DYNAMIC LOAD MODELLING OF ELECTRIC LOCOMOTIVE FOR POWER QUALITY ASSESSMENT OF
Y. Shaobing, W. Mingli
Beijing Jiaotong University, China, Beijing, 100044, Email: email@example.com; firstname.lastname@example.org
Keywords: Electric locomotives, harmonics, load modelling, probabilistic model, power quality
Abstract The AC 25 kV electric railways with DC drive locomotives are one of the main harmonic pollution sources of electric power grids in China. To assess the power quality of the railway supply networks needs to model not only the fundamental currents of the trains but also their harmonic contents. It is difficult to simulate accurately the dynamic current waveforms of an electric locomotive running on real lines because there are too many random or uncontrolled factors that can intervene the working state of the main power circuit of the locomotive. This paper presents a modelling method which can be used to imitate the probabilistic property of the current waveform and can be embedded in the train simulator. This method is based on statistical analysis of a large amount of measured data of locomotive currents that were acquired on actual railway line. A computer program has been developed for the 8K locomotive, a kind of widely used locomotives for freight services in China. The program has been applied to assess the power quality of railway power supply utilities, which shows that this modelling method can give a very realistic result from the statistics point of view and has a fast speed in the simulation.
1 Introduction With the rapid development of main line electric railways in China, more and more electric locomotives and EMUs have been taken into service. Although a transition from DC drive to AC drive has been appeared since 2000, there are still many AC-DC locomotives running on the railways. The traction loads have become an important source of harmonic pollution to the public electric power utilities, especially near the heavy freight lines. Many researches have been conducted on the modeling of the load behaviors of electric locomotives [1, 2]. The traditional method that has been widely used is to solve a series of differential equations to obtain the current waveform according to the main circuit and control strategy of the locomotive. However, due to the deficiency of control details and a lot of random factors on the operation of a locomotive, there normally are some deviations between the calculation results and the real situations. Moreover, the
calculation of differential equations makes it ill-suited for the on-line simulation. Therefore the practical application of this approach is limited.
The probability method has been introduced into the railway traction systems in order to describe the harmonic distortion of substation feeders or to predict the harmonic currents generated by the locomotives or EMUs [3-5]. Based on statistical analysis of a large amount of measured data, this paper gives a modeling method of the harmonic characteristics of the 8K locomotive. With the traction calculation software, a dynamic interactive module has been built and can be used for the study of power quality problem of traction power supply systems.
2 Harmonic property The 8K locomotive, mainly used in the area of north of China, is a representative DC drive locomotive. The electric drive system of 8K locomotive uses a single-phase rectifier circuit with high-power thyristors and diodes. Its speed control is realized mainly by the voltage regulation in virtue of phase- control technique. The harmonic contents of the current mainly include the 3rd, 5th, 7th and 9th order.
The current waveforms of the 8K locomotive are determined by (1) the topology and parameters of the main circuit; (2) the control strategy of the drive system; (3) the auxiliary electricity usage equipment; (4) the supply voltage of electric railway; (5) the operation of the train. In an actual application, there are many random or unknown factors that can influence the current of a running locomotive. It is normally difficult to give an accurate simulation of the harmonic characteristics during the running process of a locomotive. Fig. 1 gives the data of the harmonic current ratio of 8K locomotive measured on a freight line. The dispersal of the points reveals that a simple determinant model cannot describe the probabilistic property of the current waveform.
(d) Fig.1 Scatter of harmonic current ratio
(a) 3rd (b) 5th (c) 7th (d) 9th
3 Probabilistic modelling 3.1 Harmonic current ratio 3.1.1 Curve fitting Fig.1 shows some kind of decay of the 3rd, 5th, 7th and 9th harmonic current ratio with the increase of fundamental current. The points scatter around a trend curve and present a certain probabilistic distribution. To model the dynamic harmonic characteristics of load currents, a method should be established to imitate the probabilistic distribution of the harmonic ratios. This can be realized by two steps. The first step is to find the trend curve of the harmonic ratio versus the fundamental current through curve fitting technique. The second step is to append an appropriate probabilistic property to it.
For the nonlinear curve fitting, some kind of nonlinear equations normally needs to be set up and solved, or an optimization method needs to be used for obtaining the necessary parameters. The method of nonlinear least squares is widely applied for curve fitting. This paper uses the piecewise curve fitting technique to consider the impact of circuit state on the current waveform, and this can improve the usability of the result.
When the sample space is too large or the scatter of points is too dense, the reasonable division and mathematical expression of the trend curve are difficult to judge according to the original data. It is better first to simplify the input samples for curve fitting.
Any continuous curve can be approached arbitrarily closely by a piecewise linear curve as long as each section become short enough. Hence, when the abscissa section x is very small, a linear function baxy can be used within this section. So the problem of curve fitting can be first changed into searching out the parameters of a series of linear regression. This needs to resolve the minimum value of equation
i ii ybaxbaf
where (xi, yi) is the point in section x and k is the number of
points. Let 0 b f
, then we can get the stationary point
220 )( ii iiii
0 )( ii iiiii
It is clear that ),( 00 baf is the minimum value of ),( baf . We can get the average value of xi within x
i ic kxx
1 /)( (4)
Then by the linear equation, we obtain
00 bxay cc (5) The point (xc, yc) can be used to substitute the samples within x to simplify the final curve fitting for the trend line.
Based on the above method, a series of simplified data can be figured out according to the original a large number of measured data. Then we can get the trend line of harmonic ratio, as shown in Fig. 2 for the 3rd harmonic ratio. It is obvious that the trend line should be divided into three pieces for curve fitting. The fitting results are listed in table 1.
Fig.2 The trend line of 3rd harmonic current ratio
Fundamental current (A) Formula R
<24.78 y=10.10+100.29*exp((1.74-x)/7.21) 0.98 <50.58 y=4.20+0.28*x 0.93
Else y=3.20+13.50*exp((53.71-x)/6.31) -5.23*exp((53.71-x)/1.79) 0.88
Table 1: Curve fitting result for the 3rd harmonic ratio
Similarly, the trend curve formulas for the 5th, 7th, 9th harmonic current ratio can also be obtained.
3.1.2 Probabilistic distribution The measured data of harmonic ratio present, in fact, a certain statistical property around the fitting trend curve. Within each small interval of fundamental current, a probability density can be calculated. Fig.3 shows the statistical result of the 3rd, 5th, 7th and 9th harmonic ratios for fundamental current 50~60A. The Probability Density Function (pdf) can be search out from these data. In fact, the famous Gaussian formula can be used to describe the distribution, as
2x xpdf (6)
where x is the dynamic harmonic current ratio, the expect value of the harmonic current ratio, the standard deviation.
Table 2 lists the fitting parameters of the normal distribution for Fig.3. The values of R2 (coefficient of determination) are
close to 1, and the chi-squares are small, so the fitting results are satisfactory.
(c) (d) Fig.3 Probability density of harmonic current ratio
(a) 3rd (b) 5th (c) 7th (d) 9th
Fig.3 Chi2/Dof R2 a 0.0003 0.83 11.18 23.60 b 0.0002 0.84 3.38 23.74 c 0.0001 0.95 3.29 3.18 d 0.0001 0.97 1.87 5.06
Table 2: Fitting parameters of Fig.3
3.1.3 Simulation method Through a linear transformation, Equation (6) can be changed into the standard normal distribution N(0, 1). According to the Central Limit Theorem, the following formula can be used to obtain random numbers subjected to N(0, 1) from uniform distribution numbers Xi generated by a computer program.
i i (7)
where the integer n should be large than or equal to 12 for a good approximation. Then the dynamic harmonic current ratio can be created by
zx (8) 3.2 Harmonic current phase angle distribution During operation the state of the main circuit of the locomotive varies ceaselessly, and is directly influenced by the running of the locomotive, such as accelerating, constant speed running, coasting and braking. In this process, the phase angles of the fundamental current and harmonic currents vary accordingly. Furthermore, these values are also influenced by the auxiliary equipment on board and the waveform of the supply voltage. Taking the fundamental voltage as a reference, based on a large number of measured data, the trend lines at the tractive running state can be obtained, as shown in Fig. 4 for the fundamental, 3rd, 5th and 7th currents. Similarly, the lines for the braking state can also
be obtained. The result shows that it is difficult to use simple expressions to describe these lines. But we can build a fast- computing model through a look-up table method. An observation shows that the distributions of the phase angles around these trend lines are also subjected to the Gaussian formula.
Fig. 4 Trend lines of current phase angle (a) fundamental (b) 3rd (c) 5th (d) 7th
4 Modelling and verification Based on the above analysis, we can establish a probabilistic model to simulate the dynamic harmonic currents of 8K electric locomotive according to the fundamental current. This model can be used to interface with the train traction calculation and the power flow calculation of power supply system. In this way, a dynamic interactive simulation platform can be built, as shown in Fig. 5. The data of calculation can be fast exchanged among different modules.
Fig. 5 Structure diagram of dynamic simulation
The train traction calculation uses a power-source model for the locomotives to take into account the interaction between the supply voltages and train currents. Through the power flow calculation of electric power supply system that includes the railway traction networks, the voltage and the fundamental current of each locomotive can be obtained. Then the fundamental current will be used to create harmonic currents according to the probabilistic model above mentioned.
(d) Fig. 6 Comparison of current waveforms
(a) 40A (b) 96A (c) 138A (d) 200A
A computer program has been developed, which can rapidly
Module of power flow calculation of power supply system
Module of traction calculation
Probabilistic model of harmonic current
IhU P Q
exchange data with the traction calculation module since no differential equation need to be resolved during simulation. As shown in Fig. 6, the simulated current waveforms are very close to the measured ones. This model describes well the load harmonic dynamics of 8K locomotive, especially from the statistic point of view.
This modelling method is useful for analyzing the harmonics in the public power utilities caused by electric railways. When evaluating the power quality, a statistical approach can give more believable and realistic results .
5 Conclusions This paper presents a load modelling method for the 8K electric locomotive based on the measured data. Through a statistical analysis, a probabilistic model that can take into consideration of not only the relation between the harmonic current ratios and the fundamental current but also the probability of their distribution has been established. Compared to the traditional methods, which need normally to find a solution of nonlinear differential equations of the main circuit of the locomotive, the calculation speed of this method is more fast and the result is closer to the actual situation. It can better reflect the harmonic dynamic details and is not affected by the lack of parameters of the main circuit or control strategy. A simulation software embedded with this model has been developed, which can be used for the power quality assessment of traction power supply systems. This method can also be applied to other type of locomotives or EMUs.
Acknowledgements The authors wish to acknowledge the financial support from the National Natural Science Foundation of China (under grant 60776830), the Ministry of Railway of China (2009J001-E) and the Society for Electrical Engineering.
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