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1 INTRODUCTION

Methods of mathematical and computer modeling are

currently used at all stages of the life cycle of ship

power plants (SPP): in the design, operation and

assessment of the technical condition to determine the

possibility of further use.

The processes of controlled state changing of such

objects are provided by the corresponding on-board

monitoring and diagnostic systems (BMDS). For the

design of such systems, the use of computer and semi-

natural simulators is promising. The use of such

simulators can significantly reduce the subsequent

stage of bench testing and thereby reduce the cost of its

implementation.

The problematic issue of improving the simulator

stands is the need to increase the level of adequacy of

used mathematical models of the SPP state changing in

accordance with the operating conditions.

Deterministic mathematical models of processes in

SPP, usually used in simulators, do not fully reflect the

processes in real objects of control, diagnostics and

management, which leads to a discrepancy between

the obtained results and real processes. Therefore, the

study of methods and means of improving the

mathematical models of the controlled change of the

Multiparameter Approximation Model of Temperature

Conditions of Marine Diesel Generator Sets, Based on

Markov Chain Monte Carlo

V. Myrhorod, I. Hvozdeva & V. Budashko

National University “Odessa Maritime Academy”, Odessa, Ukraine

ABSTRACT: In the article we propose a multi-parameter approximation model, based on Markov chain Monte

Carlo, which describes the relationship between the temperature regime, operating conditions and

electromechanical parameters of marine diesel generator sets. The approximation model is constructed on the

basis of the analysis of experimental data of the exhaust gases temperature of marine diesel generator sets in their

long-term operation. As a statistical model of random processes of temperature deviations from the

approximation model, a Markov process model is proposed that takes into account the possible correlation of the

initial data. Since the measuring channels of modern diagnostic systems are digital, due to discretization in time

and level, the studied processes form a Markov chain, which makes it possible to establish the important features

of such processes. The use of approximation models ensures the stationarity conditions and the correctness of the

proposed Markov model in the conditions of multi-mode operation of marine diesel generator sets. The proposed

multi-parameter approximation model, based on Markov chain Monte Carlo, allows you to take into account

random perturbations that lead to a random change in the output coordinates of the diagnostic object. The

proposed improvement of the model makes it possible to ensure its adequacy to real processes of changing the

parameters of the temperature regimes of marine diesel generator sets. The proposed multi-parameter

approximation model, based on Markov chain Monte Carlo, can be used in the systems of technical diagnostics

of marine diesel generator sets in order to increase the reliability of diagnostic conclusions.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 16

Number 4

December 2022

DOI: 10.12716/1001.16.04.20

780

SPP state, which take into account the random nature

of such processes, is an actual and important scientific

and applied task.

Mathematical models of the processes of a

controlled change of the SPP state are based, as a rule,

on models of static (throttle) characteristics and

piecewise linear dynamic models (PLDM), which

sufficiently reflect the dynamics of controlled objects

for designing the corresponding BMDS.

The disadvantage of such models is their

deterministic nature, that is, the impossibility of taking

into account random influences on the change of

controlled variables. As a result, the root-mean-square

deviation of the values of the controlled coordinates

according to the results of tests on the simulator bench

and the results of bench tests on a real object can differ

significantly even in steady-state conditions. The

reason for such significant differences is that the

random nature of the processes of the SPP state

changing in real conditions is not taken into account

value [10]. At present, these scenarios are not taken in

databases of such cases, determining outside the

agreed risk criteria for the operation of protection and

assessing the nature of the risk during the operational

mode [11].

The random nature of such processes is a

consequence of the action of several factors. Firstly,

there is a random nature of external disturbances, for

example, for marine diesel generator sets (DGS),

changes in load, ambient temperature and other

factors. Secondly, there are internal perturbations

caused by the processes of energy conversion in the

control object. Thirdly, there are always noises of

measuring channels. The most significant are the first

and second factors, which require their consideration

in modeling.

Methods that take into account random

perturbations in statistical modeling are known, but

their direct application in simulators is impossible,

since SPP are multi-mode objects, which does not allow

providing the initial conditions for the use of known

methods.

Methods for constructing approximation models of

complex SPPs based on gas turbine engines (GTE) are

considered in the works of the authors [1,2], for ship

SPPs in [3]. A technique for constructing a Markov

model of a controlled change in the state of complex

power plants based on GTE is proposed in [4]. The

application of this approach to marine diesel generator

sets is proposed in [5]. The present study is mainly

based on the works [3, 4, 5] and involves combining the

approaches of constructing multi-parameter

approximation models and Markov models for

deviations of the values of the simulated parameters

from pre-formed approximation models. Such an

improvement makes it possible to increase the

adequacy of models for changing the technical

condition of the SPP, in particular, marine diesel

generator sets (DGS), under conditions of multi-mode

operation and the influence of random factors.

Materials [6 – 9] were used to solve applied

problems. Theoretical generalizations and research

methods are based on the works [10 – 15].

2 PURPOSE OF WORK

The purpose of the proposed study is to create a

statistical model of random processes of deviations in

the temperature of the exhaust gases of the cylinders of

marine diesel generator sets from the approximation

model based on Markov Chain Monte Carlo, which

makes it possible to take into account the correlation of

the initial data.

3 CONTENTS AND RESULTS OF THE RESEARCH

3.1 The Approximation Model

Under the condition of multi-mode operation of the

object, the first stage of the proposed information

technology is to construct the approximation model

corresponding to the object. Such a model reflects the

static characteristics (SC) of the control object, that is,

the dependence of the initial variables on the control

actions in steady-state conditions. If SC are known,

then their nature is presented in the passport data of a

particular object. If they are unknown, then such

dependencies are established directly from the

operation data by means of regression analysis. Thus,

we obtain the dependences of the initial variables on

the controlled actions and loads for possible operating

modes. Usually, such characteristics are polynomial in

nature. The authors also used more complex neural

network models with somewhat better results. Such a

multi-parameter model for ship DGSs was proposed

and substantiated by the authors in [3].

According to [3], the diagnostic multi-parameter

model of the change in the technical condition of the

diesel generator set in steady-state conditions during

long-term operation has the following form:

( ) ( )

, , , ,

bx ex

F T T F U I



(1)

where:

Tbx – air temperature in the ERTemperature inlet,

Tex – reduced temperature.

As justified in [3], to construct a diagnostic

statistical model, the load current and the reduced

temperature of the cylinder gases can be chosen.

However, this temperature is a thermogasdynamic

parameter, therefore, as an argument of the statistical

model, it is advisable to choose the load moment of the

DGS, which is determined from the registration data as

follows:



(2)

where:

Pe=UI – electric power,





– angular frequency,

n=RPM – DGS speed,



– efficiency.

Thus, in the analysis the additional parameters are

used in the form of DGS speed and generator voltage,

which increases the information content of the model.

The use of a regression statistical model can

significantly reduce the dispersion of residual

781

deviations for diesel generator cylinders. As follows

from the results of the analysis [3], taking into account

the inlet temperature makes it possible to reduce the

STD of residual deviations by (10-12)% on average, and

the use of the regression model by (2-2,5) times.

The authors believe that preference should be given

to such a model for which the dispersion of residual

deviations will be the smallest, and which takes into

account a large set of recorded diagnostic parameters.

The best model may be the one for which the STD of

the residual deviations is commensurate with the STD

of the measuring channels [16 – 18].

The application of an approximation model makes

it possible to take into account the multi-mode nature

of the object functioning and in the future to consider

only random deviations from the model that make up

the time series, the properties of which allow the

methods of the statistical model to be applied.

3.2 Statistical modeling

At statistical modeling of random processes, the initial

hypothesis is belonging to a sample from a general

population of independent uncorrelated random

variables with some known distribution law, usually a

normal distribution. Methods for such modeling using

a nonlinear transformation of readings of random

variables of a uniform distribution are known. But such

a hypothesis about uncorrelatedness is not satisfied in

practice [19 – 21].

Since PLDM are used in simulators, in the presence

of random perturbations described by the normal

distribution law, the resulting random process is

Markov [4]. In the steady-state regimes of SPPs, the

correlation between neighboring readings is a

consequence of the presence of a trend in the initial

data. Therefore, the adoption of the hypothesis

regarding the Markov nature of processes in SPPs has

some advantages for increasing the level of adequacy

of statistical modeling.

Measuring channels of diagnostic parameters of

SPPs are digital. Therefore, the initial data in statistical

modeling is always discretized, both in time and in

level. Sampling parameters are set by the applied

analog-to-digital conversion method, in particular, the

ADC bit depth.

With such a discretization, the model of the

generated process is a Markov chain. The dimension of

the matrix of transition probabilities can be quite large,

but at present this is not a significant factor. If

necessary, this dimension can be reduced by forcibly

dividing the range of variation of the modeled variable

into a set number of intervals, the width of which is

determined by the purpose of modeling [22 – 24].

According to the properties of the Markov chain,

the final matrix of transition probabilities is formed as

a certain step from the matrix of transition

probabilities, which makes it possible to construct a

dome of probabilities for controlled Markov processes,

the form of which can be an important diagnostic

feature.

Thus, the algorithm of the proposed statistical

modeling has the following form:

− Considers the time series of some initial SPP

variable over a long period in conjunction with the

time series of changes in the controlled variable or

load.

− A regression model of the corresponding variables

is formed by a known method.

− The hypothesis of stationarity of the resulting time

series of deviations from the regression model is

tested using well-known criteria, for example, the

Cochrein criterion.

− If it is necessary, discretization by level is

performed with a step, which is determined by the

purpose of modeling.

− For the time series of deviations, a matrix of

transition probabilities of the Markov chain is

obtained.

− A matrix of finishing probabilities is constructed

and, if it is necessary, a probability dome for a

controlled Markov process also is built.

− For simulation tasks in a simulator, the obtained

probabilistic dependencies are used to form

samples of random disturbances or simulated

variables, and for diagnostic tasks they are the basis

for the formation of statistically valid conclusions

regarding the probability of deviations of variables

from passport characteristics.

3.3 The Applied Problem

To test the proposed approach, the applied problem of

constructing an approximate Markov model for the

temperature regime of a diesel generator set during

long-term operation was solved.

Following [3], the parameters of daily recording of

the operating modes and technical condition of marine

diesel generator sets during their long-term operation

were considered as initial data. The collection of

statistical data on the technical condition of ship DGSs

of the NORDSCHELDE bulk carrier with a

displacement of 50,000 tons was carried out in the

period from 10.16.2017 to 03.27.2018 in various modes

of operation of the vessel (navigation, maneuvering,

parking with cargo operations). The investigated ship

electric power system (SEPS) consisted of three DGSs,

which included diesel engines of model 6EY18L

manufactured by YANMAR and synchronous

generators manufactured by HYUNDAI. The

parameters of the DGSs were controlled by two

monitoring systems: EPM (Enamor Power Monitor)

and Kongsberg K-Chief 600. The EPM system controls

and analyzes the electric power parameters produced

by synchronous generators (voltage, current,

frequency, power, power factor and etc.).

The transmission of the received data is carried out

according to the RS 485 standard in the NMEA

(National Marine Electronics Association) format to

other ship monitoring systems, including the

Kongsberg K-Chief 600 main monitoring system. This

system controls the diagnostic parameters that

determine the technical condition of the ship DGSs.

These parameters include: the temperature of the gases

of the cylinders, the temperature of the gases at the

inlet to the turbocharger, the temperature of the fresh

water cooling circuit at the inlet and outlet, and others

[25, 26].

782

According to [3], in order to record the values of

diagnostic parameters of ship DGSs, a special

electronic table was created, in which their current

values were entered every 24 hours. The

systematization of the obtained data made it possible

to form a common array consisting of 30 parameter

vectors, each with a length of 162 values. Data selection

made it possible to identify the change in 17

parameters of one of the DGSs with the longest

duration of operation from 101 daily slices. Fig. 1 – Fig.

6 shows the results of the implementation of the

proposed approach. The initial data and the

approximation model are presented in Fig. 1 – Fig. 3.

Figure 1. Time series of changes in the temperature of the

exhaust gases of the DGS cylinders

Figure 2. Time series of DGS’s load current changes

Figuer 3. Approximation model

Fig. 4 illustrates the initial data for building the

Markov model – a time series of deviations from the

approximation model, and Fig. 5 – the contour of the

matrix of empirical transition probabilities and the line

of equal deviations in accordance with the algorithm

for generating the statistical model of the Markov

chain. Level discretization is chosen as illustrative of 10

degrees.

Figure 4. Time series of deviations of the DGS’s exhaust gases

temperature from the approximation model for the first

cylinder

Figure 5. Contour of the matrix of empirical transition

probabilities and lines of equal deviations

To check the adequacy of the chosen approach, the

final distribution of transition probabilities was

constructed and the corresponding data sample was

generated in comparison with the original sample. The

obtained results are shown in Fig. 6.

Figure 6. The time series of actual and simulated temperature

deviations

0 20 40 60 80 100

260

280

300

320

340

360

380

400

time

T grad

T cylinders

cylinder 1

cylinder 2

cylinder 3

cylinder 4

cylinder 5

cylinder 6

0 20 40 60 80 100

100

200

300

400

500

600

700

time

I A

1000 1500 2000 2500 3000 3500 4000

240

260

280

300

320

340

360

T(M)cenzure

0 20 40 60 80 100

-60

-40

-20

time

0.1

0.2

0.3

0.4

0.5

0.6

-40 -20 0 20 40

-40

-20

-60

-40

-20

0 20 40 60 80 100

-60

-40

-20

time

dT1

Marcov dT1

783

As established by the results of a statistical

experiment, the difference between the final

distribution of transition probabilities of the Markov

model and the empirical distribution obtained by

known statistical methods is not statistically significant

at a confidence level of 0.95 according to known criteria

[27]. Therefore, we can assume that the proposed

model is adequate to the initial data in accordance with

the proposed approach.

4 CONCLUSIONS AND RECOMMENDATIONS

The purpose of the proposed study in the form of

increasing the adequacy of the models for changing the

state of the SPPs, taking into account random factors,

was achieved by using a new information technology,

which consists in sequentially performing the stages of

preliminary approximation of the time series of

deviations of diagnostic parameters from the

constructed approximation model, and the stage of

statistical modeling.

As the diagnostic parameters, the deviations of

which are considered, are selected: the temperature of

the exhaust gases of the diesel generator cylinders as

the main parameter, as well as: the air temperature at

the inlet, the voltage, the load current and the

revolutions of the diesel generator [28].

The statistical model of the formation of random

processes of such variables deviations from the

approximation model is a model of the Markov

process. The use of an approximation model expands

the possibilities of the statistical modeling method to

ensure stationarity conditions and the correctness of

the proposed model [29].

The quantitative results are that for the applied

example under consideration, the temperature

deviations of the exhaust gases of the DGS cylinders

from the established limits [3, 6, 7, 9] are not

statistically significant and, with an empirical

probability of approximately equal to 0.8, are within

±20 degrees.

In contrast to the well-known results [5], the

proposed approach to constructing a Markov model

for deviations of the temperature of the exhaust gases

of the diesel generator cylinders from the constructed

approximation model is new.

Prospects for further research may lie in the

application of mathematical models of controlled

Markov processes.

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