corresponding to the transition probability matrix of the discrete inhomogeneous Markov chain are developed. Statistically determined random transitions of system elements power consumption and the corresponding distribution laws are introduced. The matrix of transition prices, expectations for the possible states of a system price transition and, eventually, the cost of Markov process of power consumption throughout the day.
Introduction. The issue in question relates to the problem of smart house engineering for establishing controlled process of energy usage. In this research area, there are such works as [1-3]. In this problem, the leading role belongs to establishing a mathematical model of random processes of energy usage by essential appliances, the model being adequate to physical picture. In order to establish a mathematical model for the problem in question, fundamental results of probability theory and mathematical statistics [4-6], operational research [7], especially Markov random processes theories [8,9] are used. Exploring dynamics of energy usage process in residential house implies also working out an appropriate dynamic model, process scheme, method for modeling process, computation algorithm and an appropriate software package. Dynamic model for energy usage process. Typical residential house (set of rooms) is being explored. Its essential services are provided with a help of several electrical appliances. It means that a technical system consists of independently functioning subsystems (elements). For example, let us suppose for the sake of simplicity the in the considered household there are three ( ) elements: refrigerator ( ), microwave oven ( ), light source ( e3 ). It is to mention that generalizing the
10
The Authors. Published by Magnolithe GmbH. This is an open access article under the CC BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/
MMSE Journal. Open Access www.mmse.xyz
84
Mechanics, Materials Science & Engineering, December 2016
problem with bigger number of elements ( formulation volume.
ISSN 2412-5954
) is trivial and is related only to the mathematical
It is evident that the energy usage process is periodical, its period ( ) being set objectively as twentyfour hours. Thus, for the initial approximation it is logical to set the step volume ( ) of discrete time as equal to one hour ( t 1 ). That is, discrete moments of time when random system transition from one stage to the other one are found as: where
It is to mention that during the period
there should be no more than one switching on or off for must tement of technical problem being solved, it is appropriate to select the volume of step depending on discrete time , i.e., . Step volume grounding constitutes a problem apart being solved depending on a particular technical problem, either heuristically or with a help of mathematical estimation [10]. In We assume that each of three elements can be in one of two possible states: on-mode denoted with
;
off-mode denoted with
.
In a process of independent functioning of each th element of engineering system in discrete moments of time the following random transitions are taking place: On-mode is kept, that is .
This random event is defined with the probability
.
From on-mode to an off-mode, that is .
This opposite random event is defined with a probability
and, consequently,
From off-mode to an on-mode, that is . This random transition is defined with the probability
. An off-mode is kept, that is
MMSE Journal. Open Access www.mmse.xyz
85
Mechanics, Materials Science & Engineering, December 2016
ISSN 2412-5954
.
This opposite random event is defined with the possibility of non-recovery
Quantitatively, the probabilities of random transitions , result of statistical processing the possessed experimental data:
Here
a number of transitions of
element on
a number of transitions of
element on
a number of transitions for
,
, and, consequently,
,
are found as a
stage from on-mode to an on-mode; stage from on-mode to an off-mode;
element on
stage from an off-mode to an on-
mode; a number of transitions of a number of cases when
element on
stage from an off-mode to an off-mode;
element at the beginning of
stage is found in an on-
mode; is a number of cases when
element at the beginning of
stage is found in an
off-mode. Here at the beginning of
th and the following
stage, there are evident equities:
It is to mention that the volume of the main entity or the survey scope, i.e., the number of days when the genuine experiment was conducted, does not depend on discrete time element number and is a defined, whole number, constant:
MMSE Journal. Open Access www.mmse.xyz
86
Mechanics, Materials Science & Engineering, December 2016
ISSN 2412-5954
In the process of energy usage by engineering system in general, each of three independently functioning elements is passing randomly from an on-mode to an off-mode and vice versa. function [6] in quantity found as
:
The sequence of random events related to the abrupt transitions of engineering system throughout the mentioned eight possible discrete states in defined discrete time moments is a random process which happens in Markov discrete chain [8, 9]. To illustrate the dynamics of engineering
Fig. 1. Symmetric graph of probable states.
MMSE Journal. Open Access www.mmse.xyz
87
Mechanics, Materials Science & Engineering, December 2016
ISSN 2412-5954
Here the number of peaks and curves (transition probabilities) on the graph is found as The provided graph and an appropriate nonhomogeneous Markovian discrete chain constitute the dynamic model of energy usage process for the examined engineering system. Mathematical model for Markov random process of energy usage. Mathematical model for Markov random process of energy usage is made in conformity with provided above graph of conditions of nonhomogeneous Markovian discrete chain and has the form of recurrent matrix formula:
,
Here a column matrix of probability engineering system on the following
of eight conditions
for
stage is defined according to the column matrix stage and square matrix
of
transitional probabilities. It is to mention that the iteration process can begin with any reliably known step : . The elements of transitional probabilities square matrix relevant to the curves of graph of Markov discrete chain's states, are defined as transitional probabilities with the , , , , i.e.,
MMSE Journal. Open Access www.mmse.xyz
88
Mechanics, Materials Science & Engineering, December 2016
where on each
ISSN 2412-5954
step of discrete time the following conditions are met: ,
and also
,
i.e., the totals of matrix columns
are normalized.
Thus, a random process of energy usage by the examined engineering system is modeled with the nonhomogeneous Markovian discrete chain described with a recurrent matrix formula represented in detail in the following way:
.
Energy usage In a similar way, with a help of statistical method, random prices of energy usage by on th step with a time period t (depending on discrete time) are found:
Here
is a cost of energy usage by
from on to on state statistically found as is a cost of energy usage by from on to off state statistically found as
th element
th element in cases when there are random transitions ; th element in cases when there are random transitions ;
MMSE Journal. Open Access www.mmse.xyz
89
Mechanics, Materials Science & Engineering, December 2016
is a cost of energy usage by to on state statistically found as
th element in cases of random transitions from off
;
is a cost of energy usage by to off state statistically found as
ISSN 2412-5954
th element in cases of random transitions from off
.
Random transitional chains of turning energy supply on constitute the following discrete laws of distribution:
and off
house
. Respective mathematical expectations of discrete random transitions for energy usage on time interval depend on discrete time and are found as:
The elements of square matrix possible states
for are found according to a worked out square matrix of
transitional probabilities
and are the following:
c15 k
r1 k r2 k r3 k
c1 k
c2 k
c3 k
;
c17 k
r1 k r2 k r3 k
c1 k
c2 k
c3 k
;
MMSE Journal. Open Access www.mmse.xyz
90
Mechanics, Materials Science & Engineering, December 2016
ISSN 2412-5954
c23 k
r1 k r2 k v3 k
c1 k
c2 k
l3 k
;
c43 k
v1 k r2 k r3 k
l1 k
c2 k
c3 k
;
c46 k
v1 k r2 k r3 k
l1 k
c2 k
c3 k
;
MMSE Journal. Open Access www.mmse.xyz
91
Mechanics, Materials Science & Engineering, December 2016
MMSE Journal. Open Access www.mmse.xyz
92
ISSN 2412-5954
Mechanics, Materials Science & Engineering, December 2016
ISSN 2412-5954
h different states. Mathe are found with a help of mathematical expectations for energy usage transitional costs for each of three elements [6]:
or with a help of t
MMSE Journal. Open Access www.mmse.xyz
93
Mechanics, Materials Science & Engineering, December 2016
ISSN 2412-5954
The obtained mathematical ex possible state constitute a row matrix:
,
corresponding to a column matrix level. Mathematical expec states depending on discrete time
are found as:
Then, the formula
is found with a help of
The total value of Markov energy usage process on finite set of steps hours is estimated as a total of
, constituting twenty-four
Summary. The mathematical models of stochastic processes of failures, recoveries of a broad class of systems described by discrete asymmetric Markov chains were developed. The algorithms to assess the economic efficiency of systems modeled by discrete asymmetric Markov chains are proposed. Mathematical models of stochastic processes and algorithms for evaluation the economic efficiency of systems are presented in matrix dorm and adapted to use of computer technology. Generalization of the offered algorithm for bigger number of elements in the system is a trivial one. The difficulties related to the awkwardness of the required mathematical operations are overcome with a help of advanced software development and modern computing hardware usage. References [1] G. W. Hart, Nonintrusive appliance load monitoring, Proceedings of the IEEE, vol. 80, no. 12, Dec. 1992, pp. 1870-1891.
MMSE Journal. Open Access www.mmse.xyz
94
Mechanics, Materials Science & Engineering, December 2016
ISSN 2412-5954
[2] M. Weiss, A. Helfenstein, F. Mattern, T. Staake, Leveraging smart meter data to recognize home appliances, Proceedings of the IEEE International Conference on Pervasive Computing and Communications (PerCom 2012), Lugano, Switzerland, March 2012, pp. 190-197. [3] Alan P. Rossiter (Editor), Beth P. Jones (Editor), Energy management and efficiency for the process industries, AICHE Inc., John&Sons Inc., Hoboken, New Jersey, 2015, 400 p., ISBN: 978-1118-83825-9. [4] B. Ayyub, R. Mccuen, Probability, statistics & reliability for engineers, CRC Press, New York, 1997, 663 p. [5] A. Birolini, Quality and Reliability of Technical Systems: Theory, Practice, Management, Edition Springer, 2004. DOI: 10.1007/978-3-642-97983-5. [6] V. Kravets, Vl. Kravets, O. Burov, Reliability of Systems. Part 1. Statics of Failures. Lap Lambert Academic Publishing, Omni Scriptum GmbH & Co. KG., 2016. [7] E.S. Ventcel', Issledovanie operacij [Operations research], Moscow, Sovetskoe radio Publ., 1972, 552 p. [in Russian]. [8] E.S. Ventcel', L.A. Ovcharov, Theory of random processes and its engineering application, Moscow, Nauka Publ., 1991, 384 p. [9] V. Kravets, Vl. Kravets, O. Burov, Reliability of Systems. Part 2. Dynamics of Failures. Lap Lambert Academic Publishing, Omni Scriptum GmbH & Co. KG., 2016. [10] V.A. Kotelnikov, R.A. Silverman, Theory of optimum noise immunity, New York, Dover Publ., 1968, 140 p. [11] Victor Kravets, Vladimir Kravets & Olexiy Burov (2016). Matrix Method for Assessing Economic Efficiency of Systems Simulated with Asymmetric Markov Discrete Chains, Automation, Software Development & Engineering Journal, ISSN 2415-6531
Cite the paper Victor Kravets, Vladimir Kravets, Olexiy Burov (2016). System with a Help of Markov Discrete Chain. Mechanics, Materials Science & Engineering, Vol 7. doi:10.13140/RG.2.2.34948.32643
MMSE Journal. Open Access www.mmse.xyz
95