FIFTEENTH INTERNATIONAL WROCLAW SYMPOSIUM AND EXHIBITION

SQUARING AND SMOOTHING IN EMC MODELS: A STATISTICAL SOLUTION

Eduard G. Kourennyi*, Viktor A. Petrosov**, Natalia N. Pogrebnyak Fifteenth International Wroclaw symposium and exhibition: Electromagnetic compatibility 2000.- Wroclaw: National Institute of Telecommunications, 2000, part 1.- P. 322-325. *
*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 y(t) 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 Y _T(t) 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

Tç ^/_T+ç _T=y² (1)

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

The heart of the problem is definition of density of distribution ¦ (Y _T) or distribution function F(Y _T) based on which the peak value Y _Tmax of the dose is calculated (excess of Y _Tmax is possible with the given probability E_X). 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 y=x 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 [y] of the noise disturbance is standardised, and purpose of the study is determination of "an inertial maximum" .

The requirement of EMC is that y_TmaxR
²y³ .

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

F(ç _Tmax)=1-E_x , (2)

In the standard of the quality of the electric power in the countries of Commonwealth of Independent States, a value E_X = 0,05 is accepted. Since objects with different time constant of sluggishness can be connected to electric power supply network, a dependence of inertial maximum on T 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 y(t) is formed by sum of a large number of n "elementary" processes (n = 100 ¸ 1000) of the simple form. The elementary process has a mean value equal to zero and correlation function (KF) n times smaller than the desired KF k_y(t ) of the simulated process. The mean value y of a process y(t) is added to y(t) 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 N of realisations of process y(t). Fig.1 shows a sample of 5 of N = 500 realisations of normal process of negative sequence voltage changes with mean value

y=1,6%, standard deviation s _y=0,5% and exponential correlation function having time of correlation t _k =5c.

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

(3)

where x , is integration variable.

In practice, a transient comes to the end during the time
t_p = (5 – 6)T . When, due to nature of the task, it is enough to compute only a distribution function, simulation stops at t = t_p. If. in addition, it is necessary to calculate KF of doses over the range of argument from 0 to t _max, the time of simulation is equal to t_p + t _max . In Fig. 1b, five realisations of doses are shown at T = 10 S. Realisations of doses correspond to realisations in Fig. la.

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 y(t), it is necessary to check reproduction of distribution function of process y²(t), which can be determined analytically from f(y) or F(y). The additional check at T = 0 ensures quality of simulation of doses at T > 0, as the smoothing reduces the error.

time (sec)

Figure 1.

3. STATISTICAL DECISION

The desired laws of statistical distribution are found from the corresponding cross-section of the ensemble of realisations Under stationary conditions, it is sufficient to consider only one cross-section (circles in Fig. 1 b).

Only the mean value of a dose is known a priori:

. (4)

Based on ensemble of realisations, it is possible to calculate the distribution moment coefficients of any order as well as to select approximation by multi-parametrical Jonson's or Pirson's formulas. However the accuracy of definition of the moment coefficients for orders higher than two is poor. Therefore it is necessary to limit the calculation of the standard deviations of doses s _T and to select two-parametrical approximation law. Generally, the range (Y _Tm, Y _TM ) of possible values of doses is finite. The numerous simulations showed that it is possible to accept for approximation beta-distribution with parameters expressed through

quantities Y and s _T by the formulas:

, (5)

Distribution density of doses is:

(6)

where G (x) - gamma- function .

For a telegraphic signal with impulse value of ± B and the same central tendencies of duration (t₁=t₂), equation (6) coincides with the known solution, provided that Y _Tm = -B, Y _TM = B, c =2B . However there is no analytical solution at t₁ ¹ t₂. Fig. 2 shows statistical density of distribution (curve 1) and curve 2, calculated on the formula (6) for t₁/t₂ = 0,7 and a T = 4,8.

In this specific case of the normal distribution of the reaction, the doses are confined only by lower limit Y _Tm = 0. In addition to the average mean value, the standard deviation of doses is also known in this case.

Here it is possible to use gamma-distribution with parameters s=Y ²/s ²_T , l =Y /s ²_T and distribution density

. (7)

4. CONCLUSIONS

For the solution of the SS-problem, it is necessary to use methods of simulation of casual processes as an ensemble of realisations.

Generally, it is expedient to apply beta-distribution as an approximation of the statistical distribution laws for doses.

For the normally distributed reactions, acceptance of the hypothesis about gamma-distribution of doses gives the analytical solution.

The application of Edgeworth series and hypothesis about normal distribution of doses is not recommended.