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Neuro-fuzzy Modelling of a Linear Fresnel-type Solar collector System as

A Digital Twin

∗William D. Chicaiza and Adolfo J. Sánchez and Antonio J. Gallego and Juan M. Escaño

Department of System Engineering and Automatic Control, University of Seville,

Camino de los Descubrimientos s/n, 41092, Seville, Spain.

wchicaiza@us.es; asanchezdelpozo@us.es; agallego2@us.es; jescano@us.es

Abstract

One of the main components of a Digital

Twin is the modeling of the virtual entity, be-

ing this a high-ﬁdelity digital model of the

physical entity that represents the modeling

of geometry, modeling of physical proper-

ties, modeling of behavior, and modeling of

rules in the virtual world.

This paper presents a model, based on an

Adaptive Neuro-Fuzzy Inference System, of

a Fresnel linear solar collector system as

a Digital Twin, located on the roof of the

School of Engineering of the University of

Seville, which is a part of an absorption cool-

ing plant.

A distributed parameter model of the sys-

tem has been used to generate artiﬁcial data.

Real operating data were used to validate the

model.

Keywords: Digital Twin, Neuro-Fuzzy

modeling, ANFIS.

1 Introduction

Digital Twin (DT) is one of the most promising tech-

nologies applied to smart manufacturing and industry

4.0. Several DT applications have been realized in

product design, production and, prognostics and health

management (PHM) thanks to the increasing techno-

logical development, information storage, and trans-

mission speeds. The concept DT appeared in 2002

at the University of Michigan, promoting the premise

that each system has a physical entity and a virtual en-

tity. Term Digital Twin was included in Virtually Per-

fect for the conceptual model currently in use [11, 12].

The ﬁrst general standard architecture that appears is

modeling in three dimensions [11]. Similarly, [26] pro-

poses a 5-layer DT modeling architecture, being this an

extension of the three-dimensional architecture. There

is also an eight-dimension architecture that describes

the behavior and context of the DT [21].

Several are the works developed regarding DTs, in

[12] argues that DT is the digital representation of a

physical object by developing a DT of a bending test

bench. Likewise, A step-by-step process of building

a DT model is presented in [17] using a commercial

drilling rig. Also, DT modeling of a 3D printing ma-

chine was performed in [7]. Industrial companies such

as General Electric and Siemens have seen the great

potential of DT [16, 27].

Deep learning shows the ability to learn and model

large-scale data sets, unlike traditional learning meth-

ods [18]. The performance of a physical entity can

be represented through artiﬁcial intelligence (AI) tech-

niques such as fuzzy logic (FL), Artiﬁcial neural net-

works (ANN), neuro-fuzzy algorithms, or optimiza-

tion techniques such as genetic algorithms (GA), parti-

cle swarm optimization (PSO), in order to solve many

problems in different areas.

The advantages of fuzzy logic and artiﬁcial neural net-

works were concatenating in an AI technique called

Neuro-Fuzzy. The most popular is the adaptive neuro-

fuzzy inference system (ANFIS) proposed by Jang

[14], which combines the learning capacity and rela-

tional structure of artiﬁcial neural networks with the

decision-making mechanism of fuzzy logic [15, 9, 23].

Several papers use ANFIS [8] presents a neuro-fuzzy

model of the temperature of an industrial autoclave to

apply nonlinear predictive control. Used to predict the

cone index values of arable soils [1], to control the

bag process by implementing an ANFIS controller [3].

Similarly, ANFIS is used to forecast monthly water use

[28, 4]. In the energy context in [29], they investigated

the interrelationship between energy consumption and

economic growth in China. Successful applications of

the neuro-fuzzy system for modeling complex nonlin-

ear hydro-resource systems have been reported [2].

Atlantis Studies in Uncertainty Modelling, volume 3

Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society

for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP)

Copyright © 2021 The Authors. Published by Atlantis Press International B.V.

This is an open access article distributed under the CC BY-NC 4.0 license -http://creativecommons.org/licenses/by-nc/4.0/. 242

This work uses ANFIS to model an energy system,

speciﬁcally the Linear Fresnel Collector (LFC) system

of the ETSI of the University of Seville, which is part

of the solar cooling plant installed. Since the dynamics

of this kind of systems is highly nonlinear, a complex

model is needed to describe this dynamics [6]. In order

to develop optimal control and optimization strategies

which use the prediction of the solar ﬁeld temperature,

a fast model is indispensable. The use of the nonlinear

distributed parameter model to do this implies a high

computational burden and this is a great hindrance if

the problem has to be solved within a sampling time of

several seconds [5]. Faster models are very important

to be obtained. Thus the main objective of this paper

is to obtain a neuro-fuzzy model that describes the per-

formance of the real outlet temperature from the solar

ﬁeld, considering the variables that affect this process

(solar radiation, water ﬂow, ambient temperature, ﬂuid

temperature, and the local hour that will determine the

position of the sun respect to the solar ﬁeld).

The paper is organized as follows: Section 2 brieﬂy

describes the Fresnel solar ﬁeld and its mathematical

model. Section 3 describes the LFC ANFIS modeling.

Section 4 shows the training and evaluation results. Fi-

nally, some concluding remarks are given in section 5.

2 Modeling of the Fresnel solar ﬁeld

This section presents a brief description of the Fres-

nel solar ﬁeld and the mathematical model that will be

used for the simulation of the plant and the generation

of data to train the ANFIS models.

2.1 Description of the Fresnel solar ﬁeld

The Escuela Técnica Superior de Ingeniería (ETSI) of

Seville has a solar cooling plant (SCP). The SCP con-

sists of three main subsystems: (1)Fresnel-type solar

ﬁeld, see Figure 1, (2) Absorption machine chiller and

(3) a PCM storage tank. This work focuses on the

LFC subsystem. The Fresnel solar ﬁeld is installed on

the roof of the building with an East-West orientation

(Latitude =37.4108972◦,Longitude =−6.0006621◦

).

The solar ﬁeld heats up the pressurized water coming

from the chiller up to the required operation tempera-

ture (145−165◦C). The solar ﬁeld consists of 11 rows

of mirrors which focus the direct solar radiation on a

metal tube of 64 m long whereby the water is circu-

lating. A more complete description can be found in

[19].

The modeling of the Fresnel ﬁeld is developed us-

ing similar equations to the case of parabolic trough

Figure 1: Fresnel solar collector.

ﬁelds modeling [24], but differ in the way of calculat-

ing the geometric efﬁciency, the shade factor and the

thermodynamic properties of the ﬂuid. The LFC sys-

tem can be modeled using two different approaches: i)

the concentrated parameter model used in control ap-

plications because its simplicity [19, 20, 22] and ii) the

distributed parameter model which takes into account

the spatial distribution of the system providing a more

precise description [6, 10, 5]. The distributed parame-

ter model is used for simulation purposes.

2.2 Distributed parameter model

The distributed parameter model consists of a pair of

differential equations in partial derivatives describing

the energy balance [10] as follows:

ρmCmAm

∂Tm

∂t=IKo ptnoG−Hl(Tm−Ta). . .

. . . −lpHt(Tm−Tf)

ρfCfAf

∂Tf

∂t+ρfCfq∂Tf

∂l=lpHt(Tm−Tf)

(1)

where msubindex refers to metal and fsubindex refers

to ﬂuid. The description of the parameters is presented

in Table 1.

The calculation of the optical efﬁciency Kopt requires

knowledge of multiple factors such as the reﬂectivity

of the mirror, the absorptance of the metal tube, the

shape factor. The heat transmission coefﬁcient, den-

sity and speciﬁc heat coefﬁcient are obtained as poly-

nomial functions of the segment temperature and the

water ﬂow by using thermodynamic data of the heat

transfer ﬂuid (pressurized water), which can be found

in [6]. The geometric efﬁciency nois obtained using

complex trigonometric formulas, considering the ef-

fect of the cosine of the incidence angle of the solar

Atlantis Studies in Uncertainty Modelling, volume 3

243

Symbol Description Units

tTime s

lSpace m

ρDensity kg m3

CSpeciﬁc heat capacity JK−1kg−1

ACross sectional area m2

T(l,t)Temperature ◦K,

◦C

q(t)Water ﬂow rate m3s−1

I(t)Direct solar radiation W m2

noGeometric efﬁciency Unitless

Kopt Optical efﬁciency Unitl ess

GCollector aperture m

Ta(t)Ambient temperature ◦K,

◦C

HlGlobal coefﬁcient W m−2◦C−1

of thermal loss

HtCoefﬁcient of heat W m−2◦C−1

transmission metal-ﬂuid

lpLength of pipe line m

STotal reﬂective surface m2

Table 1: Description of the parameters.

beam and the shade factor [19]. The distributed pa-

rameter model is solved with an integration time of

0.25s, dividing the tube into 64 segments of 1 m each.

The distributed parameter model is used as a simula-

tion model to generate artiﬁcial data in order to be used

to train the Neuro-fuzzy model.

3 LFC ANFIS modelling

ANFIS integrates two methods of Soft-Computing: ar-

tiﬁcial neural networks (ANN) and a fuzzy inference

system (FIS) [14]. Providing a method applicable to

fuzzy modeling, learning from a data set in order to

calculate the parameters of the membership functions

(MF) that best allow the associated FIS to follow the

input-output data [23, 13]. Fuzzy logic (FL) can model

arbitrarily complex non-linear functions. Non-linearity

and complexity are handled by rules, MF and the in-

ference process, improving performance with simpler

implementation. The fuzzy Takagi-Sugeno model can

be formulated as an ANFIS [25], except that the links

indicate the direction of signal ﬂow and no associated

weights. The ANFIS architecture consists of ﬁve lay-

ers: fuzziﬁcation, product, normalization, defuzziﬁca-

tion and output. The ﬁrst and fourth layers contain

adaptive nodes (square nodes) that represent the pa-

rameter sets that are adjustable, while the remaining

layers contain ﬁxed nodes (circular nodes) that repre-

sent the parameter sets that are ﬁxed in the system [14].

ANFIS requests a training data set, (x1,x2,· · · ,y)de-

sired input-output pair that represents the system to

model. Obtaining a dynamic model is not always easy,

either because of the complexity, randomness, or lack

of knowledge of the study system. An accurate model

in many cases will consist of a high number of equa-

tions, but in some cases, it is not always adequate to

solve uncertain systems.. The use of an ANFIS to

model the qualitative aspects and reasoning processes

of human knowledge, using fuzzy rules to describe the

performance of the system, and avoiding precise quan-

titative analysis is an alternative.

ANFIS starts from previous knowledge, or the neces-

sary information can be added to improve the model.

In this work with an ANFIS, a set of rules describ-

ing the dynamics of the solar ﬁeld output temperature

has been obtained. When deﬁning its structure we are

faced with the drawback of setting the MFs, since a

greater number of MFs determines greater knowledge.

However it generates the use of numerous rules, this

can be solved by applying clustering methods that seek

to classify data into subsets.

3.1 Input/Output data set selection

The neuro-fuzzy model represents the performance of

the real outlet temperature from the solar ﬁeld, using

the variables that inﬂuence this process: solar radia-

tion, water ﬂow, ambient temperature, ﬂuid tempera-

ture, and the local hour that will determine the position

of the sun. The solar ﬁeld evolves independently in

an open-loop from an arbitrarily given starting point to

obtain data for subsequent ANFIS training.

The distributed parameter model has been used to sim-

ulate the solar plant and obtain artiﬁcial data that will

be part of the training set. Simulations have been car-

ried out, starting from the system in a permanent state.

Initially, there has only been one input change in each

variable, of the step type, while the rest of the variables

remain constant in average values. Likewise, the levels

of the amplitudes of the steps were divided into three

regions (low, medium, high) and these have been given

for the variables water ﬂow, inlet ﬂuid temperature and

solar radiation. The values of the applied steps have

thus served to better capture the dynamics of the sys-

tem, in the formation of the ANFIS.

Real and artiﬁcial input-output data from the solar ﬁeld

has been used, to prepare two data sets: training and

checking set. The training set is composed of real data

from two different days in the months of June, July

as shown in Figure 2(a) and artiﬁcial data obtained

from the distributed parameter model, see Figure 2(b).

While the checking data is composed of real data from

two different days in the month of August as shown

in Figure 3. The training set must contain all the rep-

resentative characteristics of the system to be modeled.

Atlantis Studies in Uncertainty Modelling, volume 3

244

This set is used by ANFIS to ﬁnd a spatial input-output

relationship through repetitive analysis.

Figure 2: Training data set: (a) real data, (b) artiﬁcial

data

The checking set allows to verify the generalization ca-

pacity of the obtained ANFIS model, should be differ-

ent from the training set, so that the validation process

is not trivial and avoids overﬁtting. Figures 2 and 3

shows training and checking data.

0 100 200 300 400 500 600 700 800

Time [min]

80

100

120

140

160

180

200

Temperature (ºC)

20

30

40

50

60

70

Temperature (ºC)

Tin

Tout

Ta

0 100 200 300 400 500 600 700 800

Time [min]

11

11.5

12

12.5

Water Flow (m3/h)

400

500

600

700

800

900

DNI (W/m2)

Q

I

Figure 3: Checking data set.

3.2 Data processing

A process of normalization of the input-output data

set is performed. The use of normalized values pre-

vents the different nature and magnitude of the vari-

ables from affecting the neural learning process, also

helps to reduce noise, inconsistency and leads to a bet-

ter estimation of the modeling.

The original data has been normalized in the range [0

1] using Eq. 2.

Xnorm =x−xmin

xmax −xmin

(2)

Where xis the data to be normalized, (xmax,xmin )is the

maximum and minimum of the original data respec-

tively and Xnorm is the normalized data.

ANFIS has been provided with memory through a re-

cursive structure, using past values of the inputs and

outputs to capture the solar ﬁeld dynamics. There are

no standard methods to transform human knowledge

or experience into a base of rules and data. Therefore,

it becomes the art of adjusting the ANFIS parameters

in order to minimize the error or maximize the perfor-

mance index.

4 Results

This section presents the results of the ANFIS models.

Simulations have been carried out with real data from

different days of September where the real results will

be compared with those obtained with the ANFIS mod-

els.

4.1 ANFIS training results

The training process starts with seven inputs with a

hybrid conﬁguration [14] and the subtractive cluster-

ing (SC) method with a inﬂuence range equal to 0.70.

A total of four Gaussian type MFs are obtained for

each input. The number of epochs used during train-

ing was 200 and the evaluation criterion for estimating

accuracy of the model is performed by comparing the

RMSE at each epoch. However, as can be observed in

the ANFIS modeling parameters, the inﬂuence range

of clustering, and the number of MFs have been varied

in order to obtain a more generalized learning of the

training and checking sets. The ANFIS architecture is

also modiﬁed when the RMSEChk is considerable dur-

ing training. The number of rules is changed since re-

dundant rules lead to overﬁtting. Overﬁtting indicates

that the ANFIS model obtained is ﬁtted too closely to

the training data set and the ﬁnal performance of the

model with respect to another data set will be deﬁcient.

The Figure 4 shows the overﬁtting effect considering a

high number of epochs (k=10e4), since the learning

rate also depends on the number of given epochs.

If kis small the convergence will be slow but if kis

large the convergence will be initially very fast and

will oscillate around the optimum. However, for a very

high kthe ANFIS model is ﬁtted the training data and

the RMSEChk increases.

The obtained ANFIS architecture of the model is

shown in Table 2 and Figure 5 shows the evolution of

the training and checking RMSE index at each epoch.

The RMSE checking curve has to be observed more to

evaluate the accuracy of the model, since it indicates

Atlantis Studies in Uncertainty Modelling, volume 3

245

0246810

Epoch 104

0

0.01

0.02

0.03

0.04

0.05

RMSE

RMSETrn

RMSEChk

Figure 4: Overﬁtting effect on ANFIS modeling train-

ing.

if the ANFIS learning is general1. Therefore, a middle

point is sought where the learning is general for both

sets. For the rest of the paper this model will be named

as ANFIS1.

Description RMSE min

Number MFs: 3 Gaussianty pe Normalized data

Number rules: 3 RMSET rn =0.0028

Optimization method: hybrid RMSEChk =0.0105

Output MF type: linear

Inﬂuence range: 0.72

Epoch number: 200

Table 2: ANFIS1 architecture parameters obtained

with normalized data.

4.2 ANFIS evaluation results

The results of the evaluation of the model obtained are

presented when using real data from days of operation.

Two error indexes were used to compare the models,

the RMSE and the MAPE, Eqs. 3 and 4.

RMSE =r∑n

t=1(xt−x0

t)2

n(3)

MAPE =

∑n

t=1

xt−x0

t

xt

n(4)

We performed a simulation to evaluate the ANFIS1

model. Figure 6 shows that ANFIS1 model has a good

1If RMSET rn down and RMSEChk up, the ANFIS model-

ing is learning from one set but gets more error in the other,

which indicates that learning is not general.

0 50 100 150 200

Epoch

0

0.01

0.02

0.03

0.04

0.05

RMSE

RMSETrn

RMSEChk

Figure 5: Training and checking RMSE curves for AN-

FIS1 modeling.

performance capturing the dynamics of the LFC ﬁeld

except in the last part of the day, where the error is

higher. Figure 7 shows the results of the ANFIS1

model and real data from a different day. It can show

that, although the dynamics are similar, the error in this

simulation is very high.

0 200 400 600 800 1000 1200

Samples

60

80

100

120

140

160

180

200

Temperature [°C]

Real Tout

Model

Figure 6: ANFIS1 model evaluation, model data vs

real data. Day data-set 1.

In order to obtain a model with a better overall per-

formance, a second ANFIS (ANFIS2) has been trained

with a different architecture and parameters which are

shown in Table 3. The number of epochs has been in-

creased and the range of clustering inﬂuence has been

modiﬁed to 0.70 for this ANFIS model.

It can be checked, comparing tables 2 and 3, that the

RMSE (Trn and Chk) of ANFIS2 are similar to those of

ANFIS1. The RMSE curves of training and checking

are shown in Figure 8.

Atlantis Studies in Uncertainty Modelling, volume 3

246

0 200 400 600 800 1000 1200 1400 1600

Samples

40

60

80

100

120

140

160

180

200

Temperature [°C]

Real Tout

Model

Figure 7: ANFIS1 model evaluation, model data vs

real data. Day data-set 2.

Description RMSE min

Number MFs: 4 Gaussianty pe Normalized data

Number rules: 4 RMSET rn =0.0026

Optimization method: hybrid RMSEChk =0.0106

Output MF type: linear

Inﬂuence range: 0.70

Epoch number: 1000

Table 3: ANFIS2 architecture parameters.

Figure 9 show the evolution of ANFIS2 model vs the

real data, for day data-set 1. as in the previous case

(ANFIS1), showing that ANFIS2 model also provides

a good performance in capturing the LFC system dy-

namics but with a better performance along the day.

ANFIS2 model has also been evaluated in another day

of real data day data-set 2, shown in Figure 10, where

it can be observed the good performance of the model

evolution along the day.

In order to compare the results of both models (AN-

FIS1 and ANFIS2), a set of three simulations results

are shown in Table 4 where the RMSE and MAPE in-

dexes are shown for the different test days. It can be

observed that ANFIS2 model presents lower error rates

than ANFIS1, concluding that this model presents bet-

ter results in the estimation of the outlet temperature

from the solar ﬁeld.

ANFIS2 captures the actual output temperature dy-

namics of the solar ﬁeld in a better way, with twice as

many epochs compared to the previous model. How-

ever, the beneﬁt obtained is appreciable with the ad-

vantage of not having a very high number of epochs.

The optimum number of epochs would be in this range.

Regarding the inﬂuence range, the optimum range

0 200 400 600 800 1000

Epoch

0

0.01

0.02

0.03

0.04

0.05

RMSE

RMSETrn

RMSEChk

Figure 8: Training and checking RMSE curves for best

ANFIS modeling with normalized data.

0 200 400 600 800 1000 1200

Samples

60

80

100

120

140

160

180

200

Temperature [°C]

Real Tout

Model

Figure 9: ANFIS2 model evaluation, model data vs

real data. Day data-set 1.

should be in the interval [0.7−0.72].

Finally, a simulation of the distributed parameter

model and the ANFIS2 neuro-fuzzy model has been

performed to compare the computational load of each

of them. The execution time of the distributed parame-

ter model was 35.03[ms]while that of the ANFIS2 sys-

tem was 634.98[us]. In conclusion, the computational

burden of ANFIS2 concerning the distributed param-

eter model is 55 times faster, which makes it suitable

for real-time control applications where the dynamic

model of the plant is required. Also, the number of

learning parameters of ANFIS is much lower concern-

ing the distributed parameter model.

Atlantis Studies in Uncertainty Modelling, volume 3

247

0 200 400 600 800 1000 1200 1400 1600

Samples

40

60

80

100

120

140

160

180

200

Temperature [°C]

Real Tout

Model

Figure 10: ANFIS2 model evaluation, model data vs

real data. Day data-set 2.

Error ANFIS1 ANFIS2

indexes model model

Day Test 1

RMSE 6.9890 4.42

MAPE 2.69% 2.27%

Day Test 2

RMSE 12.9572 3.6726

MAPE 8.54% 2.58%

Day Test 3

RMSE 8.58 7.5141

MAPE 5.69% 4.51%

Table 4: RMSE and MAPE indexes obtained for AN-

FIS1 and ANFIS2.

5 Conclusion

In this work ANFIS has been applied to capture the dy-

namics of real outlet temperature from a Fresnel-type

solar ﬁeld, obtaining satisfactorily a series of mod-

els which differ in the number of rules.Speciﬁcally,

the range of inﬂuence of clustering and the number of

epochs have been varied to obtain a better model with

a lower error coefﬁcient at the instant of evaluation.

The use of output feedback allows the capture of the

dynamics of the process to be modelled. The result is a

recurrent neural network, whose convergence is more

complex. The ANFIS model obtained in this work is

auto-regressive of second order.

Modifying various parameters, several ANFIS models

have been obtained. One of them has been chosen,

with an acceptable validation error, to follow the dy-

namic evolution of the process. This paper lays the

foundations for creating a framework with which to de-

sign ANFIS-based digital twins for plants of this type.

In future works, the error must be improved by learn-

ing with continuous operating data.

Acknowledgement

The authors want to thank the European Commis-

sion for funding this work under project DENiM.

This project has received funding from the European

Union’s Horizon 2020 research and innovation pro-

gramme under grant agreement No 958339.

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