The simulation solution of nonlinear problems of ergodic electric-power processes transformation

. The algorithm for the simulation solution of the stochastic electric-power processes problems (quadratic inertial smoothing (QIS) and quadratic cumulative averaging (QCA)) has been stated. It includes simulation of initial process realizations, transformation of realizations and ensemble statistical analysis of these processes, as well as simulation quality check by the test problems. Peculiarities of skewness, kurtosis and rated maxima (quantiles) calculations based on the ensemble of the stochastic ergodic process realizations have been considered. Density and cumulative distribution function of the quadratic stochastic processes with normal distribution have been analyzed. Ensemble and single realization errors for rated maxima have been evaluated. It has been proposed to use test check with distribution quantiles when searching for simulation solution of QIS on the stage of quadratic transformation. Approximation of rated maxima using the gamma distribution law is given.


Introduction
The current standard for electromagnetic compatibility (EMC) [1] involves the principle of quadratic cumulative averaging [2] to take into account the inertia of objects. The rated maximum of half-hour load [3] also implements a cumulative principle. A simulation solution for a nonlinear problem has been proposed in [4] through the example of quadratic inertial smoothing (QIS) of stochastic electric-power processes. The inertial principle [5] is free from the disadvantages inherent in the cumulative principle of EMC evaluation. The standard [6] is based on the inertial principle of EMC interference evaluation. Some approximate approaches for solution of the problem have been considered in [7].
The detailed definition of numerical characteristics, cumulative function and density distribution for a random variable is given in [8][9] and for stochastic processes in [9][10]. In practice, as a rule, single realizations of stochastic ergodic processes are considered. The simulation solution implies the statistical manipulation of characteristics in the ensemble of stochastic process realizations. The presented material is based on the example of heating an object with the constant of inertia Т.

A heat model
The superheat temperature of the object and the current are related by the linear differential equation (2) is proportional to the squared effective value e I (in short − effective, rms) or rms e z of quadratic process z(t). They don't depend on Т.
In addition to the continuous permissible the maximum permissible temperature is standardized M ϑ  .
The rated maximum M T ϑ for stochastic process ) As a rule, a marginal probability E x is taken in the range from 0.05 to 0.001 (for example, in [1] it is taken equal to 0.05 for the voltage quality indicators). If the rated maximum of the temperature M T ϑ exceeds the M ϑ  , then the object must be turned off. Let us consider a probabilistic model of load variation [11][12]. The normal distribution law of the process I(t) is characterized by the mean I c and the standard deviation σ I . An exponential autocorrelation function (CF) for the electric-power processes is proposed in [2]: The parameter α is inverse to the correlation time τ k . An exponential-cosine CF is used in [12][13]:  Figure 1. A stationary stochastic process ) (t I * is an input to the model. QIS transformation is implemented using a squarer (block 1) and the 1-st order inertial link (block 2). A quadratic inertial process is an output of block 2. In case of need square-rooting is performed to obtain a reduced inertial process I eT* (t).

Test problems
Subject to the existence of an inertia there can only be a simulation solution of the problem, so it is advisable to set high requirements for the quality of initial stochastic process realizations, as well as to verify the simulation solution on basis of test problems.
A test problem (a check) is the special case, which an exact analytic solution can be found for. In regard to QIS transformation of stochastic electric-power processes the two test problems are given in [4]: a quadratic inertialess transformation and QIS transformation when the mean value of an initial process is equal to 0. In the first case an exact analytic solution is known after transformation (the process distribution law), in the second case -solution is given for moments of distribution.

Quadratic inertialess transformation
A quadratic process z(t) occurs at the output of block 1 (Fig.1). It is possible to verify the simulation quality at this stage. The calculation is performed for the ensemble simulation density and distribution function to compare them with the exact solution. It is also possible to evaluate errors of distribution moments though it is not necessary in this case.
The distribution density of a quadratic stationary stochastic process 2 I z = is given by the exact formula in [10]: The distribution density can be expressed through a hyperbolic function Considering quadratic transformation z ≥ 0.
Kolmogorov-Smirnov test gives evaluation in Ydirection of the distribution function ) (z F z , but evaluation of maximum rated values in accordance with practical confidence principle [14] is performed in Xdirection. The distribution can have a considerable length (Fig.2) in the area of the rated values (long «tails»). This can introduce a significant error. It is proposed to increase the accuracy in addition to the evaluation with Kolmogorov-Smirnov test further to test simulation solutions for quantiles of inertialess quadratic transformation.

QIS of zero-mean initial process
The second test check is run for T > 0. In this case the exact solution can only be found for the moments of a process distribution ) (t In theory the general formulae derived from [10] allow us to define the third and fourth central moments with any mean values of I c , but the result acceptible for practical application can be received only when I c = 0. Leaving out simple but cumbersome calculations according to the formulae (34.33) and (34.27), we give the finite expressions:

Algorithm for the simulation solution
The main statements of the method to obtain the simulation solution of the QIS problem have been given in [4]. Let us state the algorithm taking into account that it is mostly common to use reduced values which are equidimensional with the initial process, in instances, when power quality factors are evaluated (a generalized model for EMC estimation is given in [15]).
1. The simulation of ensemble initial realizations of a stochastic process.
2. The verification and correction of mean values and standard deviations in the simulated initial realizations.
3. The verification of the correlation function (CF) reproduction precision.
4. The verification of the initial process distribution law.
5. The quadratic transformation of the process realizations.
6. The verification of the ensemble quadratic process distribution law and ensemble quantiles.
7. The inertial smoothing of the quadratic process realizations.
8. Obtaining a reduced inertial process. 9. The calculation of the process realizations numerical characteristics (mean value, standard deviation, variance, 3-d and 4-th order distribution moments).
10. Averaging to obtain the ensemble numerical characteristics.
11. The ensemble skewness and kurtosis calculation. 12. The calculation of the simulation density and cumulative distribution function of process realizations.
13. Averaging to obtain the ensemble simulation density and cumulative distribution function.
14. The quantiles calculation (rated maxima) of quadratic inertial or reduced inertial processes (using the ensemble cumulative distribution function).

Ensemble characteristics
The proposed algorithm contains the calculation of process numerical characteristics. The single realizations calculation (symbol ~) and ensemble averaging (symbol ^) are used. Such characteristics as skewness and kurtosis are useful for the test problem solutions, while they are not the ultimate goal of calculations. The calculation of the skewness and kurtosis is performed with ensemble averaged values of standard deviation, 3rd and 4th central moments:

Approximation by the gamma distribution law
The density can be approximated by dint of gamma distribution when Т = 0: where Γ(η) − is complete gamma function determined by a dimensionless shape parameter η. The scale parameter λ has a dimension of ( ) The skewness and kurtosis are If the mean value of the initial process is zero, the densities calculted according to the formulae (12) and (20) are exactly the same. But when I c > 0, the distribution types are different (Fig.3). For the problem under consideration the quality of the approximation is evaluated based on how accurately the inverse distribution of a quadratic process is reproduced when E x = 0.05. The percentage error is calculated using a formula : ϑ is an exact solution according to (12), is an approximate solution according to (21). The calculation results for different mean values are given in Table 1. While approximating the error when defining the rated maxima can be both negative and positive.
Let us consider the gamma distribution characteristics in case of a nonzero inertia. The distribution density when Т > 0 is The parameters are defined according to the expressions:  The maximum and minimum ratio errors for the single realization rated maxima z M were −5.94% and 5.29%. The ratio error for the ensemble was 0.104%. It confirms the necessity to use ensemble of realizations of a stochastic process in the simulation solution of nonlinear transformations of the stochastic power processes, because even inertialess nonlinear transformation of the estimated maxima for individual realizations are reproduced with considerable error.
Simulation skewness and kurtosis of a quadratic inertial process ) (t T * ϑ taken over ensemble and realizations are given in Fig.5. The exact solution is calculated according to (17).
The distribution moments based on the single realizations differ greatly from the exact values. As it can be expected the dispersion for kurtosis is greater than the dispersion for skewness. The ensembleaveraged values of these characteristics are close to the exact values.
The simulation solution of the QIS problem is given in Fig.6 in the form of * T -characteristic. The simulation is performed for every value of I c‫٭‬ . The results of the approximation with the help of the gamma distribution based on the formulae (26-28) are marked with circles.
The quantitative meanings of maxima and approximation errors are partially shown in Tables 2 and  3. It is necessary to point out that the approximation errors of the process rated maxima when E x = 0.05 do not exceed 1.2% as per their absolute value. At the same time when E x = 0.001 the approximation quality essentially deteriorates. According to the experiment data there is no steady error reduction. The maximum error is 1.176% when E x = 0.05, and −14.562% when E x = 0.001.

Conclusion
To increase the simulation solution accuracy of the nonlinear problems of ergodic electric-power process transformation it is reasonable to fulfil the statistical analysis of ensemble realizations rather than single realizations. The rated maxima of the quadratic inertial process can be approximated by gamma distribution with a relatively low error when E x = 0.05.