Xiao-Bing Zhang，Yun-Hui Li，Guo-Zhi Fang

(1.School of Measurement and Communication，Harbin University of Science and Technology，Harbin 150080，China; 2.School of Electric and Electronic Teaching and Training Center，Harbin University of Science and Technology，Harbin 150080，China)

1 Introduction

With the rapid increase of non-linear loads in power grid，Power Quality keeps deteriorating［1－2］.Therefore，the accurate measurement of electric energy with distortion signals is more required. The prerequisite for accurate measurement of electric energy is to establish an accurate mathematical model of distortion signals in power grid.Scholars have carried out extensive，in-depth discussions and research on the electric energy measurement with stationary distortion signals.They have achieved some results，such as the fundamental and harmonic watt-hour meter［3－4］.They have also explored the electric energy measurement with non-stationary distortion signals and made some progresses［5－8］.These results were based on one or a couple oftypicalnon-stationary distortion signals withoutgeneralmathematicalmodelofdistortion signals，so they can not be applied widely to the electric energy measurement with non-stationary distortion signals comparing with wavelet，Kalman filter and fractal geometry.Wiener Functional is enable to describe the composition system in unified mathematics model.So the modelofdistortion signals was established with Wiener Functional in this paper.Also modeling non-linear system with Wiener functional series has been applied in many fields and achieved very good results［9－10］.In this paper，by analyzing the principle of Wiener functional series，it established the general mathematical model of distortion signals in power grid with Wiener-G Functional.The model was validated through the Matlab simulation.

2 Modeling of Power Grid Signals

2.1 Wiener Kernel Acquisition of Non-linear Systems

2.1.1 Finding the Wiener kernel of single system

According to Refs.［11－13］，if the input signal u(t)of non-linear power system is Gaussian White noise，by the nature of the Wiener series，we have

Zero-order kernel:

where i(t)is the output of system.

First-order kernel:

where i0(t)=i(t)-k0，Rui0(τ)is cross-correlation function of u(t)and i0(t).

Second-order kernel:

where i1(t)=i(t)-G0-G1(t)，G0and G1(t)is the items of Wiener series.

Third-order kernel:

where i2(t)=i(t)-G0-G1(t)-G2(t)，G0，G1(t) and G2(t).

Nth-order kernel:

In practical engineering applications， the following numerical means is used to calculate Wiener kernel.According to the time interval Tn，the x(t)，y(t)are sampled，we can get the N sequence

Then，the time correlation function can be used as an estimate of the global correlation function，that is to say

When calculating the correlation function，if x＜0，let x=0.Because of the argument symmetry in Wiener kernel(the interchangeability)，the calculation just need to be in the range of m1≥m2≥m3…and m＞0.

Through the above step，the Wiener kernel of non-linear system can be obtained.

2.1.2 Finding the Wiener kernel of composite system

When calculating the Wiener kernel of the composite system，we give power loads different weightings to simulate the actualpowersystem.The loadsare semiconductor rectifier，electric locomotive and electric arc furnace.The method can be expressed as

where D，S and H represent the percentage of loads in power system， respectively. The loads are semiconductor rectifier， electric locomotives and electric arc furnace.

From the non-linear transfer function theory of composite system，we have known，if an output of C system is the sum of two(F and G)or more systems.Then the Nth-order kernel of system can be represented as the sum of the same order kernel of subsystem in time domain or frequency domain. And then， combining the weighting method，we can get the kernel of system C.That is

where n=1，2，…;fn，gnand hnrepresent the N-th kernel of different loads in power system，respectively.

From the above theories，we have

where n=1，2，… kdn，ktnand kfnrepresent the N-th kernel of loads in power system，respectively.The loads are semiconductor rectifier，electric locomotives and electric arc furnace.

2.2 Wiener-G Functional Expression of Nonlinear System

2.2.1 Gaussian white noise input

If the input u(t)of non-linear power system is Gaussian White Noise，the output i(t)of system can be described as a mutually orthogonal of series，that is to say

where，Gmis the items of Wiener series，which is a function of input u(t)and Wiener kernel km(τ1，…，τm).The item is:

where A is the power spectral density of input white noise.

2.2.2 Non-Gaussian white noise input

If the input u(t)of non-linear power system is non-Gaussian White Noise，the output i(t)of system can be described as a mutually orthogonal of series，that is to say

where，Lmis the items of Wiener series，which is a function of input u(t)and Wiener kernel hm(τ1，…，τm).The items are:

3 Simulation and Verification

3.1 Finding the Wiener Kernel of Power Loads

3.1.1 Finding the Wiener kernel of electric arc furnace in power system

We take an electric arc furnace of steel factory for an example.According to the non-linear characteristic of electric arc furnace［14－15］，we build the simulation module using Matlab.The randomizer with zero mean and variance 1 is used to generated the Gaussian White Noise.On the Wiener kernel calculation，We utilize time method domain.By the simulation，we take the first three orders of Wiener kernel of electric arc furnace.That is

Zero-order kernel:kd0=2×10－4.is the output expectation of system，which is the DC output part of system.

First-order kernel:kd1(m)is impulse response of system，when we think the system is a linear(as shown in Fig.1).

Second-order kernel:kd2(m1，m2)is a non-linear factor of system(as shown in Fig.2).

3.1.2 Finding the Wiener kernel of a composite power system

Based on the above kernel of electric arc furnace and the kernel of semiconductor rectifier equipment and the kernel of electric locomotive obtained in the other paper［15］.From Eq.(1)，we can find the kernel of composite non-linear power system consisting of semiconductor rectifier device，electric locomotive and electric arc furnace.That is

Zero-order kernel:h0=0.2257 is the output expectation of system，which is the DC output part of system.

First-order kernel:h1(m)is the impulse response of system，when we seem the system as a linear system (as shown in Fig.3).

Fig.1 The first-order kernel of electric arc furnace load

Fig.2 The second-order kernel of electric arc furnace load

Fig.3 The first-order kernel of composite system

Second-order kernel:h2(m1，m2)is a non-linear factor of system(as shown in Fig.4).

3.2 Wiener Series Expression of Power Loads Output

3.2.1 Simulation of Wiener series expansion of output signal of electric arc furnace

(1)Input is Gaussian white noise signal

From Fig.5(a)，we can see that the second order output of simulation is similar to actual output signal， and from Fig.5(b)，we know that the error has a normal distribute.The variance is big but the means is close to 0.The whole fit is better.This suggests that this type of arc furnace can be approximated by its second order kernel under the condition of sinusoidal input.

Fig.4 The second-order kernel of composite system

Fig.5 Fitting output of Gaussian noise for electric arc furnace and the histogram of error frequency distribution

(2)Input is power frequency signal

From Fig.6(a)，we can see that Wiener functional series can be utilized to establish the model of the electric parameters for electric arc furnace under the condition of power frequency signal input.From Fig.6(b)，we know that the error has a normal distribute.The variance is big but the mean is close to 0.The whole fit is better.

Fig.6 Fitting output of power frequency signal for electric arc furnace and the histogram of error frequency distribution

3.2.2 Simulation of Wiener series expansion of output signal of composite power load

(1)Input is Gaussian white noise signal

From Fig.7(a)，we can see that the actual output and the simulated output of combined power system have few differences， butitis practicable for modeling.From Fig.7(b)，we know the error has a normal distribute.The variance is big but the mean is close to 0.The whole fit is better.

Fig.7 Fitting output of Gaussian Noise for electric composite system and the histogram of error frequency distribution

(2)Input is power frequency signal

From Fig.8(a)，we can see that the actual output and the simulated output of composite power load system coincide on the whole.From Fig.8(b)，we know the error has a normal distribute.The variance is big but the mean is close to 0.The whole fit is better.

So utilizing the Wienerfunctionalseries to establish the model of electric parameter is practicable.

Fig.8 Fitting output of power frequency signal for electric composite system and the histogram of error frequency distribution

4 Error Analysis

When calculating the Wiener kernel and Wiener-G functional of power load，it can cause error with a little amount of acquisition.In order to limit this kind of error，we applied the Hilbert reproducing kernel space theory in Refs.［16－17］.The references show that the method greatly simplifies the data computation.It becomes an important means for the author to limit errors.

Because the Gaussian White Noise of the Matlab/ Simulink simulation models is generated in computer， it is not the ideal Gaussian White Noise，and the error can not be avoid completely.We can use the high performance Gaussian White Noise generator to reduce this kind of errors.

When calculating the Wiener-G functional，due to the limited memory of computer，we use the pruning Wiener-G functional to approximate the actual signal.It can still cause error.For the contradiction between the volume of data storage and pruning wiener wiener-G functional，the first three items were selected to get a satisfied result.

From Refs.［18－19］，we know that different power level of input bring different error when we estimate kernel of power system，so each system has the best input power level.Many scholars are working on it.We will also pay attention to it in the future.

5 Conclusions

Based on the study of the characteristics of distorted signals in power grid and the principle analysis of functional series，the general mathematical model of power grid signal was established combining the Wienerfunctionalseries.The accuracy and rationality has been tested by Matlab.For the error of the model，we had analyzed the reason and had found the methods to decrease error.The model of power grid signal was proposed under the condition of distorted signals.It could describe not only the case of single distorted power signal，but also the case of distorted power signals.The description accuracy was improved，and the unified description problem of power grid signals under the condition of distorted signals was solved.However the modeling of strong time-varying non-linear system with the method produced a larger error.The approach to reduce the error still needs further research.The research is of great theoretical value and guiding significance for the electric energy measurement.

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