This is to certify that the thesis entitled, "MODERN METHODS FOR POWER SYSTEM HARMONIC ESTIMATION (Study of Least Mean Squares (LMS) Filter, Recursive Least Squares (RLS) Filter, Kalman and Extended Kalman Filter)" submitted by Abhijit Pradhan and Bibhu Prasad Panigrahi in partial fulfillment of the requirements for the award of a Bachelor of Technology Degree in Electrical Engineering in the National Institute of Technology, Rourkela (Honourable University), is an authentic work carried out by them under my supervision and guidance. It is a matter of great satisfaction and dignity for us to present our project undertaken during seventh and eighth semester for partial fulfillment of our Bachelor of Technology degree at National Institute of Technology, Rourkela. This project could never have been possible without his supervision and much needed help amidst his busy schedule.
A complete picture of the performance of the extended Kalman filter Fundamental amplitude estimation using LMS. 3rd harmonic amplitude estimation using LMS 5th harmonic amplitude estimation using LMS Basic amplitude estimation using RLS 3rd harmonic amplitude estimation using RLS 5th harmonic amplitude estimation using RLS. Basic amplitude estimation using the Kalman filter Amplitude estimation of the 3rd harmonic using the Kalman filter Amplitude estimation of the 5th harmonic using the Kalman filter Extended Kalman filter output.
Harmonics have been around for a long time, and its presence shapes the performance of a power system. The aim is to estimate the voltage magnitude of the power system in presence distortions taking into account the noise by using different estimation methods.
INTRODUCTION
INTRODUCTION
WHAT ARE HARMONICS
CREATION OF HARMONICS
EFFECT OF HARMONICS
- TRIPLEN HARMONICS
- CIRCUIT OVERLOADING
METHODS OF ELIMINATION OF HARMONICS
- NEUTRAL CONDUCTOR SIZING
- TRANSFORMER LOADING
- K-FACTOR TRANSFORMERS
- HARMONIC FILTERS
DETERMINATION AND PRESENCE OF HARMONICS
CREST FACTOR
MEASUREMENT OF DISTORTED WAVEFORM
CHARACTERISTIC AND NON- CHARACTERISTIC HARMONICS
ESTIMATION OF HARMONICS
This extra charge creates more heat, which breaks down the insulation of the neutral conductor. If non-linear loads are a significant part of the total load on the object (>20%), there is a possibility of harmonics problem. The Crest factor of any waveform is defined as the ratio of the peak value to the RMS value.
In this case, there are harmonics in the AC current of the order of magnitude h=np+1, where p= number of pulses, n is any integer. To develop an estimate of the gradient vector∇J(n), the strategy is to substitute the estimates of the correlation matrix R and the cross-correlation vector p [4]. The estimation vector error e(n) is based on the current estimate of the tap weight vector, ŵ(n).
Therefore, we express the cost function to be minimized as ϐ(n), where n is the length of the variable data. The optimal value of the tap weight vector, ŵ(n), for which the cost function achieves its minimum value, is defined by the normal equations written in matrix form: [4]. The important feature of Kalman filtering is the recursive processing of the noise measurement data.
This filtering technique is used to obtain the optimal estimation of power system voltage magnitudes at different harmonic levels.[1] The Kalman filter is an estimator used for it. In estimation theory, the extended Kalman filter (EKF) is the nonlinear version of the Kalman filter, i.e., it can be used for state estimation in a system that is nonlinear. This is done by modifying some of the material presented in the Kalman filtering algorithm.
The basic operation of the EKF is the same as that of the linear discrete Kalman filter. On the other hand, in the RLS algorithm, this correction consists of the product of two factors: the true estimation error 𝜂(𝑛) and the gain vector k(n). The gain vector consists of the inverse deterministic correlation matrix 𝜙−1(𝑛), multiplied by the tap input vector u(n).
The convergence speed of the RLS algorithm is thus faster than the LMS algorithm [13]. The RLS algorithm requires a total of 3M(3+M)/2 multiplications, which increase with the size of M , the number of filter coefficients. One of the most common approaches for nonlinear systems is the use of the Extended Kalman Filter (EKF) [15] [16].
Where Aik, wi and 𝜑i are the amplitude, frequency and phase of the ith sinusoid respectively.
FILTERS
LEAST MEAN SQUARE (LMS) FILTERING
- LEAST MEAN SQUARE ALGORITHM
RECURSIVE LEAST SQUARES (RLS) FILTERING
- RECURSIVE LEAST SQUARES ALGORITHM
KALMAN FILTERING (KF)
- ESTIMATION OF A PROCESS
- COMPUTATIONAL ORIGINS OF THE FILTER
- KALMAN FILTERING ALGORITHM
EXTENDED KALMAN FILTERING (EKF)
- THE PROCESS TO BE ESTIMATED
- THE COMPUTATIONAL ORIGINS OF THE FILTER
An adaptive process that involves automatically adjusting the filter tap weights according to the estimation error.[4]. Second, we have a mechanism for performing the adaptive control process on the cross-filter tap weights, hence the name. For the LMS algorithm to fulfill this criterion, the step size parameter μ must satisfy some conditions related to the Eigen structure of the correlation matrix of the tap entries [4].
The second term μu(n)e*(n) on the right-hand side of (2.8) represents the correction applied to the current estimate of the tap weight vector, ŵ(n). The inner product ŵH(n-1)u(n) represents the estimate of the desired response d(n), based on the old least-squares estimate of the tap weight vector made at time n-1.[4]. However, some of the most interesting and successful applications of Kalman Filtering have been those where the process to be estimated and the measurement relationship to the process are non-linear.
In something similar to a Taylor series, we can linearize an estimate around the current estimate using partial derivatives of the process and measurement functions to compute estimates even in the case of non-linear relationships. In fact, we do not know the individual noise values 𝑤𝑘 and 𝑣𝑘 at each time step. This property makes the RLS algorithm independent of the propagation of the eigenvalues of the filter input correlation matrix.[13]
In the LMS algorithm, the correction applied when updating the old estimate of the coefficient vector is based on the instantaneous sample value of the tap input vector and the error signal. Namely, in some of the applications, the LMS filter is better at tracking non-stationary signals than the RLS algorithm [4] [19]. One approach that works considerably well with the non-stationary signals is to use a Kalman filter, which is a generalized version of the RLS filter.
The convergence of all the three algorithms can be interpreted as the convergence of several modes corresponding to the eigenvectors of the autocorrelation matrix of the reference signal. In such cases, the step size and the forgetting factors of the RLS and the LMS algorithms can be adjusted such that the filters corresponding to higher energy modes have similar bandwidths to those generated by the Kalman filter. Although the Kalman filter algorithm is one of the most widely used methods for detection and estimation due to its simplicity, robustness and optimality, its application to nonlinear systems is practically impossible.
The signal amplitude estimation (where the estimation is filtered in the case of LMS and RLS) was performed at different harmonic levels starting from the fundamental to the 5th harmonic of the signal. Since it is essential to filter these harmonics, we need an estimator to estimate the parameters of the harmonics. This is a gradient descent algorithm that adjusts the adaptive filter taps and changes them by an amount proportional to the current gradient estimate of the error surface.
The RLS algorithm performs an exact minimization of the sum of the squares of the desired signal estimation errors at each time point.
COMPARISON BETWEEN LMS, RLS, KF AND EKF ESTIMATION METHOD
COMPARISON BETWEEN LMS AND RLS ALGORITHM
COMPARISON BETWEEN LMS, RLS AND KF ALGORITHM
COMPARISON BETWEEN KF AND EKF ALGORITHM
ESTIMATION OF POWER SYSTEM HARMONICS
ESTIMATION OF A TEST SIGNAL USING LEAST MEAN SQUARE
ESTIMATION OF A TEST SIGNAL USING RECURSIVE LEAST SQUARES
ESTIMATION OF A TEST SIGNAL USING KALMAN FILTERING
EXTENDED KALMAN FILTER OUTPUT SHOWING MEAN SQUARE ERROR…
The simulation results were shown in the subsequent pages comparing between the original signal and estimated signal. A static test signal corrupted with non-linearities and Gaussian noise was used and the estimation of amplitude is done using Extended Kalman filter algorithm. The original signal and estimated signal and the comparison between the two are shown in fig 4.10.
The root mean square error for the estimated signal was found and shown in Figure 4.11. Since the harmonic content of the power circuit depends on the load, the main cause of harmonics is the presence of non-linear load and electronic converters in the system. The characteristic harmonics are integral multiples of the fundamental frequency and their amplitude is directly proportional to the fundamental frequency and inversely proportional to the harmonic order.
We have discussed the Least Mean Squares(LMS), Recursive Least Squares (RLS), Kalman Filter(KF) and Extended Kalman Filter (EKF) algorithms in this thesis. The Kalman filter is basically a recursive estimator and its algorithm is also based on the least square error. Since all the algorithms produce a noisy estimate of the filter taps, we need a low-pass filter which then processes this noisy signal.
But one limitation of KF is that it cannot be used for nonlinear systems. To work with nonlinear systems we proposed the Extended Kalman Filter (EKF). In this algorithm we have to calculate the matrix of partial derivatives (Jacobians) in order to linearize the nonlinear system about the current mean and covariance. Our main objective has been to compare the various algorithms mentioned above and decide on the most suitable algorithm to be used depending on the need of the situation.
2] Comparison of estimation techniques using Kalman filter and raster based filter for linear and non-linear system, Subrata Bhowmik (NIT Rourkela) and Chandrani Roy (MCKV Institute of Engineering), Proceeding of the International on Computing: Theory and Applications (ICCTA '07). 5] Michaels Ken, Bell South Corp., Fundamentals of Harmonics, June 1, 1999, Parts 1-3 http://ecmweb.com/mag/electric_fundamentals_harmonics/. Moo, Truncation effects of FFT on the estimation of dynamic harmonics on the power system, Power system technology, 2000, Proceedings, Power Conference 2000, International Conference on, vol.3, December 4-7 (2000), pp.
Sharaf, A Kalman filtering approach for estimation of power system harmonischen, Proceedings of the 3RD International Conference On Harmonics in Power System Nashville, Indiana, 28 september – 1 oktober (1998), pp.34-80.
CONCLUSION
