EuroWire January 2021

Technical Article

the cable. In particular, the length or duration of the impulse of the first zone matters the most, and the temperatures of the first two zones. Figure 4 shows the variables that were used as input variables for the nonlinear model. Circularity can be defined in different ways. We use the relative difference between the maximum and minimum diameters as the non-circularity. Other definitions lead to similar results and all those measures correlate strongly among themselves. The nonlinear model in the form of a feed-forward neural network with one hidden layer was developed from 44 observations. Even 30 experiments would have sufficed, but several experiments were carried out for demonstrations to customers. The statistical characteristics of the neural network model are as follows.

y

1

Output layer

Bias

Hidden layer

Bias

Input layer

x 1 x 2 x 3 x 4 x 5 x 6

rms err : 0.002371 mean |err| : 0.001707 rms % err : 2.3707 max |err| : 0.006616 Correlation : 0.7073

▲ ▲ Figure 3 : A typical feed-forward neural network

Conductor diameter [mm] Insulation thickness [mm] Impulse time [s] Impulse temperature [°C] Second zone temperature [°C] Third zone temperature [°C]

The quality of the model is better than the correlation coefficient indicates. The root mean square (rms) error of the model is 0.0024, with a maximum absolute error of 0.0066. The rms fractional error is only 2.4 per cent, which could be about the same as the repeatability of the experiments. The model was implemented in a LUMET system, a set of software components meant for easier use of nonlinear models. It has various facilities, including prediction of the output variables, plotting effects of input variables in several ways, determination of feasible solutions, optimisation, etc. Figure 5 shows the effect of impulse time on non-circularity for different values of impulse temperature.

Non-circularity

▲ ▲ Figure 4 : Variables used in the nonlinear models

nonlinear modelling, based on free-form nonlinearities, and do not require the knowledge of the type of the nonlinearities in advance. Nonlinear modelling is empirical or semi-empirical modelling that takes at least some nonlinearities into account. Nonlinear modelling can be carried out with a variety of methods. The older techniques include polynomial regression, linear regression with nonlinear terms and nonlinear regression. Feed-forward neural networks are among the newmethods that do not require a priori knowledge of the nonlinearities in the relations. Feed-forward neural networks are particularly attractive because of their universal approximation capabilities [2] , which make them very suitable for most function approximation tasks we come across in materials science and process engineering. Besides their universal approximation capability, it is usually possible to produce nonlinear models with some extrapolation capabilities with feed- forward neural networks. There are many different types of neural networks, and some of them have practical uses in process industries. Neural networks have been in use in industries for over 25 years [3] . The multilayer perceptron, a kind of a feed-forward neural network, is the most common one ( Figure 3 ). It has an input layer, usually one or two hidden layers, and an output layer. Several sectors of industries, including plastics [4] , rubbers [5] , cements [6] , concretes [7] , metals [8, 9] , medicalmaterials [10] , ceramics [11] , chemicals [12] , power generation [13] , semiconductors [14] and biotechnology [15] , benefit from them, either for materials development or for process development. The use of nonlinear models for cable production processes has also been reported [16, 17] . Nonlinear model of circularity Several variables, including conductor diameter, insulation thickness and line speed, as well as the temperatures and lengths of the different zones, affect the circularity of the cross section of

LUMET system Nonlinear Solutions Oy, Finland

Impulse set temperature

Non-circularity

Impulse time [s]

▲ ▲ Figure 5 : Effect of impulse time on non-circularity for different values of impulse temperature

LUMET system Nonlinear Solutions Oy, Finland

Non-circularity

Insulation thickness [mm]

Impulse time [s]

▲ ▲ Figure 6 : Effect of impulse time and insulation thickness on non-circularity

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January 2021