Modelling of Sliding Goertzel DFT - Core

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ScienceDirect Procedia Technology 21 (2015) 430 – 437

SMART GRID Technologies, August 6-8, 2015

Modelling of Sliding Goertzel DFT (SGDFT) Based Phase Detection System for Grid Synchronization under Distorted Grid Conditions K.Sridharana *, B.Chitti Babub , P.MuthuKannana,V.Krithikac a

Department of EEE, Saveetha School of Engineering,Saveetha University,Chennai,602105,India b Centrum ENET, VSB - Technical University of Ostrava , Czech Republic c Department of Machatronics, SRM University, Chennai, ,India

Abstract This paper presents a grid synchronization method based on the implementation of sliding Goertzel discrete Fourier transform (SGDFT). The conventional phase locked loop (PLL) exhibits large phase error, be difficult to implement, and their performance is also indistinct under conditions where the grid frequency varies or the supply is distorted with high frequency harmonics. In the proposed SGDFT show a robust phase tracking capability with fast transient response under distorted grid conditions. The sampling frequency is adjusted to correct the phase errors by the SGDFT filter whenever variation in the frequency or harmonics. © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license ©2015 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Amrita School of Engineering, Amrita Vishwa Vidyapeetham University. Peer-review under responsibility of Amrita School of Engineering, Amrita Vishwa Vidyapeetham University Keywords: sliding Goertzel discrete Fourier transform (SGDFT), phase detection, grid synchronization.

1. Introduction As the energy demand and environmental problems increase, the renewable energy sources such as solar energy and wind energy have become very important as an alternative to the conventional fossil energy sources [1, 2]. While the distributed power generation units supply power to the grid, they also bring extra pollutions into the grid if the grid-connected inverters utilize inaccurate phase information of the utility voltages to synchronize with the

* Corresponding author. Tel.: +91-9994179911 E-mail address:[email protected]

2212-0173 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Amrita School of Engineering, Amrita Vishwa Vidyapeetham University doi:10.1016/j.protcy.2015.10.065

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K. Sridharan et al. / Procedia Technology 21 (2015) 430 – 437

utility grid. With the development of power electronics, more and more non-linear loads are connected to the electric networks. Therefore, distortions and transients such as harmonics, frequency variations and phase shifts often occur to the grid [3].Several methods have been proposed for grid synchronizing and are available in the literature, ranging from simple methods based on detection of zero-crossings of grid voltage to more advanced numerical processing of the grid voltage based on PLL measurement [4]-[5]. The main idea of phase locking is to generate a signal whose phase angle is adaptively tracking variations of the phase angle of a given signal [6] The finite impulse response (FIR) filter is one among them with great interest, because of its linear phase response that leads to accuracy in phase estimation [7]. However the method based on FIR Filtering, can also be complex to understand and implement. A new adaptive notch filtering based phase detection system is simple, robust and less complexity in digital implementation [8].However, transient response is sluggish especially during grid frequency variation. Proposed recursive DFT filter for phase error correction for line synchronization by using time window and phase error correction method [9]. This paper presents a Modelling of SGDFT based phase detection system for grid synchronization under distorted grid conditions. In the proposed SGDFT based phase detection shows a robust phase tracking capability with fast transient response under adverse situation of grid. The superior performance of proposed SGDFT phase detection system is studied and the results are obtained under different grid conditions such as high harmonic injection and frequency deviation. 2. Phase Detection by SGDFT algorithm . The proposed system consists of SGDFT, Moving average filter, PI controller and Numerically Controlled Oscillator (NCO). This section describes the detailed analysis of proposed SGDFT for phase detection. Fig. 1 shows the system.

Fig.1. Block diagram of Novel SGDFT based PLL

2.1. SGDFT Fig. 2 shows the block diagram of SGDFT. The computation involves in SGDFT for N samples requires N+2 real multification and 3N+1 real addition, As a result, SGDFT filtering is more efficient as it requires small number of operations to extract a single frequency component [11]. Moreover, SGDFT is a versatile algorithm capable of extracting in phase and quadrature components of fundamental frequency from distorted grid voltages. In this work SGDFT filter is designed for the signal with the fundamental frequency of f, window width N, and the sampling frequency fs. If it is driven by another signal whose frequency is f+∆f ,the sin and cosine output show a phase shift of ∆Φ which is proportional to ∆fs. This property is highly useful for phase detection, which adjusts fs to fs+∆fs making the use of ∆Φ. In addition to that, the proposed SGDFT compute the N-point for a single bin (k), cantered at an angle θk=2πk/N rad, on the unit circle, which is corresponding to the cyclic frequency of kfs/N Hz. The bin index k is 0≤ k< N. However, for successful implementation of SGDFT, the sampling frequency fs should be equal to multiplication of signal frequency (f) and window width (N) .Thus the SGDFT phase detection is a real-time signal processing algorithm which is insensitive to harmonics distortion and robust against frequency and phase variations [10]-[11]. ሺሻ ൌ ʹ ቀ

ʹɎ

ቁ ‫ כ‬ሺ െ ͳሻ െ ሺ െ ʹሻ ൅ ሺሻ െ ሺ െ ሻ

(1)

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K. Sridharan et al. / Procedia Technology 21 (2015) 430 – 437 ݆ ʹߨ݇

‫ݕ‬ሺ݊ሻ ൌ ‫ݒ‬ሺ݊ሻ െ ݁ ܰ ‫ݒ‬ሺ݊ െ ͳሻ Where x(n) is input sequence x(n-N) is delay input y(n) is output sequence Take Z transform for equation (1) and (2) ,we get ܸሺ‫ݖ‬ሻ ൌ ʹܿ‫ ݏ݋‬ቀ

ʹߨ݇ ܰ

െͳ

ቁ‫ݖכ‬

ܸሺ‫ݖ‬ሻ ቄͳ െ ʹܿ‫ ݏ݋‬ቀ

ʹߨ݇ ܰ

ܸሺ‫ݖ‬ሻ െ ‫ݖ‬

െʹ

ܸሺ‫ݖ‬ሻ ൅ ܺሺ‫ݖ‬ሻ െ ‫ݖ‬

ቁ ‫ ݖ כ‬െͳ ൅ ‫ ݖ‬െʹ ቅ ൌ ܺሺ‫ݖ‬ሻሼͳ െ ‫ ݖ‬െͳ ሽ

݆ ʹߨ݇ ܰ

‫ ݖ‬െͳ ܻሺ‫ݖ‬ሻ ܻሺ‫ݖ‬ሻ ൌ ܸሺ‫ݖ‬ሻ െ ݁ Sub equation (4) in (5) and we get, The transfer function of the SGDFT for kth bin is

(2)

. െͳ

ܺሺ‫ݖ‬ሻ

(3) (4) (5)

݆ ʹߨ ݇

‫ܪ‬ሺ‫ݖ‬ሻ ൌ

ܻሺ‫ݖ‬ሻ ܺሺ‫ݖ‬ሻ

ቆͳെ݁ ܰ ‫ ݖ‬െͳ ቇ൫ͳെ‫ ݖ‬െܰ ൯

ൌ

(6)

ʹߨ ݇ ቁ‫ ݖכ‬െͳ ൅‫ ݖ‬െʹ ቁ ܰ

ቀͳെʹܿ‫ ݏ݋‬ቀ

The factor ͳ െ ‫ ݖ‬െܰ is a comb filter of the finite-impulse response (FIR). The real and imaginary component of H(z) can be generated from equation (6). The structure of SGDFT for kth bin is shown in Fig.2. The real and imaginary component of H(z) is given as; ܴ݁ሾ‫ܪ‬ሺ‫ݖ‬ሻሿ ൌ

ʹߨ݇ ቁ‫ ݖכ‬െͳ ቁ൫ͳെ‫ ݖ‬െܰ ൯ ܰ ʹߨ݇ ቁ‫ ݖכ‬െͳ ൅‫ ݖ‬െʹ ቁ ቀͳെʹܿ‫ ݏ݋‬ቀ ܰ

ቀͳെʹܿ‫ ݏ݋‬ቀ

‫݉ܫ‬ሾ‫ܪ‬ሺ‫ݖ‬ሻሿ ൌ

ʹߨ ݇ ቁ‫ ݖכ‬െͳ ቁ൫ͳെ‫ ݖ‬െܰ ൯ ܰ ʹߨ ݇ ቁ‫ ݖכ‬െͳ ൅‫ ݖ‬െʹ ቁ ቀͳെʹܿ‫ ݏ݋‬ቀ ܰ

ቀͳെ‫ ݊݅ݏ‬ቀ

(7)

(8)

In the phase detector, the exact cosine signal obtained from fundamental SGDFT bin (k=1) is multiplied with the input signal to yield ‘e’ (output of phase detector). The output from the phase detector is given to moving average filter in order to reduce the phase margin. The output of the moving average filter is processed by a PI controller, which provides the control input for numerically controlled oscillator (NCO) with zero steady state error. Hence NCO generates the sampling pulses at required rate. 2.2. PI Controller In order to provide the steady dc input D to the NCO, even when the phase error tends to zero at steady state, a PI controller is combined with the phase detection circuit. Thus the transfer function of the controller in the frequency domain is ‫ ܫܲܪ‬ሺ‫ݖ‬ሻ ൌ ‫ ܲܭ‬൅

‫݋ܽ݊݁ܶ ܲܭ‬ ܶ݅ ሺͳെ‫ ݖ‬െͳ ሻ

(9)

where Kp is the proportional gain, Ti integral time constant and Teno is the enabling time, for the PI block. The enabling frequency feno=1/Teno was fixed at 25.6 kHz. The output of the PI controller is limited to ±1. And the Saturation limits are allowed to prevent overflow of the integrator registers, which can incidentally limits the control input D applied to the NCO.


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Fig. 2. SGDFT structure

2.3. Frequency response characteristic The frequency response characteristic of SGDFT filtering defines the noise rejection capability and phase response of the system, and can be calculated from the equation (6). From that, pole-zero plot for SGDFT filtering is plotted and is given in Fig. 3. The extraction of single frequency component produces two poles that cancel with zeros. The poles are located on the unit circle and therefore the SGDFT filter is marginally stable [9] [11]. The frequency characteristics of a single-bin sliding filter for N=64 and k=1 is illustrated in Fig.4. It states that, the corresponding to the value of ‘k’, it allows the fundamental frequency component of grid voltage with rejection of all sub and higher order harmonic components. So it reveals neither passband nor stopband characteristics of the filter, but they adequate for coherent sampled signals [11]. 3. Implementation and Results In order to analyze the proposed study, simulations tests have been carried out in the MATLAB-Simulink environment, overall simulation diagram of SGDFT based phase detection system and sub system of SGDFT diagram as shown in Fig. 5 and Fig. 6. The following parameters are considered for the study:Input voltage V=1volt (Per Unit), f=50Hz, N=64, sampling frequency fs=3200Hz feno=25KHz, Kp=0.01 and Ki =0.026. The abnormal conditions namely; high frequency harmonic injection, frequency variation and phase variation of grid voltages are examined for accurate phase detection in this study using SGDFT algorithm.

Fig.3. Pole-Zero plot for a SGDFT filter with N=64, k=1

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Fig.4. Frequency response of the SGDFT filter, N=64,k=1

3.1. Response of SGDFT filtering during high frequency harmonic injection

(a)

(b)


435

(c)

(d)

(e)

(f) Fig.5. Response of SGDFT filtering during harmonic injection. (a) Input signal with harmonic injection. (b) Inphase and Quadrature components (c) actual and estimated phases. (d) Estimated phase and phase angle (e) THD of Actual Input signal (f) THD of Estimated output signal.

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The input signal contains fundamental frequency of 50 Hz, with 33% of 3 rd , 25% of 5th , 17% of 7th , 13% of 9th and 8% of 11th harmonics according to eqn. (10) given below: (10) ‫ݔ‬ሺ‫ݐ‬ሻ ൌ ͳ‫݊݅ݏ‬ሺ߱‫ݐ‬ሻ ൅ ͲǤ͵͵‫݊݅ݏ‬ሺ͵߱‫ݐ‬ሻ ൅ ͲǤʹͷ‫݊݅ݏ‬ሺ͵߱‫ݐ‬ሻ ൅ ͲǤͳ͹‫݊݅ݏ‬ሺ͵߱‫ݐ‬ሻ ൅ ͲǤͳ͵‫݊݅ݏ‬ሺ͵߱‫ݐ‬ሻ ൅ ͲǤͲͺ‫݊݅ݏ‬ሺ͵߱‫ݐ‬ሻ This input signal in Fig 5(a) is given to SGDFT filtering the real and orthogonal signals are extracted and the corresponding obtained results are shown in Fig.5(b).The actual and estimated phases and phase angles are extracted at accurate sampling rate by using NCO as shown in Fig.5(c) and Fig.5(d) respectively. Total Harmonic Distortion (THD) of harmonic contented input signal is 47.29% and the fundamental signal extracted by SGDFT with the THD of extracted signal is 0.56% which is shown in Fig.5(e) and Fig.5(f). 3.2. Response of SGDFT filtering during frequency deviation In order to test the transient response of proposed SGDFT filtering system, frequency variation is considered, with the frequency of the input signal changed from 49.5Hz to 50 Hz at 1 sec and from 50 Hz to 50.5 Hz at 2 sec. This input signal is given to SGDFT filtering, the Inphase and Quadrature components are extracted and the corresponding results are obtained. The actual and estimated phases and phase angles are extracted at sampling rate by using NCO. It is seen that the PI controller output settles at the new value of α from 52 to 50 at 1sec when frequency changes from 49.5Hz to 50Hz. It is observed that, the proposed SGDFT filtering can be able to accurately estimate the phase of the fundamental signal during frequency variations. Therefore, the precise detection of phase angle is realized by the proposed SGDFT filtering. 4. Conclusions The paper presents an alternative method of grid synchronization under distorted condition based on the implementation of a SGDFT. From the proposed algorithm it was concluded that the SGDFT based phase detection system can extract the fundamental frequency of the voltage signal accurately during grid voltage variation in the frequency or harmonics. Therefore, the immediate advantages of the proposed PLL are: frequency adaptability, high degree of immunity to disturbances and harmonics and structural robustness. Moreover, the proposed synchronization scheme can further be used for grid measuring, monitoring and processing of the grid signals. References [1]

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