Modeling of thermal process in the energy system “Electrical network-asynchronous motor”

The paper discusses the influence of low-quality electricity on the temperature modes of operation of an asynchronous motor. In the course of experimental and analytical studies, the heat transfer coefficients and heat capacity of a particular electromechanical converter were determined. Experimental and analytical dependences of temperature changes of an asynchronous motor on time are given when it is connected to a supply voltage with different coefficients of sinusoidal distortion and negative sequence. The resulting model is tested for its adequacy to the real process and can be used as an element in the energy-economic model of an asynchronous motor to assess its uptime. This model that can be useful for simulation of thermal processes in asynchronous motors and optimising these devices for increasing the reliability.


Introduction
The presence of low-quality electricity in the shop networks of industrial enterprises leads to a decrease in the main indicators of the operation of asynchronous motors (AM), their accelerated physical aging, and, as a consequence, the occurrence of emergency situations. It is advisable to formulate this problem in the technical and economic plane, and its solution requires a detailed consideration of the system "electrical networkasynchronous motor" with the involvement of methods of mathematical modeling and implementation of computational experiments on a computer [1][2][3].
The economic assessment of various options for restoring electricity in shop networks to standard quality indicators is the basis proposed in [4][5][6][7] for a decisionmaking method for the operation of electrical equipment, including AM, operating in conditions of poor-quality supply voltage [8].
According to this methodology, according to the current indicators of the quality of electricity in the enterprise network [9][10] and on the basis of energy models [11][12][13] of the electromechanical converter, its energy indicators are calculated and the time interval of trouble-free operation.
In case of significant deviations of the indicators calculated in this way from the specified ones, various options for technical solutions for restoring the quality of the electric energy supplied to the engine are considered. For each of the options, a cost estimate is performed and a final decision is made on the conditions for its further work.
Wide experience in researching the effect of power quality on the operation of asynchronous motors with a has been accumulated by now [14][15][16].
Poor power quality in the workshops of industrial enterprises stipulates the increase in direct industrial costs due to the growing power consumption. Moreover, indirect costs related to the reduced operating life of electric machines are increasing as well.
As is known [10,11], normative operating life of the all-purpose asynchronous motors is about ten years. However, that is true only for the cases when certain conditions are observed. The main condition here is the correspondence of the thermal mode of an electric machine to the insulation class.
Deterioration of the power quality results in the increase of heating losses and insulation temperature respectively. Combined with the overloads, that results in the considerable reduction of the operating life of the electric motors. Practice shows that in terms of 40% of all-purpose AM with nominal voltage of 0.4 kV, the operating life is 1.25-2 years [17].
The aim of the paper is to synthesize a mathematical model of an asynchronous motor, taking into account the influence of changes in the quality indicators of electricity on heating and heat transfer processes, for an economically justified choice of protective equipment.
To study the effect of the operating modes of an electric motor on its thermal conditions, so-called thermal models are applied [18][19][20][21]. They are the equivalent circuits where electric losses act as the heat sources; temperatures of structural components are within the nodes; and corresponding heat conductivities and capacities are located between them.
The considered models have different degree of detalization. A single-mass model, in which an electromechanical transducer is represented as a single homogeneous body with the overall temperature, is the simplest one. Although, the real temperature distribution is not uniform: temperature of the AM stator winding may exceed the case temperature by 15-20°С [22,23].
More detailed models have minor prediction errors; however, that requires having additional data on heat conductivities and capacities of separate structural components of a motor. As a rule, such models are used only at the design stage. Besides, while applying those models, the transient-free thermal conditions are analyzed without consideration of their dynamics.
We consider that during the operation, it is the most expedient solution to use a single-mass thermal model; moreover, it is necessary to analyze the temperature of the AM component, being critical in terms of heating, -stator end winding -as the initial parameter of the model. It is well-known that this component is under the poorest cooling conditions since its thermal efficiency is effected mainly by means of the air.
A single-mass dynamic thermal model of the asynchronous motor is described by the following differential equation: here P ' is the power of heating losses generated in the electric motor; τ is the exceedance of the motor temperature over the surrounding temperature; Δτ is the increment of the motor temperature per time t ' ; A is the coefficient of thermal efficiency, J/(sec·С) (equal to the radiation heat loss per 1 sec in terms of the difference in the indicated temperatures τ = 1 °С); C is the heat capacity of the motor, J/°С. The indicated heat capacity is equal to the amount of heat required for AM heating by 1°С in terms of the nonavailable radiation heat loss.
As is obvious, equation of thermal balance (1) has two unknown values -A and C, which may be defined with the help of experimental data by composing a system of equations relative to the unknowns. In this context, it is possible to improve the accuracy of determining a coefficient of thermal efficiency and heat capacity of a motor at the expense of the totals of parameters measured in several experiments: Corresponding experiments have been carried out in terms of experimental workshop of Ukrspetsservis Ltd.
The motor is loaded on a direct-current generator of П31У4 type (nominal parameters are as follows: U n =230 V, P n =1.0 kW, n n =1450 rot/min, І n =4.3 А, η=75%). During the experiments, AM was heated under the nominal load; the cooling took place in terms of the non-rotating rotor.
A hole was made in the motor cover to determine the temperature of winding faces with the help of laser pyrometer of Fluke 568 type. The hole was open only for a short period for measuring (5 sec); when the electric motor was operating, the hole was closed to prevent the heat exchange between the internal and external air. Currents and voltages were recorded with the help of a mobile measuring and diagnostic complex based on the current sensors of LА 25A type, voltage sensors LV100P (made by LEM, Switzerland), and AD converter Е-440 (L-CARD, Russia). Table 1 shows the characteristics of the measuring channels. To eliminate the experiment error stipulated by the increased heating during the starting, the tested electric motor is accelerated with the help of a loading machine operating under the motoring conditions. Only when the facility reaches the idling speed, source voltage is supplied to the asynchronous motor, and a loading machine is placed in the dynamic braking mode (Fig. 1). Table 2 represents the results of the experiment of test motor heating in terms of ideal supply voltage. Fig. 2 shows the experimentally obtained curve of test motor heating in terms of ideal supply voltage.
Within the period of 62 minutes, the motor temperature has reached the final value of 76.3°С. The experiment results have made it possible to compose a system of equations (2) and to calculate the parameters of a single-mass thermal model. The parameters are as follows: coefficient of the motor's thermal efficiency while rotating is А=11.2 J/(sec×°С), heat capacity of the electric motor is С -12.1 kJ/°С.  Taking into account the fact that the reference literature contains rather scarce data on thermal parameters of the electric machines (as a rule, there is only the information concerning thermal time constants for motors of certain classes and power ranges), the considered method of their determination while identifying a specific AM model is rather topical.
Further, the heating experiments were carried out in terms of different degrees of distortion of the electric motor supply voltage. The experimental results are represented in Tables 3 and 4.  Further experiments #2-4 were carried out in terms of different degrees of distortion of electric motor power supply. The quality indices of the latter (coefficient of distortion of the sinusoidal voltage curve k U , coefficient of voltage unsymmetry on the reverse sequence ε 2 ) are given in Table 5. Experience #4 corresponds to the motor operation with the temperature exceeding the admissible one for that insulation class F(105°С); AM may be in such a state only for a short period of time due to the possibility of thermal breakdown of its windings.
The considered experiments have been used to test the adequacy of the proposed AM dynamic thermal model. Figures 3-5 show the comparison of the graphs of temperature exceedance of the motor over the surrounding temperature in those heating experiments with the calculated curves obtained with the help of electrochemical and thermal model of an asynchronous motor [24][25][26].   Next, error of the predicted temperature value in the heating dynamics was calculated. Fig. 6 demonstrates the experimental and calculated (predicted) temperature values for all the performed experiments which are used to test the model adequacy according to the method represented in [27][28][29]. In this context, different format of markers belongs to the corresponding experiments. The carried out test for the adequacy supposes obtaining of the following equation of linear regression: Here, ܻ , ܻ are the average values of the predicted and effective values; ‫ݎ‬ п is the coefficient of correlation between them; ߪ п , ߪ are the mean square deviations.
The indicated parameters were calculated according to the formulas: where L=57 is the volume of statistic sampling (number of the temperature measurements in all the experiments). The mean square absolute error of measurements was determined as: where ‫ݐ‬ is the Student's coefficient for the given reliability and number of freedom degrees ݇ = ‫ܮ‬ − 1. In the case under consideration, reliability was taken as ‫‬ = 0.05. Here, ߪ * is the residual mean square deviation calculated according to the formula: The mean square relative error of prediction was determined as follows: where max n Y is the highest value of the predicted one. Finally, the obtained values are as follows: σ Yef = 21.2 ºC, σ Yп = 20.9 ºC, r YefYп = 0.99, σ* Yn = 2.34 ºC, ∆Y n = 0.28 ºC, δ Yn = 3.2%.

Conclusion
The obtained results show the adequacy of the proposed thermal model of an asynchronous motor operating in the mains with poor quality power. Taking into consideration the fact that in terms of many motor types, reference literature does not contain the required data on the coefficients of thermal efficiency and thermal capacity, and only thermal constants of time are given for certain motor types, values of the specified parameters of the model may be obtained basing on the methodology represented in the paper.