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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 The task of EMC estimation subject to capacity of a casual noise disturbance and sluggishness of object - by squaring and smoothing is considered. It is shown that distribution low of the casual process after squaring and smoothing is beta-distribution. The solution of the problem is illustrated by an example of an estimation of permissible voltage unbalance. 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 where The heart of the problem is definition of density of distribution 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 Estimated value of the dose is calculated on the acceptable probability In the standard of the quality of the electric power in the countries of Commonwealth of Independent States, a value 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 To determine the statistical solutions, it is necessary to simulate an ensemble of a large number Each realisation where In practice, a transient comes to the end during the time 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 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)
is the time constant of the sluggishness of the object.
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.
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
.
of its excess by the solving of the equation
. (2)
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.
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.
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
.
-is squared, and the corresponding realisations of doses are calculated using Duamel integral:
(3)
is integration variable.
. 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.
, 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.