IdentIfICAtIon of tHe PArAMeters of tHe CABle ProduCtIon ProCess

Trotsenko Yevgeniy, PhD, Associate Professor, Department of High Voltage Engineering and Electrophysics, National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Ukraine, е-mail: y.trotsenko@kpi.ua, ORCID: http://orcid.org/00000001-9379-0061 Brzhezitsky Volodymyr, Doctor of Technical Sciences, Professor, Department of High Voltage Engineering and Electrophysics, National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Ukraine, е-mail: v.brzhezitsky@kpi.ua, ORCID: http:// orcid.org/0000-0002-9768-7544


Introduction
Technologically, leading countries transferred or are transferring to the transmission of electricity using ultra highvoltage cables with polymer insulation. Cable pro duction is carried out on continuous technological lines under the influence of many destabilizing factors: transport delays, measurement noise, impossibility of exact coor dination of operation of all systems, fluctuations in raw material parameters and various physical and mechanical properties of cable components.
Obtaining objective information about the parameters and the course of the technological process makes it pos sible to improve the quality of the systems of optimal control, diagnostics and forecasting. A significant amount of scientific research has been devoted to the problem of parametric identification in the industrial production of cables [1][2][3][4][5][6][7][8][9][10][11][12][13]. However, in the modern theory of identi fication there is a significant gap between the theoretical part and the real situation. Owing to the natural nonsta tionarity of the processes, the large number theorem in practice is often not fulfilled, and in conditions of limited sampling, statistically optimal identification methods can lose not only optimality, but also correctness.
Thus, it is urgent to improve the methods of para metrical identification of the technological process of cable production and their introduction in adaptive optimal control systems. This will improve the quality of products in real conditions of parametric and signal uncertainty and will promote wider use of highvoltage cable with polymer insulation in the electricity industry.

the object of research and its technological audit
The object of research is the process of producing elec tric cables with polymer insulation for ultrahigh voltages.
Manufacture of cables with XLPE insulation is carried out on electrical complexes consisting of dozens of local systems interconnected via mobile cable products under the conditions of many undetermined disturbing factors [1][2][3]. The application of polyethylene insulation to a conductive core that moves at a speed of about 50 m/min is car ried out using a unit of three extruders 1 (Fig. 1). The outer diameter of each insulation layer is measured with random noise by the Xray sensor unit 2 at a distance of about 0.5 m from the exit of the extruders (Fig. 1). The thickness of each layer of insulation is regulated by automated electric drives of extruders by changing the rotation speed of worms. Improving the quality of control in the presence of noise measurements based on the concept of targeted identification [4] is an important task and for this sys tem helps to reduce the radial and axial displacement of insulation relative to metal living during its multilayer application. The scheme of the system for controlling the thickness (outer diameter) of the insulation layers under the conditions of destabilizing factors is shown in Fig. 2.
In Fig. 2: X d , X(t), X(t-t d ) -vector functions of given, actual, measured and predicted diameters; , ω(t) -vector functions of errors in the regulation of diameters, control actions and rotation speeds of worms of three extruders; N 1 (t), N 2 (t) -vector functions of the noise measure ments of the extruder speeds and the diameters of the insulation layers; β(t) -vector function of the identified parameters of the mathematical model of the connection ω(t) and X(t).
One of the most problematic places in the cable pro duction process is the presence in the loops of insula tion thickness regulation a time delay t d of about 0.5 s and noise measurements of the speeds of worms and the diameters of the insulation layers. The temporal delay negatively affects the accuracy of controlling the thickness of the layers and can lead to a loss of system stability, and the noisiness of useful signals causes a decrease in the accuracy and speed of regulation [5].

the aim and objectives of research
The aim of research is increase of the accuracy of ap plying multilayer polymer insulation of the technological process to the manufacture of a highvoltage cable in real conditions of uncertainty of the control object characte ristics on the basis of the parametric identification method.
To achieve this aim, it is necessary to perform the following tasks: 1. To develop a method of parametric identification, which in real conditions of noisy measurements of vari ables of the control object would give an estimate close to the exact values of the parameters.
2. To conduct studies on the properties of the proposed method of parametric identification in comparison with the traditional least squares method.

research of existing solutions of the problem
The identification theory establishes a mathematical model (MM) of the connection between the input vari ables of the investigated object ω(t) (causes, indepen dent regressors, control actions) and output X(t) (effects, dependent variables) by observing the behavior of the object in the passive or active experiment mode. Over the past century, there has been a significant development of methods and means of identification [6,7], which use modern electronic systems for data collection and proces sing (EDPS). This allowed to significantly increase the frequency of polling of sensors, the speed and accuracy of information processing, to increase the information content of data in timelimited samples. However, the natural properties of real objects (the continuity of matter and motion, the universal interconnection of everything with everything) do not allow to obtain MMs, which are identical (isomorphic) to the real object. The real object is not the autonomy, not the stationarity, but not the linearity of the interrelationships of the state variables, the infinite dimensionality, etc. The best EDPS is able to observe a limited set of state variables X(t) of the object in a limited range. Therefore, the MM only approximates the intercon nection of the components x t i ( ), i n = 1, , the ndimensional vector of the function X(t) and the vector function U(t) of the input (control) actions bounded in di mension m: In such conditions, the requirements to the systems of identification and management of industrial facilities have changed [4,[8][9][10][11][12][13]: the coordinated implementation of the functions of identification and management; use of current operational data of the facility; accounting features of the structure of the object; identification with obtaining esti mates of coefficients with the necessary statistical properties; multivariate identification with obtaining of several variants of differedtime estimates; general achievement of research and production goals of operation; purposeful hierarchically constructed, structuralparametric identification.
One of the ways to solve the problem can be methods, considered in [4,8,[10][11][12][13]. Under conditions of limited and the presence of the natural smoothness of the map f, the nonlinear nonstationary model (1) can be represented by an admissible error ε * t ( ) by a linear stationary system [4]: or their scalar representation: , .

X(t-t з )
Control system

Unit of extruders
Sensors of delayed diameters

Identification system
Speed sensors  (1)-(3) [4]. Then, as the best model (2), there will be one whose coefficients a b ij ik , are calculated by the least squares method for the exact data,  X t * , ( ) then formally dynamic models (2), (3) can be represented as regression ones. For example, each ith line of system (2) is represented as: in equation (3); k -the number of the discrete t k of time t, k m = 1, . Theoretically, the best estimate of the vector β of the parameters a b ij ik , will be the LS estimate for accurate measurement of variables: Thus, the results of the analysis of identification me thods allow to conclude that the achievement of the aim of the work is possible when using methods of targeted structuralparametric identification [4]. The problem con sists in constructing a parametric identification method that would give an estimate  β close to the least squares estimate (5) for accurate data in real conditions of noisy measurements X t * , ( ) Y t *( ) of object variables.

Methods of research
In practice, the least squares estimate must be ob tained from measurements disturbed by random obstac les N x and N y : Least squares estimate (5)  β of the vector β* for real data (6): Let's suppose that the obstacles N x and N y are Gaussian white noisecanceling noise and calculate the displacement Δβ of the estimate (7) with respect to the exact value (5): Let's introduce the notation: The estimate (7) is shifted relative to the true β* by the value (9). Assuming that the norm ||δA|| → 0,  β tends to β*; Δ  β → 0, under the condition ||δA|| → ∞ the estimate  β tends to zero, and Δ  β to -β*. The covariance of the estimate (7) under the conditions given above, and assuming that the norm N x T ε is much less than X *T or N Y x T * is approximately equal to [8]: where The first component of expression (10) (6) has a regularization property, similar to Tikhonov regularization. The latter consists in minimi zing the functional: where α -the regularization parameter. With the necessary minimum condition for the ex pression (11): let's obtain a somewhat lower in norm  β , but a regularized least squares estimate: Thus, comparing (8) and (13), it is possible to see that in the least squares Tikhonov parameter is equal to diag σ i 2 ⋅ M. The least squares estimate (7) are found as the coordinate of the minimum point of the functional ε Т ε. Since the functional is a square of ε, which is a mixture of a useful signal Y X rect [4]. This is due to the low accuracy of least squares estimates on short, highly noisy samples of the Y data, even for exact Х*.

research results
To improve the quality of identification, it is desirable to reduce the spread of the values of the functional I, almost without reducing its curvature and the coordinates of the extremum. This can be done by additional averaging over the set of quasistatistically independent functionals close to the mean square value for the exact data [4].
Such functional can be mean products , , let's obtain the following functional: where h q ( ) -weight function.
From the necessary condition for a minimum with re spect to β k , k =1,n of the exponent (14): let's obtain a system of equations: where A -the matrix n × n with elements a ik ; B -matrix column n × 1 1 with the elements b k : cr.

X X X
The parameters q and γ are optimized for the main (ex ternal) exponent I [6,11]. The parameter γ affects the width of the pulse h m ( ), and q its asymmetry relative to the maximum (Fig. 3).
A theoretical analysis of the unbiasedness and effec tiveness of the estimates (16) is given in [6], a numerical analysis is considered below on a concrete example.
The quality of parametric estimation, that is, the so lution of the inverse problem of mathematics [10], is af fected by the degree of interrelation of the variables x t i ( ), i n = 1, and not by their number. Therefore, let's confine ourselves to a simple example. The relation between ω*(t) and X*(t) (Fig. 2): where t d -an approximately known value of the delay in the measurements x*(t).
If the operator e e pt pt d d − = 1 is expanded in series and limited (due to a small t d with respect to the period of the change in the useful signal) by the first terms, that is: , relative to true β 1 , β 2 , 10 statistically independent realizations of noise are generated. The results of identifying the coefficients β 1 , β 2 for the least squares method and the proposed method are given in Table 1. The results of estimating the parameters in the situ ation of noisiness of only the original variables by the proposed method and least squares method are presented in Table 2.
As can be seen from Table 1, and β 1 and β 2 for least squares method are on average underestimated by 50 % (9). However, the regularization (13) takes place: the spread σ βi of estimates is 0.02 and 0.05. In the proposed me thod (Table 1), the estimates are almost unchanged: 1.005 and 0.943, but the spread is greater than in the regularized least squares method (0.15, 0.16). Reducing the spread is possible due to a compromise between displacement and dispersion by changing the parameters q and γ of the weight function h q ( ).
In the case of noise only in the original vari able (Table 2) (ideal situation for least squares method), the estimates are unbiased, but the spread of the estimates for this method (0.07 and 0.09) is greater than the spread (0.05 and 0.08) optimization of parameters q and γ of the function h m ( ). In the case that there is an opportunity to optimize h m ( ), the gain of the proposed method in the sense of unbiasedness and the effectiveness of estimates relative to least squares method is much larger.

swot analysis of research results
Strengths. The proposed method for identifying the parameters of the technological process of multilayer ap plication of highvoltage cable insulation in real condi tions of noisy measurements of control object variables in contrast to the traditional least squares method gives an estimate of parameters that is close to the exact values of the parameters. The introduction of the method will improve the accuracy of applying layers of polymer cable insulation to ultrahigh voltage.
Weaknesses. The developed method for identifying the parameters of the technological process of multilayer ap plication of insulation of a highvoltage cable has not been tested in the current production facility, which has a continu ous technological cycle. Conducting research in production is associated with additional financial and material costs.
Opportunities. A promising direction for further re search is the creation of a common method of operational identification for object control based on a purposeful hierarchically constructed identification system as an auxi liary function with respect to the main functional goal in problems of adaptive optimal control, diagnostics and forecasting. Improving the quality of the cable by sta bilizing technological parameters, estimating the values of technological indicators, predicting the values of pa rameters and technological variables, reducing the radial and axial displacement of insulation relative to the metal lived during its multilayer application, reducing the lon gitudinal vibrations of metal living in the process of dis placement will contribute to its wider implementation in electric power industry of Ukraine and other countries of the world.
Threats. The equipment and software of automated control systems for technological lines for production of highvoltage cables with polymer insulation is made by several wellknown foreign companies. The introduction of the research results is connected with the additional costs of the enterprise for the correction of the software of the existing automated control system.

Conclusions
1. A method for identifying process parameters for multilayered insulation is developed. In the actual condi tions of noisy measurements of control object variables, unlike the traditional least squares method, an ultrahigh voltage cable gives an estimate close to the exact values of the parameters. The introduction of the method will improve the accuracy of applying layers of polymer cable insulation.
2. Analysis of the research results shows that the pro posed method allows in a real situation of noisy measure ments of the input and output signals of primary conver ters to obtain unbiased estimates of parameters close to the estimates for least squares for accurate measurements. In the case of noise only in the initial variables (ideal situa tion for least squares method), the estimates are un biased, but the spread of the estimates for least squares method (0.07 and 0.09) is greater than the variance of the estimates in the proposed method (0.05 and 0.08). This makes it possible to effectively use it in systems of adaptive control of the thickness of layers of poly mer insulation in the production of cables for ultrahigh voltages. references