Entire Period Transient Stability of Synchronous Generators Considering LVRT Switching of Nearby Renewable Energy Sources

1 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < Entire Period T ransient Stability of Synchronous Generator s Considering L VR T Switching of Nearby Renewable Ener gy Sources Bingfang Li, Student Member , IEEE , Songhao Yang, Senior Memb er , IEEE , Guosong Wang, Yiwen Hu, X u Zhang, Zhiguo Hao, Senior Member , IEEE , Don gxu Chang, Baohui Zhang, Fellow, IEEE Abstract — In scenarios where synchronous generators (SGs) and grid-following renewable energy sources (GFLR) are co - located, existing research, which mainly focus es on the first-swing stability of SGs , oft en overlooks ongoing dyn amic interactions between GFLRs and SGs throughout the entire rotor swing period. To address this gap, this study first reveals that the angle oscillations of SG can cause periodic grid voltage fluctuati ons, potentially triggering low-voltage ride-through (LVRT) control switching of GFLR repeatedly. Then, the periodic energy changes of SGs under "circular" and "rectangular" LVRT limits are analyzed. The results indicate that circular limits are detrimental to SG’s first-swing stability, while rectangular li mits and their slow recovery strategies can lead to SG ’ s multi-swing instability. Conservative stability criteria are also proposed for t hese phenomena . Furthermore, an additional controll er based on feedback linearization is introduced to enhance the entire period transient stability of SG by adjusting the post-fault GFLR output current. Finally, the efficacy of the analysis is validated through electromagnetic transient simulations and controller hardware- in - the-loop (CHIL) tests. Index Terms — transient stability, synchronous generator, grid- following renewable energy sources , entire peri od dynamic , low voltage ride through, feedback linearization control. I. INTRODUCTION HE large-scale and centralized integration of wind and solar power int o the grid presents challenges like low short-circuit ratios and weak i nertia support [1]-[5] . The combination of GFLR with existing SG units in the system offers a promising solution, wh ich enhances the grid's dynamic support capability and mitigates operational challenges stemming from the inherent uncertainty of GFLR output. However, the strong power coupling between GFLRs and nearby SGs raises concerns about potent ial impacts on rotor angle stabil ity in SGs. This emerging issue warrants a tho rough investigation to ensure t he stable operation of power systems. Numerous scholars have investigated the rotor angle stability issues in co-located GFLR and SG systems. Some use detailed time-domain simulations for transient stabil ity analysis [6]-[8] , while others prefer simplified analytical models. On the electromechanical timescale of SG rotor motion, the phase- locked loop (PLL) dynamics of GFLR's grid-side converters (GSC) can be neglected[9]-[11]. This simplification allows GFLR to be modeled as a negative impedance[12], power source[13], or con trolled current source[14], enabling classical stability analysis methods such as the direct method[15],[16] , the equal area criterion[17] and phase trajectory method [18],[19] to assess system stability. Within this framework, researchers have examined how GFLR penetration, connected location, fault condition, operating mode, and vo ltage or frequency control affect system transient angle stability [5],[11],[13]. However, these studies often overlook the control switching of GSCs , especial ly the impacts of LVRT control . This oversight can lead to discrepancies between a nalytical results and actual system behavior. During severe grid faults, GSCs switch t o LVRT mode t o prevent overcurrent and provide reactive power support[20] . Ref. [21] shows that wind farms lacking conventional support may experienc e voltage oscillations due to repeated L VRT control. The underlying mechanism involves a vicious circle: LVRT switching causes sudden changes in wind farm ou tput, leading to gr id voltage shifts, which in turn trig ger f urther LVRT events. In the scenari o discussed in this paper, however, SGs provide strong short-circuit capacity support. This enhances the voltage security of the sending-end system but also highlights rotor angle in stability issues. Analyzin g the dynamic i nteraction between r otor angle swings of SGs an d the transient control of GFLRs is crucial t o addressing this challenge. Current research has explored the impact of GFLR's LVRT dynamics on the first-swing rotor angle stability of synchronous machines. First-swing stability refers to the system’s ability to maintain stable during the first swing cycle after fault clearance[22]. This primarily depends on whether the transient energy accumulated during th e fault can be effectively abso rbed and dissipated after fault clearance. Ref. [23] and [24] identify two forms of rotor a ngle instability in GFLR stations with synchronous condensers, con sidering various LVRT depths and fault types. Ref. [25] studies a similar scenario in a system where GFLR is paralleled with virtual synchronous generators (VSGs). It investigates the impact of the current-limiting angle relationship between GFLR and VSG during faults on system stability, and proposes an a dap tive current-limiting control method. However, these studies primarily focus on LVRT's impact during faults, overlooking the sustained influence of GFLRs on voltage source equipment after fault clearance. Ref. [26] reveals that increases in VSG rotor angle post-fault narrow the stability region of the PLL, thus adversely affectin g system stability during the first swing. While this study touched upon post-fault interactions , its research perspective remained T This work was suppo rted by The Key Science and Tech nology Project of CSG (GZKJXM20222178, GZKJXM202222 11, GZKJXM20222213 ). B. Li, S. Yang, Z. Hao, Y. Hu, X. Zhang, and B. Zhang are with Xi’an Jiaotong University, Xi'an, China (e-mail: {son ghaoyang, zhghao , bhzhang}@ xjtu.edu.cn, {libingfang, huyiwen }@stu. xjtu .edu.cn). G . Wang an d D . Chang are with China Southern Power G ridcsg Electric Power Research In stitute. 2 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < confined to first-swing stability in the traditional sense. Considering GFLRs’ ongoing effects on subseque nt swing periods after fault clearance, the issue of multi-swing stability in GFLR and SG co-located systems is gaining increasing attention. Multi-swing stability refers to the system’s stability during subsequent oscillation cycles, assuming first-swing stability has been achieved. This instability often stems from new coupling eff ects and disturbance injections [27] . Ref. [28] graphically illustrates the accelerati on and deceleration energy changes of SGs across multiple swings, revealing odd- numbered i ns tability tre nds due to slow GFLR power recovery. Complementing this work, Ref. [29] provides a mathematical proof of this phenomen on . However, it treats GFLR output as an independent variable, failing to accurately account for the dynamic interactions between GFLR and main grids . In a nutshell, existing research either solely focuses on first- swing stability or fails to account for the post-fault interactions between GFLR and the grid. Focusing on the interaction between SG rotor dynamics and GFLR output after fault clearance , this paper makes four main contri butions: 1) The post-fault interaction mechanism between SG rotor angle and GFLR's LVRT control is revealed. It is found that rotor angle swings of SGs can cause voltage oscillations, which may lead to the periodic activation and deactivation of LVRT . This process, in turn, affects t he rotor dynamics of SGs. 2) The entire per iod transient instability risks of SGs with different LVRT limit modes of GFLR are analyze d. The "circular limit" makes SGs more prone to first-swing instabil ity, while the "rectangular limit" may cau se multi-swing i nstability. 3) Stability boundaries in the form of crit ical ener gy are proposed for both the first-swing and multi-swing stability of SGs affected by the LVRT contr ol switching of the GFLR. 4) Based on feedback linearization, this paper proposes a stabilization control strategy applied after fault clearance to enhance the entire period tra nsient stability of SGs. II. S YSTEM M ODELING A. System Overview Fig. 1 dep icts a simplified topology of a parallel transmission system comprising GFLR and SG. They are aggregated at the point of common coupling (PCC) and connected to the receiving-end grid via A C transmission lines. The grid integration characteristics of GFLRs are primarily reflected in the interaction between the GSC and the power grid. In Fig. 1 , E s ∠ δ , U w ∠  , U pcc ∠  p , U g ∠ 0° denote the voltage phasors of the sending-end SG, the converter, and the receiving-end infinite bus, respectively. I s ∠  s , I w ∠  w , and I g ∠  g represent the corresponding branch currents, while Y s , Y w , Y g are the respective branch admitta nces. In Fig. 2, the d s - q s , d - q , and x - y refere nce fr ames rotate counterclockwise with angular speeds  s ,  p , and  g . The d s - axis, aligned with the SG rotor, leads the x -axis by an angle δ , which is the SG's ro tor angle. The d -axis of the GSC frame l eads the x -axis by an angle 𝜃 , as provided by the PLL. 𝜂 is defined as the angle by which I w ∠  w lags the d -axis. SG GFLR Grid Bu s PCC    Filter PWM PLL Outer Loop Pow er Control LVRT Contr ol Inner Loop Current Control  GSC  * dq i dq i pcc p U   s E   g 0 U  ww I   w Y s Y g Y gg I   w U   ss I   Fig. 1. Topology of the GFLR and SG c o-located system. x y d q d s q s    ww I   w  s  p  g  s w d I Fig. 2. Relationships of different frames in th e vector space. B. Transient Stability Analysi s Model The GSC employs a classic cascaded control structure [30] , as illustrated in Fig. 1 . The control dynamics of the power electronic interface can be e ffectively decoupled from the rotor motion of SGs. Typically, the inner current loop (10 0Hz and above) and PLL (10-70Hz) operate at control bandwidths mu ch higher than the SG rotor dynamics (approximately 1Hz), while the outer power control loop operates at an intermediate bandwidth (around 10Hz)[9]. Therefore, on the time scale of SG rotor dynamics, which is the focus of this paper, it can be assumed that the GSC control lo ops have reache d a quasi- steady state. This means that the outer loop control of the GSC has completed its response to grid voltage changes and pr ovid ed current refere nce values ( i * dq ). Also, the act ual GSC output currents ( i dq ) can be considered equal to i * dq . Consequently, the GSC primarily acts as a controlled current source, as determined by the outer loop or LVRT control. Under normal grid conditions, the current reference i * dq is provided by the outer power control loop. If a fault occurs causing U w to drop below the LVRT ac tivat ion threshold, U in , the LVRT is triggered. In this mode, i dq = i * dq is determine d by the LVRT control strategy to meet the requirement for ra pid current regulation. Typically, the reactive current increment of GSC should respond to voltage c hanges at the connection point: ( ) q q w N 0. 9 i K U I =− , (1) where K q is the gain coefficient of reactive current support and I N represents the rated current. Applying the reactive power priority principle, two strategies can be employed to limit GFLR active current while ensuring priority for reactive current output as per (1). The first, known as "circular limiting" and described by (2), maximizes active current while keeping the total current magnitude below I max , the maximum allowable current of the GSC[24],[31]. T his method minimizes active power losses during LVRT. The second strategy, termed "rectangular limiting" and represented by (3), directly caps the GSC active current at i ref d,lim upon LVRT activation[20]. This method offers rapid response and simpler implementation. 3 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < 22 ref d ma x q w m in , P i I i U  =−   (2) re f d d, lim ii = (3) (note: P ref is the reference power for the outer loop control.) Output c urrent of G S C Time Active current Re a ct i ve cu rre nt LVRT rec overy LVRT Ramp recovery Fig. 3. GSC 's output current under the LVRT and recovery control. Upon fa ult clearance, if U w sur passes t he LVRT deactivation threshold, U out , the GSC transitions from "support mode" (described in (1)-(3)) to "recovery mode" , marking the beginning of the LVRT recovery stage. To address the challenges of minimizing sudden act ive power change s , a common recovery strategy gradually ramps up the active current to its normal value, as depicted in Fig. 3 . This approach can be expressed as: * dd i v t = (4) where v d is GFLR's active current ramp rate. The active power of the SG ca n be expressed as [29]: s E s g sg s w si n d P E U Y I E  =− , (5) where I ds w = I w cos( δ - φ w ) is the projection of I w ∠  w onto the d s - axis; sg sg sg YY Y YY = + , ( ) s sg 0 ,1 Y YY  = + . (6) Neglecting the resistance compone nt of Y s and Y g ,  is a real number. Define P es = E s U g |Y sg | sin δ as t he electromagnetic power of the SG wit hout the influe nce of GFLR. P w =  E s I d s w represents the power cou pling term between the GFLR and the SG. P m is the mechanical power of the SG. Thus, the rotor dyna mic equation of the sending-end SG can be e xpressed as: s g g J m es w d d d d t T P P P t D         +−  = − =     = −   (7) where P m , T J , and D denote the mechanism power, inertia time constant, and the damping c oefficient of the SG, respectivel y. III. D YNAMIC I NTERACTION B ETWEEN SG R OTOR S WING A ND P OST -F AULT O UTPUT OF GFLR Af te r the c lea ra nc e of a l ar ge di st ur ba nce , SG a ng le o sci ll at io ns in duc e s yst em vo lt ag e flu ct ua ti on s, l ea din g t o ch an ges in t he GF LR c u rr ent . Th es e c ha ng es, in tur n, af fec t S G r ot or d yna mic s vi a th e pow er co u pl in g t e rm , cr ea ti ng a cl os ed- lo op i nte ra ct io n: ro to r an gl e os ci ll at i on → vol ta ge fl uct uat io n → GF LR LVR T→ GF LR out put c ha ng e → ro to r an gle o sci ll ati on . Th e fol lo win g an al ys is e xa min es t he m ec han i sm of s uch a n int er act ive p roc es s. A. Dynamic Process Analysis of GFLR LVRT Switching Induced by Rotor Angle Swings The voltage at the GFLR gri d connection point, denoted as U w ∠  i n Fig. 1, can be expressed as: ( ) ( ) w2 2 w1 1 w g s w w 1 0 90 U U U U E I            = −   +  +  +  , (8) where  = 1/( Y s + Y g )+1/ Y w . Fig. 4(a) illustrates the voltage phasor diagram corresponding to (8), where the vector U w1 ∠ 𝛾 1 starts at point o and ends on the trajectory for med by the sum of vectors (1 -  ) U g ∠ 0° a nd  E s ∠ δ . T herefore, for the vector triangle ∆ oab , the cosine law gives: ( ) ( ) ( ) 22 2 w1 s g s g 1 2 1 cos U E U E U      = + − + − . (9) Based on Fig. 4 and (8), we know that U w1 ∠ 𝛾 1 , U w2 ∠ 𝛾 2 , and U w ∠  form the vector tri angle ∆ oac , with ∠ oca =90° - η . Therefore, ∠ oda is a right angle. Using the Pythagorean theorem, U w can be expressed as: ( ) ( ) ( ) ( ) w w2 2 2 2 2 s g s g w2 sin + 1 2 1 cos cos UU E U E U U        = + − + − − .( 10 ) As in dicated by ( 10 ), the gri d connection point voltage U w of the GFLR is dynamically coupled with the rotor angle δ of the SG and t he current of GFLR. W ith in the interval [0, π ], δ varies inversely with U w in a monotonic trend. That is to say, U w decreases as δ increases. Le tting U w = U in (LVRT activation threshold) and U w = U out (LVRT deactivation threshold), the corresponding δ = δ in and δ = δ out can be solved as follows: ( ) ( ) ( ) ( ) 2 2 2 22 in w s g in s 1 ar cc os 21 U I E U E       + − − − = − , ( 11 ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 out w out s g out s sin cos 1 arc cos 21 U I I E U E          − + − − − = − .( 12 ) o b   1   a w1 1 U   s E   w U   ( ) g 10 U  −   ( ) ww 90 I   +  w cos I  c d Fig. 4. Voltage phasor of the co -located system. Let δ m denote the maximum rotor angle of the SG within a given s wing cycle, as shown in Fig. 5. If δ in < δ m , t he GFLR will trigger its LVR T control during Phase I. Subseque ntly, U w rises abruptly due to the increases in i * q and decrease in i * d . Typically, U out > U in . There will be two typical patterns after the rise of U w . Pattern 1: If U out < U w , the GFLR will immediately exit LVRT mode. However, since ∆ω >0 at this moment, δ continues to increase according to (7), potentially causing U w to drop below U in again. This could lead to repeated toggling between LVRT activation and deactiva tion. Pattern 2: If U in REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < become unstable during Phase I, it enters Phase II where δ begins to decrease along the blue trajectory and U w starts to increase until U w > U out , at which point LVRT exits. In this scenario, δ in and δ out divide one swing cycle of the SG into two subsystems: the LVRT subsystem and the LVRT recovery subsystem, as shown in Fig. 7. This analysis reveals that during one oscillation cycle after fault clearance, the converter may exhibit two switching patterns depending on the LVRT threshol d settings. Pattern 1 involves multiple rapi d LVRT switches while the rotor angle remains relatively consta nt, where LVRT switc hing and voltage oscillations mutually induce each ot her, as described in [21] . If the number of sh ort-term LVRT events exceeds the maximum allowable value, GFLR may disconnect from the grid. Pattern 2 mai ntains t he LVRT state after switching until swinging back to δ out in phase II. In this scenario, the LVRT switching frequency matches the rotor angle swing cycle of the SG . This paper focuses on transient stability under continuous dynamic interaction between the GFLR and nearby SGs, namely the latter pattern. δ   ( δ s ,0) LVRT activates Phase I: δ rises, U w drops LVRT deactivates δ m U in U out U w δ δ in δ out t t LVRT activates LVRT deactivates  in  out Phase II: δ drops, U w rises Fig. 5. Diagram of the correspondence b etween SG rotor angle swing period and LVRT switching. B . Impacts of GFLR Output Variati ons on SG Rotor Dynamics The previous analysis explains a single instance of LVRT activation and deactivation during the forward and return swing of the SG ’ s rotor angle. However, whether this phenomenon will periodically recur, and thus the underlying mechanism of GFLR ’ s impact on SG stability, still requires further analysis. To this end, we will first preliminarily analyze the impact of GFLR output variations (as disturbances) on the SG's transient energy. Following this, a systematic analysis of the SG ’ s rotor angle stability throughout a complete oscillation cycle will be provided in the next secti on. Define P w* as the value of P w corresponding to normal operation, which can be considered constant during the transient period (typically a few seconds). Let  P w =P w -P w* , system (7) is then transformed into: ( ) ( ) 2 g g J w 2 d , d ,, d d f t P tT t        = =  +     , ( 13 ) ( ) e * m w s , f P PP D   − +−  =  . ( 14 )  t is known from ( 13 ) that  P w can be regarded as a disturbance to the nominal s ystem ( 15 ): ( ) 2 g g 2 J dd , d d f tT t       = =  . ( 15 ) Define δ s as m g sg s w * s ar csin E U Y PP  + = . ( 16 ) Since f ( δ s ,0) = 0 and  P w ( t , δ s ,0) = 0, ( δ s ,0) is an equilibrium point of both the nominal system ( 15 ) and the distu rbed system ( 13 ). Small disturbance analy sis shows that this point is a stable equilibrium point (SEP). The classical energy function for the nominal system ( 15 ) can be chosen as: ( ) ( ) s 2 J es w* m g 1 1 ,d 2 V T P P P          + − − =  . ( 17 ) Eq.( 17 ) has a clear physical significance: the first term represents the system's kinetic energy, while the second represents the potential energy. To investigate how the disturbance  P w affects the system's stability, the derivative of V ( δ ,   ) along the trajectory of the syst em ( 13 ) is given by: w 2 d d d d d d V V V P t D t t       = + =  −   +     . ( 18 ) From ( 18 ) , it indicates that the inherent damping always causes the system energy to dissipate due to -D   2 ≤0 . However , the damping effect from  P w is uncertain. As shown in Fig. 6, i f  P w <0 , GFLR exhibits a po sitive damping e ffect on SG 's rotor in the half-plane w here   >0 (Phase I an d IV) of the phase plane. C onversely, a negative damping effect of GFLR is present in the half-plane where   <0 (Phase II and III). I f  P w >0 , th e effect s are reversed. Th us, changes in GFLR output have varying damping effects during different phases of the SG 's angle oscillations, necessitating further analysis of their impact on angle stabilit y. Positive damping eff ect interval of  P w (if  P w < 0) o δ   Phase I Phase IV Phase II Phase III P E Negative da mping effect interval of  P w (if  P w < 0) Phase I Phase IV Phase II Phase III δ ( δ s ,0) (a)Speed-angle plane (phase plane) (b) Acceleration-angle plane Fig. 6. Positive and negative damping effects of GFLR on SG’s dynamics. IV. I NSTABILITY M ECHANISM AND S TABILITY B OUNDARY Th e p re vi ous s e ct io n su gge st s th at S G ro to r an gle s win gs mi gh t tr ig ge r re pea te d LV RT in GF LRs af te r fa ult cl ea ran ce. The an al ys is in th e f oll owi ng wil l foc us on s cen ari os wh e re SG ro t o r an gl e st ab ili ty is th e pri ma ry co nce rn ( Ap pe ndi x A d i sc us se s the ne ce ss ar y co ndi ti ons fo r t hi s phe no men on ). The in sta bi l it y me ch a ni sm of SGs c o ns ide ri ng the ir in ter ac ti on is an al yz e d, an d th e s ta bil it y ass es s me nt c ri t e ri a a re i nve st i gat ed . A. SG Stability Considering Repeate d LVRT Switching After fault clearance, the SG's ro tor angle increases. I f U w drops below the LVRT threshold U in during this phase, the GSC will re -trigger LVRT control. This transition point, labeled as E in Fig. 7, occurs when δ ≥ δ in . As the SG rotor angle retreats from its peak ( δ = δ M ), U w climbs with the de clining δ . The GSC deactivates LVRT mode at point F when U w ≥ U out and δ ≤ δ out . 5 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < These two critical points, E and F , effectively split the SG's cyclic trajectory (denoted as  ) into two segme nts:  a , and  b , representing the trajectory during LVRT and post-LVRT. Starting from point E , the transient energy increment ∆ V over one complete swing cycle ca n be expressed as: ( ) wa wb D wab ww 0 2 w g w g g 1 1 1 d d d d d d T EME FNF VV E F F E V V V V t P P P P D t               = =  +  +  +  −        , ( 19 ) where ∆ V wa and ∆ V wb represent energy increments along δ -axis symmetric segments within  a and  b respectively. If two points symmetrical about the δ -axis belong to  a and  b respectively, the energy increment al ong this path is denoted as ∆ V wab . Next, we will discuss the periodic energy changes of the SG under the two current limiti ng strategies in Section II.B. E M F 0 o E F  V wa  V wb δ in δ out δ N δ M Trajectory in LVRT subsystem N Trajectory in LVRT recovery subsystem LVRT activates  V wab A 1 A 2 δ a  b    δ s LVRT deactiv ates Fig. 7. Phase trajectories of SG with repeate d LVRT triggering. 1) SG stability under GSC's circul ar LVRT limiting During LVRT along the E → M → E' , changes in δ influence U w . This causes the GSC's output current to adjust in response to the voltage change, which then affects ∆ P w . In the scope of this study, the influence of GFLR current on U w is limited due to a strong short circuit ratio. Therefore, the magnitude of U w mainly depends on U w1 . At this point, (1) and (2) c hange to : ( ) q q w 1 0. 9 i K U =− ( 20 ) ( )   2 22 ref d ma x q w1 w1 m in , 0 .9 P i I K U U = − − ( 21 ) According to (9) , U w1 changes monotonically with δ in the range of (0,π) . Therefore, it can be approximate d that at two points symmet ric about the δ - axis on the phase plane (such as points A 1 a nd A 2 in Fig. 7), the magnitude and angle of the GSC output c urrent are nearly equal. Consequently, along the trajectory E → M → E' , ∆ P w is approximately equal for the same δ . Since the damping effect of ∆ P w reverses wh en   changes from positive to negative, the energy changes induced by ∆ P w cancel out, resulting in ∆ V wa ≈0. From point M to F (the LVRT deactivation point) in Fig. 7 , the "circular limiting" strategy gradually reduces GSC' s reactive current support to zero while smoo thly transitioning active current to normal. This elimi nat es the need for an LVRT recovery state as shown in Fig. 4(b). Th erefore, alon g the trajectory F → N → F' , the GSC activates the outer loop control mode, where i q = 0 and i d =P ref / U w . Similarly, the energy increments caused by ∆ P w along F → N and N → F' cancel each other out, so ∆ V wb ≈0. Despite different GSC control modes in E' → F and F' → E segments, the slight variat ions in U w near LVRT activati on and deactivation points lead to small change s in GSC current. This results in similar ∆ P w values in E' → F and F' → E , making ∆ V wab negligible. Thus, LVRT reactivation minimall y impacts SG's periodic energy under the "circular li miting". Nevertheless , if the reactive current coefficient K q (in (1)) is small, the drop in U w during t he first swing prompts the GSC to increase active current output up to I max to maintain steady power as much as possible. This results in a positive ∆ P w , which, combined with   >0 at this stage, introduces negative damping from the GFLR. Thus, the SG's first swing stability is exacerbated under these condi tions. 2) SG stability under GSC's rectangul ar LVRT limiting The "rectang ular limiting" strategy imposes tighter constraints on active current, leading to a lower I ds w during LVRT. Consequently, this approach yields a significantly smaller ∆ P w th an the "cir cular limit ing". The reduce d ∆ P w enhances the positive damping effects along the trajectory E → M , which is more conducive to the S G's first-swing stability. To analyze periodic energy variations, we define ∆ P wa and ∆ P wb as the Δ P w values along t rajectories  a and  b respectively. First, we analyze the magnitude of ∆ V wa . Similar to " circular limiting", f or two points symmetrical about the δ - axis, ∆ P wa ≈∆ P wb , leading to ∆ V wa ≈0 . Consequently, the magnitude of ∆ V w is primarily determined by ∆ V wab and ∆ V wb : ( ) ( ) out N out in o u t N wab wb gg ww w wa wb b b 1 dd 1 d VV V P P P P            =  −  +  +     ( 22 ) Next, we analyze ∆ V wab . Under the constraints of the current limiting, ∆ P wa on E' → F is smaller than ∆ P wb on F' → E . It is deduced f rom ( 22 ) that if δ out < δ in , then ∆ V wab >0; conversely, if δ out > δ in , then ∆ V wab <0. Therefore, ∆ V wab primarily depends on LVRT switching conditions s pecified in ( 11 ) and ( 12 ) . For the F → N → F' interval, if the slow current recovery strategy is adopted, the GSC's current gradually returns to normal levels, making Δ P w time-dependent. During recovery, Δ P w ( t ) remains negative but increases over time, behaving monotonically with respect to δ in both F → N and N → F' intervals. A cc ording to the Mean Value Theorem for Integrals, there exist constants C 1 and C 2 , with C 1 0 under the slow recovery. If ∆ V w - ∆ V D >0, cyclic energy accumulation occurs. This, in turn, exacerbates the magnitude of its rotor angle swings. A more pronounced angle swing naturally leads to mo re severe voltage oscillation, thereby triggeri ng LVRT switching in the subsequent cycle and leading to further accumulation of 6 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < transient energy. Once this vicious cycle in Fig. 8 is established, the GFLR's LVRT is periodically triggered, an d additional energy accumulates in each cycle, resu lting in multi-swing angle instability in the SG. Fig. 8. The vicious cycle driving SG multi -swing instability. In a nutshel l, if the GFLR uses "circular limiting" with a low reactive supp ort coefficient, t he SG has a higher risk of first-swing instability. Conversely, if the GFLR uses "rectangular limiting" with a slow recovery strategy, there is a risk of multi-swing instability in the nearby SG . For a multi-machine sy stem, the sending-end SGs can be coherently aggregated. The injected current I w ∠  w at the PCC in Fig. 1 can be considered as the combined output of multiple GFL converters. Thus, the conclusi ons of this analysis also appl y. B. Stability Boundaries in the Form of Critical Energy The following discussion explores t he first-swing and multi- swing stability boundaries for SGs in the form of critical energy , considering the control dynam ics of the nearby GFLR cluste r. 1) First-Swing Stability Boundary Based on Section III.B, if P w reaches its maximum value ( P w =  E s I max ), the GFLR exerts the greatest negative damping effect during the first -swing phase (Phase I) after fault clearance. This results in the conservative first-swing stability boundary. Therefore, the first-swing critical energy of the SG, denoted as V max f , is as follows: ( ) s s s s g sg m f g ma x ma x sin 1 d V P E U I EY      − −− =  ( 26 ) The SG maintains first-swing stability if the energy at fault clearance, denoted as V c , is less than V max f . 2) Multi-Swing Stability Bound ary Sudden output variations at the moment of GFLR LVRT control switching, as well as time-varying GFLR currents during the slow recovery phase, may result in periodic energy accumulation in SG. This potentially triggers multi-swing instability. The actual multi-swing stability boundary forms an unstable limit circle (  u1 ). The proof is provided in Appendix B. Since  u1 is difficult to analyze theoretically, a more conservative stabili ty boundary is neede d, which is constructed based on two main principle s: 1) Preventing GFLR from re-entering the LVRT switching state after fault clearance. 2) Maximizing transient energy accumulation during the LVRT recovery period. To prevent GFLR from re-entering the LVRT switching state after fault clearance, SG ’s rotor angle should be less than δ in . Thus, the critical energy of SG should be less than the energy in ( 26 ) corresponding to ( δ in , 0), denoted as V max m1 : ( ) e s s sg ma x m1 ma m g x sg sin 1 d V U E E I YP      −− =  ( 27 ) where in in s e s in s , ,           −  =  −  −  ( 28 ) Even if GFLR does not re-enter LVRT mode after fault clearance, its LVRT recovery control will still introduce time- varying disturbances to SG. To account for this, the energy increment during the LVRT recovery period, ∆ V wr , should be calculated along the path of maximum transient e nergy for SG. Despite the complexity of disturbances caused by GFLR after fault cl earance, ∆ P w has up per and lo wer limits. From ( 18 ), it is known that if   <0 ( phases I and IV ), taking the minimum value of  P w maximize s ene rgy accumulation; if   >0 ( phases II and III ), taking the maximum value of  P w maximizes energy accumulation. According to (5), 0< P w <  E s I max , so t he maximum (∆ P max w ) and minimum (∆ P min w ) values of  P w can be expressed as: min m ax w * w w w s ma x w * P P P P E I P  − =      = − ( 29 ) Therefore, as shown in Fig. 9 , ∆ V wr can be maximized along path  b . Let V max m = V max m1 - ∆ V wr . This represents the equivalent stability boundary in terms of energy and can be expressed as follows: ( ) M N ma x max m m1 s max g d 1 V V E I     =−  ( 30 ) where δ M and δ N can be determined by sol ving ( 31 ) and ( 32 ): ( ) M s c s max es m g d V P EI P     −− =  ( 31 ) ( ) ( ) MZ ZN es m m es dd P P P P    − = −  ( 32 ) where δ Z is the δ value at the intersection of P min E and P m . If V c is less than V max m , SGs can be considered mult i-swing stable. δ P E P m δ M δ Z b  Phase III Phase II Phase IV Phase I δ N E es PP = E es s max P P E I  =− Fig. 9. Path of maximum energy increment during the LVRT r ecovery phase on the P E - δ plane. V. C ONTROLLER D ESIGN Next, we design an additional controller using feedback linearization. This controller adjusts the GFLR's output after fault clearance to transform its negative damping effect on the rotor motion of SGs int o positive damping. 7 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < A. Controller Design Based on Fee dback Linearization After applying an additional current controller, I add , in the post-fault period , the rotor motion equatio n of the SG becomes: g* * J m E* * s add d d d d D t T P P E I t      =     + − =−  , ( 33 ) where δ , ∆  * , and P E* represent the SG's controlled rotor angle, angular velocity deviation, and electromagnetic power , respectively. According to the feedback linearization theory, I ad d can be designed as ( ) ( ) ( ) s ad d E* m 1 * 2 * d IP E P K K    = − −  − − . ( 34 ) In ( 34 ), the non li ne ar te rm P m - P E* is e li min at ed a nd si mul ta neo us ly in tr odu ci ng a lin ea r sta te f ee dba ck ter m - K 1 ∆  * – K 2 ( δ * - δ d ), whe re K 1 and K 2 fun ct ion as the tun ing pa ra mete rs. Ac cor di ngl y, the fol lo win g li nea r clo sed- lo op sy ste m ca n be ob t ai ned: ( ) J 1 J g 2 0 e K e T D T K         =       −+      −  , ( 35 ) wh er e e = δ * - δ d a nd σ = ∆  * a re th e tra ck err ors . k A J 1 s Ts K E  +  p  g + − s + + − I max i d* i d0 i d* I add + + Additional Controller Fig. 10. Block diagrams of proposed control method. B. Stability Proof and Paramet er Selection Th e Ly ap unov func tio n, s pe ci fic al ly s el ect ed fo r t his con tro ll ed sy ste m, is est ab lis he d a s foll ows : ( ) 22 J , 0. 5 0.5 V e e T  =+ . ( 36 ) Th e onl y equi lib riu m poi nt of sys tem ( 35 ) is (0, 0) . Acc ordi ng to ( 36 ) , V ( e , σ ) is pos it ive d ef in ite be cau se i t sa tis fi es V ( e , σ )≥0 an d is e qua l to ze ro onl y whe n V (0 ,0)= 0. Th e der iv at ive of V ( e , σ ) wi th re spe ct to t im e i s: ( ) ( ) ( ) 2 21 , 1 VV V e e e K e K D     =+  − = −+ . ( 37 ) Ac cor di ng to ( 37 ) , d V /d t< 0 is val id if K 2 =1 a nd K 1 >0 . Th ere fore , the system will achieve asymptotic stability at ( δ * , ∆  * ) = ( δ d ,0) under these conditi ons. In ( 34 ) , E s can be approximat ed as a constant under excitation control . α ca n be estimated based on the system 's pre-fault parameters, which is valid because the system parameters after fault clearance do not differ significantl y from those before the fault. Due to the close electrical distance between GFLR and sending-end SG, once the GSC's PLL converges quickly after fault clearance, its frequency and phase can approximate the SG 's angular velocity and ro tor angle. Specifically, ∆  * ≈  p −  g and δ * − δ d ≈ 𝜃 − 𝜃 s , where 𝜃 s is the angle of the pre-fault PLL SEP. Neglecting the inherent damping (which results in a more conservative control approach), P m -P E* can be estimated as T J d ∆  * /d t . After fault clearance, the GFLR's reacti ve current support is withdrawn, so I add is adj uste d thr ough th e GFL R' s ac tiv e cur ren t. Le t th e comm and fo r th e adj ust me nt in acti ve cu rre nt be ∆ i d* . Th us, t he fi nal con tr ol l aw is gi ven b y ( ) ( ) * J d A 1 p g s s E Ts i k k         = − + − − −      , ( 38 ) wh er e k A i s set to be gr ea ter tha n 1 to amp lify th e c on trol ef fe ct si nce I add = ∆ i d* co s( δ - 𝜃 ) ac co rdi ng to (5) . Addi ng ∆ i d* to th e no rm al cur rent r ef ere nce i d0 giv es the cu rr en t ref ere nce i d* und er th e pr op ose d c ontr ol. Th is cont rol blo ck is s how n in Fig . 1 0. Th e co ntr ol ler a ct iva te s w he n t he LVR T con tro l qu it s. C ont rol du ra tio n c an be set to a ty pica l r oto r tra nsie nt ti me, wit hin sev era l se con ds . It is important to note that under the proposed strategy, the GFLR' s active current decreas es if  p and 𝜃 increase , stabilizing the PLL . Additionally, once the control target is achieved, it indicates that the GFLR output has returned to normal. Thus, this additional controller not only prevents adverse effects on SG power angle stability but also supports PLL grid stability and timely recovery of active current after LVRT ends. However, du e to the limited adjustment of ∆ i d* , global stabilization of the system can be achieved only if the upper and lower limits of ∆ i d* are not reached during the control pe riod. VI. N UMERICAL R ESULTS A. Test System 1: Multi-Machi ne System Simulation A simplified equivalent model of one actual grid was buil t on the PSD-BPA simulation platform. Simulation parameters are provided in Appendix C. Topology of test system 1 is shown in Fig. 11. A permanent three-phase metallic short circuit occurs on one of the three ou tgoing lines at t =1s . To verify the phenomenon of SG multi-swing instability, the fault is set near the PCC on the line for 120 ms. Test conditi ons for Cases 1 - 4 are listed in TABLE I. AC Main Grid SGs(1- 4) ( 3) f GSC 1 GSC 2 GSC 3 GSC 4 GSC 5 PCC Fig. 11. Topology of the test system 1. Fig. 12 presents the simulation results, showing SG rotor angles, GFLR gr id-connected voltages, and GFLR output current. After fault clearance, U w shows an opposite trend to δ . When U w drops below the LVRT threshold U in as δ increases, the GF LRs ’ active current decrease s while the reactive current increases . Comparing Cases 1 and 2, "rectangular limiting" enhances SG first-swing stability more effectively than "circular limiting." However, in Case 3, increasing the LVRT deactivation thresh old, and in Case 4, changing the recovery strategy from immediate to ramp, both increase the risk of SG multi-swing instability compared to Case 2. This indicates that 8 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < re peated LVRT along with specific switching conditions and slow recovery strategies can worsen SG multi-swing stability, confirming the theoretical ana lysis in Section IV. A. TABLE I. C ONDITION S AND R ESULT S OF T EST S YSTEM 1 Case 1 2 3 4 Limit type circular rectangular rectangular rectangular reactive coefficient ( K q ) 2 2 2 2 Active current limit ( i ref d,lim ) GSC1:25%; GSC2:40%; GSC3:30 %; GSC4:20%; GSC5:35% LVRT threshold ( U in , U out ) 0.9,0.91 0.9,0.91 0.9,0.95 0.9,0.95 Recovery rate ( v ) ∞ ∞ ∞ GSC1-5: 2.5, 2.5,5,1.5,2A/s Result first-swing unstable stable oscillation multi-swing unstable time (s) i d,q (p.u.) active current reactive current (c) δ (° ) U w (p.u.) Case 1 Case 2 Case 3 Case 4 (a) (b) U in Fig. 12. Simulation results of Cases 1 -4. (a) rotor angle curves of S Gs; (b) grid connection point voltage of GFLRs; (c) sum of active current of GFLRs. Next, we will verify the stability assessment criteria proposed in Section IV. B. Cases 5, 6, and 7 represent three scenarios: "circular limiting" , "rectangular limiting" with immediate recovery, and "rectangular limiting" with ramp recovery , respectively. In these cases, the ranges for K q , U in , U out , i ref d,lim , and v are [1.5, 3], [0.8, 0.9], [0.85, 0.95], [20%, 90%], and [1 A/s, 5 A/s], respectively. The fault location is set at the midpoint of the transmission line. Fa ult durations t c are se t to 20, 40, 60, and 100 ms to evaluate stability criteria. For each t c , adjust K q , U in , U out , i ref d,lim , and v within the specified ranges to observe any scenarios. The resulting rotor angle curves for representative stable and unstable scenarios are show n in Fig. 13. TABLE II. shows the critical energy calculations for first- swing ( V max f ) a nd mult i-swing ( V max m ) instabil ity. Le t V d represent the transient energy of SGs at fault clearance. If V d < V max f and V d < V max m , the system maintains first and multi-swing stabili ty; otherwise, it faces the risk of instability. As shown in TABLE II. , when the criteria indicate the system is stable at t c =20ms, simulation results confirm this stability. At t c =60 and 100 ms, both the criteria and simulation predict instability. Howe ver, at t c =40 ms, the criteria suggest first-swing stability but multi- swing instabil ity, wh ile the simulation indicates stability, revealing that the criteria are conservative. (a) (b) (c) (d) time (s) Case 5 Case 6 Case 7 δ (° ) δ (° ) δ (° ) δ (° ) Fig. 13. Roto r angle curves in Cases 5 -7. (a) t cr =20ms; (b ) t cr =40ms; (c) t cr =60ms; (d) t cr =100ms. TABLE II. S TABILIT Y A SSESSMENT AND S IM ULATION R ESULTS t cr (ms) Estimated Critical Energy ( V max f , V max m ) Fault Clearance Energy ( V d ) Stability Assessme nt Simulati on result First- swing Multi- swing Case5 Case 6,7 20 1.76 0.91 0.31 0.29 stable stable 40 1.76 0. 57 1.19 1.06 unstable stable 60 1.76 <0 2.69 2.33 unstable unstable 100 1.76 <0 8.97 3.17 unstable unstable B. Test System 2: Controller Hardw are- in -the-Loop Test GFLR1 SGs Grid (3 ) f GFLR2 Y s Y w Y g (a) Topology of the test sy stem 2 RTDS Host C omp ute r Oscilloscope DSP1 GTAO GTDI Data PWM Circuit Mo del AC Sampling DSP2 (b) GHIL test system setup Fig. 14. Con figuration of the CHIL platform. To verify the theoretical analysis, a CHIL platform is constructed, as shown in Fig. 14. The main circuit shown in Fig. 14 (a) is simulated in RTDS, while the controllers of the GFLRs are implemented on a DSP-TMS320F28335 board. Data exchange between the RTDS and DSPs is facilitated through Giga-Transceiver Analog Output (GTAO) and Giga- 9 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < Transceiver Digital Input (GTDI) interface boards. At the sending end, 10 converters are divided int o two groups (GFLR1 and GFLR2), each with identical parameters. System parameters are provided in Appendix C TABLE V. At t =1.7s, a three-phase short circuit occurs near the receiving end of one of the double-circuit transmission lines. The protection relay then isolates the faulty line, followed by a reclosing action to reconnect it. The proposed control activates when LVRT deactivates and is set to last for 4 sec onds. TABLE III. C ONDITIONS AND R ESULTS O F T EST S YSTEM 2 Case 8 Case 9 Case 10 GFLR1 GFLR2 GFLR1 GFLR2 GFLR1 GFLR2 Limit type circular rectangular rectangular Steady power 60MW 47MW 60MW 47MW 60MW 47MW U in , U out 0.87, 0.95p.u 0.86, 0.9p.u. 0.87, 0.95p.u 0.9, 0.95p.u 0.87, 0.95p.u 0.86, 0.9p.u. K q 1 .5 2 .5 1 .5 2 .5 1 .5 2 .5 i ref d,lim 1 5% 1 5% v 2p.u./s 1p.u./s 2p.u./s 1p.u./s 2p.u./s 1p.u./s k 1 , k A 2, 1.5 2 , 2 2, 1.5 2 , 2 2, 1.5 2 , 2 Fault duration 100ms 200ms 150ms Reclose After 3s After 2s After 3s Cases 8 , 9, and 10 compare and verify the effectiveness of the proposed control method in suppressing the first-swing instability and multi-swing instability of the SG. The test conditions are listed in TABLE III. T he results are presented in Fig. 15, and 16 . Fig. 15 demonst rates that the SG experiences first-swing instability under the circular L VRT limiting control. From fault clearance to loss of synchronism, Fig. 15 (g) shows that the GFLRs exert a negative damping effect, leading to transient energy accumulation in the SG, denoted as E − . However, when the proposed control method is applied, GFLRs provide a positive damping effect during the first swing, resulting in energy dissipation in the SG, denoted as E + , as illustrated in Fig. 15 (h ). (a) (c) (e) (b) (d) (f) time (s) P w   (kA· rad 2 /s) (g) (h) Fault clear Out of step Fault clear E - =0.75 E + =0 E - =1.13 E + =-13.58 time (s) (rad) (rad) (p.u.) (p.u.) (kA) (kA) Fig. 15. Simulation results of Cases 8 . (a) SG rotor angle ( δ ) without control; (b) SG rotor angle ( δ ) with control; (c) GFLR grid-conn ection vo ltage ( U w ) of without control; (d) GFLR grid-connection voltag e ( U w ) with control; (e) ac tive and reactive current of GFLR without control; (f) active and reactive current of GFLR with control; (g) GFLR energy increment witho ut control; (h) GFLR energy increment with control. (a) (c) (e) (b) (d) (f) P w   (kA· r ad 2 /s) time (s) E - =70.57 E + =-35.83 One cycle Fault clear (g) (h) time (s) E - =1.41 E + =-10.19 (rad) (rad) (p.u.) (p.u.) (kA) (kA) Fig. 16. Simulation results of Cases 9 . (a) SG rotor angle ( δ ) without control; (b) SG rotor angle ( δ ) with control; (c) GFLR grid-conn ection vo ltage ( U w ) of without control; (d) GFLR grid-connection voltag e ( U w ) with control; (e) ac tive and reactive current of GFLR without control; (f) active and reactive current of GFLR with control; (g) GFLR energy increment witho ut control; (h) GFLR energy increment with control. Fig. 16 (g) and Fig. 17 (g) reveal that compared to circular limiting, rectangular LVRT limit ing reduces the transient energy of the SG during the first swing after fault clearance, thus enhancing the first-swing stability. However, it may increase the risk of oscillations or instability in later swings. In Case 9 (Fig. 16), after the fault line is reclosed, the oscillation of δ cau ses U w to os cillate inve rsely, periodi cally triggeri ng the LVRT control of the GFLRs. Fig. 16 (g) shows that in each oscillation cyc le, the negative damping effect from GFLRs leads to more energy accumulation than the positive damping dissipates, resulting in on going oscillations despite physi cal damping. In Case 10 (Fig. 17), the SG bec omes unstable during the third swing due to delayed reclosing. With the proposed control, the SG rotor angle stabilizes in both Cases 9 and 10 , demonstrating the effective ness of the proposed control. (a) (c) (e) (b) (d) (f) P w   (kA· r ad 2 /s) time (s) time (s) Fault clear Out of step E - =13.62 E + =-11.64 (g) (h) E - =0.93 E + =-7.82 (rad) (rad) (p.u.) (p.u.) (kA) (kA) Fig. 17. Simulation results of Case 10 . (a) S G rotor angle ( δ ) withou t control; (b) SG rotor angle ( δ ) with control; (c) GFLR grid-conn ection vo ltage ( U w ) of without control; (d) GFLR grid-connection voltag e ( U w ) with control; (e) ac tive and reactive current of GFLR without control; (f) activ e and reactive curr ent of GFLR with control; (g) GFLR energy increment witho ut control; (h) GFLR energy increment with control. 10 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < VII. D ISCUSSION This section aims to deepen the understanding of this multi- swing instability problem from the dual perspectives of scenarios and solutions: first, investigating whether the observed phenomenon would persist if SG were replaced by grid forming (GFM) devices, and second, discussing more alternative strategies for enhancin g the SG’s stability. A. Similar Scenarios As GFM technology matures, co -located systems of GFLRs and GFM de vices are poise d to become mainstream in the future. Given that Virtual Synchronous Generator (VSG) -type GFM converters exhibit external characteristics similar t o those of SGs , they could simil arly manifest the multi-swing instability behavior investigat ed in this paper . Nevertheless, the highly flexible and adjustable virtual damping of GFM converters offers the potential to significantly reduce their multi-swing instability risk c ompared to SGs. However, due to their limited overcurrent capability, GFM converters may remain current-limiting even after the fault is cleared[33]. This condition not only compromises their own transient stability[34] but also causes a significant reduction in grid strength. As a result, the risk of PLL transient instability for the GFLRs would be greatly increased. During such periods, the dynamic interactio ns among GFM converters, PLLs, and GFL R’s LVRT control become considerably complex . Whether the multi-swing instability needs to be considered in such scenarios deserves furthe r in-depth investigation. B. Alternative Solutions In addition to the control proposed in this paper, modifying LVRT parameters and adjusting the SG ’ s own control strategies are also alternative solutions. 1) Modifying LVRT Param eters According to the analysis in this paper, adjusting the LVRT parameter could be considered a potential solution. However, this approach faces several practical challenges. First, LVRT strategies typically prioritize device overcurrent li mits and grid code compliance instead of stability objectives. Second, simultaneously addres sing both first-swing and multi-swing stability through such adjustments is challe nging due to their contradictory requirements. For instance, while greater active current reduction or slower recovery during LVRT may aid first-swing stability, it can concurrently increase the risk of multi-swing instability. Furthermore, modifying LVRT thresholds also introduces additional compli cations. A narrow deadband between LVRT activation ( U in ) and deactivation ( U out ) thresholds can lead to undesirable LVRT toggling, dictated by the rapid inner current control loop. Conversely, a wider deadband may, as this study shows, trigger repeated LVRT events on the SG rotor dynamic timescale that are of concern. Consequently, relying on LVRT parameter adjustments is not recommended as a pr imary solution to this specific stability problem. 2) SG Control Alternatives Alternative SG control strategies include disconnecting SG if unstable or e nhanc ing its excit ation. While disconnect ing the SG appears straightforward, this approach sacrifices the SG’s contribution to gr id stre ng th a nd risks in ducing secondary large disturbances in PLLs of GFLRs. Enhanced excitation control, on the other hand, offers a more sophisticated solution. Beyond its primary role in voltage regulation, it can be strategically used to improve angle stability. Under GFLR influence, this necessitates not only improving first-swing stability but also implementing comprehensive four- quadrant regulation (Fig. 6 ). A promising direction integrates Lyapunov-based energy dissipation criteria w ith supplementary excitation, which allows for real-time act ive damping based on rotor speed and power imbalance. Howeve r, practically implementing full-period supplementary excitation requires coordination with overvoltage limits and protection schemes (e specially transformer differential protection) to avoid undesired trips resulting from increased excitation currents. V II I. C ONCLUSION On the transient stability of the SG and GFLR co-located system, this paper extends the analysis beyond the first swing stability to encompass the entire dynamic process. It reveals a periodic forced instability phenomenon in SGs induced by a “vicious c ycle” after fault clearance. Spec ifically, the SG’s rotor angle oscillations trigger voltage fluctuations, which can re -trig ger the GFLR’s LVRT activation and deactivation. While LVRT limits can enhance SG’s first -swing stability, they may promote transient energy accumulation in subsequent swings. Ultimately, more severe rot or angle swings trigger another round of this switching process, potentially leading to SG multi- swing instability. Under worst-case scenarios, the stability margin for this GFLR-induced mu lti-swing instability may become narrower than that of the first -swing stability, demanding dedicated mitigati on strategies. To address this, the paper proposes an additional GFLR cu rrent controll er based on feedback linearization theory, which effectively suppresses GFLR-induced multi-swing instability while simultaneou sly improving first-swing stability. A PPENDIX A. System Conditions for the Stu died Scenarios  i nd ic at es t he st re ngt h of th e e le ct ric al co nne ct ion b etw ee n th e SG and t he GFL R. A s SG ca pa ci ty i nc re as es or gri d co nn ec ti o n im pe da nc e dec re as es , Y s in cre as es, re su lti ng in a la rg e r  . Ne xt , we di sc us s t he ra ng e of  i n t he sc ena ri os exa mi ned i n thi s pap er . 1) Ho w  ch an ges GF LR cur re nt ' s i mp ac t on SG ro tor sp eed Ac co r di ng t o (7 ), P w in cr ea se s as  in cre as e s . Th is in di ca te s tha t GF LR o utp ut c ha ng es ha ve a m ore si gni fic an t im pa ct on th e r oto r dy na mi cs of SG w it h a hi g he r  . 2) Ho w  c ha n ge s t he im pa ct o f δ on U w Th e s en sit ivi ty of U w t o c os δ (or δ ) ca n be exp re ss ed as : ( ) ( ) ( ) ( ) ( ) ( ) sg w 22 2 s g s g 1 cos 1 2 1 cos EU U g E U E U         −  ==  + − + − ( 39 ) From ( 39 ), we see that g (  )≥ 0, with g (  )= 0 only when  =0 or  =1. Thus, g (  ) reaches a maximum within the interval 0<  < 1. By setting g' (  )=0 assuming E s ≈ U g in per unit values, 11 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < we find that g '(  )=0 at  =0.5. At this point, U w is most sensitive to changes in δ . 3) Ho w  c han ge s t he im p ac t of G FL R Cur ren t o n U w Af te r fa ult cl ear an ce , the GF LR cur re nt is is mai nl y co mp os ed of ac ti ve cu rr en t. T he s en sit iv i t y of U w t o i d c an b e e xp res se d as: ( ) ( ) ( ) ( ) ( ) 2 wd 2 2 2 2 d s g s g d 1 2 1 c os s Ui rY i E U E U i         = − =  + − + − − ( 40 ) Di ffe re nti at i ng eq ua ti on ( 40 ) wi th re spe ct to Y s y iel ds : ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 s d g s s s g g d g s w g s w 3/2 32 2 2 3 2 2 2 2 d g s s s g g d g s w g s w 1 1 1 1 2 2 1 cos 1 1 1 1 1 2 1 cos 1 Y i Y Y E E U U i Y Y Y Y Y Y i Y Y E E U U i Y Y Y Y Y Y r                 + + + − + − − +          ++               + + + + − + − − +           ++         = ( 41 ) Fr om ( 41 ), w e kn ow th at r' >0, indicating that r increases with Y s . Since r <0, as Y s (and thus  ) increases, the impact of active cu rre nt in je ct io n f rom G FL R on sys te m v olt ag e w ea ke ns . In su mma ry , a s  inc re ase s, U w be co mes le ss sen si ti ve to ch an ge s in th e act iv e c ur re nt of GF LR s, whi le t he se nsi ti vit y of SG 's rot or a ng le d yna mic s t o G FLR cu rre nt c ha ng es in cre ase s. At  ≈ 0. 5, U w is mos t se nsi ti ve to SG' s rot or an gle cha ng es. Th e re fo re, th e s tu dy fo c us es o n t he ra nge  ∈ (0.5,1), w her e i nte rac ti ons be tw ee n SG a nd GF LR ar e mo re pr ono un ced , ma ki ng S G ro to r an gl e sta bi li ty th e p ri ma r y co nc er n. B. Proof of the Multi-Swing St ability Boundary being an Unstable Limit Circle Theorem 1 (Poincare Annular Region Theorem) [32] : If D is a n annular region that does not contain any e quilibrium point, and any trajectory that intersects the boundary of D moves in the exterior- to -interior (interior- to -exterior) direction, then there exists at least one stable (unstable) limit cycle. If the inner boundary of D shrinks into an unstable (stable) po int, the theorem can still be establis hed. E 0 δ in δ u LVRT activate δ   UEP δ s SEP Annular region D u1  u2  s  Fig. 18. Multi-swing stability boundary of the system Let  u1 be the true stability boundary of the system. Due to the control dynamics of the GFLR after fault clearance, transient energy may continue to accumulate, narrowing the SG's stability boundary compared to the first-swing boundary through the UEP. As shown in Fig. 18 , the maximum rotor angle on  u1 is less than the angle δ u correspo nding to the U EP. Thus, there exists an annular region D containing  u1 , and D does not contain the system's equilibrium points (SEP and UEP). Any trajectory crossi ng these boundaries exits D , as indicated by the red arrows in Fig. 18 . According to Theorem 1, there is at least one unstable limit cycle within D . If there is only one unstable limit cycl e,  u1 serves as the system's stability boundary. If there are multiple unstable limit cycles (  u1 and  u2 ), as shown in Fig. 18, a stable limit cycle (  s ) must exist between them. If the system stabilizes on  s , it will oscillate, which is considered unstable in terms of Lyapunov asymptotic stability and practical engineering. Therefore,  u1 remains the system's stability boundary and is a n unstable limit circle. C. Test System Parameters TABLE IV. P ARAMETERS OF T EST S YST EM 1 Parameter Name Parameter Value Wind Farms Total capacity 1500MVA Outer loop proportional/integral coefficient ( k ip , k io ) GSC 1,3,4 : 0 .2 , 4 GSC 2,5 : 0 .7 , 0.14 S CR for grid connection GSC 1-5: 2.716; 2.261; 3.928; 2.173; 1.868 LVRT reactive compensation coefficient ( K q ) 2 S Gs Capacity 2520MVA Transient d ,q -axis reactance 0.28,0.39 Inertia time constant ( T J ) 7s Grid- connected branch WF connection branch(1/ Y w ) 0.18 p.u. SG connection branch( 1/ Y s ) 0.128p.u. Branch after PCC( 1/ Y g ) 0.26p.u. Receiving- end grid Voltage level 525kV Base frequency 50Hz Base capacity 1000MVA TABLE V. P ARAMETERS OF T EST S YSTEM 2 Parameter Name Parameter Value GFLRs Outer loop proportional/integral coefficient ( k po , k io ) GFLR1: 2,10 GFLR2: 2,10 Inner loop proportional/integral coefficient ( k pi , k ii ) GFLR1: 50, 200 GFLR2: 100,250 PLL proportional/integral coefficient ( k pl , k il ) GFLR1: 10, 100 GFLR2: 15, 100 Filter inductor, capacitor, resistor 8e -3 H, 1e 6 μ F, 5e -3 Ω SG Capacity, steady state output 120MVA, 95MW Transient reactance 0.15 p.u. Physical damping coefficient ( D ) 3 p.u./p.u. Branches Y s , Y w , Y g 5, 5, 1.67 p.u. Grid Voltage level 1 00kV Base frequency and capacity 50Hz, 1000MVA R EFERENCES [1] J. Matevosyan et al., “A Future With Inverter -Based Resou rces: Finding Strength From Traditional Weakness,” IEEE PES , vol. 1 9, no . 6, pp. 1 8 – 28, Nov. 2021. [2] X. Wang, M. G. Taul, H. Wu, Y. Liao, F. Bl aabjerg, and L. 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Bingfang Li (Graduate Student Member, IEEE), received the B.S. degree from North China Electric Power University, Baoding, China, in 2022, and is currently working toward the Ph.D. degree with Xi’an Jiaotong University. Her main fields of interest include Power system stability analys is and control. Songhao Yang (S enior Member, IEEE) was born in Shandong, China, in 1989. He received the B.S. and Ph.D. degrees in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2012 and 2019, respec -tively, and the Ph.D. degree in electrical and electronic engineering from Tokushima University, Tokushima, Japan, in 2019. He is currently an Associate Professor with Xi’an Jiaotong University. His research focuses on power system stability analysis and control. Guosong Wang , born i n Guizhou Province, China, in 1978, received his bachelor's degree from Wuhan University of Hydraulic and Electric Engineering in 2001. He has long been engaged in power system dispatch and operation, security and stability analysis, and security and stability control. Yiwen Hu , received the B.S. degree from North China Electric Power University, Baoding, China, in 2023, and is currently working toward the M.S. degree with Xi’an Jiaotong University. He r main fields of intere st include Power system stability analysis and control. Xu Zhang (Graduate Student Mem ber, IEEE) , received the B.S. degree from Xi’an Jiaotong University , Xi’an , China, in 2021, and is currently working toward the Ph.D. degree with Xi’an Jiaotong University. Her main fields of interest include Power system voltage stability analysis. 13 > REPLACE THIS LINE WI TH YOUR MANUSCRIPT I D NUMBER (DOUBLE-CLICK HERE TO EDI T) < Zhiguo Hao (Senior Member, IEEE), w as born in Ordos, China, in 19 76. He received the B.Sc. and Ph.D. degrees in electrical engineering from Xi’an Jiaotong Universi ty, Xi’an, China, in 1998 and 2007, respectively. He is currently a Professor with the Electrical Engineering Department, Xi’an Jiaotong University. His research focuses on power syst em protection and control. Dongxu Chang , (Member, IEEE) was born in He -nan, China, in 1982. He received the B.S. degrees in electrical engineering from HUST, Wuhan, China, in 2006.He is currently work ing at the Electric Power Research Institute ， CSG. His research focuses on power system stability analysis and contr ol. Baohui Zhang (Fellow, IEEE) was bor n in Hebei Provi nce, China, in 1953. He received the M.Eng. and Ph.D. degrees in electrical engineering from Xi ’an Jiaotong U niversity, Xi’an, China, in 1982 and 1988, respectively. Since 1992, he has been a Professor with Electrical Engineering Department, Xi’an Jiaotong University. His research interests are system analysis, control, communication, an d protection .