SQUARING AND SMOOTHING IN EMC MODELS: A STATISTICAL SOLUTION

Eduard G. Kourennyi*, Viktor A. Petrosov**, Natalia N. Pogrebnyak*

*Chair of EPG, The Donetsk State Technical University, Artema St., 58, Donetsk 830000 Ukraine, led@donntu.ru

**Chair of EPP, The Priasovsk State Technical University, Republic alley., 7, Mariupol 87500 Ukraine

1.Problem definition

A dynamic EMC-model of an object usually includes a linear filter, a squaring and smoothing (SS) unit and a statistical analysis unit: for example the flickermeter (IEC, Publication 868, 1986). The filter simulates reaction of object to noise disturbance, and the SS unit takes into account the fact that the consequences of EMC infringement depend on capacity of reaction and sluggishness of object. We will conditionally refer to the ordinates of process after SS as doses (by analogy to a dose of flicker). The relation between the processes before and after SS is described by the differential equation

(1)

where is the time constant of the sluggishness of the object.

The heart of the problem is definition of density of distribution or distribution function , based on which the peak value of the dose is calculated (excess of is possible with the given probability). The right part of the equation is generally nonlinear, therefore the exact solution only exists for the special case of a telegraph signal. The approximate solution in expanded form of Edgeworth series is known, but the necessary initial information is inaccessible in practice.

The statistical solution of the SS problem by methods of simulation (synthetic sampling) is given in this report. For brevity the elementary EMC-model without the filter, when is considered. Such a model is applied to estimate an additional overheating of objects from non-sinusoidal and unbalanced voltage. In these cases the acceptable continuous value of the noise disturbance is standardised, and purpose of the study is determination of "an inertial maximum". The requirement of EMC is that .

Estimated value of the dose is calculated on the acceptable probability of its excess by the solving of the equation

. (2)

In the standard of the quality of the electric power in the countries of Commonwealth of Independent States, a value is accepted. Since objects with different time constant of sluggishness can be connected to electric power supply network, a dependence of inertial maximum on should be generally obtained.

2. SIMULATION OF THE DOSE

There are various methods for simulation of casual processes. In systems of power supply, which carry group of noise disturbance sources, it is expedient to use summation of individual noise disturbances with casual shifts. When realisations follow the law of normal distribution, each realisation is formed by sum of a large number of "elementary" processes () of the simple form. The elementary process has a mean value equal to zero and correlation function (KF) times smaller than the desired KF of the simulated process. The mean value of a process is added to after summation of elementary processes and before the operation of squaring.

To determine the statistical solutions, it is necessary to simulate an ensemble of a large number of realisations of process . Fig. 1 shows a sample of 5 of realisations of normal process of negative sequence voltage changes with mean value , standard deviation and exponential correlation function having time of correlation .

Each realisation -is squared, and the corresponding realisations of doses are calculated using Duamel integral:

(3)

where is integration variable.

In practice, a transient comes to the end during the time . When, due to nature of the task, it is enough to compute only a distribution function, simulation stops at . If, in addition, it is necessary to calculate KF of doses over the range of argument from 0 to , then the time of simulation is equal to . In Fig. 1b, five realisations of doses are shown at . Realisations of doses correspond to realisations in Fig. 1a.

As simulation methods provide new knowledge, the requirements for the quality of simulation must be greater than for processing experimental data. It is also necessary to take into account, that the operation of squaring increases an error Therefore, in addition to checking the fidelity of reproduction of distribution function and KF of process , it is necessary to check reproduction of distribution function of process , which can be determined analytically from or . The additional check at ensures quality of simulation of doses at , as the smoothing reduces the error.