Neural networks and fuzzy logic approaches to

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Neural networks and fuzzy logic approaches to predict mechanical properties of XLPE insulation cables under thermal aging L. Boukezzi, L. Bessissa, A. Boubakeur & D. Mahi

Neural Computing and Applications ISSN 0941-0643 Neural Comput & Applic DOI 10.1007/s00521-016-2259-y

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Author's personal copy Neural Comput & Applic DOI 10.1007/s00521-016-2259-y

ORIGINAL ARTICLE

Neural networks and fuzzy logic approaches to predict mechanical properties of XLPE insulation cables under thermal aging L. Boukezzi1 • L. Bessissa1 • A. Boubakeur2 • D. Mahi3

Received: 30 October 2013 / Accepted: 2 March 2016 Ó The Natural Computing Applications Forum 2016

Abstract The widespread use of cross-linked polyethylene (XLPE) as insulation in the manufacturing of medium- and high-voltage cables may be attributed to its outstanding mechanical and electrical properties. However, it is well known that degradation under service conditions is the major problem in the use of XLPE as cable insulation. Laboratory investigations of the insulations aging are time-consuming and cost-effective. To avoid such costs, we have developed two models which are based on artificial neural networks (ANNs) and fuzzy logic (FL) to predict the insulation properties under thermal aging. The proposed ANN is a supervised one based on radial basis function Gaussian and trained by random optimization method algorithm. The FL model is based on the use of fuzzy inference system. Both models are used to predict the mechanical properties of thermally aged XLPE. The obtained results are evaluated and compared to the

& L. Boukezzi [email protected] L. Bessissa [email protected] A. Boubakeur [email protected] D. Mahi [email protected] 1

Materials Science and Informatics Laboratory, MSIL, Djelfa University, BP 3117 route of Moudjbara, 17000 Djelfa, Algeria

2

L.R.E./Laboratory of High Voltage, Ecole Nationale Polytechnique, EL-Harrah, Algiers, Algeria

3

Laboratory of Studies and Development of Semiconductor and Dielectric Materials, LeDMaScD, Laghouat University, BP 37G route of Ghardaı¨a, 03000 Laghouat, Algeria

experimental data in depth by using many statistical parameters. It is concluded that both models give practically the same prediction quality. In particular, they have ability to reproduce the nonlinear behavior of the insulation properties under thermal aging within acceptable error. Furthermore, our ANN and FL models can be used in the generalization phase where the prediction of the future state (not reached experimentally) of the insulation is made possible. Additionally, costs and time could be reduced. Keywords XLPE insulation  Prediction  Neural network  Fuzzy logic

1 Introduction Electric cables are very important elements which ensure the transmission of electrical energy. In such cable, the electrical insulation is assured by polymeric materials [1]. Compared to some types of polymers such as polyvinyl chloride (PVC), ethylene-propylene copolymer (EPR) and ethylene-acetate copolymer (EVA) which are of common use for such purpose, the polyethylene (PE) in its crosslinked form (XLPE) is highly recommended [2]. This material is widely used in medium-voltage (MV) and highvoltage (HV) cables up to 500 kV [3] because it has excellent dielectric strength and electrical resistivity, as well as of its excellent physical properties including resistance to cracking and moisture penetration [4]. Many factors can cause the cable failure. Among others, one can quote: presence of voids and impurities in the material, incorrect handling during installation, inappropriate mechanical and electrical use and aging of polymeric insulation under service conditions. This latter presents the most important cause of the cable failure [1]. Under service

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conditions, the cable is permanently subjected to electrical, thermal, mechanical and environmental stresses [5]. Under heat conditions, thermal aging occurs and causes an irreversible damage of the cable insulation. Under temperature stress and over a period of time, chemical composition [6] and physical morphology [7, 8] of XLPE may change. Consequently, several properties may alter. For example, volume resistivity reduces, dielectric losses increase [9] and mechanical properties decrease [10]. The great importance of XLPE as insulation motivates researchers in laboratories worldwide to investigate a lot of experimental techniques in order to get more insight on the state of the insulation. The information derived from this overall work helps to understand the degradation mechanisms of the material under service conditions. On the other hand, these investigations are focused on the XLPE insulation morphological structure to know whether this material can be possibly reused [11]. Numerous recent experimental studies have attempted to diagnose the electrical tree in XLPE cables [12–15]. The effect of partial discharges on the electrical properties of the cable has been studied in a lot of investigations [16, 17]. Much work has been carried out, and the chemical modifications generated by aging and its consequences on the mechanical and electrical properties of XLPE [10, 18–21] were analyzed. All these investigations are cost-effective and time-consuming. It needs sometimes few years to obtain sufficient database to solve economical problems of energy and make maintenance in a simplest way. For this reason, efforts have been made in modelling tendency. Newest techniques are based on the use of intelligent systems. These new methods help scientists, researchers and engineers to use efficiently the database and obtain, with more precision and less time, the future state of the insulation system under aging. Over the last two decades, the different modelling and prediction methods based on artificial neural networks (ANNs) and fuzzy logic (FL) systems have become popular and have been used by many researchers for a variety of engineering applications [22–25]. In electrical engineering, especially in high-voltage field, a lot of applications of ANN can be found [26–30]. In partial discharge pattern recognition in high-voltage power apparatus, Chen et al. [26] investigated a novel extension neural network method. The prediction of breakdown voltage due to the PD in cavities for different insulating materials using multilayer feed-forward network and radial basis function network has been made by Mohanty and Ghosh [27]. Teguar et al. [28] and Genc¸og˘lu and Cebeci [29] investigated two types of neural networks (RBFG and MLP) to predict the flashover of polluted high-voltage insulators. To predict the leakage current under aging conditions of nonceramic insulators, El-Hag et al. [30] use the RBFG neural network trained

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with two-algorithm (BP and ROM) method. However, a small amount of papers dealing with applications of fuzzy logic in high-voltage engineering field can be found in the literature [31–33]. The intent of this paper is to apply the fuzzy logic approach to predict, under thermal aging, the mechanical properties of XLPE insulation used in medium- and highvoltage cables. The performances of this approach will be compared with the applied supervised neural network. This latter is based on radial basis function Gaussian (RBFG) and is trained with random optimization method (ROM) algorithm. We note that the introduction of fuzzy logic theory in high-voltage cables prediction is original and of great interest in the development of industrial tools.

2 Experimental study 2.1 Material All tests were carried out on the commercial XLPE UNION CARBIDE 4201 which is used as insulation in medium-voltage cables. This material, in its granule form, contains polyethylene blended with 2 % of dicumyl peroxide (DCP) as cross-linking agent and 0.2 % of IRGANOX 1035 as an antioxidant. The XLPE insulation is covered by two coaxial cylindrical semiconducting screens that are both extruded simultaneously during manufacturing and resulting in three concentric layers system. 2.2 Thermal aging experiments The polyethylene with its different additives is in granulating form. Plates of 2 mm thickness were molded from these granules using press machine which is heated at 180 °C under a pressure of 300 bars. The plates were then cut into samples having a dumbbell shape of 7.5 cm in length according to IEC 540 (International Electrotechnic Committee) publication. These samples were aged at four different temperatures according to the IEC 216 publication. Two temperatures are situated below the melting temperature of XLPE which are 80 and 100 °C. Two others (120 and 140 °C) are situated above the melting peak of the polymer. The aging time is 5000 h for the first two temperatures and 3500 h and 2000 h for 120 and 140 °C, respectively. The aging process has been carried out using forced air-circulating oven that could maintain the average temperature of the samples within ±2 °C. After each 500-h aging time, 10 samples were taken and then subjected to tensile stress. The experiments consist in breaking the sample, at ambient temperature, using a dynamometer which moves with speed of 50 mm/min. Elongation at rupture and tensile strength were measured simultaneously.

Author's personal copy Neural Comput & Applic

3 Intelligent prediction methods 3.1 Artificial neural network (ANN) model

The output value is given by Eq. (2). !, ! m m X X wj qj ðxi Þ qj ðxi Þ y ¼ j¼1

In addition to its ability of classification and diagnosis, ANN can be used in function estimation branch [29]. In the studied case, the variations of XLPE mechanical properties can be considered as nonlinear functions and consequently can be approximated with ANN models. To predict mechanical properties of XLPE insulation cables under thermal aging, we have used feed-forward neural network. This network is based on radial basis function (RBF). In the following, we will only present the outline of the used RBF neural network. Further detail can be found in [34, 35]. The RBF neural network used here consists of three layers, namely the input layer, the hidden layer and the output layer (Fig. 1). The input layer consists of n neurons corresponding to the experimental values of the chosen XLPE property. These n values were used to train the network and vary from each aging temperature to another. The value of n is 5 in the case of aging temperature 80 °C and 3 or 4 (according to chosen property) in the case of aging temperature 120 °C. The output layer contains one neuron corresponding to the next predicted value of the property (pn?1). To train the network, we have used random optimization method (ROM) to avoid the problem of local minima encountered with use of back-propagation algorithm (BPA). Figure 2 shows the flowchart of ROM training algorithm. The Gaussian function represented in Eq. (1) [36] is used as the activation function.  2 ! n xi  cij 1X qj ðxi Þ ¼ exp  ð1Þ 2 i¼1 r2ij where cij i = 1, n and j = 1, m are the RBFG centers, rij define the width of Gaussians. (cij,

ij)

Pi1

j

Pi2 Output (Pin+1)

Pin

Input layer

Hidden layer

Output layer

Fig. 1 A feed-forward network with a single hidden layer and a single output unit

ð2Þ

j¼1

The different prediction steps are shown in the flowchart of Fig. 3, and more details are given in our previous work [34, 37]. The proposed modelling procedure is carried out with the help of 54 sets of experimental input–output patterns which represent the different values of different properties according to the aging time and under the two aging temperatures. From these 54 sets, only 26 sets are used to train the proposed ANN model and the remaining 28 sets are used for testing purpose. The used 26 training sets are chosen to be all the first obtained results under thermal aging and for each property. For example, the used training sets in the case of elongation at rupture at 80 °C are the first 5 values obtained between 0 and 2000 h with an aging step time equal to 500 h. The testing purpose is carried out with the 6 last values situated between 2000 and 5000 h of aging. The same procedure is carried out with the other properties and other aging conditions. 3.2 Fuzzy logic model It is well known that the pioneer of fuzzy logic (FL) was Zadeh in 1965 [38]. The developed fuzzy theory is made in use in this work to predict the mechanical properties of XLPE high-voltage insulation cables, namely tensile strength (P1), elongation at rupture (P2) and Hot Set Test (P3) as a function of aging time (t) and aging temperature (T), that is Pi = f(t, T). A Mamdani fuzzy inference system (FIS) was chosen and built up with 28 sets of the same 54 experimental sets used in the case of ANN model. The general scheme of the developed FIS is presented in Fig. 4, and further details and description of its different parts are available in [30–32]. The chosen input/output membership functions are presented in Fig. 5 and have a hybrid shape (trapezoidal and triangular) with nine partitions corresponding to nine linguistic values. In the step of structure identification, the temperature changes in discreet way and takes many values. This input variable is not defuzzified in our case. We have reported in Sect. 2.2 that the carried-out experimental aging is done for four temperatures, but for the sake of brevity only two temperatures will be considered in the prediction phase. These two temperatures are 80 °C which is situated below the melting temperature of XLPE (about 105 °C) and 120 °C which is situated above the melting peak of the polymer. The set of linguistic values assigned to t and Pi i = 1, 3 is given by Eq. (3).

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Author's personal copy Neural Comput & Applic Fig. 2 Training algorithm of ROM

Initialize randomly the weights (w), the sequence average (b) and the variance (v)

Calculate the errors at the output for each case: E(w(k)), E(w(k)+ (k)), E(w(k)- (k))

E(w(k)+ (k)) E(w(k)) and E(w(k)- (k)) E(w(k))

no E(w(k)+ (k)) E(w(k))

yes

no

yes

w(k+1)= w(k)

w(k+1)= w(k)- (k)

w(k+1)= w(k)+ (k)

b(k+1)= 0.5b(k)

b(k+1)= b(k)-0.4 (k)

b(k+1) = 0.4 (k)+0.2b(k)

E(w(k))

no

k=k+1

yes

C ¼ fSmall-Small ðSSÞ; Small-Medium ðSM Þ; Small-Large ðSLÞ; Medium-Small ðMSÞ; Medium-Medium ðMMÞ; Medium-Large ðMLÞ; Large-Small ðLSÞ; Large-Medium ðLMÞ; Large-Large ðLLÞg

ð3Þ The relationship between the linguistic values and the actual values for t and Pi i = 1, 3 is presented in Tables 1, 2 and 3. The membership functions (MFs) for t and Pi i = 1, 3 are lt and lPi, respectively. Since t and each of all properties can have nine linguistic values, the rule base can be created with a maximum of 27 rules from the experimentally generated data for each temperature (Table 4). In the rule base, fuzzy variables were connected with AND operator and rules were associated using ‘‘max–min’’ decomposition technique. Also, lt and lPi would be having nine components corresponding to each linguistic value as:

As for computation, the input and output variables are introduced in the MATLABÒ environment and the fuzzy toolbox was used to establish these variables as shown in Fig. 6. Each input variable and output variable were fuzzified with membership function graphically designed with toolbox.

4 Results and discussion 4.1 Experimental results analysis

lt ¼ fltSS ; ltSM ; ltSL ; ltMS ; ltMM ; ltML ; ltLS ; ltLM ; ltLL g ð4Þ lPi ¼ flPiSS ; lPiSM ; lPiSL ; lPiMS ; lPiMM ; lPiML ; lPiLS ; lPiLM ; lPiLL g

ð5Þ

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The previous membership functions can be formulated mathematically by the following equation: 8 1 x\a > > > > ðb  xÞ=ðb  aÞ a\x\b > > < ðx  bÞ=ðc  bÞ b\x\c ð6Þ lx ¼ ðd  xÞ=ðd  cÞ c\x\d > > > > ðx  dÞ=ðe  dÞ d\x\e > > : 1 x[e

The aim of this study is the use of experimental aging data to build up a neural network and fuzzy logic models. For this reason, the experimental results have been analyzed

Author's personal copy Neural Comput & Applic Fig. 3 Flowchart for prediction with RBFG neural network

Read the experimental values yi (i=1,n) of the chosen XLPE property, the number of the neurons in the hidden layer and the wanted number of the predicted values

Distribute the centers, fix the covering rate ( ), calculate widths of Gaussians ( ij) Initiate weighs (w) Calculate the output of the network y* with the equation:

Generate a random sequence with average (b) and variance (v)

Adaptation of weighs (w) with ROM algorithm

Calculation of the RMSE

wj+1=wj/

no

yes

yp yp-1

no

wj+1=wj.

RMSE Ec yes Calculate the next output value yp+1

All the predicted values are calculated

no

yes

Fuzzy rules base

IF……THEN

Input

Fuzzification

Fuzzy input

Fuzzy output Inference engine

Defuzzification

Output

Fig. 4 A fuzzy inference system

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Author's personal copy Neural Comput & Applic Fig. 5 Membership functions of input/output system variables

1

SS

SM

SL

MS

MM

ML

LS

L

LL

a2

a3

a4

a5

a6

a7

a8

a9 a10

0.5

0

a1

Input/output variable

Table 1 Relationship between the linguistic values and the actual values for aging time t

Linguistic values

Time (h) for aging temperature 80 °C

Small Small

0–1250

Small Medium

Table 2 Relationship between the linguistic values and the actual values for different properties at 80 °C

666.66–1333.33

1250–2250

Medium Small

1750–2750

1333.33–2000

Medium Medium

2250–3250

1666.66–2333.33

Medium Large Large Small

2750–3750 3250–4250

2000–2666.66 2333.33–3000

Large Medium

3750–4750

2666.66–3333.33

Large Large

4250–5000

3000–3500

Linguistic values Small Small Small Medium

Tensile strength (P1)

Hot Set Test (P3)

510–513 510.5–515.25

101.44–109.72

513.5–520.25

105.83–113.61

17.38–18.315 17.96–18.47

Large Medium

Elongation at rupture (P2)

16.42–17.38

Medium Small

Large Small

1000–1666.66

16.725–17.96

Small Large

Medium Large

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0–1000

750–1750

Small Large

Medium Medium

Table 3 Relationship between the linguistic values and the actual values for different properties at 120 °C

Time (h) for aging temperature 120 °C

18.315–18.625 18.47–18.85

75–105.83

515.25–530

109.72–117.5

520.25–540.93

113.61–121.39

530–551.93

117.5–125.28

18.625–19.005

540.93–562.875

18.85–19.355

551.93–568.75

121.39–129.17 125.28–133.06

Large Large

19.005–21.64

568.75–606.25

129.17–135

Linguistic values

Tensile strength (P1)

Elongation at rupture (P2)

Hot Set Test (P3)

Small Small

9.87–13.69

250–394.99

80–132.5

Small Medium

11.14–16.24

298.33–491.66

127.5–137.5

Small Large Medium Small

13.69–17.64 16.24–17.88

394.99–542.56 491.66–547.69

132.5–140.83 137.5–142.49

Medium Medium

17.64–18.12

542.56–552.83

140.83–144.16

Medium Large

17.88–18.41

547.69–557

142.49–150

Large Small

18.12–18.73

552.83–560.2

144.16–160

Large Medium

18.41–19.06

557–563.4

150–170

Large Large

18.73–21.64

560.2–606.25

160–175

Author's personal copy Neural Comput & Applic Table 4 Rule base of proposed fuzzy logic model Temperature

Input Time t

Output Mechanical properties P1

80 °C

120 °C

P2

P3

SS

LL

LL

SS

SM

LM

LL

SM

SL MS

LS ML

LS ML

SL MS

MM

MM

MM

MS

ML

MS

MS

ML

LS

SL

SL

ML

LM

SM

SM

ML

LL

SS

SS

LS

SS

LL

LL

SS

SM

LM

LM

SM

SL

LS

LS

SL

MS

ML

ML

MS

MM

MM

MM

MM

ML

MS

MS

ML

LS

SL

SL

LS

LM

SM

SM

LM

LL

SS

SS

LL

and discussed in this paper briefly. For brevity reasons, only results for aging temperatures 80 and 120 °C will be presented in this section. The same behavior has been observed in the case of 100 and 140 °C [10] and can be

analyzed in a similar way as 80 and 120 °C. The mean values of mechanical properties (tensile strength and elongation at rupture) are plotted in function of aging time in Figs. 7a, b and 8a, b. The results obtained through a set of experiments have shown that mechanical properties of XLPE decrease according to the aging time. This diminution is slight at temperature of 80 °C. At elevated temperatures (above the melting temperature of XLPE), we have obtained an abrupt decrease of the studied characteristics. The elongation at rupture decreases from 600 to 250 % and the tensile strength decreases from 22 to 10 N/ mm2 after 3500 h of aging at 120 °C. The diminution of mechanical characteristics is attributed to the thermal-oxidant degradation of the material which is followed by chains scission [39, 40]. The chains scission leads to the diminution of both molecular mass and cross-linking rate, which leads to a weakness of the material. On the other hand, chains scission reactions contribute to the formation of vinylidene and vinyl groups or other unsaturated groups on the XLPE structure. These groups are the main cause of the color changes of the material under thermal stress. In general, mechanical properties are governed by morphological features, such as crystallinity degree, crystallite thickness and crystalline– amorphous ratio [41]. Andjelkavic and Rajakovic [42] reported that, under thermal aging, mechanical properties and crystallinity degree change is a similar way, which suggests that some correlation can be made. The thermal degradation of cross-linked polyethylene has been

Fig. 6 Interface of implemented FIS system with fuzzy toolbox

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Author's personal copy (a)

30

(b)

30 25

Experimental 25

Tensile strength (N/mm²)

Fig. 7 Experimental and predicted values of tensile strength: a 80 °C and b 120 °C

Tensile strength (N/mm²)

Neural Comput & Applic

ANN Fuzzy logic

20 15 10 5

Experimental ANN Fuzzy logic

20 15 10 5 0

0 0

1000

2000

3000

4000

5000

6000

0

1000

Experimental ANN Fuzzy logic

800 700 600 500 400 300 200

5000

Experimental

800

ANN

700

Fuzzy logic

600 500 400 300 200

0 0

1000

2000

3000

4000

5000

6000

0

1000

Aging time (h)

attributed to an antioxidant loss. This additive substance plays an inhibition role in the degradation mechanisms. During the process of cross-linking, volatile products are formed in small vacuoles. During thermal aging, a part of the trapped gases (hydrocarbons, ketones, alcohols) can be released and replaced by water or oxygen which is the main cause of the degradation. To study the evolution of XLPE cross-linking degree, we have conducted a set of experiments to measure the Hot Set Test (HST) elongation. The measurement of the HST was carried out on dumbbell-shaped samples suspended vertically in a ventilated oven regulated at 200 °C as described in the IEC 540 and subjected to 0.2 MPa stress in the oven. The obtained results for 80 and 120 °C are displayed in Fig. 9a, b. The maximum of cross-linking density corresponds to a minimum of elongation at heat deformation [43]. For all aging temperatures, the cross-linking degree decreases with three different speeds according to the aging time. In the first stage, the elongation increases quickly at the beginning of aging time. The second phase is characterized by a weak increase in the elongation; therefore, thermal aging does not have practically any effect on

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4000

100

100 0

3000

(b) 900

900

Elongation at rupture (%)

(a) Elongation at rupture (%)

Fig. 8 Experimental and predicted values of elongation at rupture: a 80 °C and b 120 °C

2000

Aging time (h)

Aging time (h)

2000

3000

4000

5000

Aging time (h)

the cross-linking degree. During the third one, the elongation increases rapidly, especially under high temperatures above the melting peak of the material, which results in a fast decrease in cross-linking degree. 4.2 Prediction results analysis In this study, we have used artificial neural networks trained by ROM and fuzzy logic to predict mechanical properties of XLPE insulation cables under thermal aging. Simulations have been performed many times with changing of different parameters, and the results of best choice are presented for each method. The learning time of ANN is selected after a numerous simulations with different learning times. The selected learning time should have the best learning quality. With a learning step of 500 h, we have trained the network for a learning time less than the maximum aging time for each temperature. Then the neural network predicts each property at the whole experimental interval. This procedure allows us to validate the prediction quality of the network. For aging temperature 80 °C, we have used a learning time of 2000 h to train

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(a)

(b)

180

90

Experimental

60

ANN

30

Elongation (%)

120

0

200 180

150

Elongation (%)

Fig. 9 Experimental and predicted values of Hot Set Test: a 80 °C and b 120 °C

160 140 120

1000

2000

3000

4000

ANN

80

Fuzzy logic 0

Experimental

100

60

5000 6000

Fuzzy logic 0

500

1000

Aging time (h)

RMS error for training data

all properties. For 120 °C, we have used a learning time of 1500 h to train elongation at rupture and tensile strength and 1000 h to train Hot Set Test. All these learning times give in fact the best results of prediction. To decide upon the evaluation criterion, variation of the RMS error for the training data as a function of number of iterations, as depicted in Fig. 10, is considered. As expected, the RMS error is decreasing constantly as training progresses and then remains constant. Therefore, training should be stopped at the value where RMS error does not change much with number of iterations to consider the least training time of the network. Both experimental and predicted results obtained with ANN and FL models are presented in Figs. 7a, b, 8a, b and 9a, b. Figure 7a, b illustrates the experimental and predicted values of tensile strength for temperatures 80 and 120 °C. However, experimental and predicted values of elongation at rupture and Hot Set Test are presented in Figs. 8a, b and 9a, b, respectively. As it is clear from all these figures, the predicted and experimental results are

1500

2000

2500

3000

Aging time (h)

very close. To make this result more evident, we must quantify the comparison by a statistical analysis. 4.2.1 Statistical analysis Three elements can be considered to make statistical analysis relevant. Firstly, we must plot the predicted values of each property that are obtained with each model against the experimental results. The accuracy of prediction for each model is estimated by 1:1 line which represents 100 % accuracy. In the second step, the root mean square error (RMSE), mean absolute relative error (MARE), determination coefficient (R2), obtained with least mean square method, and the index of agreement d are used as evaluation criteria. Finally, the Bland–Altman plot comparison is made for each case. The plot of 1:1 line of both experimental and predicted values obtained by ANN and FL models for tensile strength, elongation at rupture and Hot Set Test is presented in Figs. 11a, b, 12a, b and 13a, b, respectively. The linear least square fit line, its equation and the R2 values are also shown in these figures. The R2 and d measure the degree of linearity between two variables. The index of agreement was proposed by Willmot [44] to overcome the insensitivity of R2 to differences in the measured and predicted means and variances. The range of d is similar to that of R2 and lies between 0 (no correlation) and 1 (perfect fit). R2 and d are defined as follows:  Pn 2 P P n i¼1 Ti Pi  ni¼1 Ti ni¼1 Pi  P  R ¼ P P P n ni¼1 Ti2  ð ni¼1 Ti Þ2 n ni¼1 P2i  ð ni¼1 Pi Þ2 2

ð7Þ Pn

2

½ P i  Ti  d ¼ 1  Pn  i¼1   2     i¼1 Pi  T þ Ti  T Number of iteration

Fig. 10 Variation of RMS error for the training data with the number of iterations

ð8Þ

where Pi is the predicted value of property by the chosen model, Ti is the target value (measured value of the property), and T is the average of the measured values in each case.

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(b) 25

25 24 23

Y=0,95664X+0,83991

2

2

R = 0,9273

R = 0,9302

22

Fit of ANN

Fit of ANN Y=0.92331X+1.58751

Fit of FL Y=1.07926X-1.7772

Predicted values

(a)

Predicted values

Fig. 11 Linear fit of experimental and predicted values of tensile strength: a 80 °C and b 120 °C

21 20 19 18

ANN Results FL Results

17

2

R = 0,9982

20

Fit of FL Y=0,86836X+2,31259

15

R²=0,9718

ANN Results

10

FL Results

16 15

5

15 16 17 18 19 20 21 22 23 24 25

5

10

(a)

(b)

650

Fit of ANN R²=0,9163

550

Y=0,68727X+169,72633

Fit of ANN

Predicted values

Fit of FL R²=0,8438

500 ANN Results FL Results

450 400 400

450

500

550

600

R²=0,9957 Fit of FL

500

Y=0,92408X+23,86727 R²=0,9528

400 ANN Results FL Results

300 200 200

650

300

(a) 160 R²=0,9810 Fit of FL Y=0,92928X+7,4539 R²=0,9897

100 80 60

ANN Results FL Results 60

80

100

120

140

Experimental values

The RMSE and the MARE provide different types of information about the predictive capabilities of the model. The RMSE and the MARE are used as a measure of the accuracy of the model, and they are defined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1X RMSE ¼ ð9Þ ½Ti  Pi 2 n i¼1  n   1X Ti  Pi   100 MARE ¼ ð10Þ  n i¼1 Ti  in which n denotes the number of data set and Ti and Pi are the measured and predicted values of the chosen property for each case.

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600

700

Fit of ANN

180

Y=1,11141X-10,74905

Predicted values

Predicted values

120

500

(b) 200 Fit of ANN

140

400

Experimental values

Experimental values

Fig. 13 Linear fit of experimental and predicted values of Hot Set Test: a 80 °C and b 120 °C

25

Y=0,97181X+14,80391

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Y= 1,24864X-149,15578

Predicted values

Fig. 12 Linear fit of experimental and predicted values of elongation at rupture: a 80 °C and b 120 °C

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Experimental values

Experimental values

Y=0,93136X+8,67753 R²=0,9673

160 140

Fit of FL Y=0,9323X+9,60435 R²=0,9883

120 100

ANN Results FL Results

80 160

60

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100

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The obtained results of statistical parameters for ANN and FL models, and for each studied property, are listed in Tables 5, 6 and 7. A visual inspection of Figs. 11, 12 and 13 reveals that both ANN and FL models are very close to the experimental results. The linear least square line fits the points in an adequate way with a relevant determination coefficient (R2) which is close to the unit. Besides, both ANN and FL results are very close to each other; there exists a slight difference. This observation is reinforced by the close values of R2 and d obtained in the case of ANN and FL. Furthermore, the depicted statistical parameters in Tables 5, 6 and 7 reveal that in 50 % of cases the ANN

Author's personal copy Neural Comput & Applic Table 5 Tensile strength statistical parameters of proposed ANN and FL models Statistical parameters

ANN

FL

80 °C

120 °C

80 °C

120 °C

R2

0.9273

0.9982

0.9302

0.9718

d

0.9771

0.9986

0.9655

0.9875

RMSE MARE

0.3364 0.9371

0.2894 1.4821

0.4472 1.8931

0.8298 4.9634

Table 6 Elongation at rupture statistical parameters of proposed ANN and FL models Statistical parameters

ANN 80 °C

FL 120 °C

80 °C

120 °C

R2

0.9163

0.9957

0.8438

d

0.9329

0.9987

0.9371

0.9843

RMSE

22.2756

7.9608

15.9634

27.0543

MARE

2.8412

1.4287

1.8785

4.7918

0.9528

Table 7 Hot Set Test statistical parameters of proposed ANN and FL models Statistical parameters 2

R

ANN

FL

80 °C

120 °C

80 °C

120 °C

0.9810

0.9673

0.9897

0.9883

d

0.9878

0.9909

0.9956

0.9959

RMSE

3.9625

5.7491

2.1677

3.8367

MARE

1.9963

2.2483

1.5573

2.6777

model gives the best results and the FL model gives the best results in the other 50 % of cases. Considering the determination coefficient (R2), ANN model gives the best results in three cases which are: prediction of tensile strength at 120 °C and prediction of elongation at rupture at 80 and 120 °C. This model represents in these cases 2.72, 8.59 and 4.5 % as improvements over FL model. However, the FL model gives best performances of prediction as for the remaining cases. FL model has an improvement of 0.31 % in the case of tensile strength prediction at 80 °C over ANN model. In the case of Hot Set Test prediction at 80 and 120 °C, it represents improvements of 0.89 and 2.17 %, respectively. In terms of agreement index d, the obtained results are consistent with R2 and show that the two models seem to have the same

quality of prediction. From this point of view, ANN model is better than FL model in the following cases: prediction of tensile strength at 80 and 120 °C and prediction of elongation at rupture at 120 °C. In all these cases, ANN model has improvements of 1.2, 1.12 and 1.46 %, respectively. On the other hand, FL model gives the best results in the following cases: prediction of elongation at rupture at 80 °C and prediction of Hot Set Test at 80 and 120 °C. In a quantitative way, FL model represents here improvements of 0.45, 0.79 and 0.5 % for different cases, respectively. Furthermore, the obtained results for root mean square error (RMSE) follow the same trend as the index of agreement d. From Tables 5, 6 and 7, ANN model presents also the best quality of prediction in three cases which are: prediction of tensile strength at 80 and 120 °C and prediction of elongation at rupture at 120 °C. In these cases, ANN model has, respectively, 24.78, 65.12 and 70.57 % improvements over FL model. Indeed, FL model has the best quality of prediction in the rest of cases, namely prediction of elongation at rupture at 80 °C and prediction of Hot Set Test at 80 and 120 °C. In these cases, FL model has improvements over ANN model of 28.34, 45.29 and 33.26 %, respectively. For mean absolute relative error (MARE), the depicted results in Tables 5, 6 and 7 indicate that ANN model has superior performances than FL model in the majority of studied cases. ANN model has superior performances in 66.66 % of cases versus 33.33 % of cases for FL model. The ANN model achieved 50.5, 70.14, 70.18 and 16 % improvements over FL in the cases of tensile strength prediction at 80 °C and 120 °C, elongation at rupture prediction at 120 °C and Hot Set Test prediction at 120 °C, respectively, while FL model achieved 33.88 and 22 % improvements over ANN model in the cases of elongation at rupture prediction at 80 °C and Hot Set Test prediction at 80 °C, respectively. In order to evaluate more the relative performances of ANN and FL models, Bland–Altman plots are illustrated in Figs. 14a, b, 15a, b and 16a, b. Bland–Altman plot is very simple to do and to interpret. It is considered to be a substitute for correlation and regression analysis. The X-axis shows the mean values of the property, and Y-axis represents the difference between the measured and predicted values. From all these figures, it can be concluded that the Bland–Altman plots are also consistent with earlier findings. It is visible in Fig. 14a, b that variation between measured and both ANN and FL predicted values of tensile strength is within ±1 N/mm2 at 80 °C and ±1.5 N/mm2 at 120 °C. This result is in good agreement with the small finding values of RMSE and MARE in these cases (Table 5). So both models can predict tensile strength with a greater accuracy and prove to be best techniques for modelling and prediction. For elongation at

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Author's personal copy Neural Comput & Applic

(b)

2,0 ANN FL

1,5 1,0 0,5 0,0 -0,5 -1,0 -1,5 -2,0

16

17

18

19

20

21

22

Difference between the measured and predicted values (N/mm²)

(a)

Difference between the measured and predicted values (N/mm²))

Fig. 14 Bland–Altman plot of tensile strength: a 80 °C and b 120 °C

2,0 1,5

ANN FL

1,0 0,5 0,0 -0,5 -1,0 -1,5 -2,0

8

12

Tensile strength (N/mm²)

60 ANN FL

40 20 0 -20 -40 -60 450

500

550

600

650

60

20 0 -20 -40 -60 200 250 300 350 400 450 500 550 600 650

Elongation at rupture (%)

10 8 6 4 2 0 -2 -4 -6 -8 -10

(b)

ANN FL

60

80

100

120

140

160

Elongation (%)

Difference between the measured and predicted values (%)

Difference between the measured and predicted values (%)

(a)

rupture, the Bland–Altman plot (Fig. 15a, b) reveals that the error values of prediction by both ANN and FL models are ±50 % for aging temperature 80 °C and vary between -60 and ?30 % for aging temperature 120 °C. This large dispersion of values explains well the big values of RMSE and MARE illustrated in Table 6. From Bland–Altman plot in Fig. 16a, b, it is observed that measured and predicted values of Hot Set Test present about ±10 % of difference. This result is consistent with those displayed in Table 7.

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ANN FL

40

Elongation at rupture (%)

Fig. 16 Bland–Altman plot of Hot Set Test: a 80 °C and b 120 °C

20

(b)

Difference between the measured and predicted values (%)

(a)

Difference between the measured and predicted values (%)

Fig. 15 Bland–Altman plot of elongation at rupture: a 80 °C and b 120 °C

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Tensile strength (N/mm²)

15 ANN FL

10 5 0 -5 -10 -15

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5 Conclusion The prediction of accelerated thermal aging of cross-linked polyethylene properties using neural networks and fuzzy logic has been investigated in this study. From the experimental results, it can be stated that mechanical properties, which are a good indicator of XLPE thermal deterioration, deteriorate regarding the aging time. Chain session and oxidation process are the main cause of the decrease of the mechanical properties. The degradation of the XLPE was

Author's personal copy Neural Comput & Applic

highlighted by a decrease in the cross-linking degree (increase of Hot Set Test) of the material. The use of neural networks and fuzzy logic models to predict the XLPE properties under thermal aging presents an alternative solution to the laboratory experiments requiring specialized materials and expertise. The proposed ANN and FL models reproduce the same nonlinear characteristics obtained in laboratory with acceptable error and give values very close to the experimental results. The use of ANN is motivated by its ability of learning from the experimental sets, and the use of FL is due to its capability of reasoning. We have pointed out in this study that both ANN and FL models have practically same performances and can effectively predict the insulation behavior under thermal aging. Taking into consideration the results of our investigations, it is suggested, as a form of future work, to use the developed ANN and FL models to predict the mechanical properties of the insulations for aging times not reached experimentally (generalization phase) to establish the lifetime of the materials and their temperature indexes. A development of combination neuro-fuzzy approach is also a part our future investigations on the intelligent prediction methods.

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