A Novel Frequency Tracking Method Based on Complex Adaptive Linear Neural Network State Vector in Power Systems

This paper presents the application of a Complex Adaptive Linear Neural Network (CADALINE) in tracking the fundamental power system frequency. In this method, by using stationary-axes Park transformation in addition to producing a complex input measurement, the decaying DC offset is eliminated. As the proposed method uses first-order differentiator to estimate frequency changes, a Hamming filter is used to smoothen the response and cancel high-frequency noises. The most distinguishing features of the proposed method are the reduction in the size of observation state vector required by a simple Adaptive Linear Neural Network (ADALINE) and increase in the accuracy and convergence speed under transient conditions. This paper concludes with the presentation of the representative results obtained in numerical simulation and simulation in PSCAD/EMTDC software as well as in practical study.


Introduction
In some digital application systems, power system frequency tracking is an important task. Accurate power frequency estimation is a necessity to check the state of health of power index, and a guarantee for accurate quantitative measurement of power parameters such as voltages, currents, active power and reactive power, in multi-function power meters under steady states. Many researches have been done in this area. Three criteria that a frequency tracking method should satisfy is given as follows (Akke,1997): 1. Fast speed of convergence 2. Accuracy of frequency estimation 3. Robustness to noise. He compares traditional modulation with new modulation. Traditional demodulation introduces a double frequency component that needs to be filtered away. For signals with low noise, the filter to reduce the double frequency component can often limit the speed of the frequency estimation algorithm. The purpose of this section is to show that the proposed method eliminates this problem. If other filters are the bottle-neck of the estimation algorithm, we will not capitalise on the benefits. Many well-proven techniques such as zero-crossing technique, level-crossing technique, least squares error technique, Newton method, Kalman filter, Fourier transform, and wavelet transform have been used for power harmonic frequency estimation in the fields of measurement, instrumentation, control and monitoring of power systems. Besides, a comprehensive analysis of discrete Fourier transform (DFT) error is given in some researches, including the cases of synchronous sampling and error rises when sampling frequency does not synchronize with signal frequency. A frequency tracking method based on linear estimation of phase (LEP) has been introduced. Also, a processing unit for symmetrical components and harmonic estimation based on an adaptive linear combiner has been proposed. This section presents the application of a complex adaptive linear neural network (CADALINE) in tracking the fundamental power system frequency. In this method, by using stationary-axes Park transformation in addition to producing a complex input measurement, the decaying DC offset is effectively eliminated. As the proposed method uses a first-order differentiator to estimate frequency changes, a Hamming filter is used to smoothen the response and cancel high-frequency noises. The most distinguishing features of the proposed method are the reduction in the size of observation state vector required by a simple adaptive linear neural network (ADALINE) and increase in the accuracy and convergence speed under transient conditions. This section concludes with the presentation of the representative results obtained in numerical simulations and simulation in PSCAD/EMTDC software as well as in practical study. Figure 1 depicts the ADALINE structure to track fundamental frequency which is a proposed in this section.

ADALINE structure to track fundamental frequency
Equation (2) can be rewritten as: Rearranging the above equation in the matrix form, we obtain: Voltage amplitude and phase angle of N th harmonic are: Current amplitude and phase of N th harmonic are calculated as follows: To track frequency, a center frequency is assumed to be the actual value. It would be the operational frequency of the power system which is usually 50 Hz or 60 Hz. Under situations that the base power frequency changes, the k th sample of fundamental component of voltage or current signal is modeled: that can be rewritten as: where () Hi is the i th coefficient of the FIR Hamming window coefficients and 0 N is the sampling rate given by 0 . The cut frequency for the low pass Hamming window is 20 Hz and the length of filter is 40. Fig. 2 shows the impulse response of this Hamming window. By using DSOPC principle, 1 f is obtained as: 1 y′ and 2 y′ are first-order discrete derivatives defined as: Finally, the real value of fundamental frequency ( x f ) is calculated by adding the frequency deviation to the assumed center frequency as:

Complex ADALINE structure to track fundamental frequency
The proposed complex ADALINE (CADALINE) structure is based on the Widrow-Hoff delta rule, explained earlier. The improvement in ADALINE structure is made by introducing a complex observation vector. This approach reduces the number of weight updates, and so, the number of parameters to be estimated. To produce a complex vector measurement the use of the stationary-axes Park transformation is proposed. Stationaryaxes Park transformation is widely employed to study the behavior of rotating electrical machines in transient conditions. However, it can be considered a more general and powerful tool to study the behavior of three-phase systems. This transformation applied to the signals () , () ab ytyt and () c y t (voltages or currents) of a three-phase system leads to the Park components () , d y t () q y t and 0 () y t defined as: where [ ] T is the orthogonal matrix defined as: 211 366 11 0 22 In the d-q frame, it is then possible to define the Park vector as a complex quantity as: This vector is used as a desired value. The complex observation matrix Z is introduced by: 0 ω is the center angular frequency (rad/sec), defined as 00 2 f ω π = . The complex harmonic vector to be tracked at k th sample is () s kT Γ and is defined as: where 1 () s Ak T is the complex phasorial expression of center frequency in the d-q frame. According to LMS rule, weight update is: ek T is the complex error obtained as follows: where () ss Yk T is the complex estimation of the actual values of () s y kT in d-q frame.
It should be noted that under conditions where power system operates with the nominal frequency, 1 () s Ak T is a constant vector, which does not rotate with respect to the time in the complex frame. When the base frequency changes, 1 () s Ak T becomes a rotating vector. It is the result of the fact that when the base frequency changes, 1 () s Ak T components appear as modulated signals and their carrier is the occurred frequency-drift. Therefore, the rate of this rotation is the key element to track the frequency deviation from the center frequency. The frequency deviation ( 1 f ) is achieved by normalizing and differentiating 1 () s Ak T . For the types of power swing events studied here, it has been found that the non-fundamental components cannot be characterized as harmonics. A middle-filter is, therefore, required so that the signal is dominated by the fundamental component. The middle-filter, used here, is the FIR Hamming type filter as has been used in [18]. It is the same which has been used in Section 2. 1 () s Ak T passes through the FIR Hamming window and 1 () s Ah kT is obtained as: 1 () s Ahk T should be normalized to produce the rotating operator ( (  where () abs x stands for absolute value of x . By using the first-order discrete differentiator, 1 f is obtained as: It can be seen that observation matrix size and the parameters to be estimated have been reduced to ( 2 N + ) elements in comparison with the simple ADALINE which uses ( 2 2 N + ) elements. Furthermore, owing to the fact that data from three phases are combined, the most important aspect of the proposed technique is that the convergence speed is considerably improved. After all, decaying DC offset is effectively eliminated by applying stationary-axes Park transformation and using CADALINE. Fig. 3 shows the complex ADALINE structure to track fundamental frequency.
where 0 s T ψω = , 0 ω is the fundamental supply angular frequency in (rad/sec) and s T is the sampling interval in seconds.
and Q is a ( 1 (2n + 1) × ) matrix which gives the connection between the measurement ( k z ) and the state vector ( k x ). The sampled value of the signal is considered to be the sum of the real components of the harmonic phasors and the decaying DC component. Therefore, Q is given by: [1,0,1,0, ,1,0,1] Q = The harmonic components i h (RMS) are given by: ( 2 1) (2 ) 1,2, , 2 The problem of estimating the present state of the signal model (Eq. 31) from measurements ( k z ) involves the design of standard state observers [33]. The observer state can be represented by: where ˆk x denotes the estimate of the state vector k x and P is the observer gain matrix. The primary objective in choosing P is to obtain a stable observer, which is achieved by assigning the eigenvalues of the matrix FP Q − within the unit circle. The locations of the eigenvalues determine, among other things, the transient response of the observer. For the purpose of frequency tracking, the speed of response and tracking ability are of particular importance. After studying various choices, the following case is considered. www.intechopen.com To estimate fundamental frequency, the approach is based on DSPOC which has been described in Eqs. 15-20 is used.

DFT filter to track fundamental frequency
Under situation of frequency change, the k th sample of fundamental voltage or current signal is described as denoted in Eq. 15. By using a DFT dynamic window, parameters 1 x and 2 x in Eq. 16 can be achieved as follows: The fundamental frequency tracking process includes the same approach that has been expressed in Eqs. 15-20.

Adaptive linear element
ADALINE (Adaptive Linear Neuron or later Adaptive Linear Element) is a single layer neural network as the 'least mean square' (LMS) learning procedure, also known as the delta www.intechopen.com rule. It was developed by ProfessorBernard Widrow and his graduate student Ted Hoff at Stanford University in 1960. It is based on the McCulloch-Pitts neuron. It consists of a weight, a bias and a summation function.The difference between Adaline and the standard (McCulloch-Pitts) perceptron is that in the learning phase the weights are adjusted according to the weighted sum of the inputs (the net). In the standard perceptron, the net is passed to the activation (transfer) function and the function's output is used for adjusting the weights. The main functional difference with the perceptron training rule is the way the output of the system is used in the learning rule. The perceptron learning rule uses the output of the threshold function (either -1 or +1) for learning. The delta-rule uses the net output without further mapping into output values -1 or +1. The ADALINE network shown below has one layer of S neurons connected to R inputs through a matrix of weights W. This network is sometimes called a MADALINE for Many ADALINEs. Note that the figure on the right defines an S-length output vector a. The Widrow-Hoff rule can only train single-layer linear networks. This is not much of a disadvantage, however, as single-layer linear networks are just as capable as multilayer linear networks. For every multilayer linear network, there is an equivalent single-layer linear network.

Single ADALINE
Consider a single ADALINE with two inputs. The following figure shows the diagram for this network.
The weight matrix W in this case has only one row. The network output is: Equation a can be written as follows: 1,1 1 1,2 2 Like the perceptron, the ADALINE has a decision boundary that is determined by the input vectors for which the net input n is zero. For n = 0 the equation Wp + b = 0 specifies such a decision boundary, as shown below: Input vectors in the upper right gray area lead to an output greater than 0. Input vectors in the lower left white area lead to an output less than 0. Thus, the ADALINE can be used to classify objects into two categories. However, ADALINE can classify objects in this way only when the objects are linearly separable. Thus, ADALINE has the same limitation as the perceptron.

Networks with linear activation functions: the delta rule
For a single layer network with an output unit with a linear activation function the output is simply given by: Such a simple network is able to represent a linear relationship between the value of the output unit and the value of the input units. By thresholding the output value, a classifier can be constructed (such as Widrow's Adaline), but here we focus on the linear relationship and use the network for a function approximation task. In high dimensional input spaces the network represents a (hyper) plane and it will be clear that also multiple output units may be defined. Suppose we want to train the network such that a hyper plane is fitted as well as possible to a set of training samples consisting of input values p d and desired (or target) output values p d . For every given input sample, the output of the network differs from the target value p d by ( ) pp dy − where p y is the actual output for this pattern. The delta-rule now uses a cost-or error-function based on these differences to adjust the weights. The error function, as indicated by the name least mean square, is the summed squared error. That is, the total error E is denoted to be: Where the index p ranges over the set of input patterns and p E represents the error on pattern p . The LMS procedure finds the values of all the weights that minimize the error function by a method called gradient descent. The idea is to make a change in the weight proportional to the negative of the derivative of the error as measured on the current pattern with respect to each weight: Where ppp dE δ =−is the difference between the target output and the actual output for pattern p .The delta rule modifies weight appropriately for target and actual outputs of either polarity and for both continuous and binary input and output units. These characteristics have opened up a wealth of new applications.

Simulation results
Simulation examples include the following three categories. Numerical simulations are represented in Section 5.1. for two cases, simulation in PSCAD/EMTDC software is presented in Section 5.2. Lastly, Section 5.3. presents practical measurement of a real fault incidence in Fars province, Iran.

Simulated signals
Herein, a disturbance is simulated at time 0.3 sec. Three-phase non-sinusoidal unbalanced signals, including decaying DC offset and third harmonic, are produced as: After disturbance at 0.3 sec, signals are: www.intechopen.com where 0 ω is the base angular frequency and x ω is the actual angular frequency after disturbance.

Case 1
In this case, a 1-Hz frequency deviation occurs and tracked frequency using CADALINE, ADALINE, Kalman, and DFT approaches is revealed in Fig. 4; three-phase signals are shown in Fig. 5. Estimation error percentage according to the samples fed to each algorithm after frequency drift is shown in Fig. 6. Second set of samples including 100 samples, equivalent to two and half cycles, which is fed to all algorithms is magnified in Fig. 6. It can be seen that CADALINE converges to the real value after first 116 samples, less than three power cycles, with error of -0.4 %; and reaches a perfect estimation after having more few samples. Other methods' estimations are too fare from real value in this snapshoot. DFT, ADALINE and Kalman respectively need 120, 200 and 360 samples to reach less than one percent error in estimating the frequency drift. It should be considered that for 2.4-kHz sampling frequency and power system frequency of 60 Hz, each power cycle includes 40 samples. The complex normalized rotating state vector 1 () s An kT with respect to time and in d-q frame is shown in Fig. 7. It has been seen that for 1-Hz frequency deviation ( 1 1 f = Hz), CADALINE has the best convergence response in terms of speed and over/under shoot. ADALINE method convergence speed is half that in the CADALINE and shows a really high overshoot. Besides, Kalman approach shows the biggest error. in the first 7 power system cycles, it converges to 61.7 Hz instead of 61 Hz and its computational burden is considerably higher than other methods. In this case, presence of a long-lasting decaying DC offset affects the DFT performance. Consequently, its convergence speed and overshoot are not as improved as CADALINE.

Case 2
In this case, a three-phase balanced voltage is simulated numerically. The only change applied is a step-by-step 1-Hz change in fundamental frequency to study the steady-state response of the proposed method when the power system operates under/over frequency conditions. The three-phase signals are: , and values of x f are shown in Table I. The range of frequency that has been studied here is 50-70 Hz. Results are revealed in Table I and average convergence time is shown in Fig. 8 for CADALINE, ADALINE, Kalman filter and DFT approaches. The results from this section can give an insight into the number of samples that each algorithm needs to converge to a reasonable estimation. According to the fact that each power cycle is equivalent to 40 samples, average number of samples that is needed for each algorithm to have estimation with less than one percent error is represented in Table I.

Simulation in PSCAD/EMTDC software
In this case, a three-machine system controlled by governors is simulated in PSCAD/EMTDC software, shown in Fig. 9. Information of the simulated system is given in Appendix I. A three-phase fault occurs at 1 sec. Real frequency changes, estimation by use of ADALINE, CADALINE and Kalman approaches are shown in Fig. 10. Instead of DFT method, the frequency measurement module (FMM) performance which exists in PSCAD library is compared with the presented methods. Phase-A voltage signal is shown in Fig. 11. www.intechopen.com approach has a suitable response in this case, but its error and overshoot in estimating frequency are bigger than that in CADALINE. The PSCAD FMM shows drastic fluctuations in comparison with other methods proposed and reviewed here.

Practical study
In this case, a practical example is represented. Voltage signal measurements are applied from the Marvdasht power station in Fars province, Iran. The recorder's sampling frequency ( s f ) is 6.39 kHz and fundamental frequency of power system is 50 Hz. A fault between pahse-C and groung occurred on 4 March 2006. The fault location was 46.557 km from Arsanjan substation. Main information on the Marvdasht 230/66 kV station and other substation supplied by this station is given in Tables II and III, presented in Appendix II. Fig. 13 shows the performance of CADALINE, ADALINE, Kaman and DFT approaches. Besides, phase-C voltage and residual voltage are revealed in Fig. 14 (A) and Fig. 14 (B) respectively. Complex normalized rotating state vector ( 1 An ) is shown in Fig. 15.

Conclusion
This section proposes an adaptive approach for frequency estimation in electrical power systems by introducing a novel complex ADALINE (CADALINE) structure. The proposed technique is based on tracking and analyzing a complex rotation state vector in d-q frame that appears when a frequency drift occurs. This method improves the convergence speed both in steady states and dynamic disturbances which include changes in base frequency of power system. Furthermore, the proposed method reduces the size of the state observer vector that has been used by simple ADALINE structure in other references. Artificial neural networks may probably be the single most successful technology in the last two decades which has been widely used in a large variety of applications. The purpose of this book is to provide recent advances of artificial neural networks in industrial and control engineering applications. The book begins with a review of applications of artificial neural networks in textile industries. Particular applications in textile industries follow. Parts continue with applications in materials science and industry such as material identification, and estimation of material property and state, food industry such as meat, electric and power industry such as batteries and power systems, mechanical engineering such as engines and machines, and control and robotic engineering such as system control and identification, fault diagnosis systems, and robot manipulation. Thus, this book will be a fundamental source of recent advances and applications of artificial neural networks in industrial and control engineering areas. The target audience includes professors and students in engineering schools, and researchers and engineers in industries.

How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following: M. Joorabian, I. Sadinejad and M. Baghdadi (2011). A Novel Frequency Tracking Method Based on Complex Adaptive Linear Neural Network State Vector in Power Systems, Artificial Neural Networks -Industrial and Control Engineering Applications, Prof. Kenji Suzuki (Ed.), ISBN: 978-953-307-220-3, InTech, Available from: http://www.intechopen.com/books/artificial-neural-networks-industrial-and-control-engineering-applications/anovel-frequency-tracking-method-based-on-complex-adaptive-linear-neural-network-state-vector-in-po