D. Ould Abdeslam, P. Wira, D. Flieller, and J. Mercklé, "New methods for time-varying frequency estimation from distorted harmonic signals in power systems," IAR Workshop, Mulhouse, France, Nov. 16-18, 2005.
NEW METHODS FOR TIME-VARYING FREQUENCY ESTIMATION FROM DISTORTED HARMONIC SIGNALS IN POWER SYSTEMS Djaffar Ould Abdeslam ∗ Patrice Wira ∗ Damien Flieller ∗∗ Jean Merckl´ e∗ ∗ Universit´e de Haute Alsace, Laboratoire MIPS 4 rue des Fr`eres Lumi`ere, 68093 Mulhouse Cedex France {d.ouldabdeslam; patrice.wira; jean.merckle}@uha.fr ∗∗ INSA Strasbourg, Laboratoire ERGE 24 Bd de la Victoire, 67084 Strasbourg Cedex, France
[email protected]
Abstract: Two methods for on-line estimating the fundamental frequency of a sinusoidal signal whose amplitude and frequency could be either constant and time-varying are described in this paper. The first method we propose is based on the PLL (Phased Locked Loop) principle with a PI regulator. The second one is based on an Adaline neural network where the architecture and the learning are formulated based on an original decomposition of the signal. Both methods are robust and can efficiently be compared to conventional frequency estimation algorithms. The problem of detecting frequency variations in a power system is addressed and the results show that the neural frequency estimator is very efficient. The two methods are evaluated and compared through simulations and experimental examples on a real-time platform. Keywords: active filters, electric power systems, artificial intelligence, neural networks, applied neural control, adaptive control, frequency estimation
1. INTRODUCTION The frequency estimation problem is a key issue in modern applications such as noise and acoustic echo cancellation, speech analysis, transmission techniques for digital communication, and harmonic identification and compensation in power systems. Considerable effort has been made for long years to design efficient and robust algorithms and methodologies. A large number of techniques have been elaborated this last years. However, conventional and popular techniques can not always be used because of some inherent constraints that have to
be fulfilled or for some assumptions which must be used. They are generally sensitive to nonlinear transformations, to non-stationary distortions, and to correlated or non-Gaussian noises. For the sake of short computing time, numerical simplicity and real-time implementation, our goal is to cast the problem in the form of a linear equation. We first introduce a very simple and efficient method which is based on a PLL (Phased Locked Loop). We also propose a neural network approach as an alternative to all the well-known frequency measurement techniques. Based on an Adaline neural network, the second approach relies on a linear and recursive expression of the volt-
D. Ould Abdeslam, P. Wira, D. Flieller, and J. Mercklé, "New methods for time-varying frequency estimation from distorted harmonic signals in power systems," IAR Workshop, Mulhouse, France, Nov. 16-18, 2005. age signal. A recursive Least Mean Squares (LMS) algorithm carries out the weights training. This learning process allows then to acquire knowledge that is representative of the frequency. Its simplicity, as well as is efficiency, makes this approach fast and effective. Significant improvements were obtained in simulations and on real applications for the instantaneous estimation of the frequency of distorted harmonic signals. Developed to be integrated in a neural network approach of an active power filter scheme (Akagi, 1996), this method is applicable to every general frequency estimation problem. The paper is organized as follow. Section 2 provides a review of frequency estimation techniques. In Section 3, we introduce the new estimators, a PLL-based approach and an Adaline-based approach that relies on a recursive decomposition of the voltage signal. Performances, robustness with respect to disturbances and computational efficiency are discussed through computer simulations in Section 4 and through experimental tests in Section 5. The last section is a conclusion of our work.
2. POWER SYSTEM FREQUENCY ESTIMATION 2.1 Conventional Methods Parametric system identification usually concludes in the estimation of unknown parameters in a model. The estimation of the frequency parameters can be done in many different ways with Fourier techniques, least-squares estimators (Haykin, 1996), methods using automatic and adaptive filters (Xia, 2002), Kalman filters (Routray et al., 2002), etc. However, these popular and effective methods are inherently sensitive to nonlinear transformations and to noises. Indeed, most of the work on the frequency estimation problem needs a stochastic model of the plant, and of the measurement process. The models often must be approximated. Since, their implementation does not satisfy for real-time constraints, and their convergence will not be guaranteed and fast. Moreover, conventional methods assumes that the additive noise has Gaussian distribution and is stationary. This is partly because of the nice properties of the Gaussian model which allows for simplification of the theoretical work and decreases the computational complexity in signal parameter estimation. One possible way to find better estimates is to use artificial intelligence techniques such as Artificial Neural Networks (ANNs).
2.2 Frequency Estimation with ANNs Neural networks were originaly introduced to learn performance and to estimate functions. Since the last decade, they are specifically used for harmonic identification (Ould Abdeslam et al., 2005b) and for frequency estimation (Karhunen and Joutsensalo, 1991). For example, Adaline networks of Widrow (Widrow and Walach, 1996) were used in (Dash et al., 1997) to identify the parameters of a discrete signal model of the power system voltage. The learning parameters were adjusted to satisfy a stable difference error equation, rather than to minimize an error function. The technique is able to efficiently track of the power system frequency but is not immune to the presence of harmonics and random noise. More and more, with the development of hardware technologies, the use of Adaline techniques will constitute a real alternative to conventional frequency estimators used in real-time implementation such as active power filtering problems.
3. TWO METHODS FOR SYSTEM VOLTAGE FREQUENCY ESTIMATION We propose two methods to on-line extract the frequency from the power system. The first one is based on the principle of a PLL and the second one is a neural estimator that learns a recurssive expression of the voltage. The neural frequency estimator we propose is based on an original decomposition of the measured voltage signal. Thus, we first introduce a recursive expression of the power system voltage, and then we present the involved neural network approach and its learning scheme. The neural estimator is based on an Adaline neural network, because of its ability to learn linear relationships in a fast, simple and efficient way. The estimation of the frequency will be used to on-line estimate the parameters of the voltage components of a measured signal, i.e., Vd , ϕd , Vi ϕi , Vo and ϕo , respectively the amplitude and the phase of the direct, inverse and homopolar voltage components (Ould Abdeslam et al., 2005a).
3.1 System Voltage Representation A discrete time signal that describes a voltage of a three phases unbalanced power system is of the form:
D. Ould Abdeslam, P. Wira, D. Flieller, and J. Mercklé, "New methods for time-varying frequency estimation from distorted harmonic signals in power systems," IAR Workshop, Mulhouse, France, Nov. 16-18, 2005. at sample times t and t−1, while W (t)T represents the Adaline weight vector.
Fig. 1. Adaline learning scheme associated to the recursive expression of the voltage signal. √ vL1 (t) = 2 [Vd cos(ωt − ϕd ) +Vi cos(ωt − ϕi ) + V0 cos(ωt − ϕo )] (1) where Vd , Vi and Vo are respectively the amplitude of the direct, inverse and homopolar voltage components. The phases of the direct, inverse and homopolar voltage components are respectively ϕd , ϕi et ϕo . By summing vL1 (t + 1) and vL1 (t − 1) , we showed in (Ould Abdeslam et al., 2005c) that expression (1) can be written in the following recursive form:
vL1 (t + 1) = 2 cos(ω∆t)vL1 (t) − vL1 (t − 1) (2) We can see that the voltage expression given by (2) at each sample time no more depends on Vd , Vi and Vo , respectively the amplitude of the direct, inverse and homopolar voltage components of vL1 (t).
3.2 Adaline Learning for Frequency Estimation The Adaline neural network is well-suited and ideal for approximating and learning a linear relation (Widrow and Walach, 1996). It will thus be used to learn the recursive expression of the voltage signal which was previously developed. The weights, adapted with a LMS learning rule, can be interpreted; giving a non negligible advantage to the Adaline over other ANNs. Moreover, it can be noticed that 2 cos(ω∆t) in (2) is constant. The linear recursive expression (2) of the voltage signal can be rewritten with a vectorial notation: vL1 (t + 1) = W (t)T X(t + 1)
(3)
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Figure 1 synthesizes how the Adaline network is used to learn the linear expression of the voltage signal. According to (5), the first weight of the Adaline, W0 (t), computes 2 cos(ω∆t). The estimated frequency of the voltage signal, fˆ, can thus be deduced with: arccos W02(t) fˆ = 2π∆t
(6)
The Adaline weights, W (t), is a two-element vector where one is constant. Thus, for few practical implementations, an Adaline with only one weight could be used to estimate the signal frequency. In this work, we adopt two elements. The first element must converge to a value that will determine the voltage frequency. According to (5), the second element must converge to -1. This constitutes an excellent test to check and to confirm the convergence of the weight update algorithm. The Adaline weights are adapted with a modified Widrow-Hoff learning algorithm as presented in (Ould Abdeslam et al., 2005c). In order to allow the neural estimator to take into account the changes of the environment, i.e., a time-varying nonlinear load, a changing learning rate scheme was implemented. The objective of this learning scheme is to start with a high value and to decrease the learning rate in the limit of a small value. This learning algorithm is thus simple and efficient. 3.3 PLL-Based method for Frequency Estimation The second method we propose to extract the frequency is based on a PLL with a PI controller. Figure 2 gives an overview of its general principle. The first block, called ”Amplitude”, is used to norm the voltage signal to obtain a unit sinusoidal signal. A PI regulator is inserted in the PLL to cancel the error issued from the multiplier and allows to recover the phase and the frequency of the input signal. The PI regulator is tuned by using conventional methods, e.g., the Ziegler-Nichols tuning method. The integrator and cos(2πu) blocks thus elaborate a sinusoid in phase with the input signal.
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and with £ ¤ W (t)T = 2 cos(ω∆t) −1
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In (3), X(t+1) represents the Adaline input vector which is composed of values of the voltage signal
The frequency estimators are first evaluated in simulation. The objective is to estimate the frequency of the voltage signal measured in an electrical network with a three-phase power supply and a nonlinear load. Static and dynamic tests are
D. Ould Abdeslam, P. Wira, D. Flieller, and J. Mercklé, "New methods for time-varying frequency estimation from distorted harmonic signals in power systems," IAR Workshop, Mulhouse, France, Nov. 16-18, 2005.
Fig. 2. Principle of the estimation of the frequency with the PLL-based method. 400 200 (V)
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The frequency estimators are also evaluated in time-varying environments, i.e., by varying the frequency and the amplitude of the voltage signal.
4.2 Effect of Varying the Frequency of the Voltage Signal The frequency estimators are now evaluated when brutal changes of the fundamental frequency appear. We first simulate a reduction of the power system frequency from 50Hz to 48Hz although this should never happen in real industrial systems. Simulations show that the frequency estimators are very fast. Moreover, we observe that with the neural approach the error is less than 10−4 Hz in only 0.0004s after the frequency is reduced. The frequency estimated with the conventional PLL is reached in 0.002s with an error of 4.10−3 Hz. The neural frequency estimator is also fast and efficient when the frequency is suddenly changed from 50Hz to 60Hz. The results of this severe case are represented by Fig. 3. One can see from Fig. 3-b) that the frequency is as well estimated after the sudden frequency change as before. The global estimation error is less than 10−4 Hz and the frequency change is detected and well estimated in only 0.0004s. By using the estimated frequency, the voltage components are thus efficiently approximated as shown by Fig. 3-c) and Fig. 3-d).
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The neural-based and the PLL-based methods are both used to estimate the frequency of one phase. All the static tests show that the frequency is estimated with a very high accuracy. These tests also show that the performance of the neural approach is better than the performance obtained with the method based on the PLL.
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Fig. 3. Response of the neural frequency estimator to a frequency unit step from 50Hz to 60Hz at time t = 0.222s. a) 3-phase original voltages, b) estimated frequency, c) 3-phase direct voltages, and d) 3-phase inverse voltages. Other simulations show that the neural estimator is also able to follow low-changing frequencies with the same precision.
4.3 Effect of Varying the Amplitude of the Voltage Signal A balanced and equilibrated power system presents solely direct voltage components. If amplitude or phase changes occur, the power system becomes unbalanced and unequilibrated. These distortions lead to both inverse and homopolar voltage components. Expression (7) is a very general representation of a three-phase power system. It allows to take into account the different distortions observed in an electrical network, i.e., voltage sags and surges, transients, harmonics noises, as well as amplitude and phase variations and drifts.
D. Ould Abdeslam, P. Wira, D. Flieller, and J. Mercklé, "New methods for time-varying frequency estimation from distorted harmonic signals in power systems," IAR Workshop, Mulhouse, France, Nov. 16-18, 2005.
vL1 k1 V cos(ωt + α1 ) vL2 = k2 V cos(ωt + α2 − 2π/3) vL3 k3 V cos(ωt + α3 + 2π/3)
(7)
In (7), V represents the voltage amplitude, k1 , k2 , k3 are three coefficients included between 0 and 1, and α1 , α2 , α3 are the phases of the threephase voltages. In the case of a balanced and equilibrated power system, k1 = k2 = k3 = 1 and α1 = α2 = α3 = 0. We now evaluate the performance of the proposed frequency estimators with a voltage sag when the power system is originally balanced and equilibrated. Thus, at time t = 0.23s, the amplitude of the direct voltage component is reduced from 240V to 200V and the amplitude of the inverse voltage component is changed from 0 to 40V . In this case, we have k1 = 0.50, k2 = k3 = 1 and α1 = α2 = α3 = 0. The frequency is estimated by the neural approach with an error less than 5.10−3 Hz which represents a value of 0.01%. This demonstrates that the learning of the Adaline is able to rapidly compensate for any changing parameters. The error with the conventional PLL is less than 0.65Hz, i.e., less than 1.3%. The time-varying environment is considered in term of learning as a non-stationary system and the adaptability of the proposed neural method allows to compensate for the changing parameters and leads to an accurate estimation of the frequency.
5. REAL-TIME EXPERIMENTAL RESULTS Experimental results are now given to illustrate the performance proposed frequency estimators. The neural estimator has been implemented to on-line estimate the frequency of the electrical network and is compared to PLL-based estimator. The estimated frequency is then used to estimate the parameters of the direct and inverse voltage components, i.e., Vd , Vi , ϕd and ϕi . A good frequency estimation provides good estimations of the voltage components. First, the steady-state behavior of the frequency estimators is studied, and then their response to any changes of the nonlinear load.
Fig. 4. Estimation of a fluctuating frequency (in Hz). is a three-phase power supply with a low-voltage system and a nonlinear load which is a Graetz bridge of six valve functions and a RL circuit with a power variator. The load impedance is composed of RL = 37.8Ω and LL = 0.36mH. This experimental platform allows to reproduce industrial conditions and the voltage is measured with a sample frequency of 0.4mHz.
5.2 Static Experiment Strictly speaking, the frequency of a power system should be perfectly stable. In practice, however, low-frequency fluctuations can always be observed. In the following experiment, we thus on-line estimate the fundamental frequency of a power system including distorted harmonics. We use the proposed neural estimator in the case the frequency is continuously fluctuating in a range of 49.98Hz to 50.02Hz. The estimated frequency is shown by Fig. 4. This figure clearly shows that the neural estimator is efficient in following the fluctuating frequency around f = 50Hz. In order to evaluate the effect of taking into account the frequency fluctuations, we compare the neural estimation approach to the PLL. The estimated frequency is then used to approximate the parameters of the direct and inverse voltage components, i.e., Vd , Vi , ϕd and ϕi . The neural approach estimates these parameters with an accuracy of 0.6% while they are estimated with an accuracy of only 3.2% by using the PLL-based estimator. These results are illustrated by Fig. 5, which clearly shows that the voltage components are on-line well reconstructed with the neural approach.
5.3 Dynamic Experiment 5.1 Experimental Setup The effectiveness of the proposed estimators is verified with a DS 1104 dSPACE board and an experimental setup of 5 kVA. The dSPACE board is based on a Power PC 603e processor and a TMS320C31 DSP at 40 MHz. The power system
In this last experiment, the frequency is brutally changed from 50.2Hz to 49.8Hz at time t = 100s. This experiment allows to evaluate the dynamic behavior of the neural frequency estimator although such kind of frequency step a very rare in real-world environments.
D. Ould Abdeslam, P. Wira, D. Flieller, and J. Mercklé, "New methods for time-varying frequency estimation from distorted harmonic signals in power systems," IAR Workshop, Mulhouse, France, Nov. 16-18, 2005. and when sudden frequency change occurs. The stability and robustness of the proposed estimators have been established through very severe environment representative of real-world plants.
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Fig. 5. Identification of the voltage components with the proposed neural approach on an experimental real-time platform. a) original voltage on phase 1, b) direct component Vd on phase 1, c) inverse component Vi on phase 1, d) homopolar component Vo on phase 1. The fundamental frequency of the power system is perfectly estimated on-line. The neural approach is able to compensate for the frequency step and to estimate the new frequency after only 4s. The neural approach offers superior performances as well in estimating low-fluctuating frequency as in compensating for sudden frequency changes.
6. CONCLUSIONS Two approaches to estimate the power system frequency has been proposed, an PLL-based method and a method based on an Adaline neural network. Indeed, the learning has been used for approximating a recursive expression of the voltage signal, estimating thus the voltage frequency. Both methods are fast and accurate, but due to its learning capabilities, the neural approach enables faster tracking of the frequency. The Adalinebased estimator also involves less computation, which make its very attractive for real-time implementation. This estimator is developed with the objective to be inserted in a complete neural harmonic compensation scheme only based on Adaline networks. The frequency estimators were evaluated with both simulations and real-time experiments and the results appeared to be very satisfactory. Compared to the PLL-based method, the neural approach offers superior performances in all the cases, i.e, in estimating low-fluctuating frequency
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