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This paper analyzes the small-signal impedance of three-phase grid-tied inverters with feedback control and phase-locked loop (PLL) in the synchronous reference (d-q) frame. The result unveils an interesting and important feature of three-phase grid-tied inverters – namely, that its q–q channel impedance behaves as a negative incremental resistor....
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... inverters are not CPLs. They are typically controlled as current sources using single output current feedback controller or power sources using both current and power controllers as shown in Fig. 1. Instability in grid-tied inverter systems was not analyzed using the negative incremental resistor concept in the past [3]- [7]. Recently, Harnefors et al. [8] modeled the effect of PLL by introducing system and converter d-q frames. It shows that high bandwidth PLL increases the negative real part of the inverter output impedance, ...
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... this paper proposes a full analytical model for the output impedance of the three-phase grid-tied inverter in the d-q frame, considering the effect of PLL, and feedback control as shown in Fig. 1. Using the proposed model, inverter small-signal impedances are characterized with different control strategies. Specifically, impedance values are shown in the form of Bode plots, which clearly demonstrate the negative resistor behavior of Z qq . An analytical expression for the negative resistor is given. The model shows that a ...
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... be transformed to synchronous frame using the transformation provided by [42]. Modeling of the impedance of a voltage source converter with a stationary frame current regulator is shown in [43]. The influence of different current regulators will not be elaborated upon; only the current controller in synchronous frame is considered in this paper. Fig. 10 shows the small-signal model of the grid-tied inverter with current feedback control. The current controller matrix is G ci . G dei is the decoupling ...
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... the equations represented by Fig. 10, the output impedance of a grid-tied inverter system with PLL working under a closed-loop condition is Table I in the Appendix shows the parameters of the inverter prototype system. Using these parameters and (24), Fig. 11 shows the Bode plot of the inverter power stage open-loop output impedance (Z out ol ) over plotted with output ...
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... the equations represented by Fig. 10, the output impedance of a grid-tied inverter system with PLL working under a closed-loop condition is Table I in the Appendix shows the parameters of the inverter prototype system. Using these parameters and (24), Fig. 11 shows the Bode plot of the inverter power stage open-loop output impedance (Z out ol ) over plotted with output impedances open loop with PLL (Z out ol PLL ). Clearly, Z qq of Z out ol PLL is shaped as a negative resistor at low frequency. Using these parameters and (39), Fig. 12 shows the Bode plot of inverter output impedances with ...
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... of the inverter prototype system. Using these parameters and (24), Fig. 11 shows the Bode plot of the inverter power stage open-loop output impedance (Z out ol ) over plotted with output impedances open loop with PLL (Z out ol PLL ). Clearly, Z qq of Z out ol PLL is shaped as a negative resistor at low frequency. Using these parameters and (39), Fig. 12 shows the Bode plot of inverter output impedances with current feedback control and different PLL designs. The impedances are calculated up to half switching frequency where it is valid to do ...
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... it can be observed in Fig. 12, Z dd shows the current source behavior. In low frequency range, it is shaped by the current controller integrator; in high frequency range, Z dd is the impedance of the ac inductor. Z dq and Z qd are very small within the current controller bandwidth because of unity power factor control. In high frequency range, they behave as the ...
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... Hz (line frequency is set to be 400 Hz in order to allow a wide range of PLL bandwidth) are chosen to show the influence of PLL. The results show that a higher PLL bandwidth case yields a wider frequency range of negative resistance behavior. The magnitude of the resistance is related to the power rating of the inverter. Especially, as shown in Fig. 13, the current vector of the inverter is synchronized with the grid voltage vector, which is aligned in d channel to inject real power to the grid. When voltage disturbance happens in q channel, the voltage vector ¯ v s becomes ¯ v 1 s . Because of PLL, the current vector will rotate to keep synchronization with the voltage vector. ...
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... synchronization with the voltage vector. Meanwhile, the current controller keeps current magnitude in d channel unchanged. Then, current vector ¯ i L s becomes ¯ i L 1 s due to the q channel voltage perturbation. Notice that the small-signal response of q channel current is in the opposite direction of q channel voltage perturbation, so, from Fig. 13, Z qq can be calculated by (40) within the bandwidth of PLL. This conclusion is also verified by the magnitude of Z qq in low frequency as shown in Fig. 12. According to (40), an inverter with a high current rating has low impedance in q-q channel; this is shown in Fig. 14, in which impedances with three different d channel current ...
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... becomes ¯ i L 1 s due to the q channel voltage perturbation. Notice that the small-signal response of q channel current is in the opposite direction of q channel voltage perturbation, so, from Fig. 13, Z qq can be calculated by (40) within the bandwidth of PLL. This conclusion is also verified by the magnitude of Z qq in low frequency as shown in Fig. 12. According to (40), an inverter with a high current rating has low impedance in q-q channel; this is shown in Fig. 14, in which impedances with three different d channel current references are plotted. Fig. 15 shows the impedances of the inverter with and without decoupling control. With decoupling control, Z dq and Z qd are reduced. ...
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... is in the opposite direction of q channel voltage perturbation, so, from Fig. 13, Z qq can be calculated by (40) within the bandwidth of PLL. This conclusion is also verified by the magnitude of Z qq in low frequency as shown in Fig. 12. According to (40), an inverter with a high current rating has low impedance in q-q channel; this is shown in Fig. 14, in which impedances with three different d channel current references are plotted. Fig. 15 shows the impedances of the inverter with and without decoupling control. With decoupling control, Z dq and Z qd are reduced. This is expected, because decoupling control reduces the coupling between d and q channel, which is reflected by Z dq ...
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... be calculated by (40) within the bandwidth of PLL. This conclusion is also verified by the magnitude of Z qq in low frequency as shown in Fig. 12. According to (40), an inverter with a high current rating has low impedance in q-q channel; this is shown in Fig. 14, in which impedances with three different d channel current references are plotted. Fig. 15 shows the impedances of the inverter with and without decoupling control. With decoupling control, Z dq and Z qd are reduced. This is expected, because decoupling control reduces the coupling between d and q channel, which is reflected by Z dq and Z qd . The use of decoupling control does not change the negative resistance behavior of ...
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... 15 shows the impedances of the inverter with and without decoupling control. With decoupling control, Z dq and Z qd are reduced. This is expected, because decoupling control reduces the coupling between d and q channel, which is reflected by Z dq and Z qd . The use of decoupling control does not change the negative resistance behavior of Z qq . Fig. 16 shows the impedance with different PLL strategies, which indicates that use of DDSRF PLL does not change the negative resistance behavior of Z qq ...
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... grid-tied inverter can inject reactive current to support the grid. Fig. 17 shows the impedance of the grid-tied inverter with different reactive power injections. The blue line shows the impedance of the capacitive power injection case; the green line shows the impedance of no reactive power injection, and the red line shows the inductive power injection case. Notice that reactive power injection will ...
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... top of the current controller, a power flow controller can be added to control the real and reactive power accurately generated by the inverter. The small-signal model is shown in Fig. 18 for the power flow control case. G cPQ is the transfer function matrix of the power flow controller Fig. 19. Impedances of inverter with PLL and power feedback control. ...
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... top of the current controller, a power flow controller can be added to control the real and reactive power accurately generated by the inverter. The small-signal model is shown in Fig. 18 for the power flow control case. G cPQ is the transfer function matrix of the power flow controller Fig. 19. Impedances of inverter with PLL and power feedback control. ...
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... the equations represented by Fig. 18, the output impedance of the grid-tied inverter system with PLL and power feedback control is (45). Fig. 19 shows the impedance of the inverter prototype with power feedback control using the inverter power stage parameters list in Table I and the controller parameters list in Table II in the Appendix. Notice that the integrator gain ...
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... the equations represented by Fig. 18, the output impedance of the grid-tied inverter system with PLL and power feedback control is (45). Fig. 19 shows the impedance of the inverter prototype with power feedback control using the inverter power stage parameters list in Table I and the controller parameters list in Table II in the Appendix. Notice that the integrator gain of the current controller in Table II is increased compared to what is shown in Table I in order to close the ...
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... increased compared to what is shown in Table I in order to close the power feedback control loop. Once the power flow controller is applied, within its bandwidth, Z dd becomes a positive resistance, and Z qq is still a negative resistance. PLL bandwidth still influences the frequency range of the negative resistance behavior of Z qq as shown in Fig. 19. Z dq and Z qd are still low in the low frequency because of the unity power factor ...
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... line diagram representation of the inverter system under study is shown in Fig. 21. In this system, the three-phase inverter is connected to the point of common coupling (PCC) together with the three-phase passive load. This inverter is synchronized with PCC voltage by SRF PLL. Only current feedback control is applied in order to control the amount of power injection to PCC. Each phase of the passive load consists of ...
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... order to predict stable and unstable conditions of the system using the impedance-based method, an ac interface is found at the terminal of the inverter as shown in Fig. 21. As reported by [7], when using impedance-based stability analysis for a gridtied inverter system, the inverter should be treated as load, and Nyquist stability criterion should be applied to the impedance ratio between the grid-side impedance and inverter impedance as ...
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... the proposed model. Fig. 28 shows Z qq with a different SRF PLL bandwidth. The measurement results clearly show the negative impedance behavior of Z qq and its variation along with PLL bandwidth. Figs. 29 and 30 show the measurement results with inductive and capacitive reactive power under current feedback control with 100-Hz SRF PLL bandwidth. Fig. 31 shows the measurement results with power feedback control where P ref is -1150 W and Q ref is 0 Var. All the measurement results match with the results predicted by the model very ...
Citations
... Figure 5 shows the discrete admittance data points and fitting curve Bode diagram. It should be explained additionally that influence of PLL dynamics on output admittance can be ignored according to [20] and [28] when PLL bandwidth is low enough. As a result, the d-axis control and q-axis control are decoupled and identical, and the admittance of grid-connected inverter in Figure 5 is single-input single-output. ...
A multi‐inverter‐fed power system is susceptible to small‐signal instability owing to weak grid influence. This study proposes a black‐box method for small‐signal stability analysis of a multi‐inverter‐fed power system based on Bode diagrams. Not only did it consider the technical confidentiality on the engineering site, but it also took into account the presence of right‐half‐plane (RHP) pole in the aggregated impedance, achieving Nyquist circle acquisition through Bode diagrams. The stability analysis process is thus more accurate, intuitive, and convenient. First, black‐box impedance fitting for a single grid‐connected inverter and impedance aggregation for a multi‐inverter‐fed power system are discussed. Second, the principle of the proposed method is elaborated. Using the Bode diagram, the number of RHP poles of the aggregated impedance and actual number of circles for Nyquist curves are identified. Third, a multi‐inverter‐fed power system is constructed, and the effectiveness of the proposed method is verified using case analysis and hardware‐in‐the‐loop real‐time experiments based on RT‐LAB. Finally, the advantages of the proposed method are revealed by comparing it to the IF method.
... In [7][8][9], the harmonic instability is concluded to describe the interactions among different generation units. In [10,11], the instability of the phase-locked loop (PLL) is analysed, and the reason is attributed to the negative damping within the control bandwidth. In [12], it is found that the asymmetrical dq-frame control dynamics in the PLL will give rise to frequency-coupling oscillations. ...
This paper proposes an adaptive saturation module to enhance the transient stability of grid‐following inverters after voltage‐dip inception and fault‐clearance moment. The equal‐area criterion reveals that a large acceleration area will be obtained due to the step change of current reference according to grid codes, which may lead to the loss of synchronism with the grid. It is found that the saturation module used with the phase‐locked loop can enhance the transient stability performance by reducing the acceleration area. To simultaneously adapt to the two cases (the inception and clearance of the voltage‐dip fault), an adaptive saturation module that can automatically adjust the clamping mode is proposed, which is the main contribution of the paper. The selection of the threshold value for the saturation module is also presented. A comprehensive comparison is made between the proposed solution and state‐of‐art solutions. Finally, the effectiveness of the proposed control strategy is confirmed by the experimental tests.
... The authors of [8] investigated the small-signal stability in low-inertia power systems and compared various control strategies through diverse bifurcation studies. The authors of [9][10][11] explored why a PLL deteriorates the stability of the inverter system, the extent of a PLL's influence, and how to improve the stability of a weak power system. While these researchers thoroughly account for the dynamic characteristics of PLLs, overlooking the dynamic characteristics of power systems prevents revealing the dynamic interactions between the IBRs and power systems. ...
... where the details of G x ′ d , G r and G ∆P m are shown in Appendix A. Furthermore, the transmission lines and loads can be modeled as (9). The nodal current balance constraint in the network is modeled as (10). ...
... Several researchers have examined the effects of PLLs on the system to improve PLL performance to increase system stability in weak power systems [7,9,10,19]. Different from such studies, this paper provides an intuitive demonstration of the PLL's impacts on the system dynamics using the proposed interaction index, which can provide guidance for power electronic engineers to enhance PLL design. ...
In a renewable-energy-penetrated power system (RPPS), inverter-based resources (IBRs) pose serious challenges to power system stability due to their completely different dynamic characteristics compared with conventional generators; thus, it is necessary to study the dynamic interactions between IBRs and power systems. Although many research efforts have been dedicated to this topic from both power electronics and power system researchers, some research from the power electronics field treats the external power system as a voltage source with an impedance, therefore ignoring the dynamic characteristics of a power system, while most of the research from the power system field applies simulation-based methods, for which it is difficult to directly interpret the interaction mechanism of IBRs and external system dynamics. Thus, none of these studies can explore the accurate dynamic interaction mechanism between IBRs and power systems, leading to performance degradation of IBR-integrated power systems. Our study takes into account the dynamic characteristics of both IBRs and the external power system, resulting in the development of a new open-loop transfer function for RPPSs. Based on this formulation, it is observed that under certain operating conditions, the dynamic interactions between the inverter and the power system help enhance IBR-penetrated power system stability compared with the case for which the external power system is controlled as a voltage source. The study also reveals how the inverter (phase-locked loop, control parameters, etc.), external power system (network strength) and penetration ratio in an IBR-penetrated power system affect the dynamic interactions between IBRs and the external power system using the proposed quantified interaction indices.
... To achieve this necessary transformation is utilized to reorient the transfer functions across different frames. The system model, depicted in the -axis and rotating, can be characterized by a 2 2 transfer function matrix, as suggested in [16]. ...
... Since direct measurement of the bus voltage is not possible, the voltage is termed as vCf, is utilized. This approach involves the modification with the current regulation circuitry, allowing for modifications in its dynamics at low frequencies, thereby influencing stability [16]. As SSR is a phenomenon of low frequency the PLL must be accounted for representation. ...
Doubly-Fed Induction Generators (DFIGs), when integrated with wind farms and connected to grids featuring series compensation, are prone to instability caused by subsynchronous resonance (SSR). This paper proposed a resilient controller designed to boost the stability of electrical power networks and mitigate the SSR phenomenon between wind farms employed by DFIGs. A combined strategy is implemented to mitigate SSR by incorporating the rotor side converter controller (RSC) and the SSR damping controller (SSRDC) in the grid side converter controller (GSC). An algorithm is designed to locate the best position of the SSRDC. The algorithm is dependent on the input parameters of DFIG. Fast Fourier Transform (FFT) analysis is employed at a 60% compensation level with various wind speeds to obtain insights into how total harmonic distortion (THD) can affect stability. Simulation outcomes conducted on a real wind farm demonstrate that the proposed SSR elimination technique showcased a promising approach to suppressing SSR, thereby confirming the effectiveness of the proposed mitigation approach.
... The transformations between the actual system synchronizing frame and the control synchronizing frame can be given as [18], [19]: ...
This paper conducts a comprehensive analysis and comparison of the control loops of the grid-following and grid-forming voltage source converters connected to the power grid. Eigenvalue trajectories are studied in order to obtain an accurate stability analysis. A time-domain simulation model of a 1.5 kW grid-connected converter is developed by using Matlab/Simulink to investigate the stability of the grid-following and grid-forming control under different short-circuit ratios. The stability boundaries of the grid-following control and the grid-forming control are explored and compared with theoretical analysis. The result reveals that the grid-following control is better suited for a stiff power grid, while the grid-forming control is more suitable for a weak power grid. Finally, an experimental prototype is established to verify the effectiveness of the theoretical analysis.
... Though this control is straight-forward, the performance of the PLL can be negatively impacted, specifically when VSCs are connected to the weak grids, leading to reduced tracking accuracy [16]. In more severe scenarios, the limitations in performance of the PLL due to weak grid conditions can result in instability in dc-link voltage control [17]. ...
While power electronic converters, such as voltage source converters (VSCs), are crucial for the operation of converter-dominated renewables and their integration with the electricity grid, their reliance on VSCs can pose a challenge. The limited inertia of these sources can lead to a deterioration of the rate of change of frequency, potentially causing power system stability issues. A grid-forming approach utilizing dc-link dynamics is one of the attractive alternatives to achieve grid synchronization and support grid frequency. Existing grid-forming control schemes, which assume a constant or virtually constant dc source, rely on a fixed physical dc-link capacitor. Nonetheless, the inertia support from such a capacitor is brief, owing to its limited energy storage capability. Consequently, enhancing inertia becomes imperative; otherwise, it may result in an increased rate of change of voltage on the dc side, potentially leading to issues with protection, undesirable interactions, and system instability. This paper proposes a new grid-forming control strategy that considers a virtual capacitor to achieve grid synchronization while simultaneously providing the network with inertia response services during power imbalances. Moreover, including a virtual resistor in the controller effectively attenuates power and dc voltage oscillations. Simulations using Simulink and small signal stability analysis are conducted to validate the efficacy of the proposed controller.
... Little research has been done on medium and lowfrequency instability caused by the coupling of PLL and external circuits or ST LV converters. In addition, the load and output power of DER inverters have a significant impact on system stability [14], increasing the difficulty of parameter design. As the load and output power vary, the system state and oscillation frequency will change accordingly. ...
... The dq-axis and qd-axis impedances are relatively small due to unit factor control. The qq-axis impedance presents a negative resistance characteristic due to PLL, posing a potential threat to the stability of the system [14]. As the current of the DER inverter increases, the magnitude of the qq-axis impedance decreases, indicating that the negative resistance effect is enhanced, which could potentially lead to adverse effects on stability. ...
The stability of power systems is currently facing significant challenges due to the increasing penetration of distributed energy resources (DERs). To improve stability, the smart transformer (ST) can be used to reshape the grid-side impedance that interacts with low-voltage (LV) side DER inverters by adjusting the control strategy. This paper first establishes the impedance models of the ST LV converter and DER inverter and analyzes the stability of the system. Specifically, the impact of the load and output power of the DER inverter on stability is investigated. A biquad filter-based active damping scheme is then adopted to improve system stability, with filter parameters pre-designed based on the root-locus curve. However, this design method is dependent on the system model and lacks adaptability, limiting the system stability region. To address this issue, this paper proposes using reinforcement learning (RL) algorithms to achieve adaptability in stabilizing control, as RL can achieve model-free training. The results show that the proposed approach can further broaden the region of stable operation for the system. Finally, simulations and experiments are used to demonstrate the effectiveness of adaptive stabilizing control.
... The high increase in wind-power penetration has made the power system more efficient and flexible, mainly because it is coming to be dominated by high-power electronic Voltage Source Converters (VSC) [3,4]. On the other hand, many challenges for the power system's stability and synchronization to the grid have been arising, as the number of components connected to the system is being increased [5]. Therefore, extensive research has been conducted in the area of dynamic interactions between the grid-connected converters of wind turbines on wind farms. ...
In this paper, a holistic nonlinear state-space model of a system with multiple converters is developed, where the converters correspond to the wind turbines in a wind farm and are equipped with grid-following control. A novel generalized methodology is developed, based on the number of the system’s converters, to compute the equilibrium points around which the model is linearized. This is a more solid approach compared with selecting operating points for linearizing the model or utilizing EMT simulation tools to estimate the system’s steady state. The dynamics of both the inner and outer control loops of the power converters are included, as well as the dynamics of the electrical elements of the system and the digital time delay, in order to study the dynamic issues in both high- and low-frequency ranges. The system’s stability is assessed through an eigenvalue-based stability analysis. A participation factor analysis is also used to give an insight into the interactions caused by the control topology of the converters. Time domain simulations and the corresponding frequency analysis are performed in order to validate the model for all the control interactions under study.
... Besides, the stability in the weak grid caused by high impedance in the grid will be negatively impacted by PLL [47]. For this reason, the current commercially available inverters do not participate in grid ancillary services, with some exceptions [48]. ...
... On the other hand, reactive power/voltage supports are solely handled by the inverter. The GFM can be defined based on the objectives, controls, and tasks mentioned in Tables 2 and 3 [47]. ...
The increasing integration of inverter based resources (IBR) in the power system has a significant multi‐faceted impact on the power system operation and stability. Various control approaches are proposed for IBRs, broadly categorized into grid‐following and grid‐forming (GFM) control strategies. While the GFL has been in operation for some time, the relatively new GFMs are rarely deployed in the IBRs. This article aims to provide an understanding of the working principles and distinguish between these two control strategies. A survey of the recent GFM control approaches is also delivered here, expanding the existing classification. It also explores the role of GFM control and its types in power system dynamics and stability like voltage, frequency etc. Practical insight into these stabilities is provided using case studies, making this review article unique in its comprehensive approach. Lacking elsewhere, the GFMs' real‐world demonstrations and their applications in several IBRs like wind farms, photovoltaic power generation stations etc., are also analyzed. Finally, the research gaps are identified, and the prospect of GFM is presented based on the system needs, informed by GFM real‐world projects. This work is a potential road map for the GFM large‐scale deployment in the decarbonized IBR‐based bulk power system.
... The time-domain modal analysis can calculate the eigenvalues and corresponding participation factors for evaluating the system's stability margin and identifying key variables related with the instability issues [10] , which needs the detailed system's parameters. However, the modal analysis may not be suitable for widely applications in a practical multi-CBR system, since the CBRs are commonly "black-boxed" [15]. In comparison, the impedance-based analysis does not need the detailed CBRs' inner parameters and thus has been widely used for CBRinduced small-signal stability issues, since the system's impedance can be evaluated by frequency scanning. ...
... 1) The optimization problem (14) is decomposed as k iterative optimization problems (15). Then, we obtain the capacity matrix SB for n CBRs and the network impedance matrix Z, where nodes 1~n+m are remained; ...
... of jth SE in M, . By iteratively solving(15) for k times using the heuristic method, we can obtain the locations for placing k SEs.The iterative optimization problem in(14) can be solved by calculating the sensitivity of the maximal eigenvalue of 1 B SZ for SEj's capacity and choosing the minimal one to place the SE. The reason is that the sensitivity of 1 B SZ's maximal eigenvalue for SEj's capacity can be written as vj1 is the jth element of v1, wherein v1 ...
