Generalized Nyquist Criterion Research Articles

A Hierarchical Small-Signal Controller Stability Analysis Method for the MMCs

Differing from a two-level voltage source converter (TL-VSC), the internal dynamics of a modular multilevel converter (MMC) are much more complicated. Since the fluctuation of submodule capacitor voltage is unavoidable when transmitting power, the MMCs output voltages cannot track the references perfectly. The interaction between the time-varying capacitor dynamics and controllers may lead to self-instability of the MMCs. In order to address this self-instability issue, this paper first establishes the small-signal closed loop transfer function matrices (TFMs) of the external and internal control loop by the harmonic linearization method. Further, a novel design principle of MMC controllers is proposed. Then, the critical impact factors on the stability of circulating current controller and AC voltage controller are investigated based on the generalized Nyquist criterion(GNC). The proposed models and analysis methods are validated by time-domain simulations in PSCAD/EMTDC.

Stability Assessment of the PET-Based Distributed Grid

The penetration of renewables has been increasing nowadays. The traditional transformer can no longer meet the requirements of utilities. For this reason, a power electronic transformer (PET) is proposed as one of the promising alternatives. However, there are coupling issues between the PET and the connected converters in the low-voltage grid. To study the issues effectively, this article developed impedance models of the PQ node, PV node, and PET. Based on the models, the system stability under different scenarios is assessed by the generalized Nyquist criterion. The effects of the line impedance and control parameters on system stability are studied. Moreover, a comprehensive parameter sensitivity analysis was carried out to reveal the coupling mechanism between converters. Simulations are given to validate the effectiveness of the theoretical analyses.

Commutation Overlap Characteristic Modeling and Stability Analysis of LCC-HVDC in Sending AC Grid

In recent years, the doubly fed induction generator (DFIG)-based wind power generation and high-voltage direct-current (HVDC) transmission are widely utilized in practice. The stability issues of sending AC grid, which is composed of line-commutated converter (LCC)-based HVDC and DFIG system, is worthy of attention. Commutation is an important factor in the impedance model of LCC-HVDC rectifier, due to the nonnegligible leakage inductance of transformer. However, it was overlooked in previous studies because there are many difficulties in modeling commutation overlap characteristic. In this paper, the spectral models of voltage and current mapping functions are proposed by 3-D Fourier series method, in which the commutation overlap angle is considered. Then, based on the concept of harmonic linearization, an analytical model that describes the commutation process is established and validated by simulations. The mechanism that the commutation overlap characteristic affects the impedance model of LCC-HVDC is investigated, and the simplified modeling method in different frequency bands is studied. The stability of sending AC grid is analyzed applying the Generalized Nyquist Criterion (GNC). The results of stability analysis show that the effect of commutation overlap cannot be ignored. The simulation results confirm the correctness of the obtained conclusion.

Stability analysis of doubly-fed wind generation systems under weak power grid based on virtual synchronous control combined with adaptive robust control

Doubly-fed induction generator (DFIG) of wind generation system is generally located at the end of the distribution power-grid. Under the weak power grid, the grid connection of wind turbines is unstable, and the current quality is poor. Based on DFIG output impedance model, a stability analysis method combined with adaptive control method is proposed in this paper. The DFIG impedance model with power outer-loop is derived based on the virtual synchronous control of DFIG by using adaptive robust control method. Using generalized Nyquist criterion, the impacts of different power grid conditions and control parameters on the system stable operation under virtual synchronous control are analyzed and verified. The proposed control method in this paper is verified by experiments. The proposed control method improves the quality of grid-connected current.

Fast and accurate grid impedance estimation approach for stability analysis of grid-connected inverters

Interactions between grid-connected inverters and the equivalent grid impedance seen at their point of common coupling have been identified as one of the main causes of instability problems. Therefore, the need for real-time monitoring tools to identify grid impedance variations cannot be underrated. Even though existing techniques have shown to estimate the grid impedance accurately, several concerns have remained unsolved. These include the trade-offs between the estimation accuracy and the magnitude and the time duration of the injected disturbance(s). Hence, this paper proposes a fast and accurate grid impedance estimation approach for the application of online impedance-based stability analysis of grid-connected inverter systems. First, the wideband grid impedance is estimated online based on the injection of two triangular pulses, allowing fast and multiple-input multiple-output (MIMO) identification in the synchronous (dq) reference frame. Then, curve fitting of the estimated wideband grid impedances is applied to obtain their parametric models. Afterward, the stability analysis is performed in the dq reference frame based on the generalized Nyquist criterion. The proposed algorithm allows accurate stability analysis over all frequencies as it relies on the parametric impedances. Most importantly, to optimize the trade-offs between the power quality and accurate stability analysis, the proposed algorithm is executed again and injects the two pulses disturbances to the grid only if the system operating conditions change (e.g., variations of the grid impedance). It is also applicable to perform the stability analysis while the inverter is still in standby mode. The simulation and experimental results of a 3-phase grid-connected inverter are presented to validate the efficacy of the proposed technique.

Optimal Controllers to Improve Transient Recovery of Grid-Following Inverters Connected to Weak Power Grids

Leading to the enormous growth in renewable and power electronics technologies and the global drive towards environmental friendliness and sustainability, a significant number of renewable energy sources are being connected to the power system via inverter-based systems. The inverter-based generations (IBG) have no stored energy and less fault current injection capability compared to the conventional synchronous machines. Consequently, a large penetration of IBG creates challenges to maintaining the stability of the power system, especially the transient stability. The weaker the power system, the higher the significance of instability. Few solutions exist in the literature to improve the fault recovery of IBG connected to weak power systems. This paper considers the method of storing energy in sub-module capacitors of the Modular-Multi-level Converter (MMC) along with temporarily boosting the inverter’s current limit. Conversely, increasing the ratings of the inverter will result in high manufacturing costs. Hence an optimization strategy is proposed in this paper, for obtaining a robust set of inverter control parameters that enhances fault recovery without excessively increasing the manufacturing cost of MMC. A frequency scanning technique supplemented with Generalized Nyquist criteria is incorporated into the optimization methodology to constrain the search space for the optimization algorithm. This enables the optimization algorithm to converge to an acceptable solution with a reasonable computing time. Furthermore, validation of the resultant set of parameters for different system conditions is presented. Finally, IBG with optimized fault recovery controllers is integrated into a simplified real-world power system, and the applicability of the proposed optimized controllers is illustrated.

Comparison of PLL-Based and PLL-Less Control Strategies for Grid-Following Inverters Considering Time and Frequency Domain Analysis

This article presents a comprehensive comparison of three control strategies used for grid-following inverters (GFLIs). The first strategy is the phase-locked loop (PLL)-based vector current control (VCC), and the other two PLL-less controllers are the voltage-modulated direct power control (VMDPC) and the linear-parameter-varying power-synchronized control (LPV-PSC). The VCC relies on the PLL to synchronize with the grid frequency to control the exchanged real and reactive power with the grid, which may result in instability issues under weak grid conditions. To prevent this, the VMDPC and LPV-PSC are proposed recently as PLL-less approaches to overcome the difficulties of the VCC in weak grid conditions. The performance of the VCC, VMDPC, and LPV-PSC are comprehensively reviewed in this article, considering the operations of the GFLIs under both strong and weak grids. In each operation mode, the provided tests investigate: 1) step response of active and reactive power, 2) frequency jump in the grid voltage source, 3) phase jump in the grid voltage source, and 4) voltage sag in the grid voltage source. In addition to evaluating the three controllers in the time domain, the frequency domain impedance-based stability analysis based on the generalized Nyquist criterion is also considered, which confirms the time domain findings in terms of accurate predictions of the stable/unstable operations of the GFLIs that are equipped with these controllers. It is found that, compared to the conventional VCC and the VMDPC, the LPV-PSC has promising performance under various test conditions in both strong and weak grids, which may be a future solution for weak-grids connection of GFLIs.

Effect of Frequency Coupling on Stability Analysis of a Grid-Connected Modular Multilevel Converter System

Due to the internal dynamics of the modular multilevel converter (MMC), the coupling between the positive and negative sequences in impedance, which is defined as frequency coupling, inherently exists in MMC. Ignoring the frequency coupling of the MMC impedance model may lead to inaccurate stability assessment, and thus the multi-input multi-output (MIMO) impedance model has been developed to consider the frequency coupling effect. However, the generalized Nyquist criterion (GNC), which is used for the stability analysis of an MIMO model, is more complicated than the stability analysis method applied on single-input-single-output (SISO) models. Meanwhile, it is not always the case that the SISO model fails in the stability assessment. Therefore, the conditions when the MIMO impedance model needs to be considered in the stability analysis of an MMC system should be analyzed. This paper quantitatively analyzes the effect of frequency coupling on the stability analysis of grid-connected MMC, and clarifies the frequency range and grid conditions that the coupling effect required to be considered in the stability analysis. Based on the quantitative relations between the frequency coupling and the stability analysis of the grid-connected MMC system, a simple and accurate stability analysis method for the grid-connected MMC system is proposed, where the MIMO impedance model is applied when the frequency coupling has a significant effect and the SISO impedance model is used if the frequency coupling is insignificant.

Small-Signal Stability Analysis and Enhanced Control Strategy for VSC System During Weak-Grid Asymmetric Faults

Small-signal stability issues of the voltage source converters (VSC) connected weak-network during asymmetric faults is currently rarely studied. This paper develops mathematic model of VSC system during weak-grid asymmetric faults, which reveals dynamic coupling between the phase-locked loop (PLL) and both positive-sequence (PS) control system and negative-sequence (NS) control system. Based on the deduced model, the generalized Nyquist criterion (GNC) method and modal analysis are adopted to study the small-signal stability of the VSC system during asymmetric fault steady-state. The analysis demonstrates that the small-signal stability of the system during asymmetric faults is dominated by the PLL mode, and the dynamic coupling between PLL and current control loops (CCLs) will produce negative damping, which will introduce the small-signal instability risk to the system. Moreover, an improved control strategy is designed for enhancing the small-signal stability of the VSC system during weak-grid asymmetric faults, by compensating the disturbance terms related to PLL dynamic to suppress the negative damping introduced by coupling between the PLL and both the PS and NS CCLs. Finally, this paper verifies the effectiveness of the proposed control strategy through simulation and experiment.

Voltage disturbance compensation based on impedance modeling of DFIG under weak grid

This paper proposes a grid voltage disturbance compensation control for the stability of the DFIG system under weak grid conditions. A detailed impedance model of the DFIG system, including the generator, the inner and outer control loops of the rotor side converter (RSC) and grid side converter (GSC), the DC-link and phase locked loop (PLL) dynamic, is developed in the d-q synchronous reference frame. Based on the proposed model, the relationship between the outputs of the controllers and the grid voltage disturbance at the point of common coupling (PCC) can be obtained. Then, the compensation control of the grid voltage disturbance is integrated to the current controllers of RSC and GSC to improve the stability of the DFIG system connected to the weak grid. In this paper, the effectiveness of the proposed compensation control strategy is proved using the generalized Nyquist criterion (GNC), which illustrates that the compensation control strategy implemented in RSC and GSC simultaneously has a better ability to maintain the stability of the DFIG system under weak grid conditions than only one side applied. The simulation results verify the analysis.

Stability Analysis of a Grid-Tied Interlinking Converter System With the Hybrid AC/DC Admittance Model and Determinant-Based GNC

This paper is mainly focused on the stability-related issues of a grid-tied interlinking converter system. First, hybrid AC/DC admittance characteristics of an interlinking converter operating under unity and non-unity power factors are analyzed separately. Then, by using the hybrid AC/DC admittance and determinant-based General Nyquist Criterion (GNC), the stability assessment, instability root cause identification, and instability mitigation of a grid-tied interlinking converter system are studied. In addition, decoupled mitigation of the AC- and DC-side instabilities are discussed based on the hybrid AC/DC admittance characteristics. The study results show that when an interlinking converter operates under unity power factor, the stabilities of the AC and DC sides are decoupled and can be regulated independently. To be more specific, the DC-side stability is subjected to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> -axis admittance, DC admittance, and their coupling admittances while the AC-side stability is dominated by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -axis admittance. According to the hybrid AC/DC admittance characteristics, this paper further reveals that the DC-side stability is determined by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> -axis controllers. In contrast, the AC-side stability is governed by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -axis controllers and the phase-locked loop (PLL). Finally, the analysis presented in this paper is verified based on frequency- and time-domain simulations in the Matlab/Simulink environment.

Impedance analysis considering unstable subsystem poles for MMC-HVdc based wind farm integration system

Harmonic resonance issues of the wind farm connected MMC-HVdc system have been well investigated by ac/dc impedance analysis in past 5 years. These methods generally divided the studied system into two subsystems at ac or dc section. Then, the impedance of these subsystems can be used to identify the system stability by applying Nyquist criterion or generalized Nyquist criterion. However, the existing impedance analyses always assume subsystems are stable and overlook some special cases where subsystems may are unstable. In these special cases, the unstable poles of subsystems must be evaluated before applying Nyquist criterion. Based on this consideration, a simplified method for identifying unstable subsystem poles is proposed in this paper. With consideration of unstable subsystem poles, the gap of existing impedance analysis is filled. The proposed method and stability analyses are validated by electromagnetic transient simulation in PSCAD/EMTDC.

Harmonic Stability Assessment of Multi-terminal DC (MTDC) Systems Based on the Hybrid AC/DC Admittance Model and Determinant-based GNC

Growing concerns with the harmonic oscillation phenomenon of multiterminal DC (MTDC) systems calls for appropriate harmonic stability analysis methods. However, the conventional ac or dc admittance-based stability assessment methods cannot effectively protect trade secret and user privacy, distinguish different system instability root causes, and put forward corresponding instability mitigation methods. To address the aforementioned issues, this article first establishes a hybrid ac/dc admittance-based stability assessment model for a MTDC system. Then, by introducing the determinant-based General Nyquist Criterion (GNC), the system stability is assessed and three types of instability root causes, including the ac side, the dc side, and the coupling interaction between the ac and dc sides are identified. Regarding the instability caused by the ac side, the dominant interlinking converter is directly identified, in which the instability can be mitigated by reshaping its q-axis admittance. In contrast, for the instability caused by the latter two root causes, the most influential interlinking converter is recognized by an additional sensitivity analysis. The instability is then mitigated by reshaping the dc-side equivalent admittance of the most influential interlinking converter. Compared with the conventional methods, the proposed method offers advantages as follows: 1) trade secret and user privacy is effectively protected, 2) different system instability root causes can be effectively distinguished, and 3) instabilities caused by diverse root causes can be mitigated by corresponding methods. Finally, the proposed method is verified by simulations in the MATLAB/Simulink environment.

Stability Investigation of Three-Phase Grid-Tied PV Inverter Systems Using Impedance Models

With the increasing penetration of power-electronic-based renewable generations, stability issues become challenging due to interactions between converters and the grid. This article develops a framework to investigate the stability of grid-tied photovoltaic inverter systems using impedance models (IMs). IMs are developed considering the complete system dynamics for both proportional-integral (PI) control and proportional-resonance (PR) control structures. Different stability investigation methods are elaborated with a focus on the Bode plot and the generalized Nyquist criterion. Sensitivity analyses are conducted utilizing Bode plots to observe the impact of different factors in the frequency domain. Followed by that, an impedance-resonance analysis method is proposed to identify the cause of resonances according to the critical interactions of diagonal impedances and the frequency range of resonances. The performance of PI and PR controls is compared with a focus on the influence of the outer control loops. The analyses are validated through experiments in a down-scale test bench.

Admittance Modeling and Stability Analysis of Grid-Connected Inverter With LADRC-PLL

It tends to cause system oscillation when the inverter with a phase-locked loop based on proportional integral controller (PI-PLL) is connected to the weak grid. To improve the oscillation suppression ability of the grid-connected inverter, a linear active disturbance rejection controller is applied to PLL (LADRC-PLL). Considering the influence of linear extended state observer, voltage outer-loop, current inner-loop, and frequency coupling, the admittance model of the grid-connected inverter with LADRC-PLL is established. Based on the established admittance model and the generalized Nyquist criterion, the system stabilities of the grid-connected inverters with LADRC-PLL and PI-PLL are compared. The comparison results indicate that the grid-connected inverter with LADRC-PLL has better adaptability to the weak grid and shows a certain ability to suppress the sub and supersynchronous oscillation. Then, the influence of the control parameters of LADRC-PLL on the system stability is studied. It is found that the system keeps stable when the control parameter of LADRC-PLL is changed in a relatively wide range, which indicates LADRC-PLL has good robustness. Furthermore, the dynamic performances of the grid-connected inverters with LADRC-PLL and PI-PLL are analyzed. It is revealed that the grid-connected inverter with LADRC-PLL has better dynamic performance. Finally, the correctness of the analysis is verified by experiments.

Output impedance modeling and grid-connected stability study of virtual synchronous control-based doubly-fed induction generator wind turbines in weak grids

Virtual synchronous control (VSynC) technology that mimics the behavior of a conventional synchronous generator can be applied in wind turbines (WTs), where it can effectively improve equivalent inertia, damping coefficient, and wind power permeability. Given the unique partial-scale converter structure of a doubly-fed induction generator (DFIG) and its complicated DFIG-grid coupling effect, the grid-connected VSynC-based DFIG WTs have gained little attention. Moreover, under a weak grid condition, some unstable modes of VSynC can interact with the wind power system, which impacts the DFIG’s dynamic characteristics and system stability. To bridge the gap, this study proposes an output impedance model for a VSynC-based DFIG to analyze the grid-connected stability in weak grids. First, the output impedance mathematical model for a VSynC-based DFIG is derived and explained. On the basis of this novel model, the low-frequency characteristics and vital unstable factors of a VSynC-based DFIG are comprehensively investigated. The impact of different VSynC parameters and grid strengths are extensively studied. Finally, the proposed output impedance model is validated by direct current disturbing theory, frequency scanning method, generalized Nyquist criterion, and time-domain simulation (MATLAB/Simulink). All the obtained results are consistent with each other and prove the effectiveness and accuracy of the proposed impedance model.

On Consensus of Second-Order Multiagent Systems With Actuator Saturations: A Generalized-Nyquist-Criterion-Based Approach.

In this article, a frequency-domain approach is developed to deal with the global consensus problem for a class of general second-order multiagent systems (MASs) subject to actuator saturations. By employing the describing function and the generalized Nyquist criterion, the global consensus problem is thoroughly investigated for both undirected and directed topologies. First, the describing function is introduced to characterize the actuator saturations in the s -plane, and the inherent representation error is quantitatively analyzed from a frequency-domain perspective. Then, by means of the Kronecker product, the addressed consensus problem of the MAS is transformed into a corresponding stability analysis problem for a certain multi-input-multi-output (MIMO) system and, consequently, the generalized Nyquist criterion for MIMO systems is exploited to derive the condition for the global consensus of the MAS where the impact from the actuator saturation is explicitly reflected. Finally, numerical simulations are provided to illustrate the validity of the proposed theoretical result.

DQ impedance‐decoupled network model‐based stability analysis of offshore wind power plant under weak grid conditions

This study presents a dq impedance-decoupled network modelling method of offshore wind power plant (OWPP) for stability analysis. DQ impedance frequency responses of grid-connected inverters (GCIs) and long transmission cables (LTCs) are first measured by frequency scanning method, which are fitted as transfer function matrices using matrix fitting algorithm. Then, the GCIs are modelled as Norton equivalent circuits. In addition, per-unit-length electrical parameters of the LTCs are extracted from the measured dq impedance frequency responses of a specific LTC, based on which dq impedance-decoupled two port network models of the LTCs are established. DQ impedance-decoupled network model of the whole OWPP is then established based on connection relationships of these GCIs and LTCs. Finally, generalised Nyquist criterion (GNC) is performed in all the dq impedance-decoupled subsystems, and the subsystems where the GNC is not satisfied are identified as instability sources. Compared with conventional impedance-based stability analysis methods of OWPP, the proposed dq impedance-decoupled modelling method is able to facilitate the application of GNC and further instability source identification based on partitioning the whole OWPP into several decoupled subsystems. The effectiveness of the network modelling method is validated based on time-domain simulation results in Matlab/Simulink and real-time verification results in OPAL-RT.

Impedance Modeling of Modular Multilevel Converter in D-Q and Modified Sequence Domains

Modular multilevel converters (MMCs) are characterized by the multiharmonic coupling interactions among internal dynamics. In this work, multiple <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> references and multiharmonic frames, which are developed through dynamic phasor and harmonic state space, are used to derive the generalized impedance models of MMC in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> and modified sequence domain, respectively. Although some previous studies have been devoted to revealing the equivalence of different impedance models, this article attempts to elaborate the relation of different multifrequency modeling approaches and provide a more comprehensive review of MMC impedance models. A rigorous comparison of modeling procedure, impedance response, and stability analysis result between different MMC models is conducted. It is demonstrated that the MMC impedance models in the two domains can be related by a generic linear transformation matrix. Furthermore, the stability analysis results given by the generalized Nyquist criterion are identical in these two domains. Application recommendations for different impedance models are given as well. The developed impedance models and their effectiveness for stability and resonance analysis are validated by a 501-level aggregated model in MATLAB/SIMULINK.

Loop-at-a-Time Stability Analysis for Grid-Connected Voltage-Source Converters

The instability phenomena caused by converter–grid interactions can be prevented by designing controllers with adequate stability margins. Yet, the multiple-input–multiple-output (MIMO) dynamics of grid-connected voltage-source converters (VSCs) complicate the stability analysis for the controller design. To tackle this challenge, this article presents a loop-at-a-time stability analysis for grid-connected VSCs, which not only shows close correlations with the generalized Nyquist criterion for MIMO systems but also enables to quantify the stability margins of individual closed loops. Moreover, the interactions between the closed loops can be analyzed. Test cases with numerical sensitivity analysis, simulations, and field measurements of a converter validate the theory.