Dynamics Of Phase-locked Loop Research Articles

A Series Impedance Reshaping Control Method Considering PLL Dynamics for Grid-Connected Inverters Under Weak Grid Conditions

Grid-connected inverters are the key part in renewable energy power generation systems. Usually, phase-locked loop (PLL) is adopted in grid-connected inverters to achieve synchronization. However, prior-art works have revealed that the negative resistance behavior in the low-frequency band caused by PLLs may lead to system instability under weak grids. In this paper, first, a comprehensive small-signal output impedance modeling of the LCL-filtered single-phase grid-connected inverter is derived for stability analysis, where the digital control delays and PLL dynamics have been taken into account. Then, a simple but effective virtual series based impedance reshaping method is proposed. With that, the inverter system can operate stably even under a very weak grid (e.g., short-circuit ratio (SCR) is as low as 1.2). Compared with the traditional grid impedance compensation by virtually increasing grid resistance or reducing the grid inductance, the proposed method does not require grid impedance estimation and is robust against system parameter variations. Thus, its feasibility is improved. Finally, experimental tests are carried out on a single-phase 5-kW prototype, validating the effectiveness of the proposed scheme.

Accurate Stability Analysis of Three-Phase Converter System Connected to Weak Grid under Coordinate System Transformation

This paper studies the simplified impedance criterion of the three-phase converter system in different coordinate systems. In addition, the influence of small signal stability of converter units under the weak grid is discussed in three-phase converter systems. Firstly, a small signal model of a three-phase grid-connected converter system is established based on the voltage and current double closed-loop control strategy in dq real coordinate system and the dynamic characteristics of SRF-PLL (Synchronous Reference Frame Phase-Locked Loop). Then, based on the coordinate transformation, the admittance models in the dq real coordinate system and the fb complex coordinate system are derived. Through the analysis of the admittance Bode diagram, the amplitude-frequency characteristics of the three-phase converter system have conjugate symmetry in the real coordinate system and diagonal conjugate symmetry in the complex coordinate system. Therefore, the three-phase symmetrical system can be stabilized only by drawing the Nyquist curve of the open-loop transfer function GHf or GHb. The three-phase asymmetric system can also be stabilized by drawing the Nyquist curve of the modified open-loop transfer function GHfM or GHbM . Finally, the simulation results show that the converter system loses stability when the transmission line impedance Lg = 0.6 pu. In this case, the three current decoupling terms will cause the system to lose stability, but the relative stability of the current reference value is the best, and the phase angle margin is −6.8°. Also, without considering the phase-locked loop dynamics, the system is in a steady state, otherwise, the system loses stability, and the system output current has a 39 Hz oscillation, which is consistent with the theoretical analysis results.

Analytical Model of Inverter-Interfaced Renewable Energy Sources for Power System Protection

The accurate analytical fault analysis model of the emerging inverter-interfaced renewable energy sources (IIRESs) is an important basis for protection of power system with RESs. However, factors of dynamic characteristics of the control system (DCCS) such as the controller saturation and phase-locked loop (PLL) dynamics have rarely been simultaneously studied in the existing analytical models of IIRESs. As a result, the accurate analytical models can hardly be guaranteed. To address this issue, this paper proposes an analytical model for IIRESs taking DCCS into account to characterize the complex fault responses of IIRESs. Both the controller saturation and PLL dynamics are considered in the transient and steady-state phases to investigate the fault current characteristics. On this basis, the analytical fault current formulations under the influences of DCCS are yielded using the piecewise linearization method. Field short-circuit and simulation tests indicate that the proposed model is applicable to various fault conditions and its calculation accuracy can be improved by at most 55% in comparison with the existing IIRESs' analytical models.

Phase locked loop parameterisation in AC/DC interconnected system for a multi-source AGC system under open market environment

This article emphasises on deregulated AGC of a realistic multi-machine gas - thermal plant with dish-Stirling solar-thermal (DSTS) as additional power generation sources in each control area by taking into accounts of AC/DC with phase locked loop (PLL) dynamics. Eigenvalue analysis confirms the steadiness of the reasonable power system with various interconnections such as AC, AC/DC link and AC/DC with PLL dynamics. A series form of proportional-integral-derivative (PID) namely proportional gain cascaded with integral and double-derivative (PI+DD+) controller is employed as secondary controller in each control areas for robust AGC to carry out market transactions under deregulated environment. Exploring the system with different type of interconnections infer that proper tuning of PLL function parameters may synchronise the grid better than without employing PLL. The performance of PI+DD+ controller along with optimum PLL gains commensurate with wide variations in governor, turbine, fuel and compressor discharge time constant of gas turbine plant.

Stability Analysis of a Weak-Grid-Connected Voltage-Sourced Rectifier Considering the Phase-Locked Loop Dynamics

The high bandwidth of a phase-locked loop (PLL) can deteriorate the small-signal stability of a voltage-sourced converter (VSC) under a weak grid condition. This instability can occur for a VSC operating as both a rectifier and an inverter, but the root causes can differ for the two. This study, through modal analysis, demonstrates that different approaches are required depending on the power flow direction to resolve the instability. Subsequently, an analysis of the rectifier dynamics is set as the primary target, and the instability characteristic is investigated via a new approach that utilizes the open-loop transfer function of the power control loop. Consequently, it is discovered that the high-bandwidth PLL reduces the reactive power compensation for the AC system and results in a severe voltage drop. A novel parallel voltage filter technique is proposed to overcome this instability. The performance of the proposed method is verified in the frequency- and time-domain analyses. The results indicate that the proposed method effectively improves stability without major modifications to the control system.

Enhanced Virtual Inertia Control for Microgrids with High-Penetration Renewables Based on Whale Optimization

High penetration of renewable energy sources into isolated microgrids (µGs) is considered a critical challenge, as µGs’ operation at low inertia results in frequency stability problems. To solve this challenge, virtual inertia control based on an energy storage system is applied to enhance the inertia and damping properties of the µG. On the other hand, utilization of a phase-locked loop (PLL) is indispensable for measuring system frequency; however, its dynamics, such as measurement delay and noise generation, cause extra deterioration of frequency stability. In this paper, to improve µG frequency stability and minimize the impact of PLL dynamics, a new optimal frequency control technique is proposed. A whale optimization algorithm is used to enhance the virtual inertia control loop by optimizing the parameters of the virtual inertia controller with consideration of PLL dynamics and the uncertainties of system inertia. The proposed controller has been validated through comparisons with an optimized virtual inertia PI controller which is tuned utilizing MATLAB internal model control methodology and with H∞-based virtual inertia control. The results show the effectiveness of the proposed controller against different operating conditions and system disturbances and uncertainties.

DC-Link Voltage Stability Analysis of Grid-Tied Converters Using DC Impedance Models

With the integration of renewable energy sources into the power grid, a number of power electronic converters need to be connected together in parallel. Due to this interconnection among the power converters with a common DC bus, the equivalent impedance of the DC network, i.e., DC network impedance (DCNI) of these parallel converters, may vary and can cause oscillations in the DC link voltage (DCLV). In the literature, impedance models of grid-tied converters (GCs) based on the AC side are well reported without including these variations in DCNI. In addition, the dynamics of a phase-locked loop (PLL) play a significant role in GC system stability. To evaluate these stability issues, this paper proposes small signal impedance models viewing from the DC side of a three-phase GC operating under different control modes considering the PLL dynamics and the DCNI variations. Using the proposed DC impedance models (DCIM), DCLV stability analysis is evaluated for a GC. It is verified through bode plots that the interaction between the proposed DCIM and DCNI leads to unstable operation of the closed-loop converter near the PLL bandwidth when the phase difference between DCIM and DCNI is more than 180 degrees. Finally, the analytically developed models are validated using hardware in-the-loop (HIL) testing.

Analysis and Design of PLL Less Current Control for Weak Grid-Tied LCL-Type Voltage Source Converter

For a weak grid-tied <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$LCL$ </tex-math></inline-formula> -type converter, the conventional current control interacts with a phase-locked loop (PLL) through the voltage at the point of common coupling (PCC). Thus, the dynamics of PLL deteriorate converter stability, hence leads to system instability issues. To avoid the instability issues due to PLL, a PLL less current control is proposed for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$LCL$ </tex-math></inline-formula> -type grid-connected converter with capacitor-current-feedback active damping. The analysis and design of the proposed PLL less current control is presented in this article, and the performance of the proposed control is compared with conventional current control (i.e., PLL without bandpass (BP) filter) and improved conventional control (i.e., PLL + BP filter). A small-signal model is developed by considering the dynamics of the PLL and BP filter to show the impact of the current control parameters on converter output impedance. An impedance-based stability analysis is performed to show the system stability against short circuit ratio (SCR) variation for three different current control scenarios, such as proposed PLL less, PLL with a BP filter, and PLL without BP filter. The proposed current control works in both stiff grid and weak grid environments. It has strong robustness against a wide range of variation in grid impedance and more adaptable to grid frequency changes. The performance of the proposed control is verified during grid voltage harmonics, unbalanced voltage, and grid voltage variations and further validated by comparing with the conventional control. In addition to the features mentioned above, the proposed current control is able to operate stably at SCR = 1. Finally, the experiment shows the superiority of the proposed control over conventional control.

Impact of DC-Bus Voltage Control on Synchronization Stability of Grid-Tied Inverters

The impact of dc-bus voltage control (DVC) on transient synchronization stability of grid-tied inverters operating in grid-feeding mode has been assessed in this brief. A nonlinear quasi-static model magnifying the dynamics of phase-locked loop (PLL) and DVC of grid-tied inverter has been developed first. Following this, the steady-state and transient stability boundaries of the model are analytically characterized and numerically evaluated. The proposed model and the stability assessment method eliminates the inaccuracies associated with the previously developed model which considers the dynamics of PLL alone.

Improved MIMO Modeling and Enhanced Transient Performance of Phase-Locked Loop During Grid Fault

A phase-locked loop (PLL) is commonly used in grid-connected power converter for synchronization. A single input single output (SISO) linear model of PLL with the phase angle of the point of common coupling (PCC) voltage as input and the estimated phase angle as the output is generally used in the PLL analysis and design. However, an undesirable coupling between the magnitude and phase of the input voltage is present when a filtering stage is incorporated before the control loop, which a SISO model cannot capture, and could result in a significant disturbance in the PLL estimated phase during grid faults. This paper proposes a generalized multi-input multi-output (MIMO) linear model that captures the complete PLL dynamics during symmetrical faults for a PLL equipped with any prefilter. Furthermore, a compensation method that decouples the magnitude dynamics of the input voltage from the phase dynamics is proposed in this paper. The proposed compensation improves the PLL's phase tracking performance by ensuring that the prefiltered PLL acts only on the PCC voltage phase changes. A vital application of the proposed compensation method for PLL is during a grid fault case, wherein the converters are expected to contribute to the fault current, which requires an accurate measurement of the PCC voltage phase. The compensation method's effectiveness is demonstrated using power hardware in the loop simulation of a hardware voltage source converter interfaced to a distribution system simulated in real-time.

Coordinated control to achieve stability and dynamic response enhancement of weak‐grid‐connected inverters based on sequence‐admittance models

In grid-connected inverters (GCIs), the phase-locked loop (PLL) behaves as a negative admittance at the point of common coupling (PCC), composed of both the PLL controller and current controller terms. Therefore, not only the PLL dynamics, but also dynamical interactions between the PLL controller and current controller might trigger instabilities in weak grids, which complicates the controller parameters regulation. The smaller PLL bandwidth could help to mitigate the instabilities, but at the expense of sacrificing the system dynamic response. In view of this, a coordinated control composed of the q-axis voltage and PCC-voltage feedforward based on sequence-admittance models is proposed here. The q-axis voltage feedforward is designed with a simple proportional feedforward coefficient to make the PLL-induced negative admittance insensitive to the PLL controller parameters. In other words, the PLL bandwidth would not be limited by the stability requirements and the dynamic response speed could be ensured. Furthermore, the PCC-voltage feedforward is developed to coordinate the q-axis voltage feedforward by phase compensation, thus to achieve stability improvement. Finally, simulations and hardware-in-the-loop tests are carried out to verify the enhanced stability and dynamic response of the modified GCI with the proposed coordinated control in weak grid.

Multivariable State-Feedback Controller Design for PV Inverter Connected to a Weak Grid

This paper presents a robust multi-input multi-output (MIMO) state-feedback control scheme for a photovoltaic (PV) inverter connected to a weak grid. For a weak grid, the point of common coupling (PCC) voltage is very sensitive to the power disturbances and it is dynamically coupled to the PLL dynamics. So far, most of the control methods do not take into accounts these couplings. Therefore, in this paper, the MIMO controller was designed taking into account the dynamics of the phase-locked loop (PLL) and coupling effects between PCC voltage and the active power to enhance the system’s robustness. As result, the robust performance of the PV inverter interfaced to a weak grid was yielded. In addition, the sensitivity of the system to PLL was eliminated, allowing the use of larger PLL bandwidth even in a very weak grid. Based on the eigenvalues analysis method, a comparative study between the proposed control method and the conventional vector control method was performed. The proposed method is verified with simulations in PLECS and real-time simulations in the RT Box. The results show that the proposed MIMO control method preserves the system stability robustness against any change of grid strength, generated power and PLL bandwidth.

Transient Stability Analysis of Grid-Connected Converter Driven by Imbalance Power under Non-Severe Remote Voltage Sag

In the transient process of the grid-connected converter (GCC), the existing research mainly focuses on the impact of the control loops. Little attention is paid to the transient stability issues driven by the imbalance between the input power and output power of GCC. This paper shows that the transient stability issues will still exist even if ignoring the dynamics of phase-locked loop (PLL) and current loop. In this paper, the models of the AC grid and the GCC are built under the assumption that the dynamics of the PLL and current loop are ignored. Then, by analyzing the transient process of GCC under non-severe remote voltage sag, the effects of the imbalance power on the transient stability of GCC are discussed. Moreover, for the GCC to operate stably after the transient process, there should be a maximum input power limit (MIPL) for GCC, and the imbalance power equation is applied in this paper to determine the transient stability of GCC. Furthermore, the effects of the current limitation on the transient stability of the GCC are also discussed. Finally, the theoretical analysis has been verified by means of simulations.

Nonlinear Stability Analysis of the Conventional SRF-PLL and Enhanced SRF-EPLL

Phase-locked loop (PLL) systems play a crucial role in grid synchronization. Since variations on the grid voltage parameters are in general uncertain, the dynamics of PLLs are by nature nonlinear. This fact imposes a relevant challenge to guarantee their stability. Unfortunately, as argued in this paper, there is a remarkable lack of global stability tests even for the most common approaches, such as the synchronous reference frame (SRF). In this case, there is only available a small-signal—linearized—model-based approach to stability. The main issue of resorting to local conclusions is that synchronization is forced to operate under conservative conditions, where large-signal variations are forbidden. Needless to say, this has a detrimental impact on applications, since performance is limited by small disturbance assumptions. Motivated by this problem, in the present work, we provide global asymptotic stability tests for two fundamental PLL algorithms. Namely, the conventional SRF-PLL and the enhanced SRF-EPLL. To do so, we use a Lyapunov approach acting directly on nonlinear trajectories. Consequently, the provided stability proofs do not rely on any linearization technique, frequency-domain conversion, or small-signal model representation. The main contribution in this paper is a set of stability conditions that are valid in a global sense. Moreover, it is shown that these results have direct practical implications in the form of new robust PLL gain tuning guidelines.

Applying Synchronous Condenser for Damping Provision in Converter-dominated Power System

The dynamic characteristics of converter-dominated systems are governed by controlling power converters and the interactions between converter systems and conventional alternators. Frequency oscillations can appear under dynamic operation conditions caused by the phase-locked loop dynamics and interactions among the converter control systems. The oscillations may be poorly damped, which can result in reduced power generation, longer settling time, or disconnections of sensitive components. It is foreseeable that damping services will be critical for power grid stabilization in the future with high penetration of renewable generation. In this work, synchronous condensers (SCs) are evaluated and applied to provide damping services to the power grid under post-event conditions. An innovative supplementary controller for the automatic voltage regulator of SCs is proposed to improve the frequency stabilization in a converter-dominated system after disturbances. Using local and remote measurements, SCs are able to modulate the reactive power output and hence, the terminal bus voltage, which further impacts the power flow in the system; therefore, damping can be provided to the frequency oscillations. The control is implemented on an industrial-level hardware platform, and the performance is verified by the hardware-in-the-loop simulation.

Analogized Synchronous-Generator Model of PLL-Based VSC and Transient Synchronizing Stability of Converter Dominated Power System

In this article, the synchronizing dynamic issue of VSC dominated power system is focused on. First, based on the discovered analogous synchronizing dynamical process of VSC (voltage source converter) and SG (synchronous generator), the dynamics of VSC's PLL (phase locked loop) is modeled in the similar form of SG's rotor motion. The influence of control interaction on PLL is modeled in the same way as that of control dynamics on rotor motion. Then the analogized SG model of PLL-based VSC is developed. Based on the proposed model, the analysis idea jointing accelerating-decelerating area method and numerical simulation is employed to study the transient synchronizing stability of single VSC infinite bus system. It is identified that the influence of ACC (alternating current control) on PLL is to accelerate PLL's equivalent motion in first swing, which aggravates the mismatch of accelerating-decelerating area and thus deteriorates the transient stability. Furtherly, the influence factors including ACC's bandwidth, PLL's bandwidth and current references on the transient stability are studied. This article provides a useful analysis idea for transient synchronizing stability analysis of VSC with influence of control interaction considered.

Modelling and comparison analysis of grid‐connected DFIG‐based wind farm in weak grid

This study proposes a generic method for modelling and comparison analysis of grid-connected double-fed induction generator (DFIG)-based wind farms in a weak grid. A detailed model of DFIG in a weak grid is established and used as a benchmark. Specifically, the detailed model consists of the mechanical part, generator part, rotor side converter, and phase-locked loop (PLL). Two simplified models of grid-connected DFIG-based wind farms in a weak grid are compared with the detailed model to investigate the mechanism of sub-synchronous oscillation (SSO) as well as the influential factors through linear system analysis. The first simplified model ignores shafting dynamics, stator flux dynamics, and PLL dynamics. The same assumptions are used in the latter, except that PLL dynamics are considered. Time-domain simulation and eigenvalue analysis are used to verify the accuracy of the simplified models. The block diagrams of the linearised system are derived based on the proposed model, and the oscillation mechanism is explained by linear system analysis. Through eigenvalue analysis and the root locus method, the influences of the parameters of the DFIG and its controllers on the system stability are identified and analysed.

Impact of PLL dynamics on the small-signal stability of the MMC using dynamic phasor modeling in the ABC frame

This paper presents a small-signal dynamic phasor model of the modular multilevel converter including phase-locked loop (PLL) dynamics. A simple PLL model including a low-pass filter at its input and accounting for the harmonic content of the voltage at the point of common coupling is developed to assess the impact of the PLL on small-signal stability. Nonlinear models of the converter, its controllers and the PLL are separately developed and linearized. The linearized models are combined using the component connection method to obtain the overall system matrices. The developed model is verified against a nonlinear averaged model in MATLAB/Simulink, and dynamic phasor models including different harmonic content are compared in terms of their accuracy. Eigenvalue analysis is carried out to study the impact of the PLL and its filter cutoff frequency on the small-signal stability of the converter.

Impact of Current Transients on the Synchronization Stability Assessment of Grid-Feeding Converters

The synchronization instability in the presence of a fault is a main issue for the dynamic behavior and control of grid-feeding converters. In the literature, the synchronization stability assessment is carried out considering the dynamics of Phase-Locked Loops (PLL) but the transients of converter currents are neglected. The letter shows that such a simplification leads to inaccuracies and, thus, the current transients cannot be neglected. The letter proposes a model that captures the effect of such current transients on the converter synchronization. This model allows assessing the transient behavior and, hence, the stability, of power electronics converters with high accuracy, comparable, in fact, to EMT models. The fidelity of the proposed model is duly discussed in the case study.

A Modular Adaptive Robust Nonlinear Control for Resilient Integration of VSIs Into Emerging Modernized Microgrids

Nowadays, emerging modernized microgrids (MMGs) have significantly employed the voltage-source inverters (VSIs). VSIs are enabling technologies to form multi-infeed ac/dc (MIACDC) power systems integrating a variety of generation units and different loads into one coherent grid. In emerging MMGs, the frequency-dependent dynamics of multiple components affect the PQ-controlled, grid-connected VSIs (GC-VSIs), particularly when integrated into a weak network. In order to address this challenging problem, the full integration of the dynamics of the phase-locked loop (PLL) into those of GC-VSIs is accomplished in this research via an innovative modular structure to improve the VSI’s performance. This action is expected to significantly reduce the effect of the ac-side dynamics on the control of GC-VSIs. In addition, there are uncertainties associated with the parameters in the system. Accordingly, mismatched disturbances and uncertainties (both matched and mismatched ones) will appear in a nonlinear dynamic problem, and therefore, from the standpoint of control theories, mismatched disturbances and uncertainties should be overcome. Satisfying them is a difficult task in control of nonlinear systems. Therefore, one of the main contributions of this work is finding an appropriate mathematical model of GC-VSIs in the $dq$ -frame for the problem under study. Afterward, in order to control both active and reactive power independently, this article presents a novel two-degree-of-freedom (2DoF) methodology with an enhanced modular design. It combines the sliding-mode control with a “new sliding manifold” and a disturbance observer with a “new adaptation rule” taking care of uncertainties and mismatched disturbances. Those disturbance signals may be generated by the PLL dynamics or voltage signals affecting the GC-VSI’s dynamics. Through providing mathe- matical analyses (including stability assessments via various theorems using Lyapunov stability criterion), simulation results, and experiments, this article demonstrates the effectiveness of the proposed control methodology. The industrially accepted GC-VSI equipped with an $LCL$ -filter is used here.