M. R. Mosavi
Assistant Professor, Department of Electrical Engineering
Behshahr University of Science and Technology,
Behshahr 48518-78413, Iran
GPS, well known as a versatile, global tool for positioning, has also become the primary system for distributing time and frequency. Fault locator systems measure the time of arrival of a fault-generated traveling wave at the time terminals using the precise timing signals from the GPS. Therefore, GPS timing accuracy is very important. In this paper, a Radial Basis Function Neural Network (RBF NN) is proposed for GPS receivers timing error modeling and perdition. It uses successive approximation, first obtaining a number of coarse approximations, and then optimally linearly combining the coarsely defined functions to achieve an accurate end result. The method is particularly very useful for modeling and prediction of GPS receivers timing error in the presence of Selective Availability (SA) noise. The experimental results on the collected real data using a low cost GPS engine are presented. It is shown that proposed RBF NN can improve GPS timing accuracy from 340nsec and 200nsec to less than 170nsec and 40nsec, before and after SA, respectively.
The Global Positioning System (GPS) is an earth-orbiting-satellite based navigation system. GPS is an operational system, providing users worldwide twenty-four hour a day precise position in three dimensions and precise time traceable to global time standards. GPS is operated by the United States Air Force under the direction of the Department of Defense (DOD) and was designed for, and remains under the control of, the United States military. While there are now many thousands of commercial and recreational civil users worldwide, DOD control still impacts many aspects of GPS planning, operation, and use .
Primarily designed as a land, marine, and aviation navigation system, GPS applications have expanded to include surveying, space navigation, automatic vehicle monitoring, emergency services dispatching, mapping, and geographic information system georeferencing. Because the dissemination of precise time is an integral part of GPS, a large community of precise time, time interval, and frequency standard users has come to depend on GPS as a primary source of control traceable through the United States Naval Observatory to global time and frequency standards .
The need to determine the location of faults is industry wide. Equipment based on traveling wave theory has been used for many years for this purpose and also for relaying. The application to fault location, however, dos not give adequate accuracy because such systems usually rely on measuring the impedance of the line (or its voltage); a quantity which is affected by load conditions, high grounding resistance, and most notably series capacitor banks. The mentioned problems would be eliminated if the measured quantity was the time of arrival of the traveling wave rather than the encountered impedance or the produced voltage. This, however, requires that highly accurate time signals produced by a single source (i.e. having the same reference) be available at all power system locations. Traveling wave fault locator systems use the time signals available (via satellite) from the GPS .
Since GPS is a military navigation system of U. S., a limited access to the total system accuracy is made available to the civilian users. The service available to the civilians is called Standard Positioning System (GPS) while the service available to the authorized users is called the Precise Positioning Service (PPS). User current policy the time accuracy available to SPS users is 340nsec and for PPS users it is 200nsec. Additional limitation Anti-Spoofing (AS), and Selective Availability (SA) was further imposed for civilian users. Under AS, only authorized users will have the means to get access to the P-code. By imposing SA condition, timing accuracy was randomly offset for SPS users. Since May 1, 2000 according to declaration of U. S. President, SA is switched off for all users. To make effective use of GPS timing information, it is essential to model these errors and to reduce these effects [4-8].
There are generally two kinds of approaches for nonlinear time series analysis and prediction. Statisticians usually invoke stochastic approaches while physicists often concentrate on deterministic approaches. Real world time series may be comprised of a low dimensional attractor (deterministic part of the time series) and high dimensional noise (stochastic part of the time series).
By building an appropriate model, it is possible to predict the short time future of the deterministic part of chaotic and other nonlinear time series. The ability of such a prediction depends on the noise level, complexity of the deterministic dynamics, the amount of data available, and the flexibility of the model selected. Radial Basis Function (RBF) approximation is a global interpolation technique with good localization properties. It provides a smooth interpolation of scattered data in arbitrary dimension .
In this paper, a RBF NN for GPS receivers timing errors modeling and prediction is presented. A three layers RBF NN and its training algorithm in order to this purpose is proposed. The model validity is verified with experimental data from an actual data collection using a low cost GPS engine. The experimental results by using the practical implementations are provided to illustrate the effectiveness of the model. This paper is organized in the following sections. Section II presents principle of GPS traveling wave fault locators. Section III describes GPS timing errors modeling using proposed RBF NN. The experimental results are reported in section IV. Conclusions are provided in section V.
II. Traveling Ware Fault Locator Principle
Fig.1 illustrates the basic principle of operation of the Fault Locator System (FLS).
Fig.1 : Basic fault locator principle
In Fig.1, length line is L . A FL remote is coupled to each end of this line via the high frequency tap from a Capacitive Potential Transformer (CPT). A FL remote is actually a fancy electronic stopwatch synchronized to the common timing standard of UTC from GPS. When a fault occurs at distance X from an end of the line, the resulting produces traveling waves. These transients, whit 2 to 5 microsecond leading edge rise-times, emanate towards the ends of the line at the speed of light (C) . The FL remotes time tag the transient arrival times to an accuracy of one microsecond. A microsecond time tagging accuracy will allow a FL accuracy to as good as 300 meters, the typical distance between transmission line towers. By knowing the line length L and the time-of-arrival difference (tb – ta ) , one can calculate the distance X , from substation A by using the well known FL equation:
Where ta and tb are end A and B arrival time, respectively. FL error results from three basic error sources shown in table 1.
Error Type Error Time Location Error Fault Detection Error 0.5 to 5 μsec 150-1500 m Time Tagging Resolution 0 to 0.1 μsec 0-30 m GPS Timing Error < 0.3 μsec < 100 m Table 1: Sources of FL errors 
III. GPS Timing Error Modeling using RBF
Fig.2 shows the topology scheme of RBF NN. This NN consists of three layers: one input layer, one hidden layer, and one output layer. RBF NNs have only one hidden layer with radial basis functions. The linear activation functions are at the output layer . The hidden layer is a nonlinear processing layer, generally consisting of selected hidden centers determined by input training set. In general, the connection weights from the first layer to the second layer are 1s.
Fig.2: The topology scheme of RBF NN with (N,M, L) structure.
In mathematic terminology, the i – th output of RBF NN can be written as:
Where D is the desired response vector in the training set. Time series forecasting analyzes past data and estimates of further data values. In other words, prediction attempts to model a nonlinear function by a recurrence relation derived from past values. The recurrence relation can then be used to predict new values in the time series, which hopefully will be good approximations of the actual values. In Fig.1 for prediction y(t) is equal x(t +1).
IV. Experimental Results
The method described above was installed into the firmware of a commercially available low cost GPS receiver module, the MicroTracker Low Power (MLP) manufactured by Rockwell Company. The actual data were collected on the building of Computer Control and Fuzzy Logic Research Lab in the Iran University of Science and Technology. Fig.3 shows scheme of designed and implemented hardware in this research.
Fig.3: Scheme of setup test
The following description summarizes the details the implementation by describing the parameters that the user can control for his particular application, and specifics on what important system parameters are made available to the user for observation and use by the end application. In order to the testing and debugging of the proposed method, most of the code was reported to work on-line on a personal computer using the Microsoft Visual Basic 7 Development Studio Software. To evaluate proposed method performance, Root Mean Square (RMS) was used. It averages the values of prediction errors avoiding the canceling of positive and negative terms. The equation for the error is given by :
Where T is the number of records in the data set. In preparing the training data, all input and output variables are normalized in the range [0,1] to reduce the training time. Observation at time t is applied to RBF NN input and the network must predict the value of instant t +1. Fig. 4 and 5 show SSP -UTOD (Sub Seconds Portion of UTOD) predictions for 200 test data by using proposed RBF NN, before and after SA, respectively.
Fig.3: SSP -UTOD 200 predictions using RBF NN with (4,4,1) structure (SA = on)
Fig.4: SSP -UTOD 100 predictions using RBF NN with (4,4,1) structure (SA = off )
To evaluate the performance of the presented training algorithms, they were tested by collected data sets. Table 2 show prediction errors (the difference between the predicted and real values) significance characteristics for 500 test data using proposed RBF.
Table 2: 500 prediction errors significance characteristics by using proposed RBF Parameters Error Value [nsec] (SA off)Error Value [nsec] (SA on) Max 103.7 238.5 Min -120.6-381.6 RMS 39.2 169.8 Average 0.985 0.279 Variance 0.0000000031 0.0000000578 Standard Deviation 1.75 7.60 Table 2: 500 prediction errors significance characteristics by using proposed RBF
A key element in traveling wave fault locator is a source of reliable precise time. The fault location is determined by accurately time tagging the arrival of the traveling wave at each end of the line and comparing the time difference to the total propagation time of the line. The time signal is obtained via satellite from the GPS. GPS is the only system available. It also has the reliability and availability necessary for power systems. In this paper, a new approach of using RBF NN for GPS receiver timing errors modeling and prediction has been proposed. The purpose of this paper is to present a model with high predictive ability good fitting accuracy. The proposed RBF NN was implemented on collected real data. An experimental setup was designed and implemented for this purpose. The experimental results emphasize that GPS timing error RMS can reduce from 340nsec and 200nsec to less then 170nsec and 40nsec by using proposed RBF NN prediction, before and after SA, respectively.
This research is supported by Iran University of Science and Technology grants.
- M. R. Mosavi, “A Practical Approach for Accurate Positioning with L1 GPS Receivers using Neural Network”, Journal of Intelligent and Fuzzy Systems, Vol.17, No.2, March 2006, pp.159- 171.
- P. H. Dana, “Global Positioning System (GPS) Time Dissemination for Real-Time Applications”, Journal of Real-Time System, Kluwer Academic Publishers, 1997, Vol.12, pp.9-40.
- H. Lee and A. M. Mousa, “GPS Traveling Wave Fault Locator Systems: Investigation into the Anomalous Measurements Related to Lightning Strikes”, IEEE Transaction on Power Delivery, Vol.11, No.3, July 1996, pp.1214-1223.
- J. B. Bullock, T. M. King, H. L. Kennedy, E. D. Berry, and G. Zanfino, “Test Results and Analysis of a Low Cost Core GPS Receiver for Time Transfer Applications”, IEEE International Symposium on Frequency Control, 1997, pp.314-322.
- G. J. Geier, T. M. King, and H. L. Kennedy, “Prediction of the Time Accuracy and Integrity of GPS Timing”, IEEE International Symposium on Frequency Control, 1995, pp.266-274.
- W. Lewandowski and C. Thomas, “GPS Time Transfer”, Proceedings of the IEEE, Vol.79, NO.7, July 1991, pp.991-1000.
- M. A. Lombardi, L. M. Nelson, A. N. Novick, and V. S. Zhang, “Time and Frequency Measurements using the Global Positioning System”, International Journal of Metrology, 2001, pp.26-33.
- M. R. Mosavi, K. Mohammadi, and M. H. Refan, “A New Approach for Improving of GPS Positioning Accuracy by using an Adaptive Neurofuzzy System, before and after S/A Is Turned off”, International Journal of Engineering Science, Iran University of Science and Technology, Vol.15, NO.1, 2004, pp.101-114.
- X. He and A. Lanedes, “Successive Approximation Radial Basis Functions Networks for Nonlinear Modeling and Prediction”, IEEE Conference on Neural Networks, 1993, pp.1997- 2000.
- M. A. Street, I. P. Thurein, and K. E. Martin, “Global Positioning System Applications for Enhancing the Performance of Large Power Systems”, CIGRE 1994 Session, 28 Augest-3 Septembers, pp.1-6.
- M. R. Mosavi, K. Mohammadi, and M. H. Refan, “Improving of Position Accuracy on GPS Using Fuzzy Logic and RBF Neural Network”, 4th Iranian Aerospace Society Conference, 25-28 January 2003, Amirkabir University, pp.271-279.
- M. R. Mosavi, “Increasing GPS Timing Accuracy using Recurrent Neural Network”, International Symposium on Geoinformation 2005, 27-29 September, 2005, Pulau Pinang, Malaysia.