Estimation of Single-Phase Grid Voltage Fundamental Parameters Using Fixed Frequency Tuned Second-

Order Generalized Integrator Based Technique Md. Shamim Reza, Student Member, IEEE, Mihai Ciobotaru, Member, IEEE, and Vassilios G. Agelidis, Senior

Member, IEEE Australian Energy Research Institute & School of Electrical Engineering and Telecommunications

The University of New South Wales, Kensington, Sydney, NSW 2052, Australia E-mail: m.reza@student.unsw.edu.au, mihai.ciobotaru@unsw.edu.au, vassilios.agelidis@unsw.edu.au

Abstract-This paper proposes a robust technique for accurate estimation of single-phase grid voltage fundamental amplitude and frequency under harmonics. The proposed technique relies on a quadrature signal generator based on a fixed frequency tuned second-order generalized integrator. A differentiation filter is used to estimate the fundamental frequency from the instantaneous phase angle derived from the generated orthogonal voltage systems. The estimated fundamental frequency is then used to obtain the actual fundamental voltage amplitude from the orthogonal voltage systems. The proposed technique does not rely on interdependent loops offering stability and easy tuning process. The technique can reject the negative effects caused by the presence of the harmonics. Experimental results are provided to validate the performance of the proposed technique.

Keywords-Parameter estimation, quadrature signal generator, second-order generalized integrator, and single-phase voltage systems.

I. INTRODUCTION

The estimation of grid voltage fundamental parameters from a periodic waveform is relatively an easy task [1]. However, the parameters variation and harmonic pollution are commonly observed problems [2, 3] and hence the parameters estimation from a distorted non-periodic grid voltage waveform becomes relatively a difficult task. Therefore, it is necessary to have a suitable digital signal processing (DSP) technique to extract the accurate fundamental amplitude and frequency from a distorted non-periodic grid voltage waveform.

The phase-locked loop (PLL) is an efficient DSP technique for the estimation of single-phase grid voltage fundamental parameters [4-6]. However, there is less information in single-phase systems than in three-phase systems for generating the orthogonal voltage waveforms required for the single-phase PLL [7]. The PLL estimated parameters contain ripples due to the presence of the DC offset and harmonics [8-11]. The in-loop filters can be used to reject the ripples from the estimated parameters at the expense of lower bandwidth, thus leading to a slower dynamic response [12]. The tuning of the PLL controller parameters is complex due to the presence of the interdependent loops. Another drawback of the PLL is that a large frequency overshoot/undershoot is observed in the estimation during phase jumps and are reflected back on the

phase estimation and hence causes delay in the process of synchronization [13]. The large frequency overshoot/undershoot can be reduced by means of a frequency-locked loop (FLL) [14] based on a quadrature signal generator relying on a second-order generalized integrator (QSG-SOGI) [15]. The QSG-SOGI has filtering capability to reject harmonics but the lower order harmonics can introduce significant ripples into the estimated fundamental parameters [16, 17]. Moreover, due to the low-pass filter (LPF) behaviour of the in-quadrature component, the performance of the QSG-SOGI is sensitive to the presence of the DC offset [11, 18, 19]. To avoid the interdependent loops, the separate frequency estimation algorithms can be used to adaptively tune the QSG-SOGI [20, 21].

The technical literature shows that the FLL/PLL techniques based on the frequency adaptive tuned QSG-SOGI (SOGI-FLL/SOGI-PLL) require proper tuning of the controller parameters in order to achieve a trade-off between good dynamic performance and estimation accuracy. Moreover, there are interdependent loops influencing one another at the same time and hence the tuning is sensitive, thus reducing stability margins.

The objective of this paper is to propose a robust technique for the estimation of fundamental amplitude and frequency of the non-periodic single-phase grid voltage waveform. The proposed technique relies on a fixed frequency tuned QSG-SOGI and a finite-impulse-response based differentiation filter. The proposed technique does not include interdependent loops, thus offering stability and easy tuning process. The technique can estimate the fundamental voltage amplitude and frequency accurately and can also reject the negative effects caused by harmonics.

The rest of the paper is organized as follows. The proposed technique is described in section II. Section III contains the real-time experimental performance of the proposed technique Finally, the conclusions are summarized in section IV.

II. PROPOSED FIXED FREQUENCY TUNED QSG-SOGI BASED TECHNIQUE

The grid voltage fundamental amplitude and frequency can be estimated using the frequency adaptive tuned QSG-SOGI

QSG-SOGI

FrequencyAdaptive Tuned

'1v

'1qv

Frequency-Locked

Loop

1v

(a)

= r 2 2(.) (.) 1A

QSG-SOGI

FrequencyAdaptive Tuned

'1v

'1qv

Phase-LockedLoop

1v

(b)

= r 2 2(.) (.) 1A

1v

QSG-SOGI

Fixed FrequencyTuned

'1v

'1qv

Amplitude &FrequencyEstimation

1v

(c)

r1A

Fig. 1. Single-phase grid voltage fundamental amplitude and frequency estimation using (a) Frequency adaptive tuned QSG-SOGI relying on FLL. (b) Frequency adaptive tuned QSG-SOGI relying on PLL. (c) Proposed fixed frequency tuned QSG-SOGI based technique.

based techniques such as SOGI-FLL [16, 17] and SOGI-PLL [6, 7], as shown by the block diagrams in Figs. 1(a) and 1(b), respectively, where v1 is the input fundamental voltage waveform, r is the tuning frequency of the SOGI, is the gain of the SOGI, '1v is the estimated in-phase component of

v1, '1qv is the estimated in-quadrature component of v1, 1A is the estimated fundamental voltage amplitude, and is the estimated fundamental angular frequency. As it can be seen, the estimated frequency is feedback ( )r to tune the QSG-SOGI adaptively in both SOGI-FLL and SOGI-PLL techniques. The presence of the interdependent loops makes the tuning process complex and reduces the stability margin. The proposed fixed frequency tuned QSG-SOGI based technique for the estimation of fundamental voltage amplitude and frequency is shown in Fig. 1(c). As it can be noticed, the proposed technique does not include interdependent loops, thus makes the tuning process easy and increases the stability margin as compared to QSG-SOGI based technique including interdependent loops.

A. QSG-SOGI The QSG-SOGI to track the orthogonal waveforms of a

single-phase voltage system is shown in Fig. 2, where 1v

e is the error signal of the grid voltage fundamental component. The transfer functions of the SOGI, in-phase and in-quadrature components of the QSG-SOGI can be expressed by (1), (2) and (3), respectively [7, 14, 16, 17, 20].

1'1

2 2r

v r

v ssSOGI s e ss

(1)

'1

2 21

ri

r r

v ssT s v s ss

(2)

' 21

2 21

rq

r r

qv sT sv s ss

(3)

The Bode plots of the transfer functions (2) and (3) are shown in Figs. 3(a) and 3(b), respectively, where the tuning frequency is constant (r=250 rad/s) and the gain is varied (=0.5, 1.0 and 1.5, respectively). As it can be seen, the transfer functions, as given by (2) and (3), behave like a band-pass filter (BPF) and a LPF, respectively. The tuning frequency sets the resonance frequency of the SOGI and the gain determines the bandwidth of the in-phase component and the static gain of the in-quadrature component [7, 14, 16, 17].

1ve1v

'1v

'1qv

r

'1v

SOGI

Fig. 2. QSG-SOGI to track the orthogonal waveforms of the single-phase grid voltage fundamental component.

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e (d

eg)

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nitu

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B)

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101

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0

45

90

Phas

e (d

eg)

Frequency (Hz)

1.5 1.0 0.5

1.5 1.0 0.5

(a)

(b) Fig.3. (a) Bode plots of the in-phase transfer function {Ti(s)} of the QSG-SOGI for r=250 rad/s and different values of . (b) Bode plots of the in-quadrature transfer function {Tq(s)} of the QSG-SOGI for r=250 rad/s and different values of .

It can also be noticed that a trade-off is required between good dynamics and harmonics rejection capability when choosing the value of , where =2 and represents the damping factor of the QSG-SOGI. The Bode plots of the transfer functions (2) and (3) for constant gain (=1) and different values of tuning frequency (r=250 rad/s, 260 rad/s and 270 rad/s, respectively) are also shown in Figs. 4(a) and 4(b), respectively. As it can be noticed, the higher tuning frequency increases the bandwidth of the in-quadrature component. However, the harmonics rejection capability of both orthogonal components is reduced for higher tuning frequency. It can also be seen that the amplitudes of the orthogonal waveforms are different and unequal to the input voltage amplitude when the tuning frequency is not equal to the input frequency. Moreover, the tuning at higher or lower

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10

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nitu

de (d

B)

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90

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e (d

eg)

Frequency (Hz)

=2 50r =2 60r =2 70r

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nitu

de (d

B)

100 101 102 103-180

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-90

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0

Phas

e (d

eg)

Frequency (Hz)

=2 50r =2 60r =2 70r

(b)

(a)

Fig.4. (a) Bode plots of the in-phase transfer function {Ti(s)} of the QSG-SOGI for =1 and different values of r. (b) Bode plots of the in-quadrature transfer function {Tq(s)} of the QSG-SOGI for =1 and different values of r.

frequency than the input frequency will introduce a leading or lagging phase angle, respectively, in both orthogonal waveforms, as can be seen in Figs. 4(a) and 4(b), respectively.

If the grid voltage fundamental component, v1(n)= A1(n)sin{(n)nTs+1}, at the nth sampling instant is used as the input of the QSG-SOGI, where A1, , Ts and 1 are amplitude, angular frequency, sampling time period and initial phase angle, respectively, the steady-state expressions of the orthogonal waveforms obtained by the QSG-SOGI can be expressed by (4) and (5), respectively [14, 16, 17].

'1 1 1sini s iv n A n T j n n nT T j n (4)

'1 1 .riqv n A n T j n n

1cos s in nT T j n (5) where

22 2 2r

i

r r

nT j n

n n

2 2

1tan rir

nT j n

n

It can be observed from (4) and (5) that the estimated in-phase component ( '1v ) always leads the estimated in-quadrature component ( '1qv ) by 90

o irrespective of the gain and tuning frequency. For the condition r=, it can be seen from (4) and (5) that 1iT j n and 0,iT j n then the amplitude and phase angle of (4) and (5) are equal to the input amplitude and phase angle, respectively. However, for r, the amplitudes and phase angles of both (4) and (5) are not equal to the input voltage amplitude and phase angle, respectively. If the grid voltage fundamental frequency is known, the fundamental voltage amplitude can also be estimated from (4) and (5), respectively, and is given by

2' 2 ' 21 1 121ri

nA n v n qv n

T j n

(6)

The settling time (Tset) of the QSG-SOGI under dynamic conditions can be expressed by [16, 17, 22]

10set

r

T (7)

It can be seen from (7) that the settling time depends on the value of and r. Based on a constant value of i.e. for a constant damping factor, the higher value of r can reduce the settling time. Therefore, the tuning at a fixed frequency higher than the grid frequency (r>) can improve the settling time as compared to the frequency adaptive tuning (r=) of the QSG-SOGI [23]. The change of the settling time for fixed frequency tuning with respect to the frequency adaptive tuning can be obtained by

Change of settling time % = 100r rr

set set

set

T T

T

(8)

where r

setT and rsetT are the settling times of the QSG-

SOGI at tuning frequency r and r=, respectively. The percentage change of the settling time for different fixed frequency tuning with respect to the adaptive tuning at grid frequency is shown in Fig. 5, where =250 rad/s. As it can be seen, the settling time of the QSG-SOGI decreases as the tuning frequency is higher than the grid frequency. The settling time is reduced by 50% when the tuning frequency is two times of the grid frequency. On the other hand, the settling time increases if the SOGI is tuned by a frequency less than the grid frequency. It can be observed from Fig. 5 that the settling time increases 100% when the tuning frequency is half of the grid frequency.

Let us assume that a grid voltage waveform contains only fundamental component at frequency =250 rad/s and +20% amplitude step is occurred. Fig. 6 shows the estimation of fundamental voltage amplitude using (6) under r=250 rad/s and r=275 rad/s tuning conditions, respectively, where =1.

0 50 100 150 200 250-100

-50

0

50

100

150

200

Tuning Frequency (Hz)

Cha

nge

of S

ettli

ng T

ime

(%)

=r

=2r

=2r

Fig. 5. Percentage change of the settling time of the QSG-SOGI for different fixed frequency tuning with respect to the adaptive tuning at grid frequency, where =250 rad/s.

As it can be seen, the tuning of the SOGI at higher frequency (r=275 rad/s, =250 rad/s) provides faster amplitude estimation as compared to the tuning at grid frequency (r==250 rad/s). Therefore, based on the same gain i.e. same damping factor, the faster amplitude estimation can be achieved by tuning the SOGI at a higher frequency than the adaptive tuning at grid frequency.

B. Estimation of Fundamental Frequency The grid voltage fundamental frequency is not always

constant. It varies mainly due to the mismatch between the power generation and load demand. The actual fundamental frequency needs to be tracked for accurate estimation of the fundamental voltage amplitude using (6). In the proposed technique, the actual fundamental frequency is estimated by

0 n n

(9)

where 0 and are the nominal and deviation of fundamental angular frequency, respectively. The frequency deviation can be estimated by differentiating the instantaneous phase angle deviation and is given by [24-26]

1 ( )dt tdt

(10)

where

1 1 0( )t t t 1( )t is instantaneous phase angle deviation, t denotes

continuous time and is discretized by t=nTs. However, due to the condition r, the instantaneous phase angle estimated from the orthogonal waveforms generated by the QSG-SOGI will contain two kinds of phase angle error i.e. an error introduced by iT j and an estimation error occurs due to the factor r/. Moreover, the presence of the lower order harmonics will introduce significant harmonic distortions into the estimated phase angle.

B-1. Error introduced by iT j The plots of iT j for different values of and r are

shown in the phase responses of Figs. 3(a) and 4(a),

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.040.95

1

1.05

1.1

1.15

1.2

1.25

Actual

Time (s)

Fund

amen

tal V

olta

geA

mpl

itude

(p.u

.)

= =2 50r =1.5 =2 75r

Fig. 6. Fundamental voltage amplitude estimation using (6) for r=250 rad/s and r=275 rad/s, respectively, where =1 and =250 rad/s.

respectively. Due to the differentiation operation, the estimation frequency error ( )

iT introduced by iT j can

be expressed by

iT id T j tdt

2 2

22 2 2 2 2

0 for

for

r

r rr

r r

t d tdtt t

(11)

It can be seen from (11) that iT

is zero for frequency adaptive tuning i.e. for r=. On the other hand, the plots of

iT for r and different input frequency variation cases are shown in Fig. 7. As it can be seen,

iT is also zero when

=constant i.e. the rate of the fundamental frequency change is zero. However, the continuous variation of input fundamental frequency will introduce an error whose magnitude depends on the rate of the fundamental frequency change. The grid fundamental frequency varies slowly and hence the estimated frequency error introduced by iT j is small and can be neglected.

B-2. Error introduced by the factor r/ The relation, as expressed by (12), can be obtained from

the orthogonal voltage waveforms as given by (4) and (5), respectively.

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0.05

Frequency Change Rate= -10 Hz/s

Frequency Change Rate= -5 Hz/sFrequency Change Rate= 0 Hz/sFrequency Change Rate= +5 Hz/sFrequency Change Rate= +10 Hz/s

Freq

uenc

y Er

ror

(Hz)

Frequency (Hz) Fig. 7. Frequency error caused by iT j for different input frequency variation cases, where =1 and r=275 rad/s.

'1 1'1

tan s ir

v n nn nT T j n

qv n

(12)

The estimated instantaneous phase angle using (4) and (5) can also be expressed by

'11

1 '1

tans i errorv n

n nT T j nqv n

(13)

where error is the estimated phase angle error introduced by the factor r/. The following relation can be obtained from (12) and (13).

1tan s ir

nn nT T j n

1tan s i errorn nT T j n (14)

For calculating the estimated phase angle error error, expression (14) can be simplified using the following trigonometric function relation

tan( ) tan( )tan1 tan( ) tan( )

x yx yx y

where 1s ix n nT T j n and y=error. After some mathematical calculations, error can be expressed by

error

11

1

0.5 1 sin 2 2 2tan

1 0.5 1 1 cos 2 2 2

s ir

s ir

nn nT T j n

nn nT T j n

(15)

The plots of error for different values of r are shown in Fig. 8, where =1, =250 rad/s and 1=0. As it can be seen, error is zero when r=. On the other hand, error is a second harmonic oscillation when r. The value of iT j determines the phase angle difference among the error plots

iT j is different for different values of r}, as can be observed in Fig. 8. Therefore, a LPF or an adaptive notch filter (ANF) is required to reject the second harmonic oscillation generated by the factor r/ from the estimated instantaneous phase angle. B-3. Frequency estimation using differentiation filter

In the proposed technique, the fundamental frequency deviation is estimated by differentiating the instantaneous phase angle deviation. However, the frequency estimation using differentiation operation is sensitive to the high frequency disturbances present in the instantaneous phase angle [26]. Therefore, a LPF will be required to combine with a differentiation filter (DF) in order to reject the high frequency disturbances from the instantaneous phase angle. A finite-impulse-response (FIR) based DF relying on a modulating function (DF-MF) is reported in [27] to track the fundamental frequency deviation from the instantaneous phase

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-30

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0

10

20

30

Phas

e A

ngle

Erro

r(d

eg)

Time (s)

=0.50r =0.75r =r =1.25r =1.50r

Fig. 8. Phase angle error (error) introduced by the factor (r/), where =1, =250 rad/s and 1=0.

angle deviation and is expressed by

1 10

m

ln n l l

(16)

where

1

10

0

1Km

Kr

m ll

r

l=0,1,2,,m-1, 1K is the 1st order derivative of a spline type modulating function (K) with maximum derivative order K2, m=Tw/Ts=number of coefficients of the DF-MF, and Tw=window size of the modulating function. The ith derivative of the spline function { ,iK l i=0,1,2,,K-1} with characteristics time Tc=Tw/K can be expressed by [21, 28-30]

0

0

1 , =0,1,..., -1

1 , =

K jji s cj

iK

K js cj

Kg lT jT i K

jl

KlT jT i K

j

(17)

where

11 , ,

1 !0, otherwise

K is c s c w

ji s c

lT jT lT jT TK ig lT jT

and (lTs) is the Dirac delta function. The frequency response of the DF-MF is shown in Fig. 9. As it can be seen, the DF-MF has notch characteristics at the nominal values of fundamental and harmonic frequencies. The position of the notch frequencies are determined by the inverse of the characteristics time Tc (1/Tc=1/0.02=50 Hz for Fig. 9) [27]. As it can be noticed from Fig. 9, the notches are observed at the multiples of 1/Tc Hz. Moreover, the high frequency disturbance rejection capability is improved as the value of m is increased [27]. However, the high value of m will degrade the dynamic response and also increase the computational burden of the DF-MF. The DF-MF can be used to reject the unwanted fundamental and harmonic oscillations present in the instantaneous phase angle. However, the ripples of the estimated frequency due to fundamental and harmonics

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nitu

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Frequency (Hz)

DF-MF( =20ms, =400, =40ms, =2)c wT m T KDF-MF( =20ms, =600, =60ms, =3)c wT m T KDF-MF( =20ms, =800, =80ms, =4)c wT m T K

Ideal DF

Fig. 9: Magnitude responses of the ideal DF and DF-MF for constant value of Tc and different values of m, where K and Tw are varied simultaneously.

oscillations into the instantaneous phase angle increases as the deviation of frequency increases from the nominal values.

The implementation of the proposed fixed frequency tuned QSG-SOGI based technique is shown in Fig. 10. As it can be seen, the fundamental frequency deviation is estimated by differentiating the unwrapped instantaneous phase angle deviation and is then added with the nominal frequency to obtain the actual frequency. The estimated frequency is also used to obtain the actual fundamental voltage amplitude using (6), as can be noticed in Fig. 10. The estimated fundamental voltage amplitude is also filtered by a moving average filter (MAF) in order to reject harmonic ripples. Moreover, the window size of the MAF is updated using the estimated fundamental frequency. It can be seen from Fig. 10 that the proposed technique is open loop system based and hence increases the overall stability and eases the tuning process.

III. EXPERIMENTAL RESULTS

In this section, the performance of the proposed technique is tested on a real-time experimental setup. The laboratory setup, as shown in Fig. 11, consists of hardware and software parts. The hardware part contains a programmable AC power supply, a voltage sensor, a dSPACE1103 (DS1103) control board and a personal computer (PC). The programmable AC power supply is used to emulate the real-time single-phase grid voltage (vLN, subscript LN indicates line-to-neutral) under different conditions such as harmonics, frequency step, frequency sweep, amplitude step and amplitude excursions. The voltage sensor measures the emulated grid voltage and

TABLE I PARAMETERS OF THE PROPOSED TECHNIQUE

QSG-SOGI DF-MF =1.0 r=275 rad/s Tw =40ms m=400 K=2 Tc=20ms

sends it to the 16 bit analog-to-digital converter of the DS1103 control board. On the other hand, the software part contains MATLAB/Simulink, DS1103 Real-Time Interface (RTI) and Control Desk Interface. The proposed technique is implemented in Simulink with the parameters given in Table I and is uploaded to the DS1103 control board using automatic code generation. The Control Desk Interface running on the PC is used to control the parameters in real-time and also to monitor the estimated values.

The performance of the proposed technique is carried out under the following real-time case studies:

i. Steady-state with harmonics (Case-1) ii. Frequency step and harmonics (Case-2)

iii. Frequency sweep and harmonics (Case-3) iv. Amplitude step and harmonics (Case-4) v. Amplitude excursions and harmonics (Case-5)

The nominal grid voltage fundamental frequency and sampling frequency are chosen as 50 Hz and 10 kHz, respectively. The fundamental component of the grid voltage waveforms presented in all the case studies are distorted by 5% of 3rd, 4% of 5th, 3% of 7th and 2% of 9th harmonic, leading to a total harmonic distortion (THD) of =7.35%.

Case-1: Steady-State with Harmonics A real-time distorted grid voltage waveform including

7.35% THD is shown in Fig. 12(a). The steady-state estimations of the real-time fundamental voltage amplitude and frequency are depicted in Figs. 12(b) and 12(c), respectively. As it can be seen, the proposed fixed frequency tuned QSG-SOGI (tuned at 75Hz) based technique can track the actual amplitude and frequency accurately. Moreover, the estimated parameters do not contain ripples under harmonic condition.

Case-2: Frequency Step and Harmonics

In this case, a +1Hz step change of fundamental frequency is introduced into the distorted grid voltage waveform containing 7.35% THD. The real-time estimations of the fundamental voltage amplitude and frequency step using the proposed technique are shown in Fig. 13. As it can be observed, the proposed technique can track the fundamental voltage amplitude and frequency step accurately and also not disturbed by the presence of the harmonic content.

1v nr

'1v n

'1qv nTuned

Fixed Frequency

QSG-SOGIPhase

Unwrapping 1 n Differentiation Filter '1 1 '

1

tanvqv

2 2. .1/ r

n

1 n

0

n

0 1A nMoving 1A n

0 snT

n

r

iT j nModulating Function

Based on AverageFilter

Fig. 10. Proposed fixed frequency tuned QSG-SOGI based technique for the estimation of single-phase grid voltage fundamental amplitude and frequency.

Electrical

~Grid

Programmable ACPower Supply

VoltageSensor

dSPACE1103

PersonalComputer

LNv

Fig. 11. Laboratory setup for real-time experiment.

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.u.)

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amen

tal

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uenc

y (H

z)

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Proposed Actual

Fig. 12. Case-1: Steady-state with harmonics. (a) Grid voltage waveform. (b) Fundamental voltage amplitude. (c) Fundamental frequency.

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olta

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itude

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.)

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amen

tal

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uenc

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z)

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Proposed Actual

Fig. 13. Case-2: Frequency step and harmonics. (a) Fundamental voltage amplitude. (b) Fundamental frequency.

Case-3: Frequency Sweep and Harmonics The grid voltage fundamental frequency varies slowly due

to the large inertia of the rotating shaft of the power generators. A -10Hz/s fundamental frequency variation with duration of 0.1s is introduced in the grid voltage waveform containing 7.35% THD. The real-time estimations of the fundamental voltage amplitude and frequency sweep by the proposed technique are shown in Fig. 14. As it can be noticed, the proposed technique can estimate the fundamental frequency sweep and amplitude accurately while not being affected by the presence of the harmonic content.

Case-4: Amplitude Step and Harmonics In this case, +20% amplitude step is introduced into the

distorted grid voltage waveform containing 7.35% THD. The estimations of fundamental voltage amplitude step and

0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.980.95

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itude

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.)

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(b)Time (s)

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amen

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uenc

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z)

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Proposed Actual

Fig. 14. Case-3: Frequency sweep and harmonics. (a) Fundamental voltage amplitude. (b) Fundamental frequency.

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amen

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olta

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mpl

itude

(p.u

.)

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50.1

50.2

(b)Time (s)

Fund

amen

tal

Freq

uenc

y (H

z)

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Proposed Actual

Fig. 15. Case-4: Amplitude step and harmonics. (a) Fundamental voltage amplitude. (b) Fundamental frequency.

frequency using the proposed technique are depicted in Fig. 15. As it can be observed, the proposed technique can track the amplitude step accurately. However, the proposed technique provides a little overshoot/undershot in the frequency estimation under the step change of amplitude and harmonic condition, as can be noticed in Fig. 15(b).

Case-5: Amplitude Excursions and Harmonics In this case, the grid voltage waveform contains amplitude

excursions and 7.35% THD. The frequency and the range of the amplitude excursions are 2.5 Hz and 5%, respectively. The real-time estimations of the fundamental voltage amplitude excursions and frequency using the proposed technique are shown in Fig. 16. As it can be seen, the proposed technique can track the amplitude excursions accurately and also not affected by the harmonics present in

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.9

0.95

1

1.05

1.1

(a)

Fund

amen

tal V

olta

geA

mpl

itude

(p.u

.)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.549.8

49.9

50

50.1

50.2

(b)Time (s)

Fund

amen

tal

Freq

uenc

y (H

z)

Proposed Actual

Proposed Actual

Fig. 16. Case-5: Amplitude excursions and harmonics. (a) Fundamental voltage amplitude. (b) Fundamental frequency.

the grid voltage waveform. The estimation of fundamental frequency is also accurate under the amplitude excursion and harmonic condition, as can be noticed in Fig. 16(b).

IV. CONCLUSIONS

A robust technique has been proposed in this paper for the estimation of single-phase grid voltage fundamental amplitude and frequency. The proposed technique consists of a fixed frequency tuned quadrature signal generator based on a second-order generalized integrator and a differentiation filter. The estimations of the fundamental voltage amplitude and frequency by the proposed technique are accurate and also not affected by harmonics. Moreover, the proposed technique does not create any interdependent loops, thus offering stability and easy tuning process. The experimental results are presented to verify the performance of the proposed technique for real-time applications.

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