Synchronization Stability of Grid-Connected VSC With Limits of PLL

Limits of phase locked loop (PLL) are necessary to make sure the transient converter frequency bounded in a safety range. This leads to the grid-connected voltage source converter (VSC) with limits of PLL as one switched system. Stability of switched dynamic system is fully discussed in this paper. One conservative analytical stable region of the unbounded autonomous system is derived by Lyapunov method. It is proved that when bounds are touched, any trajectory of the studied switched system deviates from these bounds at two fixed switching Points. Moreover, the switched system energy is non-increasing when trajectories lie inside the derived analytical stable region. Based on these properties of switched system, it is proved that to ensure the globally asymptotically stability, the fixed switching Points should be placed inside the derived stable region of the unbounded autonomous system. Finally, stabilization method is proposed via placement of the fixed switching Points so that the globally asymptotically stability is ensured.


I. INTRODUCTION
V SC has become an indispensable component of the modern power system due to its fast controllability and high flexibility [1]. Keeping synchronism with the main grid is one necessary and critical condition for the normal operation of VSCs [2], [3]. Recent studies mainly focus on the synchronization stability of the grid-connected VSC without controller limits, and good results are obtained based on the stability theory of autonomous system such as phase plane method [4], equal area criterion (EAC) [5], and direct method of Lyapunov [6]. However, controller limiters are normally activated during the large disturbances, which leads to the system as one switched dynamics system. The stability analysis of such system remains one unsolved challenge. Manuscript

A. Related Works
The general way to analyze the stability of the high-order nonlinear system is via numerical calculations [7]. Time response of VSC velocity and angle are obtained and it is easy to check whether VSCs lose or maintain synchronism with the main grid. Reference [8] indicated that wind farm might trip off under the severe short-circuit fault due to PLL's incapability of tracking the grid phase. In [9], [10], [11], it was revealed that high grid impedance, improper setting of PLL gains, and heavy load conditions might induce instability.
To determine the main factors that influence the stability of VSCs, [5], [12] proposed a reduced two-order model that the fast inner current control of VSC and the network dynamics are reduced to the algebraic equations, and only dynamic equations of PLL are reserved. Phase portrait method is one effective numerical method to analyze the plane nonlinear system. The core idea is to map VSC velocity and angle information in the phase plane and the system stability can be clearly judged to see whether system trajectory is escaping from stable equilibrium point (SEP) or not. Reference [4] investigated the impacts of PI gains of PLL on the system stability in the phase plane. The stability of wind farm considering current limiting mode using this method is reported in [13]. Even though the plane system stability can be identified, system trajectories are still obtained from the numerous calculations. Therefore, it is hard to provide the analytical results as well.
EAC and Lyapunov Theorem are two mainstream approaches to obtain the analytical stable regions of the nonlinear system. Reference [5] firstly applies EAC to the stability analysis of VSC. However, EAC can only be feasible to analyze the plane conservative system with no damping term involved. The analytical conservative stable region can be derived without developing time response by Lyapunov method. For two synchronous machine system, Reference [14] has proved the results from EAC and Lyaupnov methods are identical, and the stable region is composed of the stable manifold of UEP. Proper Laypunov functions are founded in [15] and [16] based on the similar dynamic equations of grid-forming converter and synchronous machine. References [6] and [17] utilize the Lyapunov Theorem to study the stability of grid-following converter and the conservative stable region are derived. In [18], the estimation of domain of attraction for the grid-following converter is transformed into the semi-definite programming so that larger system stable region can be obtained.
The synchronization stability of the grid-connected VSC with PLL is discussed by the classical EAC, Lyapunov method, and phase portrait method. Above methods are well-effective for the autonomous system with no limits of state variables. In fact, controllers have self-limits due to the physical constraints. In addition, the limits of output of PI controller of PLL are necessary to make sure the transient converter frequency bounded in a safety range. As a result, the studied dynamic system becomes one switched system. This leads to the above methods invalid for such dynamic system.
The stability of the switched system is determined by not only each subsystem but also the switching mechanism [19]. Reference [20] pointed out that switched system is asymptotically stable under the arbitrary switching if there exists a common Lyapunov function. However, in many switched systems, restricted switching may arise from the physical constraints. Therefore, results obtained from a common Lyapunov function are rather conservative. The stability analysis with constrained switching is usually pursued by multiple Lyapunov functions [21] that are concatenated by Lyapunov function related to each single subsystem. Therefore, switched system stability only requires non-positive Lie-derivatives for some subsystems in certain regions of state space, instead of being negative globally. Another important property of the switched dynamic system is the dwell time that is defined as the time interval between any two consecutive switchings. It has been proved in [22] that system stability can be maintained when all subsystems are stable and switching is slow enough, that is, the dwell time is sufficiently large. Reference [23] proved that there exists one minimal dwell time such that the linear switched system is exponentially stable if the system average dwell time is larger than this value. Most literature focus on the stability of the linear switched dynamic system. However, how to investigate the stability of the nonlinear switched system, especially the grid-connected VSC with limits of PLL is not discussed in the previous studies

B. Main Contributions
In this paper, theory of "fixed switching Points" is established to analyze the stability of the grid-connected VSC with limits of PLL. It is interesting to find that the switched system trajectories leave away from the upper or lower bound at two fixed switching Points once they touch these bounds. Furthermore, system energy is non-increasing when trajectories lie inside the derived conservative stable region (positive damping region) of the unbounded autonomous system. More importantly, to ensure globally asymptotically stability, it is proved that two fixed switching Points should be placed inside the derived stable region. Based on these properties of switched dynamic system, analytical method of placement of fixed switching Points are proposed to realize the globally asymptotically stability.
II. SYSTEM MODELLING The studied power system consists of one VSC connected to the infinite bus through the transmission line. As shown in Fig. 1, where U sL is the rms value of the line-to-line voltage of u abc s . Generally, the inner current control dynamics of VSC are much faster than that of PLL. When studying the synchronization stability of VSC with PLL, it is regarded that the fast state variables of the inner current control marked as i d c and i q c are reduced to their steady-state value [4].
Accordingly, the network equations are as follows.
As shown in Fig. 2, the aim of PLL is to sustain u q p as zero so that the phase angle of u q p can be accurately tracked during disturbances. Non-wind up limits are adopted in this paper, and the output variable y is limited. It comes off the limit as soon as dy/dt changes sign. As a result, PLL model with the limiting action is expressed as follows.
dy dt = 0, y = y min , (y ≤ y min , dy/dt < 0) (8) where K PLL P , and K PLL I are proportional and integral gain of PLL. y min , and y max are two limits of the state variable y. When the limiter of PLL is not activated, replacing u q p of (3) with (6), it provides that, One equilibrium point of the final system after disturbance is directly obtained from (5) and (9).

B. Switched System Model
Under the large disturbance, outputs of PI controllers of PLL normally reach their limits. As a result, the studied system becomes a nonlinear switched system. The switched system is a dynamic system that consists of a finite number of subsystems and a logical rule that connects switching between these subsystems. Generally, a continuous-time switched nonlinear system can be modelled as where the state x ∈ R n , the infinite set I is an index set and stands for the collection of discrete modes (subsystems). Accordingly, by substituting (3) into (6) with the transformation of x = θ − θ s , the studied switched system described by (5)-(8) is re-written in the form of (12).
d dt d dt d dt Equilibrium point is switched to origin. h 1 (x, y), h 2 (x, y), h 3 (x, y) are three subsystems of the studied switched system.

III. STABLE REGION OF AUTONOMOUS SYSTEM
When the limiter of PI controller is not activated, the subsystem of (13) is an autonomous system and its stability is studied in this section.

A. Stable Equilibrium Point
To ensure the origin as one stable equilibrium point, smallsignal stability of the studied system at the origin should be satisfied. Linearizing the system model of (13) around origin, The characteristic function of (16) is obtained as follows, Accordingly, the condition to make sure the small signal stability of the origin is derived as follows.
Inequality of (18) relates to two conditions to ensure system small signal stability.
where u d s(0) = U sL cos θ s relates to the initial value of u d s . Combining (19) and (20), the range of the proportional parameter of PLL for ensuring system small signal stability is derived as follows.
It is founded that both large and small proportional gains of PLL may lead to the system instability. Inequality of (21) only ensures the system small signal stability. In fact, we expect the stable region of the studied system as large as possible, so that the system stability can be also ensured during the large disturbances.

B. Stable Region
Subsystem h 1 (x, y) is the classical Liénard Equation. One proper energy function for Liénard system of (13) is defined as follows, where x 1 and x 2 are the angle of UEPs of Subsystem h 1 (x, y). The defined energy function is positive-definite over the specified interval indicated by (23). Based on the Lyapunov theorem, the derivation of the defined energy function with respect to time should be kept as nonpositive.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
It implies that, (21) and (25) that there is one trade-off of proportional gain of PLL in the perspective of system stability. Large gain of PLL may enlarge the stable region of the studied system, but might also damage the system small signal stability.
Based on the conditions of (23) and (25), the analytical stable region of subsystem h 1 (x, y) by Lyapunov method is provided by (25). The derived stable region has clear physical meaning. It requires the system damping coefficient f (x) ≥ 0 under large disturbances, so as to dE/dt ≤ 0 always held.
Notably, it is not necessary to do the numerical simulations to determine the system stability if system trajectory lies inside the stable region determined by (25). The range of values for E(x, y) are from 0 at origin to a maximum value at x = x 4 , y = 0 marked as E max , that is, If the value of E(x, y) at the instant of the last switching action is less than E max calculated by (26), the system is stable with the relative degree of stability.

IV. SYSTEM STABILITY ANALYSIS WITH LIMITS OF PLL
The limits of output of PI controller as shown in Fig. 2 are normally activated to constraint the transient converter frequency bounded in a safety range. Accordingly, the studied system becomes one switched system. The stability of such system is determined by the three subsystems of (13), (14), and (15) noted as h 1 , h 2 and h 3 , and their switching signals.

A. Fixed Switching Points
are the switching Points of any two consecutive subsystem. K is an index set and stands for the switching times of each subsystem. As for the studied switched system, there are totally two switching processes. Fig. 3(a) shows the first switching process that begins from h 1 to h 2 , and returns back to h 1 , which is noted as h 1 → h 2 → h 1 . The second switching process that starts from h 1 to h 3 , and switches back to h 1 (marked as h 1 → h 3 → h 1 ), as indicated by Fig. 3(b).
As for the first switching process, suppose that the system trajectory touches the lower bound at j − th time, and the system is switched from h 1 to h 2 at Point (x(t 2,j ), y(t 2,j )). Based on (14), on the switching Point, it yields that y(t 2,j ) = y min , dy/dt < 0,  2 (27) As for the subsystem h 2 , state variable x moves along the Line x = x(t 2,j ) + y min t until to the switching Point (x(t 1,k ), y(t 1,k )), where the system is switched back to h 1 from h 2 . Clearly, the switching point is on the both subsystem h 1 and h 2 . Based on (13) and (14), on the switching Point, it provides that y(t 1,k ) = y min , dy/dt = 0. As a result, the following Equality holds. 1 (28) As for the second switching process as shown in Fig. 3(b), suppose that the system trajectory touches the upper bound at p − th time, and the system is switched from h 1 to h 3 at Point (x(t 3,p ), y(t 3,p )). Based on (15), on the switching point, it yields that y(t 3,p ) = y max , dy/dt > 0, and the following Inequality holds. 3 (29) Accordingly, the system moves along the Line x = x(t 3,p ) + y max t and switches back to h 1 from h 3 at Point (x(t 1,q ), y(t 1,q )). Similarly, the switching point is on both subsystem h 1 and h 3 . Based on (13) and (15), on the switching Point, it provides that y(t 1,q ) = y max , dy/dt = 0. As a result, the following Equality holds. 1 (30) It should be noted that when the bounds of the system (y min , and y max ) are specified, the roots of the following Equality noted as x l and x u are certain.
Based on (28), (30), and (31), it implies that, Points of l (x l , y min ), and u (x u , y max ) are fixed, and they are irrespective of the switching time, which are defined as the fixed switching Points of the studied plane switched system. It is indicated from (32) that if the system touches the lower or upper bound at any points, system trajectory will depart from bounds and return back to the subsystem h 1 at the fixed switching Points.

B. Properties of Fixed Switching Points
Obviously, any switched system trajectory is bounded in the region of [y min , y max ], and the fixed switching Points are the highest Points of system trajectories. Proof: Define differentiable continuously functions noted as T 1 (x), and T 2 (x) as follows.
Due to T 1 (x) as differentiable continuously function, there exists at least one Point that makes T 1 (x) = 0 hold. Moreover, these Points locate inside the region of (0, x 4 ]. Similarly, based on −f (0)y max < 0, and f (x 3 ) = 0, it implies that, It is also proved that there exists at least one Point that makes T 2 (x) = 0 hold. Moreover, these Points lie inside the region of [x 3 , 0).
In fact, f (x) and g(x) are periodical functions, and the property can be extended to the fixed switching Points of other periods. Thus, fixed switching Points of period k are inside the region of [x 3 + 2kπ, 2kπ), and (2kπ, 2kπ + x 4 ], k ∈ Z.
Property 2: The fixed switching Points are unique inside the region of [x 3 , 0), and (0, Proof: One differentiable continuously function as z : R → R is defined as If z(x) ∈ (−∞, ∞) is monotonously decreasing inside the region of [x 3 , 0), and (0, x 4 ], the following Equality of has unique roots (fixed switching Points) inside the region of [x 3 , 0), and (0, x 4 ]. To ensure the monotonous property of the defined function z(x) over above range, the following Inequality should be hold.
(36) The numerator of above fraction is divided by the term of C 2 U sL , and it provides that, Based on the relation of sin θ s = C/U sL and cos θ II = C 1 /C 2 , Inequality of (37) converts to, The maximum value of the function d(x) can be obtained by setting ∂d(x)/∂x = 0, and it provides that, Based on (38) and (39), the maximum of function d(x) is Small signal stability condition of f (0) > 0 should be satisfied, and it provides that, Combining (40) and (41), it yields as, d(x) max = sin 2 θ s + cos 2 θ II < sin 2 θ s + cos 2 θ s = 1 Based on Property 1 and 2, it is indicated that the studied plane switched system has two unique fixed switching Points l (x l , y min ), and u (x u , y max ), which locate inside the region of (0, x 4 ], and [x 3 , 0) separately. Similarly, the property can be extended to the fixed switching Points of other periods. Therefore, fixed switching Points of period k are unique inside the region of [x 3 + 2kπ, 2kπ), and (2kπ, 2kπ + x 4 ], k ∈ Z.
As discussed previously, the fixed switching Points are the roots of equations of As for two fixed switching Points of first period, they are unique inside the region of (0, x 4 ], and [x 3 , 0). As a results, the two roots (x l and x u ) can be calculated by using Newton's method with the initial value of x 3 and x 4 . Accordingly, the fixed switching Points of the period k are (x u + 2kπ, y max ) and (x l + 2kπ, y min ).

C. Energy Change for Subsystem 2 and 3
Lemma: When the trajectories of subsystems h 2 and h 3 lie inside the region of [x 3 , x 4 ](f (x) ≥ 0), dE(x, y)/dt ≤ 0 holds.
Proof: For the trajectory of subsystem h 2 , the derivation of E(x, y) with respect to time becomes as with the dwelling condition (the condition that makes system trajectory remain at the specified subsystem) of Based on (43) and (44), it implies that, It shows that if f (x) ≥ 0, dE/dt ≤ 0 holds for the trajectory of the subsystem h 2 . Similarly, the derivation of E(x, y) with respect to time for the trajectory of subsystem h 3 becomes, with the dwelling condition of As a result, It is proved that under the condition f (x) ≥ 0, the energy of the trajectory of subsystem h 2 and subsystem h 3 is nonincreasing.

D. Stabilizing Condition for Switched Dynamic System
Theorem: When the energy of two fixed switching Points l (x l , y min ), and u (x u , y max ) satisfy E u,l ≤ E max = G(x 4 ), the studied switched system is stable.
Proof: Without loss of generality, suppose that the system trajectory touches the lower bound and leaves the lower bound at the fixed switching point of (x(t 1,k ), y(t 1,k )) = (x l , y min ). Two situations are considered. If no switching action triggers after t 1,k , the autonomous subsystem h 1 is kept stable under the condition of E l ≤ E max , which is clearly indicated by the solid blue line in Fig. 4.
Corollary: The plane switched system is globally asymptotically stable only if the fixed switching Points of the system are inside the derived conservative stable region of h 1 , that is, Phenomenon that the grid-connected VSC is stabilized on the SEP of other periods is defined as "periodical stability", which is new finding for the power system embedded by VSCs.
This phenomenon does not happen for traditional SG due to no limits for the rotor speed of SG. Instability occurs once the rotor angle of SG cannot return back after first swing. However, the periodical stability always happens for the grid-connected VSC, since the stability is entirely determined by controls. Once the limits of controls are activated, and limits of controls are properly settled based on the above Theorem, the globally stability is ensured, and the grid-connected VSC can be stabilized on SEP of other periods. In other words, out of steps will not happen for the grid-connected VSC irrespective of fault clearing time only if the limits of controls are properly settled based on the proposed Theorem, which is the prominent advantage of our work.

E. Stabilization by Placing Fixed Switching Points
Based on Corollary, to make sure the studied switched system globally asymptotically stable, the fixed switching Points should be placed so that E u,l ≤ E max = G(x 4 ) holds.
Accordingly, the range of limits of y is derived based on the following steps to ensure globally stabilization.
Step 1: Solve the equation of to obtain two roots of x a and x b , x a < 0 < x b .
Step 2: y a and y b are calculated based on Step 3: The limits of y are set based on Then, the switched system is globally asymptotically stable.

V. NUMERICAL RESULTS
The case of one VSC connected to the infinite bus through one transmission line as shown in Fig. 1 is utilized to analyze the large disturbance stability of VSC with limits of PLL. Base capacity of the studied system is 100 MVA, and the nominal system voltage is 110 kV. The equivalent line resistance and inductance are 0.05 pu and 0.5 pu. PI parameters of PLL are set as 0.045 and 20. The delivered power from VSC is 80 MW, and the d-axis current reference i dref c is 0.7664 kA. Accordingly, θ s is 0.4351 rad based on (11). The other related parameters for the case study can be referred to Table I. The disturbance is the infinite source U sL suddenly drops to 0.3 pu at time t = 0.01 second, and returns to U sL after t c seconds.   Fig. 5(a). When the fault duration time of t c reaches 28 ms, the system trajectory does not exceed the dashed grey line (x = 0.498), and the system stability is ensured. In addition, the system trajectory under t c = 50 ms marked as blue solid line converges to origin as shown in Fig. 5(a), which indicates the conservation of the derived stable region by Laypunov method. When the fault duration time increases to 69 ms, the system trajectory marked as red solid line presents oscillatory instability, and finally converts to the classical aperiodic instability when the system trajectory crosses over the angle of UEP (x = 2.271).
Exact stable boundary of such dynamic system can be obtained from the backward (reverse-time) starting from UEP as shown in Fig. 5(b). The resulting trajectory finally converges to an limit cycle, which indicates the exact system boundary is one unstable limit cycle. The derived conservative stable region in solid black line is composed by the equal energy surface of Point E (0.498, 0) as shown in Fig. 5(b), and it is apparently inside the exact stable region (unstable limit cycle) of studied system, which verifies the conservation of the Lyapunov method. Fig. 5(c) shows the energy change of the system trajectories with respect to time under the different fault duration time. Based on (18), the system critical energy is the potential energy of E (0.498, 0), which is calculated as 223.9, as shown in dash gray line in Fig. 5(c). When the fault is cleared, system trajectory reaches to Point A, and its energy is calculated as 209.1, and it is less than the system critical energy. This makes sure the energy decreasing property of the trajectory under the fault duration time of 28 ms, and the system stability is ensured. When the fault duration time reaches to 69 ms, Energy of Point C at the instant of fault clearance is increased to 1799.4, which is much larger than the critical energy of the studied system, and finally the oscillatory instability happens. Although the stability can be maintained under the fault duration time of 50 ms, the system trajectory energy marked in blue solid line cannot be kept as monotonically decreasing with respect to time. This again proves the conservation of the stability judgement by Lyapunov method that only if the energy at the instant of the fault clearance is less than the critical system energy, the system energy is   Table II shows the placement of the fixed switching Points inside the stable region of the subsystem h 1 by the proposed method. The black solid line presents the equal energy surface of Point E (0.498, 0), which is the derived stable boundary by Lyapunov method. To ensure system stability, the limits of y are set as ±15.7, which are inside the limits of the stable boundary indicated by Table II. As shown in Fig. 6(a), under t c = 70 ms and t c = 80 ms, the trajectories in dash blue and red lines without the limits of PLL cross over the angle of UEP, and the aperiodic instability appears. As for the studied switched system with limits of PLL, system trajectories start to move at Point A and C under t c = 70 ms and t c = 80 ms, respectively. Then, the system trajectories in solid blue and red lines touches the lower bound at Point B and D respectively. Accordingly, both trajectories move along Lines BL and DL, then finally deviate from the bound at the same Point L (fixed switching Point) and returns back to the subsystem h 1 . It is proved that system trajectories depart from bounds and return back to the subsystem h 1 at the fixed switching Point. Since Point L is inside the derived stable boundary, and no switching event triggers when system trajectory leaves away from Point L, the system trajectory will converge to the origin. It can be verified that the switched system can be locally asymptotically stable if the fixed switching Points are inside the derived conservative stable region of h 1 . Fig. 6(b) shows the energy change of the system trajectory with respect to time under the different fault duration time. Point A and B are the starting (initial) Points of the system trajectories under t c = 70 ms (blue solid line) and t c = 80 ms (red solid line), separately. Both system trajectories touch the lower bound at the Point C and D, and the system is switched to the subsystem h 2 from h 1 . Accordingly, both system trajectories stay subsystem h 2 and begin to move along the Line CF and DF until to the Point F where the system trajectories are inside the region of f (x) ≥ 0. It can be seen from Fig. 6(b) that in the region of f (x) ≥ 0, the energy of system trajectory (Line F L) with respect to time is decreasing. Then, the system switches back to the subsystem h 1 at fixed switching Point L. As indicated by Fig. 6(b), the energy of Point L (154.3) is less than the critical energy (223.9) marked as dashed black line, which implies the fixed switching Point is inside the derived conservative stable region of h 1 . As a result, the system trajectory energy is monotonically decreasing with respect to time and the system stability is ensured. Fig. 7 shows the impacts of limits of PLL on the stability of the studied switched system under the fault duration time of 80 ms. The minimal value of limit of PLL (y min ) is set as −15.7, and the maximum values of limit of PLL (y max ) are set as 12.6 (2 Hz) and 9.42 (1.5 hZ), separately. As shown in Fig. 7(a), system trajectory in blue solid line under y max = 9.42 and in red solid line under y max = 12.6 starts to move at Point A and B when the fault is cleared. Both trajectories touch the lower bound at Point C and D, and deviate from the lower bound at the fixed switching point L. Accordingly, both system trajectories again touch the upper bound at Point G and H, respectively, and the system switches to the subsystem h 3 from h 1 . Finally, both system trajectories switch back to the subsystem h 1 at the fixed switching Points U 1 and U 2 . It can be seen from Fig. 7(a) that no switching event is triggered when system trajectories leaves away from U 1 and U 2 , and the system stability is ensured since the fixed switching Points are inside the stable region of subsystem h 1 noted as solid black line in Fig. 7(a). It is again proved that once the fixed switching Points lie inside the derived stable region of h 1 , locally asymptotically stability can be guaranteed. Fig. 7(b) shows the energy change of the system trajectory in blue line under under y max = 9.42 and in red solid line under y max = 12.6. Both system trajectories will touch the lower bound and switch to the subsystem h 2 . Accordingly, both system trajectories reach to Point F where system trajectories begin to enter the region of f (x) ≥ 0, and the system energy starts to decrease from Point F until to the fixed switching Point L. Due to Point L lying inside the derived stable region of subsystem h 1 , and the energy of trajectories of LG and LH with respect to time is decreasing as indicated by Fig. 7(b). Both system trajectories touch the upper bounds at Point G and H. Due to Lines of GU 1 , and HU 2 inside the region of f (x) ≥ 0, the energy of trajectories of GU 1 and HU 2 with respect to time is decreasing as clearly shown in Fig. 7(b). No switching event happens after both system trajectories leaves away from the upper fixed switching Points. In addition, the energy of Points U 1 and U 2 are less than the system critical energy, which indicates the studied switched system can be locally asymptotically stable. Fig. 8(a) shows the system dynamic response when the larger fault duration time is imposed. The limits of PLL are set as ±15.7 (±2.5 Hz), and the system fixed switching Point of the next period noted as Point U is (6.26, 15.7). Apparently, it is inside the derived stable boundary of subsystem h 1 in next period, which is marked as the black solid line in Fig. 8(a). Both trajectories under t c = 132 ms and t c = 140 ms in solid blue and red lines cross over the angle of UEP (x = 2.271), then they touch the upper bound at Point B, and D, separately. Accordingly, both trajectories depart from the upper bound at the fixed switching Point U , and finally converge to the SEP of the next period with the limits of PLL as depicted in Fig. 8(a). It can be concluded that the globally asymptotically stability is guaranteed only if the fixed switching Point are placed inside the derived conservative stable region of the subsystem h 1 . Fig. 8(b) indicates the energy change of system trajectories in blue and red solid lines under t c = 132 ms and t c = 140 ms. Both system trajectories cross over the angle of UEP and touch the upper bound of the system at Point B and D, respectively. Accordingly, the system energy is sharply decreased until to the system fixed switching Point U . It should be noted that the fixed switching Point U lies inside the derived conservative stable region of subsystem h 1 , and the energy of both system trajectories is monotonically decreasing when they depart from the system fixed switching Point U as shown in Fig. 8(b). As a result, the globally asymptotically stability is guaranteed when the fixed switching Point U lies inside the derived stable region of subsystem h 1 . It should be noted that due to the limits of the angular speed of VSC, speed of VSC is constrained. When the system trajectory under disturbances crosses over the angle of UEP, the system trajectory leaves away the limits from the fixed switching Points of next period. As a result, the out of step will not happen, and the system trajectory will finally converge to the SEP of the next period, and the globally stability of the studied system is ensured. Fig. 9 shows the system dynamic response under different PI control parameters of the outer power control loop, noted as K o P , and K o I . The dash black and blue lines in Fig. 9 represents system trajectories under t c = 80 ms with the fast (K o P = 0.004, K o I = 0.02) and the slow K o P = 0.0004, K o I = 0.002 outer power control loops. With slow power control loop, d-axis ac current nearly keeps constant during transient, which verifies the correctness of the assumption that ac current can be regarded as constant for the system transient stability analysis. In addition, the system trajectory in blue indicates the system stability is ensured under the large disturbances. With the fast power control loops, d-axis ac current shown in Fig. 9(a) touches the limits (1.2 p.u.) during disturbances, and then gradually decays after faults. It can be clearly seen from Fig. 9(b) that the system stability can be maintained with fast outer power control loops. It is well-verified that the proposed method can be applied for the grid-connected VSC with the faster outer power control loop.   the line parameter change of L l = 0.425 p.u., and L l = 0.575 p.u., respectively. Both system trajectories with line parameter changes are stable under the large disturbance with the proposed limits setting method, which verifies the robustness of the proposed method under line parameter changes. It should be pointed out the system stability becomes worsen when the actual line inductance is larger than the estimated one. This can be explained by the fact that system stability is severely deteriorated when the electrical distance becomes longer.   Fig. 11(a), the system trajectory with t c = 70 ms touches the lower bounds of PLL, then returns back to the subsystem h 1 at the fixed switching Point L. It is indicated that the system stability is ensured even the fixed switching Point L lies outside the derived stable boundary in grey, which well verifies the conservativeness of the proposed limits setting method. In addition, when the fault duration time increases to 80 ms, the system trajectory as shown in Fig. 11(b) will cross over the UEP of the first period, and leaves the upper bounds of PLL at the fixed switching Point U . Similarly, the fixed switching Point U lies outside the stable boundary in grey while the system globally stability is maintained. As a result, the conservativeness of the proposed method is again verified.

VI. CONCLUSION
The stability of the grid-connected VSC with limits of PLL is investigated. The results of case studies reveal that when the studied plane switched system trajectories touch the bounds, they will leave these bounds at two fixed switching Points. In addition, the energy of switched system is non-increasing when the system trajectories lie inside the derived conservative stable region of the unbounded autonomous system. Furthermore, it is interesting to find that once the fixed switching Points are placed inside the derived stable region of the unbounded autonomous system, trajectories of the studied switched system will asymptotically converge to SEP globally. Accordingly, one globally stabilization method is proposed by simply placement of the fixed switching Points, which is well-beneficial for the stable operation of future power system with high-penetrations of renewables.