FACTA UNIVERSITATIS

Series: Mechanical Engineering Vol. 18, N
o
3, 2020, pp. 453 - 472

https://doi.org/10.22190/FUME200306032P

Original scientific paper

FUZZY MODEL OF THE OPERATIONAL POTENTIAL

CONSUMPTION PROCESS OF A COMPLEX TECHNICAL

SYSTEM

Michał Pająk

University of Technology and Humanities in Radom,

Faculty of Mechanical Engineering, Poland

Abstract. During the operation process of a system its technical state is changed. The

changes take place because of the wearing factors impact. The impact depends on the

flow and intensity of the operation process what is characterized by the time histories of

the working parameters. Simultaneously, the changes of the technical state are correlated

with the changes of the amount of the operational potential included in a system. In order

to avoid the inability state occurrence the amount of this potential should be higher than

the boundary value. The amount of the operational potential included in a system is

determined by the values of the cardinal features of it but in the case of the real technical

system the values cannot always be measured. Therefore, the amount of the operational

potential and the technical state of the system cannot always be determined online. To

solve this problem the model of the operational potential consumption process was

created and presented in the paper. The model uses artificial intelligence techniques to

calculate the change of the operational potential amount by determining the changes of

the cardinal features of the system on the basis of the time histories of the working

parameters. The verification of the model quality was performed using the pulverized

boiler OP-650k-040 operating in the power plant. The description of the conducted

research and the results of the verification were presented in the end of the paper proving

the adequacy of the model implementation in the case of industrial objects.

Key Words: Fuzzy Model, Operational Potential Consumption, Complex Technical

System, Operation, System Feature, Working Parameter

Received March 06, 2020 / Accepted August 30, 2020

Corresponding author: Michał Pająk

University of Technology and Humanities in Radom, Faculty of Mechanical Engineering, Stasieckiego St. 54,

26-600 Radom, Poland

E-mail: m.pajak@uthrad.pl

454 M. PAJĄK

1. INTRODUCTION

The object under consideration is a crucial technical system of the strategic importance.

In the case of such systems the main objective of the operation and maintenance control is to

perform the operational tasks and avoiding the failure occurrence [1,2]. This is a very

important issue because the consequences of a failure can be very costly or can pose a health

risk. In order to avoid failure the operation processes should be stopped and the service

activities should be performed. Unfortunately, when the operation processes are stopped

the technical system user does not obtain any effect of the system operation. Thus, the

service activities should be started at the proper moment. The main question of the

research is how to determine that moment.

The failure is defined as a transition of the technical state of the system from the area

of the ability states to the area of the inability states [3]. The technical state is determined

by the values of the cardinal features of the system [4] and is correlated with the amount

of the operational potential included in it. The inability state occurs when the amount of

the operational potential included in a system is lower than the amount corresponding to

its boundary state [5]. In order to avoid the inability state occurrence it is necessary to

know the amount of the operational potential of the system at every moment of the system

operation. This amount, similarly to the technical state, is determined by the values of the

cardinal features of the system. But not always it is possible to measure the values online.

The changes of the values take place during the operation processes because of the forcing

factors influence. The forcing factors can be divided into two groups dependent on and

independent of the system operation [6]. The first group of the factors acts as a result of the

operation process execution while the second characterizes the impact of the environment

[7]. Thus, the level of the forcing factors influence is determined by the way of the operation

process execution and the intensity of the environment impact. If the intensity of the

environment impact is assumed as an independent variable then the forcing factors influence

is determined only by the operation process flow which is described by the values of its

parameters. In fact, the intensity of the operation process is described by the time histories

of the working parameters. So, it was stated that in order to avoid the inability state

occurrence it is necessary to apply the model of the operational potential consumption

process which can calculate the change of the potential as a function of the time histories of

the working parameters. Therefore, the analysis of the existing models of the operation and

maintenance processes was performed [8-11] in order to identify their adequateness in the

case of crucial, complex technical systems of the strategic importance. As a result it was

stated that in the case of complex technical objects changes of operational potential amount

could not be given with proper accuracy thanks to analytical models usage. Therefore, in this

paper, the system fuzzy model is proposed to model the process under consideration. In

Chapter 2 the main idea of fuzzy model application is presented. Its structure and the way of

operation are described in details in Chapter 3. Subsequently, in Chapter 4 the industrial

research is presented which was performed in order to identify the model parameters values

(described in Chapter 5) and to verify prepared model accuracy (described in Chapter 6).

Finally, in Chapter 7, some conclusions can be found.

The main contribution and the novelty of the proposed solution are the universal

creation method of the system model of the operation potential consumption process. The

result of the proposed method is the fuzzy model. Thanks to the fuzzy sets theory

Fuzzy Model of the Operational Potential Consumption Process of a Complex Technical System 455

implementation the uncertainty of the initial technical state identification and the approximation

of the time histories of the operation parameters are taken into consideration.

2. THE IDEA OF THE FUZZY MODEL OF THE OPERATIONAL POTENTIAL

CONSUMPTION PROCESS

It was decided that the designed model will estimate the levels of the forcing factors

influence on the basis of the time histories of the working parameters. Subsequently, on

the basis of the levels of the forcing factors influence the changes of the cardinal features

values will be identified. Eventually, on the basis of the changes of the cardinal features

values the change of the operational potential amount will be determined Eq. (1)

Puxx

t
tff

t
tpo

t
tpoPuM

idid

aop







),...,(

))(,...,)(,)(,...,)((

))(,...,)((:

(1)

where: MPu - the model of the operational potential consumption process, poi - i-th
parameter of the operation process i = 1,...,aop, aop - the working parameters amount,

poi(t) - the time dependent value of i-th working parameter,
1

2
)(
t

t
tpo

i - the time history of

i-th working parameter in period [t1, t2],
j

d
f f - j-th forcing factor depended on the system

operation j = 1,...,n,
k

id
f f - k-th forcing factor independent of the system operation

k=1,...,r, xl - the change of the value of l-th cardinal feature of the system l = 1,...,s,

Pu - the change of the amount of the operational potential of the system.
The application of the constructed model will be used to determine the technical state

of the system in t2 moment taking into account the technical state of the system in t1 moment

and time histories of the parameters of the operation process in period [t1, t2] Eq. (2).

)())(,...,)(),((
2

11
ts

t
tpo

t
tpots

PuM

aop
 


(2)

where: s(t) - the technical state of the system for time t.

The first input value of the model is the technical state of the system in moment t1. It

results from the Markov characteristic of the technical state [12]. According to it, the state

of the system in moment t2 is unequivocally determined by the state in moment t1 and

known input values for t⸦[t1,t2] and is independent of the system state and input values

before t1 moment.

The operating point of the technical object is expressed by the values of the object

features. In the case of complex technical systems operating in the industry the values of

the part of the measurable features still can be identified only by destructive testing [13].

Therefore, they cannot be identified at given moment in which the operation processes are

456 M. PAJĄK

performed. Simultaneously, because of immeasurable features presence, it is impossible

to accurately determine the technical state of the system. Even though, the moment of the

object installation in the operation place is treated as initial moment t1, the technical state

depends on conditions during the production phase and pre-operation processes execution.

In the course of the technical object designing calculations isotropic structure of the material

is assumed. In reality this assumption is fulfilled only approximately. For example, as far as

metal is concerned, its non-uniform structure causes up to 20% differences in the mechanical

characteristics values. Additionally, during the assembling the designing calculations

requirements can be violated. What is more, because of the lack of the equipment required to

perform non-destructive testing, the majority of the producers of the machinery, devices and

constructions do not support the full quality tests at the end of the production phase [14,15].

Usually, the partial tests are executed (about 5%) and on this basis the decision about the

correctness of the whole series of the products is made [16]. As a result, the devices of

approximately identified technical state are designated to operation. Subsequently, during

the pre-operation processes execution the devices are under the influence of forcing

factors. This impact depends on the storage conditions and the intensity of the environment

influence, which are tested periodically [17]. Therefore, it was stated that the technical state

of the system at the start moment of the operation process can be identified only

approximately.

Next group of the input data of MPu model are the time histories of the parameters
describing performed operation processes. Its values, just like those of the system features, can

be measurable or immeasurable. And similarly to the system features, the values can be

determined only approximately.

Inaccuracy of the working parameters values and the approximation of the technical

condition of the system make it impossible to accurately transform the initial state of the

system s(t1) in resultative state s(t2). Therefore, the designed MPu model has to take into
consideration the approximate character of the input data.

This type of inaccuracy consists in the lack of the possibility to categorize the given

phenomenon as true or false [18]. To model this type of inaccuracy the fuzzy sets theory is

widely used [19,20]. The application of fuzzy logic is especially appropriate in the case when

the mathematical model describing the phenomenon does not exist, the existing model is

strongly nonlinear, the existing model is irresolvable or the calculation time required to obtain

the resolution is too long to fulfill the demands about the model response time [21].

Modeling of the operational potential consumption process can be treated as a case

when adequate mathematical model does not exist and tries of the model construction face

the problems arising from not clear enough process description and approximate character

of the input data. In such cases the fuzzy modeling implementation can be found in

industrial application [22-25] as well as in theoretical research papers [26-28]. Thus, on

the basis of the research conducted by the author and the literature analyze it was decided

that the MPu model will be constructed by the fuzzy modeling implementation.
The formulated model of the operational potential consumption is the model of the

Multi Input Single Output (MISO) type. The calculation process performed by the model

consists of three sequential transformations given in Eq. (1) where the last one is executed

according to Eq. (3) [5]:

Fuzzy Model of the Operational Potential Consumption Process of a Complex Technical System 457





n

i
ii

txtxtPu
1

21
))()(()( (3)

where: Pu(t) - the change of the operational potential amount due to the operation

process execution in period t = [t1, t2], xi(t1) - the value of i-th feature for time t1, xi(t2) -
the value of i-th feature for time t2.

Therefore, the task of creation of the fuzzy MPu model was decomposed into
creation the ncf fuzzy models performing transformation Eq. (4), where ncf is the amount

of the cardinal features of the system.

lidriddnd

aopl

x
t

t
tff

t
tpo

t
tpoxM





))(,...,)(,)(,...,)((

))(,...,)((:

(4)

where: Mxl - the model of the change of l-th cardinal feature value,
1

2
)(
t

t
tpo

i - the time history

of the i-th working parameter in period [t1, t2],
j

d
f f - j-th forcing factor depended on the system

operation j = 1,...,n,
k

id
f f - k-th forcing factor independent of the system operation

k = 1,...,r,xl - the change of the value of l-th cardinal feature of the system l = 1,...,s.
The input data of the models are the time histories of the working parameters in period

of the operation process execution while the output is the change of the value of the

system cardinal feature.

It was decided that the considered models will be constructed and verified using the

measurement collections recorded during the conducted operational test, subsequently

analyzed and transformed to the form of Eq. (5) according to the method proposed by the

author in the papers [29,30].

1 2( ) :[ , ,..., , ]j l nipo lmmc x po po po x   (5)

where: xl - l-th cardinal feature of the system, poi - i-th working parameter i = 1,...,nipo(xl),

nipo(xl) - the number of the important working parameters for feature xl, mmcj - j-th major

measurement collection, poi - the distance between the time history of i-th working
parameter and its zero time history (ZTH - the time history of the working parameter for

which the difference between the initial and final value of the feature is the minimum

one), xl - the change of the value of l-th feature of the system.

3. STRUCTURE AND THE OPERATION OF THE MODEL OF THE OPERATIONAL POTENTIAL

CONSUMPTION PROCESS

Each of the models of the system feature value change is of MISO type describing the

unknown in shape solution space. Major measurement collections expressed in the form

presented above Eq. (5) describe the relation between the models output data (change of

the feature value) and the models input data (distances between time histories of the

458 M. PAJĄK

significant working parameters and their zero time histories). So, the collections are the

inputs-output samples of the considered models. The inputs-output models are

constructed employing the fuzzy sets theory. They can use inference of Mamdani type

[31], of Takagi-Sugeno-Kang type [32] or be of relational type [33]. In order to solve the

issue under consideration the fuzzy models with use of the Mamdani type inference are

chosen because thanks to their generalized structure the accurate knowledge about the

structure of the modeled object is not required [34]. Moreover, the inference rules of the

models are not uniformly distributed along the solution space. Regions where the solution

space changes its shape are modeled by a bigger number of the rules than the regions

where the space is of fixed shape. Owing to this, in the Mamdani models the amount of

the inference rules can be decreased. The structure and the general scheme of the worked

out models operation are presented in Fig. 1.

Fig. 1 The general scheme of the fuzzy model operation

The input data of the models are crisp values of the distance between the working

parameters time histories and their zero time histories:

]...[
21 nipFM

popopopo   (6)

where: poFM - the input data of the fuzzy model, poi - the crisp value of i-th input
parameter, nip - the number of the input parameters of the fuzzy model.

The first step of the fuzzy model operation is the fuzzification process. It is

transformation of the crisp values to the fuzzy ones. Fuzzification is accomplished using

knowledge base of the model [35]. The knowledge base consists of groups of the fuzzy

sets defined for each input and output parameter of the model. These fuzzy sets cover the

domain of the parameter in a way presented in Fig. 2.

Fig. 2 The exemplary group of the fuzzy sets for the input parameter of the fuzzy model

Fuzzy Model of the Operational Potential Consumption Process of a Complex Technical System 459

The fuzzy sets which cover the domain of the parameter are their linguistic values.

First of the sets is of L type, Eq. (7) while the last one is of  type, Eq. (8). The

remaining sets are of  type, Eq. (9).
























rrsx

rrsxrrk
rrkrrs

xrrs

rrkx

xFS
L

)( (7)

where: FSL(x) - the membership function of the L type fuzzy set, x - the argument of the

fuzzy set, rrk - the right range (the maximum value) of the kernel of the fuzzy set, rrs - the

right range (the maximum value) of the support of the fuzzy set.

























lrkx

lrkxlrs
lrslrk

lrsx

xFS

)( (8)

where: FS(x) - the membership function of the  type fuzzy set, x - the argument of the
fuzzy set, lrs - the left range (the minimum value) of the support of the fuzzy set, lrk - the

left range (the minimum value) of the kernel of the fuzzy set.






























rrsxlrk
lrkrrs

xrrs

lrkxlrs
lrslrk

lrsx

rrsxlrsx

xFS

)( (9)

where: FS(x) - the membership function of the type fuzzy set, x - the argument of the
fuzzy set, lrk - the left range (the minimum value) of the kernel of the fuzzy set, lrs - the

left range (the minimum value) of the support of the fuzzy set, rrs - the right range (the

maximum value) of the support of the fuzzy set.

During the fuzzification process for each crisp value of the considered input parameter

the values of the membership functions of each of its linguistic values are determined:

])(...)()([ 21 iLiLiLi popopoz nlv
ipoipoipo




 (10)

where: zi - the fuzzy value of i-th input parameter of the model, poi - the crisp value of
i-th input parameter of the model, )( iL poj

ipo




- the value of the membership function of

j-th linguistic value of i-th input parameter for poi value,
j

poi
L
 - j-th linguistic value of

i-th input parameter of the model, nlv - the amount of the linguistic values defined for the

parameters of the model.

460 M. PAJĄK

The second step of the models operation is the fuzzy inference. It is performed using

the rule bases of the models. The rule base of each model consists of the inference rules

described by Eq. (11), which should be read according to the notation from Eq. (12):

popo

poi FMniponip
LyLpoLpoLpoFMR





 ...:
21 21

(11)

where: FMRi - i-th inference rule of the fuzzy model, poi - the crisp value of i-th input

parameter of the model,
j

poi
L
 - j-th linguistic value of i-th input parameter of the model,

yi - the fuzzy output value of i-th inference rule,
n

xFM
L
 - n-th linguistic value of the output

parameter of the model, xFM - the output parameter of the fuzzy model.












ni
y

mzkzjzFMR

out

nipi

0)(...0)(0)(:
21

(12)

where: zi - the fuzzy value of i-th input parameter of the model, iout - the number of the

linguistic value of the output parameter of the model.

The fuzzy inference consists in the premises aggregation, implication and resulting

sets accumulation [36]. For each of the rules present in the rule base of the model the

premises aggregation is performed according to the formula:

))(),...,(),(min()(
21

mzkzjzFMRAGR
nipi

 (13)

Subsequently, the implication is executed according to formula:

nvliiyFMRAGRiy
outoutiiouti

,...,2,1))(),(min()(  (14)

where: FMRi - i-th inference rule of the fuzzy model, AGR(FMRi) - the value after

premises aggregation of i-th inference rule, yi(iout) - the value of the membership function

of the linguistic value on position iout for fuzzy value of the output parameter of i-th rule,

nvl - the amount of the linguistic values defined for the parameters of the model.

The final step of the fuzzy inference process is the resultant sets accumulation. The

step consists in determination of the fuzzy value of the output parameter according to

formula:

nvliiyiyiyiy
outoutnrbroutoutout

,...,2,1))(),...,(),(max()(
21

 (15)

where: y(iout) - the value of the membership function of the linguistic value from position

iout for fuzzy value of the output parameter, yi(iout) - the value of the membership function

of the linguistic value from position iout for fuzzy value of the output parameter of i-th

rule, iout - the number of the linguistic value of the output parameter of the model,

nrbr - the amount of the rules in the rule base of the model.

Thus, the fuzzy value of the output parameter can be expressed in the following form:

]...[ 21 nvl
FMxlxlx

LLL
yyyy



 (16)

where: y - the fuzzy value of the output parameter of the model, i
lx

L
y

 - the maximum

value of the membership function of i-th linguistic value of the output parameter,
i

xFM
L
 - i-th linguistic value of the output parameter, nvl - the amount of the linguistic

values defined for the parameters of the model.

Fuzzy Model of the Operational Potential Consumption Process of a Complex Technical System 461

The expression above should be interpreted as a fuzzy set with support equal to the

output parameter domain. The value of the membership function of this set ought to be

calculated according to the formula:

)),min(

...

),,min(

),,max(min()(

nvl

FMx
nvl

FMx

FMxFMx

LLFM









(17)

where: (xFM) - the membership function of the output parameter, , xFM - the output
parameter of the fuzzy model, i

FMx
L

y


- the maximum value of the membership function of i-th

linguistic value of the output parameter, i
FMx

L
 - the value of the membership function of i-th

linguistic value of the output parameter, i
xFM

L


- i-th linguistic value of the output parameter,

nvl - the amount of the linguistic values defined for the parameters of the model.

The output of the fuzzy model is the crisp value of the output parameter. It is

calculated in the defuzzification process. In the process the fuzzy value of the output

parameter determined in the inference process is defuzzified by the centre of gravity

defuzzification operator application:








max_

min_

max_

min_

)(

))((
FM

FMFMFM

FMFM

xdxx

xCOGx



 (18)

where: xFM - the output parameter of the fuzzy model, COG - the centre of gravity

defuzzification operator, (xFM) - the membership function of the output parameter,

xFM_min - the minimum value of the output parameter domain, xFM_max - the maximum

value of the output parameter domain.

4. THE PERFORMED INDUSTRIAL RESEARCH

In the described studies it was decided to construct the considered fuzzy model using

learning data registered during the industrial research. There are two main groups of the

learning techniques which can be used to solve this issue. First of them is the connection

of artificial neural networks and fuzzy systems which is widely applied decision-making

and data classification problems [37-39]. The second one is connection of genetic

algorithms and fuzzy logic which is implemented in automatic fuzzy model generation

and data classification problems solutions [40,41].

The task of creating models as described by Eq. (4) was solved using the automatic

iterative method of the Mamdani fuzzy model generation [42]. The applied method

generates the models using the sample data in input/output form. Exactly, the sample data

were prepared in the form depicted by Eq. (5). In order to collect the data the operational

tests were carried out on the biggest Polish hard coal fired power plant in Kozienice. The

production system of the plant includes 8 - 200MW power units, 2 - 500MW power units

462 M. PAJĄK

and 1 - 1000MW power unit [43]. The tests were limited to 8 - 200MW production units.

Main parts of the units were OP-650k-040 pulverized coal fired boilers [44] and 13K215

steam turbosets. The performed operational tests lasted fifteen months and consisted in

collecting the working parameters of the units. After the preliminary analysis of the data

correctness 23 parameters were selected for further considerations (Table 1). As a result

of the collecting process 5440008 values for each parameter were obtained.

The technical state of the OP-650k-040 boiler is described by the values of the system

features. These features also describe the quality of the whole system operation. Therefore, to

identify the cardinal features of specified element of the system the TKE method was taken into

consideration. TKE is a common analysis method of an operation quality of power units’

devices. As a result of the completed analysis deviations q3 (the deviation of the heat

consumption caused by the temperature of the reheated steam [kJ/kWh]), q4 (the deviation of

the heat consumption caused by the pressure in the secondary reheater of the reheated steam

[kJ/kWh]), q5 (the deviation of the heat consumption caused by the water injections to the

reheated steam [kJ/kWh]) and q8 (the deviation of the heat consumption caused by the reduced

efficiency of the boiler [kJ/kWh]) were chosen as cardinal features determining the technical

state of the boiler [1]. On the basis of the measurements sets collected during the operational

tests, the momentary values of the deviations were calculated. Applying the polynomial

approximation the time functions of each deviation were formulated.

Table 1 The set of the selected working parameters of the research object (RAP - rotary

air preheater, HP - high pressure, MP - medium pressure)

No Value Medium Side Unit Symb.

1. Temperature Main steam left °C t0l

2. Temperature Main steam right °C t0p

3. Pressure Main steam both MPa p0

4. Pressure Outgoing steam of HP turbine both MPa psahp

5. Temperature Outgoing steam of HP turbine left °C tahpl

6. Temperature Outgoing steam of HP turbine right °C tahpr

7. Temperature Reheated steam left °C tssl

8. Temperature Reheated steam right °C tssr

9. Pressure Incoming steam of MP turbine both MPa tsbmp

10. Temperature Feed water both °C tfw

11. Temperature Flue gasses 1 °C teg1

12. Temperature Flue gasses 2 °C teg2

13. Contents O2 in flue gasses before RAP 1 % o2bp1

14. Contents O2 in flue gasses before RAP 2 % o2bp2

15. Contents O2 in flue gasses after RAP 1 % o2ap1

16. Contents O2 in flue gasses after RAP 2 % o2ap2

17. Amount Injected water left t/h wsil

18. Amount Injected water right t/h wsir

19. Amount Burned mazut both t/h ma

20. Amount Burned coal both t/h ca

21. Amount Active load both MW P

22. Amount Main steam left t/h m0l

23. Amount Main steam left t/h m0r

Fuzzy Model of the Operational Potential Consumption Process of a Complex Technical System 463

Using the calculated values of the boiler’s cardinal features as well as recorded

measurements sets 874 intervals of equal time length (24 hours) were obtained for each

distinguished deviation. The measurement collection was formulated for each interval.

The collection consisted of the time histories of 23 working parameters in range from the

start to the end time of the interval and the difference between the values of the specified

deviation for the start and the end time of interval.

Among the formulated measurement collections for each deviation one measurement

collection was selected. It was the collection corresponding to the smallest change of the

deviation value. On this basis, according to the method presented in literature [29] the

measurement collections were transformed into the form described by expression Eq. (5)

but comprising all 23 parameters.

Subsequently, according to the method proposed by the author [30], the most significant

working parameters in term of the operational potential change have been selected. For

deviation q3 parameters 3, 12 and 17 have been selected and so have, for deviation q4,

parameters 1, 13, 14 and 15; for deviation q5 parameters 8 and 10 are selected and so are,

for deviation q8, parameters 17 and 18. The numbers are used according to Table 1.

Thus, as a result of the performed studies, for each of 4 models of the cardinal feature

change as a function of the most significant working parameters time histories, 874 major

measurement collections in form described by Eq. (5) were obtained.

5. THE IDENTIFICATION OF THE PARAMETERS OF THE OPERATIONAL POTENTIAL

CONSUMPTION PROCESS MODEL

In order to generate the models of the working parameters time histories impact on the

system features values, for each deviation the learning and testing data sets were prepared.

Each of the sets consisted of the major measurement collections which included input and

output data of the model. The form of the learning and testing data for each deviation is

presented in tables (Table 2 - Table 5).

Table 2 The structure of the major measurement collections for q3 deviation (Names of

the fields according to the symbols from Table 1)

No. Field name Symbol Unit

1. The distance from ZTH - p0 p0db MPa
2. The distance from ZTH - teg2 teg2db °C
3. The distance from ZTH – wsil wsildb t/h
4. The change of the deviation value - object dq3r kJ/kWh
5. The change of the deviation value - model dq3e kJ/kWh

Table 3 The structure of the major measurement collections for q4 deviation (Names of

the fields according to the symbols from table 1)

No. Field name Symbol Unit

1. The distance from ZTH - t0l t0ldb °C
2. The distance from ZTH - o2bpl o2bp1db %
3. The distance from ZTH - o2bp2 o2bp2db %
4. The distance from ZTH - o2apl o2ap1db %
5. The change of the deviation value - object dq4r kJ/kWh
6. The change of the deviation value - model dq4e kJ/kWh

464 M. PAJĄK

Table 4 The structure of the major measurement collections for q5 deviation (Names of

the fields according to the symbols from Table 1)

No. Field name Symbol Unit

1. The distance from ZTH – tssr tssrdb °C

2. The distance from ZTH – tfw tfwdb °C

3. The change of the deviation value - object dq5r kJ/kWh

4. The change of the deviation value - model dq5e kJ/kWh

Table 5 The structure of the major measurement collections for q8 deviation (Names of

the fields according to the symbols from Table 1)

No. Field name Symbol Unit

1. The distance from ZTH – wsil wsildb t/h

2. The distance from ZTH – wsir wsirdb t/h

3. The change of the deviation value - object dq8r kJ/kWh

4. The change of the deviation value - model dq8e kJ/kWh

Table 6 The values of the parameters of the fuzzy model generation process

No. Parameter Value

1. Coverage degree of the measurement collection 1.0

2. Compatibility limit of measurement collection and inference rule - z 0.05

3. Incompatibility limit of inference rule and learning data set - kicFMR 0.1

4. Threshold value of the compatibility degree - sFM 0.25

5. T-norm used in generation process MIN

6. Amount of the genetic algorithm generations in one iteration 50

7. Amount of useless mutations of evolution strategy until the process stop 25

8. Amount of the chromosomes exposed to the evolution strategy 20%

9. The parameter of the value mutation used in evolution strategy 0.9

10. Amount of generations of simplification step 500

11. Amount of generations of tuning step 1000

12. Population size of simplification step 61

13. Population size of tuning step 61

14. Parameter of non-uniform mutation 5

15. Crossover probability of generation process 0.6

16. Crossover probability of simplification process 0.6

17. Crossover probability of tuning process 0.6

18. Mutation probability of generation process 0.004

19. Mutation probability of simplification process 0.003

20. Mutation probability of tuning process 0.005

21. Coefficient of arithmetic MIN-MAX crossover operator - g 0.35

22. Aggregation operator of the fuzzy model MIN

23. Implication operator of the fuzzy model MIN

24. Accumulation operator of the fuzzy model MAX

25. Defuzzification operator of the fuzzy model COG

The learning data set comprised 582 measurement collections (first two-thirds of all

measurement collections) while the testing data set comprised remaining 292 measurement

Fuzzy Model of the Operational Potential Consumption Process of a Complex Technical System 465

collections. Subsequently, on the basis of learning data set the automatic generation of the

model was performed. The generation procedure consisted of preliminary generation,

simplification and tuning steps. The adopted values of the model generation process can be

found in Table 6.

As a result of the calculations four models of the working parameters time histories

influence on the values of the system features were obtained. Each of the model in the

aggregation process used MIN operator, in implication process MIN-MAX operator, in

accumulation process MAX operator and in defuzzification process COG operator, Eq.

(18). The rule bases of the obtained models included the groups of fuzzy sets defined for

each input and output parameter of the model. Below, as an example, the group of fuzzy

sets defined for p0db - the first input parameter of the model for q3 deviation is presented

(Fig. 3), Eqs. (19-25).

The model obtained for q3 deviation consisted of 63 inference rules, the model obtained

for q4 deviation consisted of 82 inference rules, the model obtained for q5 deviation

consisted of 20 inference rules and the model obtained for q8 deviation consisted of 40

inference rules. The exemplary form of the obtained inference rule is presented below Eq.

(26). In all expressions describing the rule and knowledge bases of the fuzzy models the

symbols were used in accordance with tables (Table 2 - Table 5). The linguistic variables of

input and output parameters of the models are marked by subscript meaning the parameter

and superscript meaning the number of the linguistic variable.

Fig. 3 The group of the fuzzy sets of first input parameter p0db of the model for q3 deviation






















000146,0p0db 0

000146,0p0db0,000115
000031,0

0000146,0

000115,0p0db 1

dbp
L

dbp
(19)

























000245,0p0db000145,0
0001,0

0000245,0

000145,0p0db000111,0
000034,0

000111,00

000245,0p0db000111,0p0db 0

dbp

dbp
L

dbp

(20)

466 M. PAJĄK

























000293,0p0db000258,0
000035,0

0000293,0

000258,0p0db00019,0
000068,0

00019,00

000293,0p0db00019,0p0db 0

dbp

dbp
L

dbp

(21)

























000395,0p0db000305,0
00009,0

0000395,0

000305,0p0db000198,0
000107,0

000198,00

000395,0p0db000198,0p0db 0

dbp

dbp
L

dbp

(22)

























000463,0p0db000379,0
000084,0

0000463,0

000379,0p0db000285,0
000094,0

000258,00

000463,0p0db000285,0p0db 0

dbp

dbp
L

dbp

(23)

























000527,0p0db000454,0
000073,0

0000527,0

000454,0p0db00038,0
000074,0

00038,00

000527,0p0db00038,0p0db 0

dbp

dbp
L

dbp

(24)






















0005,0p0db 1

0005,0p0db000489,0
000011,0

000489,00

000489,0p0db 0

dbp
L

dbp
(25)

1
3

1 1

0
3 20

edqwsildbdbtegdbp
e=L=>dqwsildb=Ldb=Ltegdb=Lp  (26)

6. THE ANALYSIS OF THE MODEL ACCURACY AND THE RESULTS OF SIMULATION EXPERIMENTS

Finally, the verification of the generated fuzzy models was performed. To do it, for each

model the following measures of the operation quality were determined: the relative

minimum error, Eq. (27), the relative maximum error, Eq. (28), the relative mean error,

Eq. (29), the relative mean square error, Eq. (30), and the value of the correlation function,

Eq. (31). The values of the measures were determined taking into consideration the values

calculated using the real collected data and the values calculated by the models for testing

data set.

Fuzzy Model of the Operational Potential Consumption Process of a Complex Technical System 467

ammci
mmcdqmmcdq

mmcdqmmcdq
dq

irir

irie
,...,2,1100

))(min())(max(

))()(min(
[%]

min





 (27)

where: dqmin - the relative minimum error [%], dqr(mmci) - the value of the output
parameter of the model for i-th major measurement collection, dqe(mmci) - the value of

the output parameter of the object for i-th major measurement collection, mmci - i-th

major measurement collection, ammc - the amount of major measurement collections.

ammci
mmcdqmmcdq

mmcdqmmcdq
dq

irir

irie
,...,2,1100

))(min())(max(

))()(max(
[%]

max





 (28)

where: dqmax - the relative maximum error [%].

ammci
mmcdqmmcdq

mmcdqmmcdq
ammc

dq
irir

ammc

i
irie

mean
,...,2,1100

))(min())(max(

)()(
1

[%] 1 







 (29)

where: dqmean - the relative mean error [%].

mmci
mmcdqmmcdq

ammcammc

rmvdqrmvdq

dq
irir

ammc

i
irie

mean
,...,2,1100

))(min())(max(

)1(

))()((

[%]

2













(30)

where: dq
2

mean - the relative mean square error [%].

100

)()()()(

)()(

)(

1
























ammc

i
ieie

ammc

i
irir

ammc

i
irie

mmcdqmmcdqmmcdqmmcdq

mmcdqmmcdq

dqcorr (31)

where: corr(y) - the value of the correlation function [%].

The values of the operation quality measures of each model are presented in Table 7.

Table 7 The values of the operation quality measures of the models

Measure of the operation quality model -

deviation q3

model -

deviation q4

model -

deviation q5

model –

deviation q8

Relative minimum error [%] 0.2498 1.2772 1.2606 1.3084

Relative maximum error [%] 155.3265 92.9864 92.0827 84.574

Relative mean error [%] 13.8125 5.129 9.4532 11.4643

Relative mean square error [%] 1.1368 0.6205 0.8314 0.9089

Correlation function value [%] 63.2596 60.7267 60.7023 82.1394

468 M. PAJĄK

Analyzing the calculated values of the operation quality measures of the models of the

working parameters time histories influence on the values of the system features it was

decided that the proposed method of the models generation is efficient enough to apply it

to the conducted research. It was also stated that the loss of the amount of the operational

potential would be calculated as a length of the vector given by Eq. (3) and the estimated

changes of the deviations values would be the vector components.

In order to verify the adequacy of the constructed model of the operational potential

consumption process in case of the real complex technical system the simulation experiments

were carried out. The experiments covered 14 days period of OP-650k-040 steam boiler

operation. The calculations were performed for 25 selected initial states spread uniformly in the

range of the disposed amount of the operational potential calculated for all initial states.

The time histories of the most significant working parameters for the considered 14 days

period and the initial values of the deviations determined on the basis of the data recorded

during the operational tests were used as input values of the models of the working

parameters time histories influence on the values of the system features. As a result of the

models operation the estimated values of the deviations for the end of the simulated

operation period were obtained. Having determined real initial values of the deviations and

estimated final values of them the estimated change of the disposed amount of the

operational potential was calculated according to expression Eq. (3). Similarly, using real

initial and real final values of the deviations (obtained as a result of the operational tests) the

real change of the disposed amount of the operational potential was calculated.

In Table 8 for each one of the 25 initial states considered during the simulation

experiments the initial values of the deviations, the initial value of the disposed amount of

the operational potential and real and estimated final values of the disposed amount of the

operational potential can be found. Additionally, in the table the values of the relative

error of the model operation calculated according to formula Eq. (32) are presented.

%100)( 





real

realest

rel
Pu

PuPu
PuMErr (32)

where: Errrel(MPu) - the relative error of the operation of the MPu model [%],

Puest - the change of the operational potential amount estimated by the model, Pureal -
the real change of the operational potential amount.

Analyzing results of the model operation in case of 25 selected periods of the OP-650k-

040 steam boiler operation it was stated that the accuracy of the estimation of the operational

potential consumption process by the generated model is very high (up to 99%).

Thus, it was proved that the presented model can be successfully used to estimate the

loss of the operational potential in the case of complex technical systems. Therefore, it is

planned to apply the proposed solution to the control of the complex technical systems

operation in order to decrease the loss of not exploited operational potential and avoid the

failure of the system.

Fuzzy Model of the Operational Potential Consumption Process of a Complex Technical System 469

Table 8 Results of the MPu model operation (P - disposed amount of the operational

potential [kJ/kWh])

No.

Initial values of the deviations

[kJ/kWh] [q3;q4;q5;q8]
Initial P Final P -

estimated

value

Final P -
real value

Relative

error [%]

1. [-19.3074;-5.8577;-8.5227;-405.9765] 638.8881 128.118 120.1735 6.61

2. [-39.0666;4.4073;56.2565;-263.0552] 490.3015 137.7455 138.8941 0.83

3. [-27.5759;8.9035;59.6207;-223.0393] 449.131 103.6373 99.0042 4.68

4. [-36.0886;8.9555;57.9187;-156.9477] 384.6197 111.7565 111.1584 0.54

5. [-17.9859;4.1348;27.6282;-103.3115] 335.065 115.4199 115.7112 0.25

6. [-13.8601;1.3529;42.9267;-90.7771] 319.9943 115.3871 114.5998 0.69

7. [-9.9071;1.5013;44.2801;-79.7246] 308.6944 118.2544 116.392 1.60

8. [-30.7651;11.8714;70.5773;-64.6891] 291.6223 116.3791 114.7306 1.44

9. [-22.9398;-21.7261;36.5607;-18.2813] 255.517 114.4198 113.0098 1.25

10. [-10.7811;-19.3661;36.2806;17.0488] 220.7192 113.5734 115.2506 1.46

11. [-24.7443;-16.0588;62.7121;39.1986] 195.0298 112.5225 112.5837 0.05

12. [-23.6838;-15.0915;63.2691;48.5913] 185.6962 113.0006 112.6729 0.29

13. [-15.2574;25.7487;75.0135;70.5717] 155.21 112.9432 112.3101 0.56

14. [-10.5887;21.7895;42.7999;78.6968] 154.5821 112.3773 111.9687 0.36

15. [-9.8674;24.6442;44.6542;94.1526] 139.3418 99.1767 98.7857 0.40

16. [-9.5275;25.5339;45.6591;101.7391] 131.903 94.7355 95.0247 0.30

17. [-9.2861;25.9666;46.4051;106.8706] 126.8706 87.563 87.0652 0.57

18. [-18.5995;9.2839;46.0432;118.2013] 120.3831 76.1608 75.7769 0.51

19. [-7.7767;26.9908;50.5532;124.9305] 108.5901 76.0057 76.0297 0.03

20. [-28.3647;19.5421;74.9466;130.5547] 101.4431 72.5974 72.6728 0.10

21. [-1.2134;9.3858;70.4222;138.4202 91.0723 70.2585 70.1301 0.18

22. [-15.15119.2709;67.0263;151.5864] 82.7258 56.9301 56.825 0.19

23. [-0.5087;10.4719;74.4923;157.4561] 72.1249 53.7195 53.81 0.17

24. [-0.3174;10.7947;75.6293;162.7304] 66.9231 50.0816 50.0096 0.14

25. [-0.04222;11.2884;77.2998;170.4669] 59.3524 44.0658 43.9252 0.32

7. SUMMARY AND CONCLUSIONS

In the paper the studies in the field of the complex industrial technical systems

operation and maintenance were presented. As a result of the studies the model of the

operation process of the power boiler was prepared. Below some conclusions and

highlights from the research can be found:

1. The prepared model of the operational potential consumption process should be

designed to take into consideration inaccuracy of the working parameters values

and approximated knowledge of the technical condition of the considered object,

2. The modeling of the operational potential consumption process is the issue when

the adequate mathematical model does not exists and attempts of the development

the model come across the problems arising from not enough clearly defined

description of the process and the approximate character of the input data,

3. The fuzzy modeling was used to develop the system model MPu of changes of

the operational potential included in a technical object,

470 M. PAJĄK

4. The task of the fuzzy MPu model creation was decomposed into creation of ncf

fuzzy models of the influence of the working parameters time histories on the

changes of the values of the system features, where ncf is the amount of the

cardinal features of the system,

5. The set of the working parameters can be very large; thus the set of the parameters of

the highest influence on the change of the specified feature value should be defined,

6. During the operational tests of the OP-650k-040 steam boiler the list of the most

significant parameters in term of the operational potential changes was specified,

7. The fuzzy models of the influence of the working parameters time histories on the

values of the features of the considered boiler were generated using the inputs-

output samples according to the iterative method of the automatic generation of the

Mamdani models,

8. The quality of the models operation was verified using the set of the major

measurement collections recorded during the conducted operational tests,

9. Analyzing the values of the measures of the models operation quality it was

decided that the iterative method of the models generation is efficient enough to

implement it into the conducted research,

10. Considering the results of the simulation experiments performed it should be

stressed that thanks to the implementation of the developed MPu model in the

case of the selected technical system very high accuracy (mean error about 1%) of

the process estimation was obtained.

The major benefit that accrues from the described method application to the complex

technical system is the development of a very accurate model of the operational potential

consumption process (mean efficiency about 99%) according to the universal manner. Its

novelty consists in the universal system approach which can be used in the case of a wide

range of the technical systems. Prior to the application of the proposed method the

operation parameters time histories should be registered. Additionally, all cardinal

features should be identified and their values at the beginning and the end of the operation

parameters collecting period have to be accessible for measurement. These two elements

are the main limitations of the described solution.

The research on the problems of complex technical system operation and maintenance

control will be continued. The first direction of future studies will cover the verification of the

proposed method in the case of different technical systems. It is planned to construct the model

of the operational potential consumption process for systems where the number of the cardinal

features is significantly higher than four and to check the method usefulness in the case of the

systems with a limited number of cardinal features (less than 4).

The second issue is the application of the created model to control the complex technical

systems operation in order to decrease the loss of not exploited operational potential and avoid

the failure of the system. It is supposed to be achieved by using the model described in the

paper and the reverse model of the operational potential consumption process.

Fuzzy Model of the Operational Potential Consumption Process of a Complex Technical System 471

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