Thermal processes of the isolating materials in problems of nondestructive diagnostics of the main and power supply systems

Methods of calculation of thermal processes of the isolating materials are considered, in problems of nondestructive diagnostics of the main and distributive power stations. Theoretical justification of calculation of thermal processes of the multilayered isolating structure was given. On the basis of the analysis of results of operation, service life, physical processes of "ageing" and use of diagnostic parameters of the isolating structure characteristics of resistance of the equipment to refusal are offered. Time of refusal and a residual service life allow to predict using of dependences of refusal resistance.


Introduction
Temperature is one of the major factors defining wear and destruction of the isolating materials, power stations.If temperature becomes nearer to the most admissible, begins process of intensive thermal wear of the isolating material, its thermal aging.As a result can occur thermal breakdown of the isolating material.Factors of environment and electromagnetic processes complicate a task of the analysis of the thermal modes, from the point of view of heating engineers, on some orders.

Model of thermal processes
The mathematical model of thermal processes of the isolating material [1], doesn't consider a number of factors of the isolating materials influencing heating -dielectric losses, superficial effect in veins, losses in the screen, a way of screen grounding.In the work is considered the mathematical model of thermal processes of the power cable line (PCL) which isn't using empirical coefficients and dependences, but allowing a message calculation of temperature of the most heated point of isolation of a cable with a sufficient accuracy.Write down the system of the differential equations for thermal processes of single-core PCL considering specific factors where: θ1-θ5 -temperatures of uniform bodies, picture.2, °C; C1-C4 -thermal resistance of the corresponding uniform bodies; P1-P4 -losses of active power, respectively, in a current carrying vein, isolation of a vein, the screen and a protective cover.
On the basis of system (1) is synthesized the thermal equivalent circuit of the isolating material, Figure 1.We can note that θ 2 temperature of the most heated point of isolation of a cable (Figure 2).Thermal capacities of uniform bodies (Figure 2), the APV power cable of Pu of -1х30/25-10 C1-C3 can be found on formulas: Thermal resistance of uniform bodies of PCL R1-R3 pay off on formulas:

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Heat conductivity of metals considerably exceeds heat conductivity of dielectrics therefore the ratio is fair: R 1 ≪ R 2 ; R 1 ≪ R 3 .Losses of power, taking into account dependence of resistance of a vein of a cable on temperature, are equal where I -the current proceeding on a vein of a cable (A): Relcable vein resistance length of l (Ohm): ρ20 -the specific resistance of material at a temperature θ1 = 20 °C (Om.m); αthe temperature coefficient of resistance (TCR) of material (1/°C).
Losses in a cable vein taking into account the phenomenon of superficial effect which is shown in increase resistance of the conductor on alternating current of Rω in comparison with resistance on a direct current of R0, are equal: where I1 -the current proceeding on a cable vein, A; l -cable length.
The superficial effect in uniform cylindrical conductors is described as: where h -the radius of the conductor, especially cable vein radius; μ -relative magnetic permeability of material of a vein; γ -specific conductivity of material of a vein.Dielectric losses of P2 and P4 pay off on a formula P i = 2πU i 2 f • C i tgδ i , where i = {2; 4};   -isolation voltage; fworking frequency;   -isolation capacity; tgδ i --a tangent of angle of dielectric losses of isolation.Value of tension of   depends on a way of grounding of the screen of a cable.Generally it is possible to write down U 2 ̇= U ф ̇− U 3 ̇; U 4 ̇= U 3 ̇, where U3 -complex value of voltage on the cable screen concerning the earth.Isolation capacity for cylindrical capacity   = ( •   •  0 • )/((ℎ ./ℎ .), where hext.i,hint.i-external and internal radiuses of the screen; εi -relative dielectric permeability of material.Losses of power in the P3 screen depend on a way of grounding of the screen of a cable.If the screen is grounded on the one hand, current through it is approximately equal to zero.Losses of power in the screen pay off on the operating value of current in the screen I3,  3 =  3 2 •  20 (1 +  3 ( 3 − 20))(/ ((ℎ .3 ) 2 − (ℎ .3 ) 2 )) , where ρ20 -the specific resistance of the screen at a temperature of 20 °C; α3 -screen material PCL.Having considered a thermal equivalent circuit (Figure 1), we will receive expression for the established value of the temperature of cable veins: From comparison of ratios ( 4), ( 5), ( 7) and ( 9) it is understand that values of length of l also don't influence calculations for the set thermal mode.

Modeling of thermal processes
Data on the geometrical sizes of elements of the AP PU of -1х30/25-10 and their help data [2,3,5].Are reduced in the table 1. Design of a single-conductor power cable as system of thermal uniform bodies, it is presented on the Figure 2.

Results of modeling
Modeling of non-stationary thermal processes in isolation of a power cable was carried out in the system of Mathcad.On the picture 3 is taken dependence of process of heating of a cable, on layers.

Figure 3. Settlement chart of process of heating of the cable
The considered method of calculation of thermal processes can be applied to cables with any number of veins by increase in quantity of the equations in system (1).

Research of service life
Research of distribution of temperature and dynamics of its change it is executed in the program COMSOL MULTIPHY-SYCS complex on the basis of a method of final elements.Basic data for calculation are the parameters specified in Table 1.We received the gradient of temperatures on the section of a cable, picture 4.

Figure 4. Gradient of temperatures on the section
Results of calculation of the set mode (Figure 5) caused by the different current proceeding on a vein distinctly show weak, almost zero, dependence of distribution of temperature of highheat-conducting elements − a vein and the screen of a cable (horizontal regiments on schedules).Thus in volume of isolation is observed the big gradient of temperatures that leads to an essential difference of temperatures of the screen of a cable where temperature sensors and the central vein are installed.

Figure 5. Distribution of temperature in cable section at different temperatures on a vein
That is why , service life of the isolating materials of cables can be less than established 25 -30 years, as is confirmed in practice; the capacity calculated on design stages and actual can significantly differ.

Resistance to equipment failure under operating conditions
The operational resource time of life of t of the main and distributive power stations feeling a complex of various influences is described by dependence [2].
where the constant defining an initial condition of system, b -the basis of an exponential function, a-describes changing character of function in time.
Service life of power stations it is described by dependence of a look (11) Nature of change of the function described by a formula (11) depends on value a.If to accept the characteristic of resistance of object to refusal in the form of the  parameter, function (11) can be presented in the form where  -resistance to refusal in an initial state, t -flowing time;   -operating time to the full object;   -statistical service life in the field of the maximum values: -by commissioning object when t=0, value is equal 1, -by the ending of operation when t = t n , ot value is equal 0.
That is why, ()is Thermal processes of the isolating materials in problems of nondestructive diagnostics of the main and power supply systems Formation of defect in a time point of t1 and t2 (Figure 7) leads to reduction of resistance to refusal that will be expressed in decrease in  value on .Elimination of part of defects in the period of t3 leads to increase in relative resistance to refusal.Formation of defects at the time of t4 leads to reduction of resistance to refusal and at an invariance of regularity of aging -to definition of the predicted refusal time.where   -value of the measured diagnostic parameter,  rated value of diagnostic parameter, m -the indicator determined by a type of physical process in controlled object.[6] By the formation of several defects which values of diagnostic parameters are equal or exceed rated level, the indicator of  is defined as the sum of indicators   of each defect:  =  1 +  2 + ⋯ +   = ∑   , (18)

Conclusion
1.The considered mathematical model can be used as basic at calculation of thermal processes of the isolating materials in problems of nondestructive diagnostics of the main and distributive power stations in real time.
2. The mathematical model taking into account service conditions can be used in devices and systems of continuous diagnosing of power stations in thermal parameters.

Figure 2 .
Figure 2. System of uniform thermal bodies

Figure 6 .
Figure 6.Functions of resistance to refusal at formation of defect and a residual resource respectively

Figure 7 .
Figure 7. Functions of resistance to refusal by formation of defects and their partial elimination