Grid-Synchronization Stability of Converter-Based Resources—An Overview

This paper presents an overview of the synchronization stability of converter-based resources under a wide range of grid conditions. The general grid-synchronization principles for grid-following and grid-forming modes are reviewed first. Then, the small-signal and transient stability of these two operating modes are discussed, and the design-oriented analyses are performed to illustrate the control impact. Lastly, perspectives on the prospects and challenges are shared.


I. INTRODUCTION
HE past few years have witnessed an exponential growth of power-electronics-based resources in electric grids [1].Differing from synchronous machines, the converterbased resources have no inherent 'swing equation' governing their synchronizations with the grid, but rather rely on digital control algorithms [2].Therefore, the grid-synchronization dynamics of converter-based resources can be fundamentally different from synchronous machines.
A wide variety of grid-synchronization methods have been developed for converter-based resources [3]- [7].The focus of research on the grid synchronization has been shifted from the rejection of disturbances, such as voltage harmonics and imbalances, phase jumps, and frequency deviations [3]- [5], to the synchronization stability in weak and faulty grids [6], [7].This change has been driven by the ever-increasing share of converter-based resources in the power system, where the decreased inertia and the lowered short-circuit ratio (SCR) deteriorate the stability of converter-based resources [8].
Generally speaking, the grid synchronization methods can be classified into two categories with respect to the operating modes of converter-based resources: 1) The voltage-based grid synchronization that measures or estimates (voltage sensor-less) the frequency and phase of the voltage at the point of common coupling (PCC) of gridconnected converters.The detected phase is then used with the vector current control [4] or the direct power control [3] for regulating the active and reactive power exchanged with the grid.Such a voltage-based grid synchronization control is named as the grid-following control, since they follow the phase of grid voltage [9].
2) The power-based synchronization dictates directly the phase of the PCC voltage by regulating the active power of the converters [10], where the general idea is to utilize the active power-frequency (P-ω) droop control, which is widely used with synchronous generators, to synchronize converters with the grid [11].To track the generated phase of the PCC voltage, the voltage control is required, which, thus, differs from grid-following converters.Such voltage-controlled converters are recently known as the grid-forming converters [9], [12].
The stability implications of these two categories of grid synchronization methods are different.The synchronization stability of converter-based resources can be categorized into two types: small-signal stability and large-signal (transient) stability, similarly to the rotor angle stability of synchronous generators in legacy power systems [13].
The small-signal synchronization stability entails the ability of converter-based resources to maintain synchronism with the grid under small disturbances [6].The disturbance is so small that the system dynamics can be linearized around the equilibrium point of interest, and thus the linear systems theory can be used for the dynamic analysis [13].The smallsignal analysis is widely used to evaluate the robustness of synchronization dynamics with different grid strengths.It is reported that the voltage-based synchronization methods, e.g. the phase-locked loop (PLL) that is commonly found in grid-following converters, can induce sideband oscillations around the grid fundamental frequency in low SCR grids [6], [8], [14], [15].In contrast, the power-based synchronization algorithms used with grid-forming converters can result in sideband oscillations in stiff grids [16]- [20] and series compensated grids [21].
During large disturbances such as severe grid faults and loss of a large generation/load, the small-signal analysis will be inadequate to characterize the synchronization dynamics of converter-based resources, since the equilibrium points of the system may be changed or even lost in those scenarios.It is, thus, necessary to check first the presence of equilibrium points after the disturbance.Certainly, converters will lose the synchronism with the grid if no equilibrium points exist.However, even if there are equilibrium points after the large disturbance, converters may still lose the synchronism with the grid, since the system dynamics may not converge to the stable equilibrium point, which is highly dependent on the used grid-synchronization methods [22]- [26].Therefore, the presence of equilibrium points is only a necessary condition for the transient stability of converter-based resources [7].The nonlinear systems theory is required to evaluate if the system dynamics can converge to a stable equilibrium point when subjected to large disturbances.It has recently been found that the PLL essentially introduces a second-order nonlinear swing equation to grid-following converters, and a voltage-angle curve, instead of the conventional power-angle curve, is resulted for the transient stability analysis [22].In contrast, the droop-controlled grid-forming converters can be characterized as a first-order nonlinear system, which can significantly improve the transient stability [23].Yet, the reactive power droop control loop can adversely affect the transient stability of grid-forming converters [24].Differing from synchronous generators (SGs), the limited overcurrent capability of power converters necessitates the use of current limiting control [12], which imposes another constraint to the transient stability behavior of grid-forming converters [25].
This paper intends to provide a systematic overview of the grid-synchronization stability of converter-based resources.The basics of grid-synchronization control are reviewed first in Section II, where the typical control topologies of gridfollowing and grid-forming converters are discussed, and the static power transfer limitations are analyzed to identify the presence of equilibrium points.Then, both the small-signal and transient stability of grid-following converters, as well as stabilization methods are addressed in Section III.This is followed by the stability characteristics and enhancement methods for grid-forming converters in Section IV.Section V concludes the paper with the challenges on grid codes and the prospects for future research.

II. BASICS OF GRID-SYNCHRONIZATION CONTROL
A. Grid-Following Converters Fig. 1 shows a control diagram of a grid-following converter [26].The typical vector current control is adopted.The PLL plays a vital role in the synchronization of the grid-following converter; its diagram is shown in Fig. 1(b) [27].With the transformation of the voltage vector in the dq frame and the regulation of the q-axis voltage by a proportional + integral (PI) controller, a feedback control loop is used to obtain the phase angle of the PCC voltage ( ).
With the phase angle of  , the vector current control can also be oriented into the voltage dq-reference frame.The Direct Voltage Control (DVC) and the Alternating Voltage Control (AVC) can be utilized in the outer loops to generate the active-power and reactive-power current references by  and  , respectively.Consequently, the magnitude and phase of the current reference can be determined, and the current control can be implemented in the inner loop in either the αβ-reference frame or dq-reference frame.Also, the io Voltage-Source Converter voltage feedforward (VFF) control is usually used through a low-pass filter and added to the output of the current controller, to enhance the dynamic performance.

B. Grid-Forming Converters
Fig. 2(a) shows a general system diagram of the grid-forming converter [10].The output active and reactive powers of the grid-forming converter are regulated through the active and reactive power controls (APC and RPC) to generate the phase angle and voltage magnitude of the voltage reference,   .The inner voltage control (VC) loop is used to regulate the output voltage of the converter, which is further cascaded with the current control (CC) loop for the overcurrent Δω limitation and the LC filter resonance damping [28].Differing from grid-following converters, a grid-forming converter behaves as a voltage source, which synchronizes with the power grid through the APC, as shown in Fig. 2(b).Hence, the PLL is not used for the grid-synchronization of grid-forming converters during normal operations [10], [29].
Fig. 3 shows four typical schemes of APC and RPC, which are 1) the power synchronization control (PSC) [10], 2) the basic droop control [28], 3) the droop control using low-pass filter (LPFs) [30], and 4) the basic virtual synchronous generator (VSG) control [31], [32].The PSC and the basic droop control are equivalent to each other, and they can be categorized as the first-order power control [24].The socalled "virtual inertia" can be synthesized by adding the LPF into the basic droop control, which has been proven equivalent to the basic VSG control [33], and they are categorized as the second-order power control.Further, since the first-order power control can be seen as a special case of the second-order one without inertial emulation, a general second-order power control model can be used to characterize the dynamics of grid-forming converters [24].Hereafter, the droop control with LPFs is selected as the representative for the further analysis.
It is worth mentioning that many alternative VSG control schemes have been recently introduced, in addition to the basic VSG control illustrated in Fig. 3(d).For example, an additional damping correction loop is reported in [34], which provides one more degree of freedom to adjust the damping ratio of VSG without affecting its steady-state frequency droop characteristic.In [35], the power system stabilizer (PSS) is added in the reactive power control of VSG for the stability enhancement.While these additional control loops bring in add-on benefits, their core control principle still follow the second-order swing equation of SGs, whose dynamic plays a critical role in the transient stability analysis.

C. Static Power Transfer Limitation
During weak and faulty grid conditions, the synchronization instability will be inevitable when the converter violates the maximum static power transfer between its terminal bus and the grid.This limitation is present for any control structure (grid-following and grid-forming) and is a condition for the existence of equilibrium points in the system.Therefore, this limitation is a necessary stability condition, ensuring that the converter operation represents a stable operation point [7], [36].Considering the grid-following converter in Fig. 1, the q-axis component of PCC voltage, which is used for gridsynchronization, can be expressed as sin( ) sin( ) where Z  is the angle of the grid impedance.For stable operation, the PLL must be able to control the q-axis voltage component to zero.It should be noticed that the second righthand term behaves as a destabilizing positive feedback term in the PLL control since the PLL is only able to regulate the first term [27], [37].Therefore, if the grid is weak (large  ) or during a severe fault where  is small, the PLL may not be able to control the q-axis voltage to zero, causing the synchronization instability.By setting  0 , the stability condition with an inequality can be formulated as From (2), it is noted that only a limited current magnitude ( ) can be injected, given a current injection angle, a grid impedance, and a grid voltage magnitude.For a fully active current injection ( 0), the limit is reduced to the ratio between the grid voltage magnitude and the grid reactance.However, during severe grid faults where a fully reactive current provision is often required, the current injection limit reduces to the ratio between the grid voltage magnitude and the grid resistance.This is exemplified in Fig. 4 where the stable area highlighted is the region where the injected current vector satisfies the expression in (2).Here, it can be seen that the more severe the grid fault is, the more limited is the stable region.As shown in Fig. 4(b), the voltage may drop so low that the 1 p.u. reactive current provision cannot be achieved without overloading the converter by injecting additional active current.Further, during normal operations, it can be appreciated from (2) that the minimum short-circuit ratio (SCR) for an inductive grid is 1 p.u., considering the static power transfer limitation.It is important to note that the power flow direction also influences the stable operating area.As can be seen in Fig. 4, the stability limit is not only influenced by the direction of the active power flow, but is also highly affected by the direction and magnitude of the provided reactive power.
For grid-forming converters, its output active power can be calculated as which can be visualized by the well-known Po-δ curve, as given by Fig. 5, where point c and e are defined as the stable equilibrium point (SEP) and the unstable equilibrium point (UEP), respectively.The maximum active power that can be transferred from the grid-forming converter to the grid is defined as  in Fig. 5, and it is clear that   is required for the existence of an equilibrium point.

A. Small-Signal Synchronization Stability
A conceptual explanation of the synchronization stability of grid-following converters in low SCR grids is given first.With a low SCR, the PCC voltage of the converter is severely affected by the current injected into the grid.The variation of the PCC voltage is then propagated through the PLL in Fig. 1, impacting the converter current.Thus, a selfsynchronization loop is formed, which can deteriorate the synchronization stability of grid-following converters.

1) Sideband Oscillations
In low SCR grids, the synchronization instability of gridfollowing converters can result in sideband oscillations in the voltage and current waveforms, whose waveforms are given in Fig. 6.From the spectrum of  , it is seen that the sideband oscillations around the grid fundamental frequency, i.e. f1±f (88 Hz and 12 Hz), are generated, where f 1 is the fundamental frequency [14].
The sideband oscillation is induced by the asymmetrical control dynamics in the dq-reference frame [38], which can be illustrated by Fig. 7.If the converter only has symmetrical control loops, the dq frame oscillation at the frequency (ω) only results in a frequency-shifted response (ω 1 +ω) when transformed into the αβ frame.However, the grid-following converters have asymmetrical control loops.For instance, the synchronization unit, i.e., PLL, only regulates the q-axis PCC voltage, as shown in Fig. 1(b).Consequently, the asymmetrical dynamics in the dq frame (±ω) are inevitable, which propagate through the inverse Park transformation, resulting in sideband oscillations in the αβ frame (ω 1 ±ω).

2) Small-Signal Modeling and Analysis
Small-signal modeling and stability analysis can characterize the sideband oscillations induced by the PLL of gridfollowing converters in low SCR grids.To briefly illustrate the negative damping mechanism of the PLL, a simplified small-signal model based on the feedback control loop of  can be derived, as shown in Fig. 8.The control dynamics are modeled in the dq-reference frame, where the impacts of CC, DVC, and AVC are neglected.It can be seen that the PLL itself forms a negative feedback control loop, as shown in the grey box.Additionally, the interaction with the grid impedance leads to a positive feedback control loop when I dref > 0 (i.e., the inverter mode).The PLL is in cascade with the positive feedback control loop, and hence, adversely affects the synchronization stability of grid-following converters.More specifically, with higher values of I dref and Lg, corresponding to a lower SCR, the gain of the positive feedback control loop is increased.With a higher control bandwidth of the PLL, the effective frequency range for the positive feedback control loop with a high gain is widened.Therefore, both the high PLL bandwidth and the low SCR of the grid can deteriorate the synchronization stability of gridfollowing converters [40].
The eigenvalue-based analysis and the impedance-based analysis are two common approaches to analyze the synchronization stability of grid-connected converters.The eigenvalue-based analysis relies on building an explicit statespace model of the entire system, then the sensitivity analysis of PLL control parameters can reveal its negative damping to the oscillation modes [41].
The impedance-based analysis is more suitable to analyze so-called "black-box" systems.The converter-grid interaction can be modeled by equivalent impedances seen from the PCC.Such impedance models form an open-loop gain of the system, where subsequently the stability is evaluated using the Nyquist stability criterion [42].The impedance-based analysis enables us to provide physical insight into the converter-grid interaction from the passivity perspective [38].The negative damping contributed by the PLL has been interpreted as a negative resistance (nonpassive) in the low-frequency range, which jeopardizes the converter-grid stability [27], [43].It is noted that due to the asymmetrical dq-frame dynamics induced by the PLL, the converter impedance model has to be characterized as a multi-input multi-output (MIMO) matrix.Thus, the passivity needs to be evaluated through the passivity index [38].To simplify the MIMO-based analysis, two equivalent singleinput single-output (SISO) impedance ratios are derived from a closed-loop MIMO model including the grid impedance [44], [45].Thus, the synchronization stability can be evaluated by using the SISO impedance-based analysis [42].Yet, the equivalent SISO impedances of converters are composed by impedance terms of converter and grid, which fails to shed clear insight into the passivity of converter impedance.

3) Stabilizing Control: Controller Tuning and Additional Damping
Many research efforts have been made to mitigate the sideband oscillations of grid-following converters operating in low SCR grids, which are summarized in Table I.The most common practice is to tune the PLL control parameters.According to Fig. 8, since the PLL works like a LPF, by reducing its control bandwidth, the effective frequency range of the positive feedback loop is reduced, such that the synchronization stability can be improved.It is also found that reducing the PLL bandwidth also results in a  reduced frequency range of the negative resistance in the converter output impedance, which explains the stability enhancement from the passivity perspective [46].In addition to the controller tuning, several active damping control methods have been reported to improve the dynamic performance of grid-following converters in low SCR grids.
Since the PLL is the root cause that destabilizes the system, some efforts are directly put on the modifications of the PLL, whose control diagrams are summarized in Fig. 9.
A damping controller with a band-pass filter (BPF) is proposed in [47], where the negative resistance caused by the PLL can be minimized by tuning the BPF.In [48], a virtual impedance is embedded with the PLL, such that the VSC is synchronized with a remote stronger voltage behind the virtual impedance.The negative impact of grid impedance on the synchronization stability can thus be reduced.The two methods directly apply active damping by modifying the PLL, yet the controller tuning depends on the grid condition.A symmetrical PLL is further proposed in [39], which establishes a pair of complex phase angles (  ) through the symmetrical control dynamics.Although the PLL itself is not intended for active damping, its symmetrical structure enables a SISO-based impedance shaping for stability enhancement.
Besides modifying the PLL, the feedforward-based control is also commonly used for active damping.One general approach is based on the virtual impedance control, where a feedforward control loop from the PCC voltage can be added to the current reference, to realize impedance shaping and reduce the negative resistance caused by the PLL [47], [49], as shown in Fig. 10(a).It is noted that the control implementation of the virtual impedance can be flexible in either the αβ or dq frame.Also, the voltage feedforward control can be alternatively added to the modulation voltage [50], which results in a similar damping effect.These virtual impedance control designs still rely on the grid condition, thus a parameter tuning effort is needed.Alternatively, a recent work reported in [51] designs a feedforward loop from the PLL to the current control, based on establishing symmetrical dynamics in d and q axes, whose control diagram is given in Fig. 10(b).This method mitigates the frequency couplings caused by the PLL, which contributes to the synchronization stability in low-SCR grids.
The above stabilizing methods mainly tackle the negative damping caused by the PLL.In recent years, some studies have revealed that the grid-forming control achieves superior stability in weak grids compared to the grid-following control [52], since the PLL is not needed for grid-forming control, and thus the corresponding positive feedback control loop can be avoided.However, the grid-forming control introduces other challenges for synchronization stability, which will be further discussed in Section IV.

B. Large-Signal Synchronization Stability
Besides the synchronization stability in low SCR grids, large-signal synchronization (transient) instability is a large threat to the power system security during grid faults and large load-generation imbalances.In the following, the loss of synchronization (LOS) phenomenon during grid faults is  visualized first.The large-signal dynamic model for transient stability is then discussed, which is followed by a designoriented analysis and countermeasures by control to enhance the transient stability.

1) Loss of Synchronization Phenomenon
The transient instability may occur for two reasons: i) the necessary stability condition in ( 2) is violated, i.e. a stable operating point does not exist during the fault, and ii) the synchronization dynamics do not possess sufficient damping to be attracted to the stable operation point during the fault.Fig. 11 shows experimental waveforms of LOS caused by the insufficient damping during a severe symmetrical grid fault, although, the necessary stability condition given in (2) is satisfied.As can be seen, the injected currents seen from the PLL rotating frame are following the current references, prioritizing reactive current support during the fault.However, when referring the injected currents to the actual phase-angle of the PCC voltage, large low-frequency oscillations are observed, which distorts the PCC voltage towards a voltage collapse.

2) Large-Signal Modeling
For the transient stability assessment, the PLL-synchronized grid-following converter can be represented as a secondorder nonlinear equation as [53] where [ sin( ) , and the grid impedance magnitude and phase are based on the real-time PLL frequency.For the model in (4), the converter is represented as an ideal controllable current source, which is oriented by the output phase of the PLL.Such an approximation is justified by the fact that the timescale of PLL-induced LOS is much slower than that of the inner current control [53], [54].
Since the model in ( 4) is second-order and nonlinear, no known analytical solution exists.Therefore, to assess the stability, either an approximation of the model has to be made or it has to be computed using numerical integration.In [55], [56], it is revealed that the second-order nonlinear model in ( 4) can be further rearranged to the same form as the power-angle swing equation of a synchronous machine, which is given by where  and  are the q-axis components of the voltage across the line impedance and the grid voltage, respectively.Eq. ( 7) forms the nonlinear voltage-angle "swing" equation, where the virtual moment of inertia is inversely proportional to the PLL integral gain, and the system damping is proportional to the ratio between the PLL controller gains.It is evident that the transient stability can be enhanced by increasing the proportional gain or decreasing the integral gain of the PLL.With this analogy to the synchronous machine swing equation, analytical assessment methods employed in the field of synchronous generation can be used for grid-following converters.Fig. 12 shows the voltageangle curve of grid-following converters, where the equalarea criterion (EAC) is adopted in [7], [56] to assess the transient stability of the system.With the EAC, the decelerating and accelerating areas depicted in Fig. 12 can be analytically determined and instability is concluded when   [7].However, the system damping is neglected in the EAC formulation, which is achieved by setting the PLL proportional gain to zero.This approximation is, however, not realistic in a practical design of the PLL [43], which causes the EAC to be conservative in its assessment.
Besides the EAC criterion, the Lyapunov theory has also been used to assess the transient stability of the system.In [57], the grid-following converter is modeled as an electrostatic machine where the Popov method is used to find a Lyapunov function to assess the large-signal stability of the system.This method also assumes a damping-less system, which gives a conservative stability assessment.In [58], a catastrophic bifurcation phenomenon is identified for gridfollowing converters under grid fault conditions.Here, it is observed that the converter cannot synchronize with the grid and that voltage saturation limit imposed by the available dclink voltage causes a structural change to the system during saturation, which causes instability that cannot be restored after fault recovery.Using the above direct methods, the damping ratio is overlooked, which is infeasible for PLLsynchronized converters.Hence, a design-oriented analysis considering the damping effect of the PLL is needed.

3) Design-Oriented Analysis of Transient Stability
As mentioned above, the impact of controller parameters of the PLL needs to be considered for the accurate transient stability assessment.The damping ratio (ζ) and the settling time (ts) of the PLL are defined as For the second-order nonlinear differential equation given by ( 4), the phase portrait can be adopted for obtaining a graphical solution, which is intuitive and facilitates the design-oriented analysis.
Fig. 13 shows the phase portraits of a grid-following converter with different ζ and ts.Points a and c represent the system's SEPs before and after the fault, respectively.The system is stable when the phase portrait converges to the SEP c after the fault and is unstable when the phase portrait diverges.From Fig. 13, it is clear that the transient stability of grid-following converters can be enhanced by increasing the ζ and ts of the PLL.Using phase portraits, the critical system damping and area of attraction of the nonlinear system can be estimated by sweeping a set of initial conditions for PLL controller gains [59].This is exemplified in Fig. 14, where the PLL integral gain is swept from the initially designed value to a lower value, which provides higher damping and, therefore, enhanced transient stability.As can be noticed in Fig. 14, for a decreasing fault voltage magnitude, the critical PLL damping needs to be significantly increased for the phase trajectory to be attracted to the stable operating point during the fault.
4) Stabilizing Control There are, in general, two categories of control methods for enhancing the transient stability of grid-following converters.

Table II. Stabilizing control to mitigate transient instability based on changing the injected current vector
Ref.
Proposal [60] Reduced active current proportional to voltage drop during fault [36] Increase active current reference based on the PLL frequency error [61] Increase active power reference based on the PLL frequency error [62] Align current vector angle with line impedance angle [63] Set reference power equal to actual power to eliminate ac-/de-celerating areas in EAC a

i) Modified Active Current or Active Power
The first category of control methods is to modify the injected active current during the fault.These methods are listed in Table II.In addition to these, it is also proposed to reduce or cancel the current injection to improve stability during severe faults [64].However, these methods do not comply with the grid codes and are, therefore, not included here.The block diagrams of the proposed methods from Table II are shown in Fig. 15.During faults where the reactive current is limited to 1 pu and the converter possesses short-term overloading capability, providing room for injecting active current, instability may occur as the active current component pushes the current vector into an unstable operating area [60].To address this issue, it is proposed in Fig. 15(a) to reduce the active current in proportional to the experienced voltage drop during the fault.This method has a straight-forward implementation but suffers from the fact that the active current during the fault is low in many applications, and a pure reactive current cannot be injected.In such cases, it is desired to increase the active current rather than decreasing it.This is considered in Fig. 15(b), where the active current is increased based on the PLL frequency error.This method, as proposed in [36], contains a closed-loop controller where the PLL frequency signal is used as a feedback to detect LOS and regulate the active current, such that the injected current vector moves to a stable operating area.This method has the advantage that it can address the problems with either too low or too high active current during the fault.However, it introduces an additional closedloop control where its stability is not discussed or proven.
A similar approach is proposed in Fig. 15(c), where the active power reference is modified instead of the active current.From (2), it can be seen that if the injected current vector is aligned with the negative grid impedance angle, a stable operating point will always result [62].This fact is utilized in the proposal in Fig. 15(d), where the injected current vector is aligned in this way.This method has the advantage of always proving a stable operation and its transient stability can be mathematically proven.However, this method needs a rapid estimation of the impedance angle when the fault occurs.In [65], such an estimation of impedance angle is reported, which can be utilized with the control scheme in Fig. 15(d) for stability enhancement.Lastly, another approach similar to those discussed in [38], [64] is proposed in Fig. 15(e) [63].Here, the stabilizing idea is based on the principles of the EAC.If the reference power is set equal to the measured active power (  ), then ideally, the accelerating and decelerating areas are eliminated, which means that stability can be achieved with any positive damping term.However, as shown in (7), the damping is a nonlinear function of  and can, therefore, not be guaranteed positive.To address this issue, Fig. 15(e) uses a PI controller on the PLL frequency error.With this, it is shown that the proportional gain of this controller introduces additional damping, which can be designed to always compensate for potential negative damping in (7).(d) Alignment of the current vector into a stable operating area [62].(e) Cancellation of ac-/de-celerating areas in EAC and damping provision [63].

ii) Modified PLL Parameters
The second branch of stabilizing control methods modify the PLL parameters to enhance the transient stability.These are listed in Table III and their proposed control structures are depicted in Fig. 16.The simplest method is the PLL freezing method visualized in Fig. 16(a), which nullifies the PLL control error when a predefined severe fault is detected [66], [67].This effectively makes the PLL-synchronized converter able to operate in a stand-alone mode based on its pre-fault frequency and phase estimations.This method has the advantage that it allows the converter to operate under any condition, including zero-voltage conditions, and will always operate at a stable operating point, no matter the fault severity.As this method freezes its internal states, it is unaware of phase-angle jumps and frequency drifts during the fault.In the second method in Fig. 16(b), the proportional gain and integral gain of the PLL are increased and Increase PLL damping ratio [59], [68].(c) Gain schedules the PLL integral gain depending on estimation frequency change [69]. is the initial designed integral gain.(d) Switch to a first-order PLL when maximum ROCOF is exceeded [22], [70].

Table III. Stabilizing control to mitigate transient instability based on modifying the synchronization dynamics
Ref.
Proposal [66], [67] Freeze the PLL during the fault [59], [68] Increase damping ratio of PLL [69] Gain schedules  during fault [22], [70] Switch PLL to a first-order loop during fault decreased, respectively, to improve the system damping, which enhances the transient stability.This method is simple to implement, yet it only works when the converter has two equilibrium points during the fault and lacks parametertuning guidelines.The third method ( [59], [68]) shown in Fig. 16(c), proposes gain scheduling of the PLL integral gain based on the transient dynamics in the estimated PLL frequency.This effectively decreases the integral gain towards zero if instability occurs.Δ is the maximum frequency band around nominal considered, here 10 Hz.Compared to the method in Fig. 16(b), this has the advantage of increasing the PLL damping sufficiently during transients without having to select a pre-defined specific value.Nevertheless, it suffers from not being able to address faults where only one equilibrium point exists during the fault or in cases where the two operating points are very close to the necessary stability condition.In either case, the overshoot from the second-order dynamics may still cause instability.This issue is addressed in [22], [70], Fig. 16(d), where the PLL is switched to a first-order loop during the fault, eliminating the integral gain.The activation of the first-order loop is based on a rate-of-change-of-frequency (ROCOF) threshold rather than a fault signal based on the voltage magnitude.Hence, if instability does not occur, the proposed method will not remove the integral gain.The proper selection of the ROCOF values is discussed in detail in [22].This method has the advantage that it can stabilize any system with a least one operating point.During a severe fault, where no operating points exist, one still needs to switch to, e.g., a PLL freezing method or some of the methods presented in Fig. 15 to remain stable.

A. Small-Signal Synchronization Stability
Differing from grid-following converters, the grid-forming converter may suffer from small-signal instability in stiff grids [16].Since grid-forming converters regulate their PCC voltages directly, the voltage regulation tends to be more difficult if two voltage sources are electrically close to each other.There are two types of synchronization instability phenomena for grid-forming converters: 1) sideband oscillations and 2) synchronous oscillations.

1) Sideband Oscillations
The first type of instability is the sideband oscillation of the grid fundamental frequency, similar to that of gridfollowing converters in low SCR grids.This phenomenon is due to the asymmetrical dynamics of grid-forming converters in their outer-loop power control.
Fig. 17 shows experimental waveforms for grid-forming converters operating in stiff grids.The sideband oscillations can also be observed in the voltage and current waveforms, which turn into low-frequency oscillations in the power waveforms.
Several studies have been reported recently to characterize the sideband oscillations.It has been found that the innerloop voltage control of grid-forming converters also plays a vital role in the negative damping [17].It is also revealed that the voltage feedback decoupling loop added at the output of the innermost current control loop has a detrimental impact on the stability of grid-forming converters in stiff grids [18].Besides, the power control parameters, e.g., the droop control coefficient [19], the converter operating point [20], and the line dynamics [16], also impact the sideband oscillations.
In addition to stiff-grid scenarios, grid-forming converters can also induce sideband oscillations in series-compensated weak grids [21].In such cases, the grid impedance has an abrupt change from inductive to capacitive behavior near the fundamental frequency, which tends to interact significantly with the grid-forming converter and lead to the sideband oscillations.It is found that the stability becomes worse as the series compensation level increases since the resonant frequency of the grid impedance moves towards the grid fundament frequency.

2) Synchronous Oscillations
The second type of small-signal synchronization instability of grid-forming converters is the synchronous oscillation [10].Fig. 18 shows the measured synchronous oscillation waveforms of grid-forming converters with a step change of active power reference, where oscillations at the fundamental frequency appear in the power waveform (C1), corresponding to the second-order harmonics and dc offsets in the voltage and current waveforms (C3 and C4).This is in principle induced by the power transfer through electrical networks.The small-signal model of the plant in the power synchronization control loop can be derived as [72]  It is seen that the inductive line parameters result in a synchronous resonance peak in the plant models, and the line resistance adds damping to the synchronous resonance.Therefore, such an issue is more severe in systems with low R/X ratios.The synchronous resonance was overlooked in conventional synchronous generators, due to the slow electromechanical dynamics [73].However, it becomes more evident in grid-forming converters, since the power is controlled much faster.

3) Stabilizing Control: Controller Tuning and Additional Damping
Stabilizing control methods have been developed to address the sideband oscillations and the synchronous oscillations of grid-forming converters, which are summarized in Table IV.The sideband oscillations can be mitigated by tuning the control parameters, such as reducing the powerloop controller gains [19], or reducing the voltage feedback decoupling gain [18].
Table IV.Stabilizing methods for small-signal stability enhancement of grid-forming converters.

Issues
Stabilizing methods

Sideband oscillations
 Reducing power-loop controller gains [19]  Reducing voltage feedback decoupling gain [18]  Multi-loop damping controllers in APC [74], [75]  Lead-lag compensation in APC and RPC [76]  Active power feedforward [76] or frequency dependent voltage control [77], [78]  Virtual impedance control for seriescompensated grids [21] Synchronous oscillations  Increasing virtual inertia (reducing power control bandwidth) [73]  Virtual impedance control [10]  Phase compensation in power loop [72]  Cross-coupling control schemes [71], [72]  (a) Multi-loop controllers in APC [74] with artificial intelligence [75].(b) Lead-lag compensation in power loop [76].(c) Active power feedforward [76] or frequency dependent voltage control [77], [78].(d) Virtual impedance control [21], [76].Some additional damping control schemes of sideband oscillations are developed for grid-forming converters, whose diagrams are provided in Fig. 19.Multi-loop controllers can be embedded into the APC to dampen the low-frequency oscillations in the power waveforms [74], as shown in Fig. 19(a), where the center frequencies (ω n ) of the BPFs are tuned at the oscillation modes and the LPFs are used to emulate swing equations.The oscillation modes can be predicted by artificial intelligence [75].This method is simply designed within the APC loop.Another method applies lead-lag compensators (Gllp and Gllq) in both the APC and RPC loops [76], as shown in Fig. 19(b).Similar to the previous one, the design of the lead-lag compensators relies on the oscillation modes.In addition, since the compensators are in cascade within APC and RPC loops, they may affect the power control bandwidth.Alternatively, the active power feedforward control [76] or the frequency dependent voltage control [77], [78] can be applied for stabilizing the system.However, those controllers strengthens the dynamic couplings between the APC and RPC, and their design are dependent on grid conditions [76].In addition to the shaping of power control loops, the virtual impedance-based control can be designed with the inner loop dynamics, whose principle in active damping is similar to that for gridfollowing converters [79].This concept has also been applied to dampen the sideband oscillations in a seriescompensated grid, where a current feedforward control that is directly added to the modulation voltage [21].
For the synchronous resonances, it can also be damped by controller tuning.With larger virtual inertia, a mechanical Reference Generator damping can be emulated to damp the resonance [73].However, the method adds a low-pass filter in the APC, which sacrifices the power control bandwidth.Some additional damping approaches have been reported, as shown in Fig. 20.An easy way is to implement the virtual impedance control [10], which emulates electrical damping in series with the line impedance in (10).This approach is in principle the same as Fig. 19(d) for the sideband oscillation suppression.Alternatively, the phase compensation filter can be directly deployed in the power loop [72], which avoids the phase crossing 180° at the synchronous resonant peak, thus contributes to the stability.The aforementioned two damping approaches can be simply designed based on the single APC or RPC loop, which are thus unable to address the crosscouplings in (10).Such cross-couplings can intensify the power oscillations between the two control loops [73].A cross feedforward compensation method is thus utilized in [72], where two proportional damping terms are tuned to suppress the cross-couplings.Although the synchronous resonances can be mitigated more effectively with larger damping terms, the instability risk still exists since the crosscouplings cannot be fully compensated by the proportional damping terms.A phase-amplitude cross-regulation method is further proposed in [71], which not only fully eliminates the synchronous resonance, but also remarkably reduces the couplings between the APC and RPC.However, this decoupling control is merely designed assuming that δ0≈0 in (10), which means that the steady-state phase angle difference between the converter PCC voltage and the grid voltage is negligible

B. Large-Signal Synchronization Stability 1) Phenomenon
Similar to the large-signal synchronization stability issues of grid-following converters presented in Section III.B, the grid-forming converters may also lose synchronization with the power grid when subjected to large disturbances.The root causes of the LOS are:  Lack of equilibrium points when P ref is larger than the maximum power that can be transferred from the grid-forming converter to the grid (Pmax in Fig. 5). The crossover of the UEP (point e in Fig. 5 ) due to the overshoot in the dynamic responses.Since Pref > P o always holds after the UEP, the APC will continuously increase the output frequency of the grid-forming converters, and finally leads to the LOS.Fig. 21 shows the measured waveforms for grid-forming converters with LOS after a fault clearance [80].In this test, the equilibrium points of grid-forming converter are restored by clearing the fault.Although the controller parameters are tuned to assure the small-signal stability, the grid-forming converter still loses the synchronism with the power grid, due to the crossover of the UEP.

2) Large-Signal Modeling and Analysis Methods
The large-signal dynamics of grid-forming converters are determined by the APC and RPC loops, whose nonlinear differential equations can be derived based on Fig. 3(c) sin 3 2 where ωp and ωq are cutoff frequencies of LPFs in the APC and RPC, respectively.It is difficult to obtain analytical solutions of ( 11) and ( 12) due to their nonlinearity.As an alternative, Lyapunov's direct method can be adopted for the transient stability assessment.Yet, it requires the derivation of a candidate Lyapunov function, which is challenging to obtain in practice [81].In contrast, phase portraits that provides a graphical solution of first-and second-order nonlinear systems, are more intuitive and can be readily implemented.

i) First-Order Power Angle Control
In this part, the transient stability of grid-forming converters with the first-order power control, i.e., PSC or the basic droop control, will be For simplicity, the dynamics of the RPC are neglected at first by assuming a constant Vo.Thus, the power angle dynamics can be simplified as [23] Based on (13), the phase portrait of grid-forming converters with the first-order power control can be plotted, as shown in Fig. 22(a).Due to the overdamped response brought by the first-order control dynamics, the grid-forming converter can always be stabilized at the SEP when there are equilibrium points during the disturbance, as shown in Fig. 22(a).Hence, the risk of crossing over the UEP is fully prevented by the first-order power control, which yields an attractive transient stability performance [23].
Moreover, in the case that there is no equilibrium point during the fault, the critical clearing angle (CCA) is fixed as δ u , which is the power angle corresponding to the UEP, as shown in Fig. 22(b).Differing from the SG, where the LOS is inevitable if the fault clearing angle (FCA) is larger than CCA, the grid-forming converter with the first-order power control can still re-synchronize with the grid after around one cycle of oscillation, even if FCA> CCA, as shown in Fig. 22(c).This superior feature prevents the system collapse, which is traditionally caused by the delayed fault clearance.
Fig. 23 further shows the V o -δ curve of the grid-forming converter by considering the impact of the first-order RPC.It can be seen that Vo is reduced with the increase of δ due to the droop characteristic, which lowers the maximum power transfer capability of the grid-forming converter.Hence, the grid-forming converter is more prone to the loss of equilibrium points with the RPC, and thus, the transient stability is also degraded.However, the abovementioned Phase portraits of the grid-forming converters with second-order APC [24].benefits of the first-order power control still remain when there are equilibrium points, as demonstrated in [24].

ii) Second-Order Power Angle Control
By adding the LPF into the basic droop control, a secondorder power angle dynamics is yielded, as seen in (11), [33].The second-order APC leads to an underdamped response of the grid-forming converter, which is similar to SGs.Fig. 24 shows the phase portrait of the grid-forming converter with the second-order APC, in which the overshoot in the dynamic response can be observed.The smaller ωp (or equivalently, larger virtual inertia) enlarges the overshoot, which poses a higher risk of crossing over the UEP, as the red dashed line given by Fig. 24.Hence, although large virtual inertia is supposed to be beneficial to the frequency response of the power system, it jeopardizes the transient stability.Therefore, a tradeoff exists in selecting the virtual inertia [24].Fig. 25 shows the Vo-δ curve of the grid-forming converter with the second-order RPC, it can be seen that transient voltage drop introduced by the RPC is alleviated by the LPF therein.The smaller cutoff frequency ωq of the LPF helps to slow down the transient voltage dynamic, which is beneficial for the transient stability of the system.
From the above analysis, it can be concluded that the two LPFs in the active and the reactive power loops take opposite effects on the transient stability, i.e. the former degrades the stability while the latter improves it.The better transient stability performance demands an LPF with a high cutoff 3 sin . 2 frequency (low inertia) in the active power loop and an LPF with a low cutoff frequency in the reactive power loop [24].

3) Stabilizing Control Methods
There are, in general, two categories of control methods for enhancing the transient stability of grid-forming converters, which are summarized in Table V.

i) Change Active and Reactive Power References
Since grid-forming converters with the second-order power control loops exhibit similar dynamics as SGs, the existing methods for stabilizing SGs, e.g., controlling the governor to reduce the accelerating power and/or injecting additional reactive power to boost the output voltage during grid faults, can be directly adopted.pointed out in [81], [82], the transient stability of grid-forming converters can also be enhanced by reducing the active power reference and/or increasing the reactive power reference during grid voltage sags, as shown in Fig. 26(a)-(b).The typical challenge of these methods lies in how to quantify the changes of power references, i.e., the selections of np and nq in Fig. 26(a)-(b), which requires the prior knowledge of grid impedance.

ii) Modify Control Loops or Controller Parameters
It has been pointed out that the better transient stability performance of grid-forming converter is yielded with the first-order power control.Yet, the inertia emulation is often required, which leads to a second or even higher-order power control loops.In those cases, it is still possible to coordinate the design of virtual inertia and damping terms, in order to approximate the first-order power control dynamics of gridforming converters, even if higher-order power control loops are used [24].Besides using the fixed virtual inertia and damping terms [24], the gain-scheduled virtual inertia and damping control for grid-forming converters is reported in [83], where the power control loops only approximate the first-order dynamics during the transient disturbance, as shown in Fig. 26(c).
Differing from the approximated first-order power angle control during the fault, Fig. 26(d) shows the block diagram of a mode-switching control, which is reported in [80], where a control gain k is inserted in the forward path of the control loop, and is switched between +1 and -1 based on operating scenarios, such that the positive-feedback dynamics after crossing over the UEP can be avoided.The advantage of the mode-switching control is that no approximation of secondorder power control to the first-order one is needed, and the virtual inertia can be tuned in a wide range.Yet, the modeswitching control does require a reliable detection of the operating scenarios.

4) Current Limiting Control
The current limit of grid-forming converters can be triggered during severe faults.In this case, the grid-forming converter will be operated in the current limiting control mode, and its synchronization can be realized by the backup PLL [10], or still based on the APC [84].In the former case, the grid-forming control turns into the grid-following mode and its transient stability characteristic is detailed in Section III-B.Further, during the fault recovery stage, switching Δω  [81].(b) Increase the reactive power reference [82].(c) Gain scheduled virtual inertia and damping terms [83].(d) Modeswitching control [80].

Categories
Proposal Change active and reactive power references Reducing the active power reference [81] Increasing the reactive power reference [82] Modify control loops or controller parameters Coordinate the design of virtual inertia and damping term [24] Gain scheduled inertia and damping [83] Mode-switching control [80] back to the grid-forming control may also cause instability issues [85].In the latter case, the maximum power transfer capability is reduced due to the current limitation, and the grid-forming converter is prone to the LOS due to the higher risk of loss of equilibrium points [84].
To avoid the switching between the grid-forming and gridfollowing modes, it is increasingly required to keep the gridforming mode even during severe grid faults, and to provide an effective current limitation.Several control methods have been to meet this requirement, which are listed in Table VI.
It is clear that directly limiting the converter current will leave the converter voltage uncontrolled, resulting in control errors in the outer loops.Hence, the effective current limiting control is often accomplished in the outer loops.In general, the current limitation can be accomplished by using a virtual impedance controller or a modified droop controller.One implementation of this is given in [86], where a grid-forming fault-mode controller including converter current limitation is reported.The power references for outer droop controllers are replaced with the attainable power references based on the measured voltage drop during the fault.In this way, the power-synchronization control still works, and the converter can limit its injected currents without controller saturation and integrator windup.This has the advantage of accurate current limitation but is only developed under symmetrical fault conditions.In [87], the current limitation is achieved by using a virtual resistor, yet the design of the virtual resistance is cumbersome, due to its voltage dependency.In [88], the outer power loops are disabled during faults, and the current is limited through a virtual admittance.This method can limit the converter current during faults, but has the disadvantage of disabling the outer control loops that are needed for the grid-forming operation.
The large-signal stability of the post-fault synchronization process of grid-forming converters is analyzed in [25], which uses an adaptive virtual impedance for the current limitation.To enhance the transient stability, the adaptive droop gains based on the voltage amplitude is proposed.A droop control scheme that accounts for the current limitation is reported in [89], which shows improved performance compared to the virtual impedance approach.A current limitation approach through imposing voltage limits is presented in [85].Despite those methods being able to limit the current and retain the grid-forming mode during the fault, the transient instability may still occur, due to the adoption of second-order APC in those cases.Using the first-order APC together with the current limiting control may yield a better transient stability Proposal [10], [90] Switching to a grid-following converter [86] Modify outer power references during fault [87] Current limitation through virtual resistance [88] Disable outer power controllers and limit the current through virtual admittance [25] Adaptive parameters for droop controller and virtual impedance [89] Enhanced droop control for current limitation [85] Using voltage limitation to provide current limitation behavior, as explained in Section IV-B 2. However, the impacts of current limiting control schemes on the transient stability of grid-forming converters still remain as an open issue.Further, the analysis and estimation of the CCA for grid-forming converters with the current limiting control need to be addressed.

V. CONCLUSIONS
This paper has given a comprehensive review of the gridsynchronization stability of converter-based resources under weak, stiff and faulty grid conditions.Systematic discussions on instability phenomena, modeling and analysis methods, as well as active stabilization schemes, in regards to both the small-and large-signal synchronization stability, have been presented.In each category, the advantages and drawbacks of grid-following and grid-forming converters are discussed.

A. Challenges with Grid Codes
With the increasing share of converter-based resources in power grids, the grid code needs to be continuously updated to maintain the synchronization stability of grid-connected converters.First, it has been identified that the fully reactive current injection during severe grid faults may cause the LOS of converter-based resources.Second, the requirements for voltage and frequency ride-through have been specified in grid codes.Yet, how to ride through the rate of change of the voltage angle in the future converter-dominated power grids still remains an open issue [91].Third, although grid-forming converters are increasingly demanded by power transmission system operators, there are no grid codes published on the grid-forming capability of converter-based resources [92].

B. Prospects of Future Research
To avoid causing the sideband oscillations in low SCR grids, grid-forming converters are increasingly preferred over grid-following converters.However, grid-forming converters tend to be unstable in the stiff and seriescompensated grids.More research efforts on the small-signal synchronization stability of grid-forming converters are expected.Further, while the analogies between synchronization dynamics of converters and that of SGs have been established, the unique challenge with converter-based resources is that their synchronization dynamics are highly dependent on their control structure and controller parameters, and hence, a design-oriented analysis plays a critical role in stabilizing converter-based resources.Such design-oriented analysis works well in single converter systems, yet tends to be complicated in power systems with multiple converters.Addressing this challenge requires the development of reduced-order modeling methods for future converter-dominated power grids.

Fig. 1 .
System diagram of a grid-following converter.(a) Main circuit and control system structure; (b) Phase-Locked Loop (PLL).

Fig. 2 .
Fig. 2. System diagram of a grid-forming converter.(a) Main circuit and control system structure.(b) Active Power Control (APC). dq

Fig. 4 .
Fig. 4. Depiction of the stable area of injected current vector during grid faults.(a) A grid fault where  / 1pu, (b) A severe grid fault where  / 1 pu and nominal capacitive reactive current cannot be injected as requested by the grid code.Both cases are shown for X/R=2.5. denotes the maximum amount of reactive current provision from (2).

Fig. 8 .
Fig. 8. Simplified small-signal model of synchronization control loop considering the PLL and grid dynamics.

Fig. 11 .
Fig. 11.Experimental waveforms, visualizing loss of synchronization during a severe symmetrical grid fault where the grid voltage drops to 0.045 pu.a) Unstable PLL frequency estimation.b) DQ-axes currents relative to the PLL phase angle.c) DQ-axes currents relative to the actual phase angle of the PCC voltage.and d) Three-phase faulted voltage at the PCC [53].

Fig. 12 .
Fig. 12. Nonlinear voltage angle curve shown for a pre-fault condition and during a fault.The q-axis voltage terms can be interpreted as equivalent mechanical and electrical power.The accelerating area ( ) and maximum allowed decelerating area ( ) is used in the EAC formulation to assess the transient stability of the system.

Fig. 14 .
Fig. 14.Phase portraits for varying  during a severe symmetrical fault where the critical damping and area of attraction can be extracted.The grid voltage magnitude during the fault drops to (a):  0.05 , (b):  0.045  .The grid impedance is  0.04 0.1  and  denotes the initial design for the PLL integral gain.Each subplot contains 200 different initial conditions used to estimate the area of attraction [59].

Fig. 15 .
Fig. 15.Stabilizing control through modifying the active current component during the fault.(a) Active current reduction based on voltage drop [60].(b) Active current injection based on PLL frequency error [36].(c) Active power injection based on PLL frequency error [61].(d)Alignment of the current vector into a stable operating area[62].(e) Cancellation of ac-/de-celerating areas in EAC and damping provision[63].

Table VI .
Methods for current limitation for grid-forming convertersRef.