An Adaptive Phase-Locked Loop Algorithm for Single-Phase Utility Connected System Sérgio Augusto Oliveira da Silva, Vinícius Dário Bacon, Leonardo Bruno Garcia Campanhol, Bruno Augusto Angélico FEDERAL TECHNOLOGICAL UNIVERSITY OF PARANÁ – UTFPR Department of Electrical Engineering Av. Alberto Carazzai, 1640, CEP. 86.000-300 Cornélio Procópio – PR, Brazil Tel.: +55 / (43) – 3520.40.00. Fax: +55 / (43) – 3520.40.40. E-Mails:
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Acknowledgements The authors gratefully acknowledge the financial support received from Araucária Foundation.
Keywords «Adaptive Control», «Single phase system», «Harmonics», «DSP»
Abstract This paper proposes a novel algorithm employed to detect both the phase-angle and the frequency of the single-phase utility grid voltage. The proposed algorithm uses an adaptive notch filter (ANF) operating in conjunction with a Phase-Locked Loop (PLL) system, yielding an algorithm called ANFαβ-pPLL. The αβ-pPLL structure is based on the instantaneous active power theory for three-phase power systems. Thus, for single-phase systems, the proposed PLL algorithm can be analyzed into a fictitious two-phase stationary reference frame (αβ-coordinates), in which a normalized fictitious quadrature voltage ( ´ ) is obtained from the estimated PLL phase-angle. On the other hand, an adaptive filter is used to extract the fundamental component of the utility voltage, allowing the rejection of voltage harmonic disturbance. In order to validate the theoretical development, the performance of the single-phase ANF-αβ-pPLL algorithm is evaluated by means of both simulation and experimental tests under utility grid disturbances, such as voltage harmonics, voltage sags/swells, phase-angle jumps and frequency variations.
Introduction The most of power electronics applications involving power conditioning equipment have demanded synchronization with respect to the phase-angle of the utility grid voltage. Normally, the utility information, such as frequency and phase-angle, have been estimated by means of phase-locked loop (PLL) algorithms [1-16]. However, in many cases the signals used for synchronization can suffer deviations due to utility disturbances, such as voltage sags/swells, harmonics, phase-angle jumps, frequency variations, voltages imbalances, and others, causing poor performance and/or malfunction in power conditioning equipment. Therefore, PLL structures must to maintain accurate phase-lock with respect to the utility voltages. Additionally, they must be able to lock-in as fast as possible even under utility disturbance conditions, ensuring the quality and precision in which the utility voltage information is obtained. Besides, high noise immunity and robust performance should be taking into account.
Many phase-angle and/or frequency detection algorithms have been proposed for synchronizing the PLL output signals with respect to the fundamental component of the single-phase utility voltage [2-6, 13]. For three-phase systems the synchronization occurs with respect to the utility positive sequence voltage components [1, 7-12, 14-16]. In the present study, a novel single-phase PLL algorithm is proposed for utility phase-angle estimation, which uses a PLL based on the three-phase instantaneous active power theory [5,6,16], operating in conjunction with an adaptive notch filter (ANF) [17,18], yielding an algorithm called ANF-αβ-pPLL. Basically, the ANF is used to extract the fundamental component of the utility voltage allowing high voltage harmonic rejection. This novel algorithm intends to overcome the drawbacks related to low harmonic rejection found in the single-phase pPLL algorithms presented in [3-6], as well as ensure frequency-adaptive ability. In most PLL algorithms [3-6], the rejection of harmonic components can be increased by reducing the bandwidth of the system, which is obtained by performing an appropriate tuning of the PI controller gains. On the other hand, the system can become slower. Additionally, frequency-adaptive ability is necessary in order to avoid phase-angle errors when the utility frequency varies. In this paper the proposed single-phase ANF-αβ-pPLL algorithm is implemented and tested in order to validate the presented theoretical development. Its performance is evaluated by means of simulation and experimental tests by using of a DSP (TMS320F28335) under utility grid disturbances, such as voltage harmonics, voltage sags/swells, phase jumps and frequency variations. This paper is organized as follows: In Section II the principle of operation of the PLL algorithm, which is based on the electrical power theory (pPLL) is described, and the problems related to its implementation are discussed. The adaptive notch filter used in this paper is presented in Section III, while the proposed algorithm of synchronization is presented and described in Section IV. The proposed algorithm is investigated in Section V, in which the performance of the ANF-αβ-pPLL algorithm is evaluated and both simulation and experimental results are presented. Finally, the conclusions are presented in Section VI.
PLL Algorithm based on the instantaneous active power theory for threephase systems (αβ-pPLL) The conventional structure of PLL systems is shown in Fig. 1 (a), which is basically composed by three distinct parts: the phase-detector (PD), the loop filter (LF) and the voltage controlled oscillator (VCO). In most cases, the PLL algorithms differ from each other in the way that the PDs are implemented. The conventional single-phase pPLL based on the instantaneous active power theory for single-phase systems is shown in Fig. 1 (b) [4], while the αβ-pPLL algorithm, which is based on the instantaneous active power theory for three-phase systems, is shown in Fig. 1 (c) [5, 6]. Supposing that the voltages of the fictitious two-phase system ( ´ and ´ ) of the αβ-pPLL presented in Fig. 1 (c) are sinusoidal and balanced, the problem related to the double-frequency oscillation found in the pPLL scheme of Fig. 1 (b) does not appear in the PLL output signals. On the other hand, both the pPLL schemes presented in Fig. 1 (b) and (c) suffer with limited harmonic rejection. Besides, an appropriate scheme must be introduced in the αβ-pPLL shown in Fig. 1 (c) in order to adapt its frequency.
Principle of operation (αβ-pPLL) The input signal of the αβ-pPLL system is the measured single-phase voltage ( ), which is assumed to be equal to that of the coordinate ‘α’ ( ´ ) that represents the two-phase stationary reference frame (αβ coordinates). The ‘β’ coordinate, represented by the orthogonal voltage ( ´ ), is obtained by introducing a phase delay of π/2 rad on the voltage ´ . The operation principle of the presented PLL structure is to cancel the dc component ´ of the fictitious instantaneous power p´ as shown in Fig. 1
(c). Notice that the PLL angular frequency reference ( =2π ) is given by the Proportional-Integral (PI) controller output, where is the utility nominal frequency. The angle is obtained by the integration of the angular frequency reference that will be identical to the utility frequency ( ). is used to calculate the feedback fictitious currents ´ and ´ . Thus, the angle
vf
vd
vs
vo
(a)
p*
=0 p,
LPF ps vs
ωff
Kp
ω
^ ω
Ki s is
1 s
^
θpll
sin(θ^pll )
(b)
(c) Fig. 1: Single-Phase PLL: (a) Basic scheme; (b) Conventional power-based pPLL for single-phase systems; (c) αβ power-based pPLL (αβ-pPLL) for a fictitious three-phase-systems In order to cancel the dc component of p´ ( ´), the fictitious currents ´ and ´ must be orthogonal to the voltages ´ and ´ , respectively. A feed-forward frequency is used to improve the initial dynamic performance of the αβ-pPLL system. The two-phase fictitious input voltages ( ´ , ´ ), the fictitious currents ( ´ , ´ ), and the fictitious instantaneous power (p´) of the αβ-pPLL are defined by (1), (2), (3), (4) an (5), respectively. ´ ´
(1) ⁄2
(2)
´
(3)
´ ´
Where
⁄2 ´ ´
´ ´
(4) ´
´
is the single-phase input voltage, and
(5) is the peak of the input voltage.
Substituting (1), (2), (3), and (4) into (5) the fictitious instantaneous active power is obtained by (6): ´
(6)
Where ⁄2
(7)
A simplified αβ-pPLL model is shown in Fig. 2, which replaces the pPLL control diagram of Fig. 1 (c). For small values of , the term behaves linearly, such that . As can be observed in (6), different from the conventional pPLL algorithm of Fig. 1 (b), it is not necessary the use of a low-pass filter in order to extract the dc component of p’, because the double-frequency term with high amplitude does not exist. Supposing that the utility frequency and the estimated frequency are approximately equals, i.e. , and considering and , (6) can be rewritten as: ´
Where
(8) is a small perturbation on the utility phase-angle.
Thus, in steady state, for small difference between the phase angles ( ), the signal ´ is equal to zero and converges to its reference [5]. Thereby, the closed-loop transfer function can be written as: (9) cos is equal to 1pu in steady state; Where and integral gains, respectively.
and
are the αβ-pPLL proportional
Fig. 2: Control model of the single-phase pPLL
Adaptive Notch Filter (ANF) In most power electronics applications involving power line conditioning, the harmonic components existent in the utility grid voltage should be rejected. Once the αβ-pPLL algorithm shown in Fig. 1 (c) has limited harmonic rejection capability, an ANF has been used as a way of improving the αβ-pPLL robustness associated with the utility harmonic distortion.
The ANF employed in this paper uses an adaptive noise cancelling approach presented in [17], which is based on the Wiener theory. When compared with the non-adaptive notch filters, it offers a bandwidth easily controlled, as well as the ability to adaptively track the exact frequency of noise interference. In [17], the ANF was used to eliminate sinusoidal interference found in distorted signals. In this work, the ANF is used to find out the fundamental component from the distorted utility voltage. Due to its adaptive behavior, the ANF tracks the exact fundamental component even when changes occur in the harmonic composition of the utility voltage. The ANF algorithm implemented in this work is shown in Fig. 3 (a). It is composed by a finite impulse response (FIR) filter, which uses the least mean square (LMS) algorithm to self-adjust its weights and . The input signals of the ANF are , which represents the distorted and that represent the orthogonal signals obtained from single-phase utility voltage, and the αβ-pPLL. Additionally, the ANF output signal represents the fundamental component of , as given by (10). (10) Therefore,
that represents the harmonic components of
can be obtained by (11). (11)
The procedure for updating the ANF weights
Where
and
are given as follows:
1
2
(12)
1
2
(13)
is the step size parameter, which controls the convergence rate of the ANF algorithm.
Proposed ANF-αβ-pPLL The proposed PLL scheme called ANF-αβ-pPLL is shown in Fig. 3 (b), which the adaptive notch filter discussed in section III is used in conjunction with the αβ-pPLL presented in section II. The ANF is used to extract the positive sequence components of a fictitious three-phase system represented by the orthogonal and normalized signals ´ and ´ (Fig. 3(b)). Thus, considering the scheme shown in Fig. 3 (b), the output ANF signal can be rewritten as follows: ´
, represented by (10),
´
(14)
Once represents the fundamental component of the input voltage ( rewritten as:
´
), (15) can be (15)
Where is the utility phase-angle; angle and the ANF output angle , and
is the difference between the estimated utility phaseis the maximum amplitude of .
cos From (14) and (15) it is possible to conclude that in steady state and considering that is approximately equal to
and sin 0 , the amplitude (
. Thus, ) of
the filtered signal is obtained by (16), which is used to normalize the α-input signal of the pPLL ( ´ ), as shown in Fig. 3 (b). (16) Therefore, for small difference between the utility and the estimated phase angles fictitious instantaneous power (p´), which is represented by (6), can be rewritten as follows: ´
0
, the
(17)
(a)
(b) Fig. 3: ANF schemes: (a) Adaptive notch filter; (b) Proposed ANF-αβ-pPLL algorithm
Simulation and Experimental Results The ANF-αβ-pPLL algorithm proposed in this work was evaluated by means of numerical simulations using MATLAB environment, as well as experimentally by using the DSP TMS320F28335 floatingpoint (150 MHz), operating with a sampling frequency of 60 kHz. The ANF uses a first order FIR with a step size parameter = 0.005, while the proportional and the integral controller gains of the αβ-pPLL are equal to 247.5092 and 10,969, respectively.
The following utility disturbances were taking into account for presenting both the simulation and the experimental results: phase-jump, frequency variation, voltage sag, voltage swell, and voltage harmonics. Fig. 4 shows the PLL estimated frequency ( ) and the PLL phase-angle error ( ) for a phase-jump of 40o obtained by simulation and experimental tests. Similarly, the estimated frequency and the phase-angle error are shown in Fig. 5 for a frequency variation of 5 Hz. In both disturbances, the ANF- -pPLL locks in about 50ms (3 cycles) to the new steady state condition with zero error.
Phase Error (deg)
Detected Frequncy (Hz)
The results related to the phase-error of the ANF-αβ-pPLL submitted to voltage sag (30%) and voltage swell (42.8%) are shown in Fig 6. As can be seen in Fig. 6, the peak phase error was found around 2 degrees in both simulation and experimental results. Fig. 7 (a) and (b) shows the ANF- -pPLL signals (normalized input voltage , PLL phase-angle and the fictitious current ´ for the input voltage without harmonic contents. The same signals are shown in Fig. 7 (c) and (d) for the input voltage with total harmonic distortion (THD) of 12.5%. For this utility condition the THD of the fictitious current ´ (sinusoidal αβ-pPLL output) was equal to 0.4%. As can be noted, the ANF- -pPLL algorithm has high harmonic rejection. ^ 70 60 50
0
-40
0.48
0.5
0.52
0.54
0.56
0.58
Time(s)
(b) (a) Fig. 4: Detected frequency ( ) and phase error ( ) for phase jump of 40º: (a) simulation results; (b) experimental results
Phase Error (deg)
Detected Frequncy (Hz)
80
70
60
10 0 -10
0.48
0.5
0.52
0.54 Time(s)
0.56
0.58
(b) (a) Fig. 5: Detected frequency ( ) and phase-angle error ( ) for frequency variation of 5Hz: (a) simulation results; (b) experimental results
0 -120 -180
Phase Error (deg)
Amplitude (V)
180 120
4 0 -4
0.4
0.45
0.5 Time(s)
0.55
0.6
(b)
(a)
Fig. 6: Input Voltage ( ) and phase-angle error ( ) for voltage sag (30%) and voltage swell (42.8%): (a) simulation results; (b) experimental results
vs i
(a)
(b)
v^s
(c)
(d)
Fig. 7: Experimental results: (a) input voltage (60V/div, 5ms/div) and the normalizaed ANF-αβpPLL output ( ´ (1A/div, 5ms/div); (b) normalized input voltage (1V/div, 5ms/div), and ANF(3rad/div, 5ms/div); (c) distorted input voltage (60V/div, 5ms/div) pPLL phase-angle ( ´ and the output signal (1A/div, 5ms/div); (d) normalized distorted voltage (1V/div, 5ms/div) and ) (3rad/div, 5ms/div) ANF- αβ-pPLL phase angle (
Conclusions This paper proposed a novel algorithm for estimation of the phase-angle in single-phase systems. An adaptive notch filter was used to extract the positive sequence voltage components of a fictitious threephase system represented by orthogonal voltages in the two-phase stationary reference frame. The ANF was connected to a based-power PLL system, resulting in the algorithm called ANF-αβpPLL, in which both high harmonic rejection, as well as frequency-adaptive ability was carried-out. The performance of the proposed ANF-αβ-pPLL algorithm was evaluated under utility grid disturbances, such as voltage harmonics, voltage sag/swell, phase-angle jump and frequency variation. It has been shown from simulated and experimental results the robustness and the effectiveness of the proposed PLL algorithm.
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