=Paper=
{{Paper
|id=Vol-2740/20200400
|storemode=property
|title=Modeling of the TV3-117 Aircraft Engine Technical State as Part of the
Helicopter Power Plant in the Form of the Markov Process of Death and Reproduction
|pdfUrl=https://ceur-ws.org/Vol-2740/20200400.pdf
|volume=Vol-2740
|authors=Serhii Vladov,Yurii Shmelov,Tetiana Shmelova
|dblpUrl=https://dblp.org/rec/conf/icteri/VladovSS20
}}
==Modeling of the TV3-117 Aircraft Engine Technical State as Part of the
Helicopter Power Plant in the Form of the Markov Process of Death and Reproduction==
https://ceur-ws.org/Vol-2740/20200400.pdf
Modeling of the TV3-117 Aircraft Engine Technical State
as Part of the Helicopter Power Plant in the Form of the
Markov Process of Death and Reproduction
Serhii Vladov1*[0000-0001-8009-5254], Yurii Shmelov2*[0000-0002-3942-2003], and Tetiana
Shmelova3*[0000-0002-9737-6906]
1Kremenchuk Flight College of Kharkiv National University of Internal Affairs,
Peremohy Street, 17/6, 39005, Kremenchuk, Ukraine
ser26101968@gmail.com
2Kremenchuk Flight College of Kharkiv National University of Internal Affairs,
Peremohy Street, 17/6, 39005, Kremenchuk, Ukraine
nviddil.klk@gmail.com
3
National Aviation University, Liubomyra Huzara av.,1, 03058, Kiev, Ukraine
shmelova@ukr.net
Abstract. In this work, the actual scientific and practical problem of modeling
the technical state of the TV3-117 aircraft engine as a part of the helicopter
power plant in the form of the Markov process of death and reproduction, the
results of which are applicable in the form of a stochastic network of the GERT
type (Graphical Evaluation and Review Technique) for formalizing the behav-
ioral activities of the helicopter crew in emergency situations. The results ob-
tained allow us to simulate the development of flight situations in the direction
of complication and vice versa. The resulting information model for determin-
ing the failure of the TV3-117 aircraft engine is a graph of the standard devia-
tions of the state vector of the probabilities of its state, obtained as a result of
flight tests, with a graph for its ideal state.
Keywords: Aircraft engine, power plant, modeling, technical state, GERT-
systems, information criterion, Markov process
1 Introduction
The probability of TV3-117 aircraft engine failure at present level of reliability of
the power plants is very small. Engine failure during take-off and climb stages is an
even less likely event, since, firstly, these stages last for a relatively short period of
time, and secondly, an aircraft thorough inspection and inspection of all engines is
carried out immediately before takeoff. However, it is impossible to underestimate,
though negligible, the possibility of engine failure at the specified stages.
For the formalization of the behavioral activity of an aircraft crew in emergency
situations and to simulate the corresponding development of flight situations, models
in the form of stochastic network of GERT type (Graphical Evaluation and Review
Technique), which allow to model the development of flight situations in the direction
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
of complication and vice versa. GERT is an alternative probabilistic network planning
method used in case of organization activities, where follow-up actions may start only
after some previous actions have been completed, therefore allowing for cycles and
loops. There are several possible consequences of an emergency situation, its elimina-
tion, localization and the development of an emergency situation in the direction of
deterioration, so the use of GERT-type networks is advisable.
The use of GERT-type networks in the model of decision support systems will al-
low us to model the prediction of the emergencies development. Here is a transition
from one flight situation related to the failure of the TV3-117 aircraft engine as a part
of the helicopter propulsion system to another in the form of Markov process of death
and reproduction. In order to determine the probabilities of a system state at any given
time, it is necessary to use mathematical models of Markov processes with continuous
time (continuous Markov processes). If two continuous chains of a Markov network
have the same state graphs and differ only in intensity values, then it is possible to
find the limit probabilities of the states for each graph separately [1–5].
The homogeneous Markov process of flight situations development with continu-
ous time can be interpreted as the process of change of states under the influence of
some flow of events – environmental factors. That is, the probability of a transition
can be interpreted as the intensity of the flow of events that translate the system from
one state to another – the influence of environmental factors, factors affecting the
aircraft crew decision making to be professional or unprofessional.
2 Application of Markov process equations for control and
diagnostics of technical condition of TV3-117 aircraft engine
as a part of helicopter power plant and mathematical
description of its failure process
Given the fact that the helicopter has two engines as parts of the power plant, it is
believed that the helicopter is a system that has reliable operation (P ≥ 0.9), which
consists of i units of engines. Let the random operating time of one engine have a
positive probability distribution and does not depend on the state of other units of
engines [6, 7]. In a more general mathematical model of the system, consisting of i
units, which takes into account the relationship between the units of engines, we can
assume [7] that the exponential distribution has a random time τi of joint work before
the failure of one of the available units of engines, that is:
P i t = 1 − e −i t ; (1)
where φ0 = 0, φi > 0 at i = 1, 2, …
In general case Pij(t) represents the probability of the presence at the time t of ser-
viceable j units of engines, provided that at the initial time t = 0 there were i units of
serviceable engines. The first step in diagnosing and predicting the probability of
failure of one of the helicopter engines is to obtain an equation for the transition prob-
abilities of failure of one of the helicopter engines.
The mathematical model under development is the Markov process of failure of
one of the helicopter engines ξt, t ∈ [0, ∞), on many states N = 0, 1, 2, …, in which
transition probabilities Pij(t) = P{ξt = j | ξ0 = i}, i, j ∈ N are presented at t → 0+ as [5]:
Pi,i−1(t) = φit + o(t); (2)
Pii(t) = 1 − φit + o(t). (3)
Jumps in the process of simple engine operating states change ξt are presented on
fig. 1. Assume that at t = 0 the process is in its initial state i. At time τi
P i t = 1 − e −i t ; the process transitions to the state i – 1 etc.
Absorbing state
0 1 2 i– i–2 i–1 i
Fig. 1. Graphs changes in the state of the TV3-117 aircraft engine as part of the helicopter
power plant
The first (reverse) system of Kolmogorov differential equations for transition prob-
abilities in the case of the transition of the aircraft engine TV3-117 as part of a heli-
copter power plant from one state to another takes the form [7]:
dPoj ( t )
= −0 Poj ( t ) ; (4)
dt
dPij ( t )
= i Pi −1, j ( t ) − i Pij ( t ) ; (5)
dt
where i = 1, 2, … with initial conditions Pii(0) = 1, Pij(0) = 0 at i ≠ j.
The second (direct) system of Kolmogorov differential equations for transition
probabilities in the event of a change in the state of operation of the TV3-117 aircraft
engine as a part of the helicopter power plant is:
dPi 0 ( t )
= − Pi 0 ( t ) 0 + Pi1 ( t ) 1 ; (6)
dt
dPij ( t )
= − Pij ( t ) j + Pij +1 ( t ) j +1 ; (7)
dt
where j = 1, 2, … with initial conditions Pii(0) = 1, Pij(0) = 0 at i ≠ j.
According to [8], after performing a series of mathematical transformations, we ar-
rive at a closed expression for the double generating function
z
(1+( s −1)e )
− t
F ( t ; z; s ) = e . (8)
From here and from the definition F(t; z, s) we get:
Fi ( t; s ) = (1 − e−t + se−t ) ;
i
(9)
at i ∈ N.
Relation (9) means that the random operating times of each of the available i en-
gine units are independent of each other; this independence property holds only for a
linear process type.
For applications in the mathematical theory of reliability [6, 7], it is of interest to
find a similar (9) closed integral representation for the generating function Fi(t; s), as
solutions to the Kolmogorov equations (4)–(7) for the failure process of one of the
engines (by summation of the Fourier series), under particular assumptions about the
function φi = φ(i) [8].
In the case of a quadratic type process [8], we have a system of equations:
dF dF d 2 F
= z2 − 2 ; (12)
dt dz dz
dF
dt
(
= s − s2
d 2F
ds 2
)
; (13)
with initial condition F(0; z; s) = ezs.
Thus, the obtained equations for describing the failure of one of the engines show
that the value of φi depends on the intensity of occurrence of the event λ, despite the
type of process (quadratic, polynomial, power-law or Poisson), which makes it possi-
ble to use standard methods for constructing an algorithm for determining continuous-
ly Markov process when simulating the operation of the TV3-117 aircraft engine as a
part of the helicopter power plant.
3 Algorithm for determining a continuous Markov process
when simulating the operation of TV3-117 aircraft engine
We simulate the operation of a helicopter engine, which can be in the following
states: S0 – engine operation in idle mode; S1 – engine operation at rated operation
mode; S2 – engine is serviceable, reconfiguration λ02 < λ21; S3 – the engine is faulty,
repair is underway λ13 < λ30.
Let us set the values of the parameters λ, using the experimental data obtained dur-
ing the flight tests of the helicopter: λ01 – engine work flow at nominal operation
mode (without reconfiguration); λ10 – engine maintenance flow; λ13 – engine failure
flow; λ30 – stream of updates.
The main task of modeling is to obtain a continuous Markov process when simulat-
ing the operation of a helicopter engine during its operation in real operating modes,
that is, determining the time of transitions from one engine operating mode to another
in all possible variants, the parameters of which are set S and λ [9].
The simulation algorithm will have the form shown in fig. 2, where
1 1 1
01 = − ln R; 13 = − ln R; 20 = − ln R;
01 13 20
1 1 1
02 = − ln R; 10 = − ln R; 21 = − ln R.
02 10 21
Start
Data input
λ01 λ30, Tk
S := 0
T := 0
τ01 := … τ30 :=
S=0 No
Yes
Yes S=1 No
Yes τ01 τ02
No τ10 τ13 No S=2 No
Yes Yes
S := 1 S := 2 S := 0 S := 3 S := 1 S := 0
T := T + τ01 T := T + τ02 T := T + τ10 T := T + τ13 T := T + τ21 T := T + τ30
Data output
T, S
Yes T Tk
No
Finish
Fig. 2. The flowchart of the modeling algorithm for the continuous Markov process on the
example of simulating the operation of the aircraft engine TV3-117 as part of the helicopter
power plant
Since it is known that a helicopter engine in flight test conditions can be in four
states: S0 – engine idle, S1 – engine runs in rated mode, S2 – serviceable engine, recon-
figuration is in progress, S3 – engine defective. According to these data, one can ob-
tain a discrete Markov network based on the following possible transition states: S0 –
S1, S1 – S0, S0 – S3, S1 – S3, S2 – S0, S2 – S1, S2 – S3, S3 – S2, in this case, it is necessary
to take into account the probability of the engine staying in each of the above states
(fig. 3).
P11 P22
P21
S0 S1
P12
P31 P 14 P32 P24
P33 S P43 S3
2 P44
P34
Fig. 3. Markov discrete network showing all kinds of changes in the state of TV3-117aircraft
engine as a part of a helicopter power plant during flight tests
According to fig. 3, to find the probabilities of a Markov chain staying in certain
states as n → ∞ (final probabilities), we solve a system of equations that will have the
form:
P11 1 + P21 2 + P31 3 = 1 ;
P12 1 + P22 2 + P32 3 = 2 ;
(14)
P33 3 + P43 4 = 3 ;
P14 1 + P24 2 + P34 3 + P44 4 = 4 .
Deciding which, we get:
− P21 P32 P43 − P31 P43 ( P22 − 1)
1 =
( P11 − 1) ( ( P22 − 1)( P33 − P43 − 1 + P32 P43 ) ) − P21 ( P12 ( P33 − P43 − 1 + P32 P43 ) ) +
;
+ P31 ( − P12 P43 + P43 ( P22 − 1) )
P31 P43 ( P11 − 1) − P12 P31 P43
2 =
( P11 − 1) ( ( P22 − 1)( P33 − P43 − 1 + P32 P43 ) ) − P21 ( P12 ( P33 − P43 − 1 + P32 P43 ) ) +
;
+ P31 ( − P12 P43 + P43 ( P22 − 1) )
− P43 ( P11 − 1)( P22 − 1) + P12 P21 P43
3 =
( P11 − 1) ( ( P22 − 1)( P33 − P43 − 1 + P32 P43 ) ) − P21 ( P12 ( P33 − P43 − 1 + P32 P43 ) ) +
;
+ P31 ( − P12 P43 + P43 ( P22 − 1) )
( P11 − 1)( P22 − 1)( P33 − 1) − P12 P21 ( P33 − 1)
4 =
( P11 − 1) ( ( P22 − 1)( P33 − P43 − 1 + P32 P43 ) ) − P21 ( P12 ( P33 − P43 − 1 + P32 P43 ) ) +
.
+ P31 ( − P12 P43 + P43 ( P22 − 1) )
As a result of solution (14), expressions are obtained for determining the elements
of the state probability vector π(π1, π2, π3, π4) (shows the probability that the engine
will be in the i-th state), which allows, if there are transition probabilities P11…P33 to
obtain prognostic indicators of changes in the state of TV3-117aircraft engine as a
part of a helicopter power plant. As can be seen from the system (14), to determine
the probability of engine failure π4, it is enough to know the transition probabilities
for the other three states, as well as the transition probabilities P34, P43, which can be
obtained as a result of technical diagnostics of the helicopter and other statistical data.
It is assumed that in perfect condition (the engine is flawless), the probability of its
operation in idle mode and in nominal mode is 1.0, that is, P11 = P22 = 1. The proba-
bility of transition from idle mode to nominal mode and vice versa is also 1, 0, since
the operator can at any moment of time freely change its operating modes, that is, P12
= P21 = 1. Reconfiguring the engine does not make operational sense, however, point
changes can be corrected, that is, P33 = 0.1. Consequently, the transition after recon-
figuration to both idle and nominal mode will be 0.9, that is, P31 = P32 = 1 – 0.1 = 0.9.
The probability of engine failure is assumed to be 0.1% – this is due to production
fault or a professional mistake of engineers when assembling the engine structure (P44
= 0.001) [6], and, consequently, the transition probabilities associated with reconfig-
uring the engine after detecting its failure or malfunction is P43 = 1 – 0.001 = 0.999.
The transition probabilities associated with the transition to the engine idling and / or
to the nominal operating mode are zero, since after a malfunction which led to engine
failure, is detected the engine is sent for repair, reconfiguration, etc., i.e. P41 = P42 = 0.
The transient probabilities associated with engine failure during its operation both
in idle mode and in nominal mode are 99.9 % (the defect described above is taken
into account when assembling the engine), i.e. P14 = P24 = 1 – 0.999 = 0.001.
Given the human factor in the maintenance of aircraft engines [10], it is accepted
that after reconfiguring the engine due to professional mistakes by the maintenance
personnel, the probability of failure can be 1 % P34 = 0,01.
Absolute and relative errors of the obtained values of the state probability vector
π(π1, π2, π3, π4) in numerical modeling of the Markov chain and analytical solution
amounted to:
0,9196 − 0,9208
Δπ1 = | 0,9196 – 0,9208| = 0,0012; 1 = 100% = 0,13%;
0,9196
0,9196 − 0,9208
Δπ2 = | 0,9196 – 0,9208| = 0,0012; 2 = 100% = 0,13%;
0,9196
0, 0107 − 0, 0108
Δπ3 = | 0,0107 – 0,0108| = 0,0001; 3 = 100% = 0,93%;
0, 0107
0, 0029 − 0, 00293
Δπ4 = | 0,0029 – 0,00292| = 0,00002; 4 = 100% = 0, 69%.
0, 0029
As can be seen from the obtained results, the relative error of the obtained values
of the state probability vector during numerical simulation of the Markov chain and
the analytical solution did not exceed 1 %, which confirms the adequacy of the re-
sults.
4 Conclusion
1. In this work, the equations for describing the failure of one of the TV3-117 air-
craft engines, which are part of the helicopter power plant, are obtained, which makes
it possible to use standard methods for constructing an algorithm for determining
continuously Markov process when simulating the operation of the TV3-117 aircraft
engine as a part of the helicopter power plant.
2. In this work, we obtain a graph of the standard deviations of the state probability
vector π(π1, π2, π3, π4) for the ideal state of the engine (a new engine exited from the
factory), which is an information model for determining failure when comparing it
with the graph obtained as a result of flight tests.
3. Both the algorithm for constructing this graph and the algorithm for determining
the state probability vector π(π1, π2, π3, π4) are implemented in the MatLAB software
package, it shows the probability that the engine will be in the i-th state. The follow-
ing value of this vector π(0.9196 0.9196 0.0101 0.0029) was obtained in the work, the
value of the element π3 is explained by a possible point reconfiguration, and π4 due to
the human factor during assembly and maintenance of the engine.
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