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Block diagram of the pPLL. v i is the grid voltage signal, ωn is the nominal frequency, ˆ θ andˆωgandˆ andˆωg are the estimated phase angle and frequency, respectively, and k i and kp are the integral and proportional gains of the proportional-integral (PI) controller, respectively.
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The grid synchronization unit, which is often based on a frequency-locked loop (FLL) or a phase-locked loop (PLL), highly affects the power converter performance and stability, particularly under weak grid conditions. It implies that a careful stability assessment of grid synchronization techniques (GSTs) is of vital importance. This task is most o...
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... are nonlinear feedback control systems with three distinct elements: phase detector (PD), loop filter (LF), and voltage-controlled oscillator (VCO) [13]. Fig. 1 illustrates a standard single-phase PLL, often called the power-based PLL (pPLL), in which these parts are highlighted [14]. This standard PLL, however, suffers from large double-frequency oscillatory ripples in its output signals [14], [15]. Solving this problem has been the main motivation behind devel- oping more advanced ...
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... are also like PLLs have a nonlinear closed-loop nature, but they are (most often) implemented in the stationary Fig. 2. Block diagram of the EPLL. ˆ V is the estimated amplitude and kv is the gain of the amplitude estimation loop. Other parameters have been defined before in the caption of Fig. 1. Throughout this manuscript, it is assumed that kv = kp. frame. Designing FLLs may be carried out in different ways. In single-phase applications, which are focused on in this letter, they are often constructed using a second-order general- ized integrator (SOGI). Fig. 3 illustrates a SOGI-FLL, which is a standard single-phase FLL ...
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... are nonlinear feedback control systems with three distinct elements: phase detector (PD), loop filter (LF), and voltage-controlled oscillator (VCO) [13]. Fig. 1 illustrates a standard single-phase PLL, often called the power-based PLL (pPLL), in which these parts are highlighted [14]. This standard PLL, however, suffers from large double-frequency oscillatory ripples in its output signals [14], [15]. Solving this problem has been the main motivation behind devel- oping more advanced single-phase PLLs [7]. For example, in [16], an enhanced PLL (EPLL) is proposed to deal with the pPLL double-frequency problem. The EPLL, which its block diagram is shown in Fig. 2, has been designed using an optimization procedure. The complete elimination of the aforementioned double-frequency oscillations in the steady state, estimating the grid voltage amplitude which makes the amplitude normalization possible, and achieving a higher filtering ability and a more smooth transient performance in detecting the grid voltage frequency 1 while maintaining the implementation simplicity are the main features of the EPLL. A review of other advanced single-phase PLLs can be found in ...
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... are also like PLLs have a nonlinear closed-loop nature, but they are (most often) implemented in the stationary Fig. 2. Block diagram of the EPLL. ˆ V is the estimated amplitude and kv is the gain of the amplitude estimation loop. Other parameters have been defined before in the caption of Fig. 1. Throughout this manuscript, it is assumed that kv = kp. frame. Designing FLLs may be carried out in different ways. In single-phase applications, which are focused on in this letter, they are often constructed using a second-order general- ized integrator (SOGI). Fig. 3 illustrates a SOGI-FLL, which is a standard single-phase FLL [10]. This structure includes a SOGI in a unity feedback control loop and a frequency estimation algorithm for adjusting the SOGI resonance fre- quency. The SOGI outputs are estimations of the grid voltage fundamental component and its quadrature version, which are used for estimating the grid voltage amplitude and ...
Citations
... The dynamic of the SOGI-FLL has been analyzed in previous studies [23,26,28], which have demonstrated that the structure can be vulnerable to distortions caused by various factors such as DC-offset voltage, harmonics, subharmonics, faults, voltage sags, and voltage swells. Specifically, research in [26] revealed that voltage sags have the greatest impact on the SOGI-FLL, causing high levels of peak distortion in the estimated frequency of the FLL response, while voltage swells have a similar but less significant impact. ...
... The dynamic of the SOGI-FLL has been analyzed in previous studies [23,26,28], which have demonstrated that the structure can be vulnerable to distortions caused by various factors such as DC-offset voltage, harmonics, subharmonics, faults, voltage sags, and voltage swells. Specifically, research in [26] revealed that voltage sags have the greatest impact on the SOGI-FLL, causing high levels of peak distortion in the estimated frequency of the FLL response, while voltage swells have a similar but less significant impact. ...
... The small-signal model of the SOGI-FLL can be obtained by assuming a quasi-locked operating point for amplitude voltage, frequency, and phase [22,24,26]. This model is described by equations (5) and (6) and is shown in Fig. 2. ...
The Second Order Generalized Integrator–Frequency Locked Loop (SOGI-FLL) is a widely used and popular adaptive filter for estimating grid voltage parameters with minimal computational burden. However, it is vulnerable to voltage sag and swell faults, especially voltage sags that can significantly distort the estimated frequency. In this paper, we propose an error-and-hold algorithm for the SOGI-FLL that can quickly detect faults and hold the estimated frequency during these perturbations. The algorithm uses the absolute value of the SOGI's error, its average, and the average of the FLL's estimated frequency to operate. It reduces induced errors in the SOGI-FLL's quadrature outputs, improves the FLL's transient response, holds the estimated frequency, and restores the phase to its previous value before the fault. The proposed algorithm is a straightforward and low computational burden algorithm that can be executed on a low-cost processor. We validate the effectiveness of the proposed error-and-hold algorithm through simulations and experimental results.
... These oversimplifications will limit the closed-loop dynamic performance of the system. Linear time periodic models of a SOGI-FLL were proposed for accuracy [15]. However, this model still cannot deal with the nonlinearity in power calculation part caused by multiplication. ...
... Since the LTP model is approximated as an LTI MIMO model of a certain order, the poles can be determined in the same way as with MIMO LTI theory [31]. 6 For method 2, the LTI and LTP MIMO matrices can be found similarly, leading to M LTI2+ , M LTI2-, M LTP2+ and M LTP2-. In M2, however, it is only necessary to check the poles of the positive-sequence sub-system, since only θ PLL+ is fed back. ...
... The K has to be increased until K = 2.45 in order to predict instability (not shown in the figure). The pole maps of M LTP1+ and M LTP1-are also shown in Fig. 12, and they 6 This means that the same method for pole determination will be used for the LTP system as for the LTI system, which is to plot the poles of each individual transfer function in the matrix. This method is also used in [32]. ...
In this article, the Linear Time Invariant (LTI) and Linear Time Periodic (LTP) models of two different implementations of the DDSRF-PLL in the presence of voltage imbalance are derived analytically. The accuracy of the models is investigated with time domain simulations, frequency scans, and stability analysis. On top of this, a guideline for properly choosing between LTI and LTP models for stability assessment of the DDSRF-PLL according to the degree of grid voltage imbalance is proposed. Furthermore, it is revealed that, depending on the DDSRF-PLL implementation, the positive-sequence voltage might also cause LTP dynamics, rendering the LTI model inaccurate even when the imbalance is low.
... Consequently, the application of the FFT to the experimental data of SMG as shown in Fig. 1 is not suitable and results in spectral leakage, as the FFT is a frequency-dependent technique. For online applications, this issue can be solved by adapting the window size of the FFT using the frequency/phase-locked-loop (FLL/PLL) [30]. However, the FLL/PLL is distinguished by a negative feedback closed-loop system, which can be performed only online; therefore, the application of the FFT in offline analysis-based short-term protective actions becomes a challenge. ...
Modern maritime microgrid systems are witnessing a revolutionary advancement by integrating more renewable energy sources and energy storage systems. The integration of these sophisticated systems is achieved, however, through the power electronics converters that cause severe harmonic contamination. This problem becomes more serious when some of these harmonics that are non-integer multiples of the fundamental (inter-harmonics) exist concurrently with both system frequency drifts and large-power transients, which is a commune issue in maritime microgrid systems such as shipboard microgrids. Hence, the performance of the widely signal processing algorithms applied in the measurement and communication systems such as the smart meters and power quality analyzers tends to worsen. To address this problem this paper proposes an effective method based on the eigenvalue solution to estimate the harmonics and inter-harmonics of modern maritime microgrid systems effectively. This method, which is a system frequency independent technique can work effectively even under large frequency drifts with short window width. The proposed method is evaluated under MATLAB software, and then the experimental validation is carried out via analyzing the electrical power system current of a bulk carrier ship.
... Recently, some efforts for deriving analytic linear models and analyzing the stability of single-phase SOGI-based signal conditioning/synchronization systems have been made in the literature. For instance, in [19], an analytic linear time-periodic (LTP) model for a simple SOGI-FLL (including only one SOGI centered at the fundamental frequency) is presented, and its accuracy in predicting the stability and dynamic behavior of the SOGI-FLL is compared with that of its linear time-invariant (LTI) model. Although more accurate than the LTI one, this LTP model is not completely accurate as it neglects the amplitude estimation dynamics of the SOGI-FLL and, therefore, results in optimistic predictions about the SOGI-FLL stability. ...
... The above equation and (12) have a common nonlinear term highlighted in bold. Considering (16), which is the result of the lineraization of (12), (19) can be linearized as ...
... If we compare the above equations with (11), (12), and (19), it is immediate to conclude that they are the same equations if λ = γ and k1h1ω = µ1 are considered. 4 It implies that the EPLL and SOGI-FLL and, therefore, their extended versions (the MSOGI-FLL and MEPLL system) are approximate equivalent and have the same LTP model. ...
A SOGI is a resonant regulator with a pair of complex-conjugate poles, and therefore, with infinite magnitude at its center frequency. Thanks to this property, one can put together a set of m SOGIs in an elegant way and decompose a single-phase signal into its constituent frequency components and detect their amplitude and phase angle. Such a configuration, which is often referred to as the multi-SOGI (MSOGI) structure, requires a frequency estimator to adapt the center frequency of SOGIs to frequency changes. This frequency estimation is often provided by interconnecting the MSOGI structure with a basic frequency-locked loop (FLL) or phase-locked loop (PLL). The resulting structures are known as the MSOGI-FLL and MSOGI-PLL. These structures and their close variants are mathematically difficult to analyze probably because of the lack of a linear model for these systems. This paper aims to bridge this research gap. First, it is shown how linear time-periodic (LTP) models of the MSOGI-FLL and MSOGI-PLL can be obtained. The model verification, obtaining open-loop harmonic transfer function (HTF) from the LTP model, and LTP stability assessment of MSOGI-based synchronization systems are the next parts of this study. Finally, some close variants of the MSOGI-FLL/PLL are considered and their modeling and stability assessment are briefly discussed.
... Based on (1), Gv′ (s) is a second-order bandpass filter with unity gain and a phase angle of zero at the resonance frequency, 0. Also, Gqv′ (s) is a second-order low-pass filter with unity gain and a phase angle of 90º at 0 [5]. In other words, when 0 is set to the grid frequency, the amplitude of v′ and qv' is equal to the input voltage, v, and their phase shift regarding v is zero and 90º, respectively. ...
... As seen, it takes almost 100 msec to reach the steady-state conditions after the phase angle transient. This fully agrees with (5) and shows that the proposed QSG has a negligible effect on the dynamics of the ordinary SRF-PLL. The same frequency step scenario, i.e., from 50 Hz to 47 Hz, was also studied in [8], where a thorough theoretical study was carried out to design the conventional SOGI-PLL parameters for the best dynamic response. ...
Suppressing the negative effects of grid voltage harmonics on the estimated frequency of a PLL is a challenge in literature. This paper proposes an easy-to-implement quadrature signal generator (QSG) to attenuate the oscillations on the estimated frequency in single-phase grid-connected inverters which use SOGI-PLL. It is shown that the second-order generalized integrator (SOGI) exerts a stronger filtering effect in generating the quadrature signal than the in-phase signal. Against this background, in this work, a modified integrator is introduced into the path of the in-quadrature signal to generate another in-phase component with much lower harmonic content. The proposed method imposes only a small computational burden on the existing SOGI-PLL compared to the previously presented methods that address the input voltage harmonic problem. Moreover, this method can work properly within the allowable range of grid voltage frequency deviations. The proposed integrator benefits from an error-decaying mechanism to overcome dc drift in pure integrators. The integrator has been designed based on theoretical equations. The validity of the proposed QSG and theoretical equations is evaluated using simulations and experimental studies. A fixed-point representation of the proposed QSG is also provided for implementation on low-cost microcontrollers
... In [8], Orillaza has used the LTP theory to model a threephase thyristor controlled reactor (TCR) in voltage control mode, including the control systems. Golestan in [9], has modeled SOGI-frequency-locked loop and enhanced PLL (EPLL) using the LTP theory and compared the results with the conventional LTI modeling approach. It is found that the LTP modeling approach leads to more a more accurate model. ...
... In the modeling process, the time-periodic dynamic variables of PLL are replaced by rated values to simplify the modeling of SOGI-PLL. For instance, a linearized model is developed for SOGI-PLL (hereinafter referred to as LTI model A) in [2] and a linearized model is derived for enhanced PLL (EPLL) in [19]. ...
... Therefore, the modeling accuracy of SOGI-PLL is low and stability analysis of SOGI-PLL is also not exact. For improving modeling accuracy, in [19], the LTP model for EPLL is derived. Compared to LTI models, the phase/ 0278-0046 (c) 2021 IEEE. ...
... Citation information: DOI 10.1109/TIE.2021.3090699, IEEE Transactions on Industrial Electronics 2 frequency dynamics of the EPLL are accurately depicted by this LTP model, since the frequency coupling, which is caused by periodic dynamics in the system [19]- [20], is taken into account in the LTP model. However, this basic LTP model ignores the coupling characteristic between the amplitude and phase/frequency of the grid voltage estimated by PLL. ...
This paper focuses on the modeling and stability analysis of the typical single-phase phase-locked loop (PLL) based on second-order generalized integrator (SOGI). Traditionally, SOGI-based PLL (SOGI-PLL) is linearized as the linear time-invariant (LTI) model for the stability analysis and parameter design. Yet, the PLL has periodical nature due to the periodical variation of the grid voltage. The traditional LTI model for PLL thereby reduces the precision of stability analysis. For addressing this problem, an accurate linear time-periodic (LTP) model of the single-phase SOGI-PLL is proposed in this paper. The proposed model can precisely characterize the coupling nature of phase, frequency, and amplitude estimated by SOGI-PLL. A precise stability region of SOGI-PLL thereby can be derived from the proposed LTP model. The effectiveness of the proposed LTP model for SOGI-PLL is verified by simulation and experimental tests.
... Examples and Applications: LTP models can be more accurate than LTI models to describe CDPS dynamics. In [189], the accuracy of the LTI and the LTP frameworks were compared using a grid-following converter considering different grid-synchronization methods. It was demonstrated that the LTP framework is not only able to more accurately describe the instantaneous dynamics, but also is capable of describing the double-frequency oscillations in the transient response of the converter. ...
In response to national and international carbon reduction goals, renewable energy resources like photovoltaics (PV) and wind, and energy storage technologies like fuel-cells are being extensively integrated in electric grids. All these energy resources require power electronic converters (PECs) to interconnect to the electric grid. These PECs have different response characteristics to dynamic stability issues compared to conventional synchronous generators. As a result, the demand for validated models to study and control these stability issues of PECs has increased drastically. This paper provides a review of the existing PEC model types and their applicable uses. The paper provides a description of the suitable model types based on the relevant dynamic stability issues. Challenges and benefits of using the appropriate PEC model type for studying each type of stability issue are also presented. INDEX TERMS Average models, data-driven models, dynamic phasor models, inverter-based resources, large-signal models, positive-sequence models, power electronic converters, power system modeling, power system simulation, power system stability, small-signal models, switching models.
... This means that FLLs and PLLs are virtually the same control systems as those implemented in different reference frames. In addition, higher penetration of renewable energy resources or the weak grid condition, the dynamic interaction between a PLL or FLL and the converter make the PLL or FLL and, therefore, the converter unstable [27] if the conventional vector current control is applied. The PLL introduces negative incremental resistance at low frequencies and [28] has shown the frequency coupling dynamics of the converter introduced by the PLL. ...
This paper presents an effective quasi open-loop (QOLS) synchronization technique for grid -connected power converters that is organized in two different blocks. The first block is a new flexible technique for extracting the positive and negative sequence voltage under unbalanced and distorted conditions. It is a decoupled double self-tuning filter (DD-STF) or multiple self-tuning filters (M-STF) according to the conditions. The main advantages of this technique are its simple structure and the fact of being able to work under highly distorted conditions. Each harmonic is separately treated and this allows for selective compensation in active filter applications. The second block is the frequency detector; we propose a neural approach based on an ADALINE for online adaptation of the cut-off frequency of the DD-STF and M-STF considering a possible variation in the main frequency. The main advantage of this method is its immunity to the voltage signal amplitude and phase. In order to improve the performance of the frequency estimation under distorted source voltage, a pre-filtering stage is introduced. Experimental tests validate the proposed method and illustrate all its interesting features. Results show high performance and robustness of the method under low voltage ride through.
