Transient Synchronizing Stability Analysis of VSG Considering Voltage/Current Source Mode Switch Induced by Current Saturation Limit

1. Introduction

With the accelerated transformation of energy structure globally, the proportion of renewable energy generation with voltage source converters (VSCs) as grid interfaces is rapidly increasing in modern power system [1]. Currently, grid-following (GFL) control is commonly employed in VSCs, which track terminal voltage through phase-locked loop to synchronize with AC grid [2]. Yet, the high penetration of converter interfaced generation results in the grid presenting low inertia and weak damping. Grid-forming (GFM) converters such as virtual synchronous generators (VSGs) are considered as a viable solution, generating synchronization phase based on power control and actively participating in grid regulation [3]. VSG imitates the dynamic characteristics of synchronous generator (SG) and may suffer from similar synchronizing stability issues under disturbances, which has attracted increasing attention in industry and academia.

In the last decade, the small-signal stability of GFM converters has been extensively studied and many scholars have made outstanding contributions [4,5,6,7]. In [4], a linearized model is developed to assess the power oscillation risk of VSG. Based on eigenvalue analysis, increasing damping coefficient and grid impedance is more beneficial to improve the small-signal stability. In [5], the dq-frame impedance model of VSG is established. Small-signal stability is explored from source-load interactions in the low-frequency range. In [6] and [7], an analogized Heffron-Phillips model of VSG-based GFM converter is developed and complex torque method is used to reveal the synchronizing oscillation mode damping. It indicates that control loops may introduce negative damping torque deteriorating the VSG’s small-signal stability. Extensive literatures carry out such studies, which give a deeper comprehension on the small-signal stability of GFM converters. Yet, the analysis method based on linearized model is concerned only with the synchronizing dynamics around steady-state operating point.

In general, disturbances, large or small, occur in the power grid all the time. GFM converters are at risk of losing synchronizing with AC grid under large disturbance conditions [8,9,10,11]. In [8], a comparative study of GFM converters based on different control schemes is carried out. It reveals that VSG-based GFM converter may undergo transient instability due to insufficient damping of inertial response, even if the equilibrium points exist after fault. In [9,10], the Lyapunov-based stability criterion is employed to evaluate the VSG’s transient stability. The attraction domain is closely related to grid strength and control parameters. In [11], the energy function model of GFM converter is developed. The stability boundary under different grid conditions is investigated from the perspective of energy conversion. Yet, GFM converters are essentially power semiconductor devices with poor overcurrent capability unlike SGs. Consequently, current limiting methods are incorporated into the control system to prevent overcurrent damage, which may deeply change the VSG’s transient synchronizing behaviors [12]. This vital factor is ignored in the stability analysis of above studies.

Currently, common current limiting strategies are classified into two types: current reference limiting and virtual impedance. Virtual impedance is an indirect strategy to achieve current limiting by reducing voltage reference [13]. It may not be effective in suppressing current peaks if the parameters are not properly preset [14]. In contrast, the current reference limiting strategy has better steady-state performance [12] and is easy for implementation, which is employed and focused on in this paper. Recently, some scholars have conducted studies on the transient stability of GFM converters with current reference limiting [15,16,17,18,19,20,21,22,23]. It has been reported in [15] that the synchronizing dynamics of GFM converters with current limiting is deeply changed due to current saturation under grid fault conditions. In [16,17,18], based on droop-controlled GFM converter, it focuses on the existence of post-fault equilibrium points and then reveals the negative impact of current limiting on transient stability. Yet, for VSG-based GFM converter, sufficient deceleration area after fault clearing is also a necessary condition for maintaining transient stability. In [21], the transient stability assessment of VSG with current limiting is experimentally carried out. Complementarily, the transient stability of VSG is quantitatively studied but the damping effect is neglected [22]. In [23], the transient instability mechanism of VSG with current limiting during grid fault is investigated, but the synchronizing dynamics after fault clearing is not considered. Thus, the transient stability analysis of GFM converters with current limiting is significant but inadequate, which urgently needs to be further explored.

In this paper, the transient stability of VSG-based GFM converter is focused on, with the effect of transient voltage/current source switching mode (TVCSM) induced by current limiting considered especially. Compared with existing works, the main contributions and novelties of this paper are in three aspects:

(1)

An equivalent motion model with switching logic is developed to depict the VSG’s transient synchronizing behaviors, where TVCSM reshapes power angle curve and adjusts equivalent driving force.

(2)

Based on the acceleration/deceleration area method, the impact mechanism of TVCSM on the VSG’s transient stability is explored, which is manifested as increasing acceleration area and decreasing deceleration area.

(3)

A novel analysis idea combining phase portrait method and inverse trajectory method is proposed to quantitatively assess the VSG’s transient stability with damping considered, and the main influencing factors are investigated.

The rest of this paper is organized as follows. In Section 2, a synchronizing dynamic model of VSG considering TVCSM is developed. Based on the developed mode, the impact mechanism of TVCSM on the transient stability is clearly revealed based on the acceleration/deceleration area method in Section 3. In Section 4, the transient stability quantitative assessment is carried out. A novel analysis idea combining phase portrait method and inverse trajectory method is presented to accurately estimate critical clearing angle (CCA) and critical clearing time (CCT). In Section 5, the influencing factors are investigated based on acceleration/deceleration area method and CCA/CCT quantitative assessment. Finally, Section 6 concludes this paper.

2. Transient Switching Dynamic Model of VSG

2.1. Equivalent Motion Model of VSG

The typical topology of VSG connected to AC grid is shown in Figure 1. A multilayer cascade control structure is employed by VSG to mimic the dynamics of synchronous generator (SG). The outer control loops including active power control based on virtual rotor motion and alternating voltage control imitate SG’s inertia/damping response and voltage regulation, which usually have low bandwidth. The outer control loops generate dq-axis component order and phase information of command voltage for inner control loops. As actuator, the inner control loops including dq-axis current control and dq-axis voltage control are used to improve dynamics and protect against overvoltage or overcurrent during generating the command voltage vector, which usually have high bandwidth. To prevent overcurrent in case of grid fault, current saturation limit is applied between the dq-axis voltage control and dq-axis current control. When the command current generated by the dq-axis voltage control is within the limit range, current saturation limit does not take effect. Yet, when the command current exceeds the limit range, the real current references of dq-axis current control are fixed by the current saturation limit and do not respond to the dynamical regulation of outer control loops. VSG operates in current source mode in this situation.

Virtual rotor motion that imitates SG’s rotor motion characteristics for grid synchronization and inertia emulation is the core feature of VSG, which dominates VSG’s synchronizing dynamics. In the p.u. system, the angular frequency ω_vsg and phase θ_vsg of VSG are expressed as

$${\omega }_{\mathrm{vsg}}=\left[P_{\mathrm{ref}}-P-D\left({\omega }_{\mathrm{vsg}}-{\omega }_0\right)\right]/(Js)+{\omega }_0$$

(1)

$${\theta }_{\mathrm{vsg}}={\omega }_{\mathrm{b}}{\omega }_{\mathrm{vsg}}/s$$

(2)

P_ref is regulated by primary frequency regulation in slow timescale, which can be assumed to be constant in the virtual rotor motion analysis.

The power angle δ of VSG is generally defined as the difference between the converter synchronizing phase θ_vsg and grid voltage phase θ_g, and is given as

$$\delta ={\theta }_{\mathrm{vsg}}-{\theta }_{\mathrm{g}}={\omega }_{\mathrm{b}}\left({\omega }_{\mathrm{vsg}}-{\omega }_0\right)/s={\omega }_{\mathrm{b}}{\omega }_{\mathrm{p}}/s$$

(3)

where ω_p denotes the angular frequency deviation between VSG and AC grid.

VSG’s multilayer cascade control loops present multi-timescale dynamic characteristics. For the concerned synchronizing dynamic timescale that is dominated by VSG’s virtual rotor motion, the fast response control loops including alternating voltage control, dq-axis voltage control and dq-axis current control can be simplified to a quasi steady state model by neglecting controllers’ dynamics. The slow response control loops that adjusts P_ref and U_t^ref can be assumed to no enough time to respond, indicating that P_ref and U_t^ref can be assumed to be constant. Then VSG can be simplified to a controlled voltage source vector as shown in the upper part in Figure 2 when the current saturation limit does not take effect. Under disturbance, the amplitude of the controlled voltage source vector keeps constant at U_t^ref. The phase of the controlled voltage source vector rotates with θ_vsg.

However, unlike SG, VSG is the power semiconductor device with poor overcurrent capability. Current limiting strategy is usually employed to prevent overcurrent in case of grid fault. In this paper, a current saturation limit strategy in 24 is employed and defined as

$$\left\{\begin{array}{l}{I}_d^{*}={\displaystyle \frac{I_d^{ref}}{\left|I_d^{ref}\right|}}\times \min \left(I_{\max },\left|I_d^{ref}\right|\right)\hfill \\ I_q^{*}={\displaystyle \frac{I_q^{ref}}{\left|I_q^{ref}\right|}}\times \min \left(\sqrt{（I_{\max }{）}^2-（\left|I_d^{*}\right|{）}^2},\left|I_q^{ref}\right|\right)\hfill \end{array}\right.$$

(4)

where I_dq^ref are the current references generated by the inner dq-axis voltage control. I_dq^* are the real current references rectified by the current saturation limit. I_max is the maximum allowable current determined by the current capacity of VSG.

From (4), the current limiting strategy does not work when the amplitude of I_dq^ref is smaller than I_max. Then the output voltage of VSG is driven to follow the command voltage by the voltage control loops and dq-axis current control. VSG operates in controlled voltage source mode in this situation. Yet in case of grid voltage sag fault, the generated current references I_dq^ref may be larger than I_max for the output voltage of VSG to follow the command voltage. The current saturation limit takes effect, indicating that the voltage control loops lose effectiveness and the real current references I_dq^* are up to the current saturation limit in (4). VSG operates in controlled current source mode in this situation. The amplitude of the controlled current source vector keeps constant at I_max. The phase of the controlled current source vector rotates with θ_vsg in the same way. As a result, current saturation limit brings voltage/current source switching mode (TVCSM) for VSG as shown in Figure 2.

In voltage source mode, the active power P delivered by VSG can be expressed as

$$P=\mathrm{Re}({\mathit{U}}_{\mathrm{g}}\overline{\mathit{I}})={\displaystyle \frac{U_{\mathrm{t}}{}^{ref}U_{\mathrm{g}}}{X_{\mathrm{g}}}}\sin \delta$$

(5)

where U_g is the equivalent grid voltage vector; I is the output current vector of VSG; $\overline{\mathit{I}}$ is the conjugate of I. Note that only grid inductance is considered here, since in power transmission scenarios there is usually X_g >> R_g.

The voltage and current vector diagram in voltage source mode is shown in Figure 3. According to the above analysis, VSG operating in voltage source mode needs to satisfy certain vector constraints, which is shown as

$$\left\{\begin{array}{l}\left|{\mathit{U}}_{\mathrm{t}}\right|=U_t^{ref}\hfill \\ \left|\mathit{I}\right|=\left|{\displaystyle \frac{{\mathit{U}}_{\mathrm{t}}-{\mathit{U}}_{\mathrm{g}}}{j{X}_{\mathrm{g}}}}\right|\le I_{\max }\hfill \end{array}\right.$$

(6)

where U_t is the terminal voltage vector.

In voltage source mode, the output current magnitude |I| increases with the increase of power angle δ. When δ reaches the critical value δ_s, the output current of VSG increases to the maximum allowable current I_max, as shown in Figure 3. If the voltage and current vectors satisfy the triangle rule, δ_s can be calculated by the constraint equation. If a severe grid voltage dip makes the triangle rule unsatisfied, the magnitude of δ_s is 0. Thus, δ_s can be obtained by

$${\delta }_{\mathrm{s}}=\left\{\begin{array}{cc}\arccos \left({\displaystyle \frac{{\left(U_t^{ref}\right)}^2+U_{\mathrm{g}}^2-I_{\max }^2{X}_{\mathrm{g}}^2}{2U_t{}^{ref}U_{\mathrm{g}}}}\right)& \left|U_t{}^{ref}-U_{\mathrm{g}}\right|\le X_{\mathrm{g}}{I}_{\max }\\ 0& \left|U_t{}^{ref}-U_{\mathrm{g}}\right|>X_{\mathrm{g}}{I}_{\max }\end{array}\right.$$

(7)

Then the constraint in (6) can be identified by comparing δ and δ_s. When δ ≤ δ_s, the constraint I ≤Imax is satisfied and the VSG operates in voltage source mode.

If δ > δ_s, the output current exceeds the maximum allowable current and thus triggers the current limiting. In this case, the VSG can be considered as a controlled current source, as shown in Figure 2. In current source mode, the phase dynamics of current vector is still dominated by rotor swing equation. Differently, the output current amplitude I_max is directly determined by (4). In synchronizing dynamics, the output active power in current source mode is given as

$$P={\mathit{U}}_{\mathrm{g}}·{\mathit{I}}_{\max }=U_{\mathrm{g}}{I}_{\max }\cos \delta$$

(8)

where I_max is the maximum output current vector; • means the dot product operation of vectors.

According to (1), (3), (5) and (8), the synchronizing dynamic differential equation of VSG can be described as

$${\displaystyle \frac{J}{{\omega }_{\mathrm{b}}}}{\displaystyle \frac{d^2\delta }{d{t}^2}}+{\displaystyle \frac{D}{{\omega }_{\mathrm{b}}}}{\displaystyle \frac{d\delta }{dt}}=P_{\mathrm{ref}}-P$$

(9)

where

$$P=\left\{\begin{array}{cc}{\displaystyle \frac{U_{\mathrm{t}}{}^{ref}U_{\mathrm{g}}}{X_{\mathrm{g}}}}\sin \delta & \delta \le {\delta }_{\mathrm{s}}\\ U_{\mathrm{g}}{I}_{\max }\cos \delta & \delta >{\delta }_{\mathrm{s}}\end{array}\right.$$

(10)

The equivalent motion model of VSG considering TVCSM can be developed as shown in Figure 4. The critical value δ_s is referred to as the TVCSM angle. Obviously, this is a synchronizing motion model with switching logic. With normal operating conditions, the VSG is equivalent to a controlled voltage source and its active power is determined by (5). When δ increases above the threshold δ_s, the VSG is considered as a controlled current source and its active power is given by (8). In the transient process, VSG presents the power characteristics switching between voltage source mode and current source mode due to current limiting. Thus, TVCSM reshapes the P-δ curve of VSG, which deeply affects the VSG’s synchronizing dynamics.

2.2. P-δ Curve Reshaping Analysis

Generally, the power angle characteristics of VSG ignoring TVCSM are consistent with those of SG and can be described by (5). The P-δ curve is determined by the sine function of voltage source mode only, as shown by the blue dotted curve in Figure 5. For VSG considering TVCSM, if δ > δ_s, the VSG will switch to current source mode as in (8). Quite differently, the P-δ curve is constructed jointly by the sine function of voltage source mode and the cosine function of current source mode, which is described by (10), as shown by the black solid curve in Figure 6. Similar to SG, the VSG’s synchronizing dynamics is driven by unbalanced torque. The intersections of equivalent mechanical torque P_ref and equivalent electromagnetic torque P are stable equilibrium point (SEP) and unstable equilibrium point (UEP), whose angles are δ_se and δ_ue, respectively. After the grid fault is cleared, the system will regain transient stability if the operating point can return to SEP. Conversely, if the operating point crosses UEP, the system will be transiently unstable. Based on Figure 6, TVCSM reshapes the P-δ curve of VSG and thus modifies equivalent electromagnetic torque, which inevitably affects the transient stability, as discussed in the next section.

3. Transient Synchronizing Stability Analysis

Transient stability is closely related to the acceleration and deceleration area, which is determined by the unbalanced torques P_ref and P. Based on the previous analysis, the equivalent mechanical torque remains constant in both operating modes. Yet, TVCSM reshapes the P-δ curve of VSG and thus modifies equivalent electromagnetic torque. Driven by different unbalanced torques, the synchronizing dynamics of VSG considering TVCSM differs significantly from that of VSG ignoring TVCSM. In this section, the influencing mechanism of TVCSM on the transient stability is analyzed based on the acceleration/deceleration area method.

Considering a large grid voltage dip, the acceleration areas of VSG considering and ignoring TVCSM are shown in Figure 7(a) and (b), respectively. Assuming that initial operating angle δ₀ and fault clearing angle δ_fc are the same for both cases, the difference in acceleration area is mainly related to the unbalanced torque. Ignoring TVCSM, the equivalent electromagnetic torque during fault is determined by the sine function of voltage source mode, as shown in Figure 7(a). The kinetic energy E_ki accumulated during fault is equivalent to acceleration area S_a1, which can be calculated as

$$E_{\mathrm{ki}}={\displaystyle {\int }_{{\delta }_0}^{{\delta }_{\mathrm{fc}}}\left(P_{\mathrm{ref}}-\frac{{U_{\mathrm{t}}}^{ref}{U}_{\mathrm{g}}^{\mathrm{res}}}{X_{\mathrm{g}}}\sin \delta \right)}d\delta$$

(11)

where V_g^ref is the residual voltage of the AC grid after a fault.

Considering a severe voltage dip, i.e., satisfying $\left|{U_t}^{ref}-U_{\mathrm{g}}^{\mathrm{res}}\right|>X_{\mathrm{g}}{I}_{\max }$, TVCSM angle δ_s during grid fault is 0 according to (7). More specifically, VSG will switch to current source mode directly after a fault occurs due to δ₀ > δ_s = 0.

Thus, when TVCSM is considered, the equivalent electromagnetic torque during fault is determined by the cosine function of current source mode only, as shown in Figure 7(b). The kinetic energy E_kc accumulated during fault is equal to the sum of acceleration area S_a2 and S_a3, which can be calculated as

$$E_{\mathrm{kc}}={\displaystyle {\int }_{{\delta }_0}^{{\delta }_{\mathrm{fc}}}\left(P_{\mathrm{ref}}-U_{\mathrm{g}}^{\mathrm{res}}{I}_{\max }\cos \delta \right)}d\delta$$

(12)

It is seen that acceleration area S_a2 is approximately equal to acceleration area S_a1. Yet, S_a3 is the additional acceleration area introduced by TVCSM. Thus, TVCSM changes the unbalanced torque, resulting in the VSG with a larger acceleration area. That is, the VSG considering the effect of TVCSM will accumulate more kinetic energy during grid fault, which is not conducive to maintaining transient stability.

The comparison of feasible deceleration area is shown in Figure 6 (c) and (d). For VSG ignoring TVCSM, the equivalent electromagnetic torque after fault clearing is still determined by the sine function of voltage source mode. Quite differently, for VSG considering TVCSM, the equivalent electromagnetic torque after fault clearing is determined by both the sine function of voltage source mode and the cosine function of current source mode. When δ is smaller than δ_s, the equivalent electromagnetic torque depends on the sine function. And if δ > δ_s, the equivalent electromagnetic torque is given by the cosine function. It is seen that the sine function of voltage source mode is significantly higher than the cosine function of current source mode, which results in a significant reduction of feasible deceleration area, i.e., S_d2 << S_d1. The reduced deceleration area due to TVCSM is S_d3, as shown by the gray region in Figure 6 (d).

Based on the above analysis, the transient stability comparison of VSG considering and ignoring TVCSM with the same fault clearing angle δ_fc are shown in Figure 6 (e) and (f). Since TVCSM reshapes the P-δ curve, the acceleration and deceleration areas of VSG change significantly with different unbalanced torques. Ignoring TVCSM, there is still a large deceleration area after the fault is cleared, and thus the system is less prone to transient instability. According to Figure 6 (e), the maximum convertible potential energy E_pi is equal to deceleration area S_di and is given as

$$E_{\mathrm{pi}}={\displaystyle {\int }_{{\delta }_{\mathrm{fc}}}^{{\delta }_{\mathrm{ue}}}\left(\frac{{U_{\mathrm{t}}}^{ref}{U}_{\mathrm{g}}}{X_{\mathrm{g}}}\sin \delta -P_{\mathrm{ref}}\right)}d\delta$$

(13)

Considering TVCSM, the additional acceleration area is introduced resulting in more kinetic energy accumulated during fault, i.e., S_ac1 > S_ai. Moreover, the feasible deceleration area is significantly reduced due to the fact that the cosine function of current source mode is smaller than the sine function of voltage source mode after fault clearing. From Figure 6 (f), the UEP angle δ_ue of VSG considering TVCSM is less than fault clearing angle δ_fc. This means that the VSG does not move into the deceleration area instead the next acceleration area S_ac2 after fault clearing. Even if the SEP exists in the system after fault clearing, the VSG is doomed to transient instability due to the absence of deceleration area. As a result, TVCSM changes the unbalanced torque of VSG, which increases the acceleration area and decreases the deceleration area, worsening the transient stability of the system.

The transient response of angular frequency ω_vsg in both cases is shown in Figure 7. A severe grid voltage dip fault is set at 1.0s with residual voltage of 0.2p.u.. The grid fault lasts for 200ms and then is cleared. It is seen that the VSG ignoring TVCSM can regain synchronizing stability after fault clearing due to its larger deceleration area. Based on the previous analysis, TVCSM introduces additional acceleration area and reduces deceleration area due to P-δ curve reshaping which significantly changes equivalent electromagnetic torque. Thus, the VSG considering TVCSM undergoes transient instability for the same fault duration as shown in Figure 7.

4. Quantitative Assessment of Transient Stability

In this section, the transient stability of VSG considering TVCSM is quantitatively assessed furtherly based on two core indexes, CCA and CCT. On the one hand, ignoring the effect of damping, CCA and CCT are calculated analytically based on equal area rule. The transient stability of VSG considering and ignoring TVCSM is quantitatively compared. On the other hand, a novel analysis idea combining phase portrait method and inverse trajectory method is presented to approximately estimate the transient stability of VSG with the damping considered. In the phase plane, the intersection of forward acceleration state trajectory and backward deceleration state trajectory identifies the CCA.

Neglecting the damping effect, the VSG system is energy conservative. In the transient process, kinetic energy E_k and equivalent potential energy E_p are converted to each other, and the total energy remains constant. According to equal area rule and Figure 8, the CCA of VSG considering TVCSM satisfies

$$\begin{array}{l}{\displaystyle {\int }_{{\delta }_0}^{{\delta }_{\mathrm{cc}}}\left(P_{\mathrm{ref}}-U_{\mathrm{g}}^{\mathrm{res}}{I}_{\max }\cos \delta \right)}d\delta ={\displaystyle {\int }_{{\delta }_{\mathrm{cc}}}^{{\delta }_{\mathrm{s}}}\left(\frac{{U_{\mathrm{t}}}^{ref}{U}_{\mathrm{g}}}{X_{\mathrm{g}}}\sin \delta -P_{\mathrm{ref}}\right)}d\delta \\ +{\displaystyle {\int }_{{\delta }_{\mathrm{s}}}^{{\delta }_{\mathrm{ue}}}\left(U_{\mathrm{g}}{I}_{\max }\cos \delta -P_{\mathrm{ref}}\right)}d\delta \end{array}$$

(14)

Among them, δ₀, δ_s and δ_ue are calculated separately as

$${\delta }_0=\arcsin {\displaystyle \frac{P_{\mathrm{ref}}{X}_{\mathrm{g}}}{U_{\mathrm{t}}{}^{ref}U_{\mathrm{g}}}}$$

(15)

$${\delta }_{\mathrm{s}}=\arccos \left({\displaystyle \frac{{\left(U_{\mathrm{t}}^{ref}\right)}^2+U_{\mathrm{g}}^2-I_{\max }^2{X}_{\mathrm{g}}^2}{2U_{\mathrm{t}}{}^{ref}U_{\mathrm{g}}}}\right)$$

(16)

$${\delta }_{\mathrm{ue}}=\arccos {\displaystyle \frac{P_{\mathrm{ref}}}{U_{\mathrm{g}}{I}_{\max }}}$$

(17)

Then, combing (14)-(17), the analytical expression of CCA for VSG considering TVCSM is obtained as

$${\delta }_{\mathrm{cc}}=\arccos {\displaystyle \frac{C}{\sqrt{A^2+B^2}}}+\phi$$

(18)

where

$$A={U_{\mathrm{t}}}^{ref}{U}_{\mathrm{g}}/X_{\mathrm{g}}$$

(19)

$$B=U_{\mathrm{g}}^{\mathrm{res}}{I}_{\max }$$

(20)

$$\begin{array}{l}C=P_{\mathrm{ref}}{X}_{\mathrm{g}}{U}_{\mathrm{g}}^{\mathrm{res}}{I}_{\max }/\left({U_{\mathrm{t}}}^{ref}{U}_{\mathrm{g}}\right)+\left({\left({U_{\mathrm{t}}}^{ref}\right)}^2+U_{\mathrm{g}}^2-I_{\max }^2{X}_{\mathrm{g}}^2\right)/\left(2X_{\mathrm{g}}\right)\\ +U_{\mathrm{g}}{I}_{\max }\left(\sin {\delta }_{\mathrm{s}}-\sin {\delta }_{\mathrm{ee}}\right)+P_{\mathrm{ref}}\left({\delta }_{\mathrm{ee}}-{\delta }_0\right)\end{array}$$

(21)

$$\phi =\arccos {\displaystyle \frac{A}{\sqrt{A^2+B^2}}}$$

(22)

Further, the numerical method 26 is employed to calculate CCT. Similarly, the transient stability of VSG ignoring TVCSM can be quantitatively assessed based on CCA and CCT. With different power reference P_ref, the comparisons of the transient stability for VSG considering and ignoring TVCSM are shown in Table 1.

The results indicate that both CCA and CCT of VSG considering TVCSM are much smaller than that of VSG ignoring TVCSM. That is, TVCSM greatly reduces the stability domain and thus deteriorates the transient stability of VSG. The quantitative study well validates the conclusions of mechanism analysis.

Quantitative Evaluation of CCA and CCT with Damping Considered

With the damping considered, there is not only energy transformation between kinetic energy E_k and equivalent potential energy E_p but also energy dissipated by damping in the transient process. Thus, the results of CCA and CCT calculations neglecting damping are conservative due to the neglect of energy dissipation. In order to more accurately quantify the transient stability of VSG considering TVCSM, the analysis idea combining phase portrait method and inverse trajectory method is employed.

First, the whole grid fault process is divided into two stages: fault duration and fault recovery. The state trajectories inside the two stages are discussed separately. In fault duration stage, the synchronizing dynamic differential equation with the damping considered is given as (23) based on the developed model in Figure 4.

$$\left\{\begin{array}{l}{\displaystyle \frac{d{\omega }_{\mathrm{p}}}{dt}}=\left(P_{\mathrm{ref}}-U_{\mathrm{g}}^{\mathrm{res}}{I}_{\max }\cos \delta -D{\omega }_{\mathrm{p}}\right)/J\hfill \\ {\displaystyle \frac{d\delta }{dt}}={\omega }_{\mathrm{b}}{\omega }_{\mathrm{p}}\hfill \end{array}\right.$$

(23)

By numerical calculation, the time forward integration of (23) from δ₀ yields the forward acceleration state trajectory in fault duration stage, as shown in Figure 9.

Then, the synchronizing dynamic differential equation with the damping considered in fault recovery stage is represented as

$$\left\{\begin{array}{l}{\displaystyle \frac{d{\omega }_{\mathrm{p}}}{dt}}=\left(P_{\mathrm{ref}}-P_{\mathrm{rec}}-D{\omega }_{\mathrm{p}}\right)/J\hfill \\ {\displaystyle \frac{d\delta }{dt}}={\omega }_{\mathrm{b}}{\omega }_{\mathrm{p}}\hfill \end{array}\right.$$

(24)

where

$$P_{\mathrm{rec}}=\left\{\begin{array}{cc}{\displaystyle \frac{U_{\mathrm{t}}{}^{ref}U_{\mathrm{g}}}{X_{\mathrm{g}}}}\sin \delta & \delta \le {\delta }_{\mathrm{s}}\\ U_{\mathrm{g}}{I}_{\max }\cos \delta & {\delta }_{\mathrm{s}}<\delta \le {\delta }_{\mathrm{ee}}\end{array}\right.$$

(25)

The backward deceleration state trajectory can be easily obtained based on the inverse trajectory method [27]. The time backward integration of (24) is equivalent to the time forward integration of the following system.

$$\left\{\begin{array}{l}{\displaystyle \frac{d{\omega }_{\mathrm{p}}}{dt}}=-\left(P_{\mathrm{ref}}-P_{\mathrm{rec}}-D{\omega }_{\mathrm{p}}\right)/J\hfill \\ {\displaystyle \frac{d\delta }{dt}}=-{\omega }_{\mathrm{b}}{\omega }_{\mathrm{p}}\hfill \end{array}\right.$$

(26)

Nonlinear systems (24) and (26) have the same state trajectory in the phase plane but with opposite trajectory direction. (26) is the inverse trajectory system of (24). The backward deceleration state trajectory in fault recovery stage obtained by numerical integration of (26) from δ_ue is shown in Figure 9. The x-axis is power angle δ. And the y-axis is angular frequency deviation ω_p, which reflects the kinetic energy E_k = (Jω²_p)/2 of the state point. The intersection of forward acceleration state trajectory and backward deceleration state trajectory implies that the difference between accumulated kinetic energy and damped dissipation energy in fault duration stage is equal to the sum of consumed kinetic energy and damped dissipation energy in fault recovery stage. Thus, the power angle δ_cc corresponding to the intersection of state trajectories is CCA. Based on the CCA obtained from phase portrait, the CCT can be similarly calculated by numerical method. According to the parameters given in the Appendix, the CCA and CCT with the damping considered are calculated as 0.5rad and 90ms, as shown in Figure 9. Compared with the results 0.39rad and 61ms in Table I, it is seen that the transient stability assessment neglecting the effect of damping is more conservative and the damping term significantly increases the transient stability margin of VSG.

5. Simulation Validations

In this section, the effects of power reference, maximum allowable current and grid impedance on the transient stability of VSG considering TVCSM are investigated based on acceleration/deceleration area method and CCA/CCT quantitative assessment. Besides, detailed simulations are carried out in MATLAB/Simulink to verify the correctness of theoretical analysis.

Figure 10 shows the influence of power reference P_ref on the acceleration and deceleration area of VSG considering TVCSM. Obviously, P_ref directly determines the equivalent mechanical torque. Thus, the acceleration area decreases and the maximum deceleration area increases with the decreasing of P_ref.

Figure 11 shows the transient stability assessment results with different power reference P_ref. From (15) and (17), both initial power angle δ₀ and UEP angle δ_ue are related with P_ref. With the decreasing of P_ref, δ₀ decreases while δ_ue increases. Since P_ref determines the magnitude of equivalent mechanical torque, decreasing P_ref slows down the acceleration of forward state trajectory and speeds up the deceleration of backward state trajectory. Thus, both CCA and CCT increase as P_ref decreases, and the transient stability is enhanced. This means that the improved control strategy of enhancing transient stability by reducing power reference during fault is still applicable to the scenario where TVCSM is considered.

The simulation results are given in Figure 12, where the main parameters are kept constant as in Appendix, except for the studied influencing factor. A severe grid voltage dip fault is set at 6s with the residual voltage of 0.2p.u. It is seen that VSG with P_ref of 1.0p.u. can regain synchronizing stability if the fault is 0.09s, while transient instability occurs if FCT is 0.1s. Thus, the CCT of EMT model is between 0.09s and 0.1s, which is very close to the CCT of 90ms as the quantitative estimation in Figure 11. Similarly, the CCT of EMT model with P_ref of 0.8p.u. is between 0.17s and 0.18s, and the CCT with P_ref of 0.6p.u. is between 0.3s and 0.31s, both of which are close to the results of theoretical analysis in Figure 11. The simulation results verify the correctness of transient stability quantitative assessment.

5.1. Influence of Maximum Allowable Current

Figure 13 shows the influence of maximum allowable current I_max on the acceleration and deceleration area of VSG considering TVCSM. The equivalent mechanical torque is only depended on P_ref, which does not change varying with I_max.

According to (10), the sine function in voltage source mode is independent of I_max while the cosine function in current source mode is closely related to I_max. Thus, the equivalent electromagnetic torque tends to rise as I_max increases in the fault

duration and recovery stages. The acceleration area decreases and the maximum deceleration area increases with the increasing of I_max.

Figure 14 shows the transient stability assessment results with different maximum allowable current I_max. (15)-(17) indicates that I_max does not affect δ₀ but significantly changes δ_ue and δ_s. Based on the previous analysis, it is known that the equivalent electromagnetic torque increases with increasing I_max while the equivalent mechanical torque remains constant. More specifically, a larger I_max will correspond to a smaller unbalanced torque difference in the fault duration and recovery stages. Thus, increasing I_max will slow down the acceleration of forward state trajectory and speed up the deceleration of backward state trajectory. The CCA is obtained from the intersection of state trajectories and then the CCT is estimated as in Figure 14. It can be seen that as I_max increases, both CCA and CCT increase and the transient stability is improved. This illustrates that the VSG with strong overcurrent capability has better transient stability and is easier to maintain synchronizing after fault clearing.

The simulation results are given in Figure 15, with the same fault settings as in Figure 12. It is seen that the CCTs of EMT model are 0.05s-0.06s, 0.15s-0.16s and 0.21s-0.22s when I_max is 1.2p.u., 2.0p.u. and 3.0p.u., respectively. The CCTs based on quantitative assessment are 46ms, 138ms and 215ms, respectively. The theoretical analysis and simulation results are in good agreement.

5.2. Influence of Grid Impedance

Figure 16 shows the influence of grid impedance X_g on the acceleration and deceleration area of VSG considering TVCSM. Similarly, the equivalent mechanical torque is not influenced by X_g. Contrary to I_max, the sine function in voltage source mode is related to X_g, while the cosine function in current source mode is independent of X_g. Thus, in fault duration stage, the equivalent electromagnetic torque keeps unchanged with the increasing of X_g. The equivalent electromagnetic torque tends to drop as X_g increases in fault recovery stage. Yet, initial power angle δ₀ is strongly influenced by X_g. The acceleration area variation needs to be discussed in the specific scenario. The maximum deceleration area decreases with the increasing of X_g.

Figure 17 shows the transient stability assessment results with different grid impedance X_g. Based on (15)-(17), it is known that δ₀ and δ_s are closely related to X_g while δ_ue does not depend on the magnitude of X_g. When X_g is 0.2p.u., 0.3p.u. and 0.5p.u., the quantitatively calculated CCA and CCT are 0.48rad and 97ms, 0.52rad and 83ms, 0.64rad and 59ms, respectively. It can be seen that the CCA of VSG considering TVCSM increases but the CCT decreases as X_g increases. The reason for this difference is that increasing X_g reduces the maximum deceleration area, but it also significantly increases initial power angle δ₀. Thus, although the variation Δδ decreases in transient process, the magnitude of δ_cc = δ₀ + Δδ still increases due to the larger δ₀. Since X_g has little effect on the unbalanced torque in fault duration stage, CCT depends mainly on the increment Δδ, which decreases with increasing X_g due to the decreasing of equivalent electromagnetic torque in fault recovery stage. As a result, as X_g increases, CCA increases and CCT decreases instead. It is easy to misjudge the effect of X_g on the transient stability of VSG considering TVCSM based on the CCA index alone. In practice, the CCT index is commonly used to guide relay protection design and should be taken into account even more.

The simulation results are given in Figure 18. It is seen that the CCTs of EMT model are 0.1s-0.11s, 0.09s-0.1s, and 0.06s-0.07s when X_g is 0.2p.u., 0.3p.u. and 0.5p.u., respectively. The simulation results are close to the quantitative assessment in Figure 17, thus verifying the validity of theoretical analysis.

6. Conclusions

In this paper, the transient stability of VSG-based GFM converter is studied, with the effect of TVCSM induced by current limiting considered especially.

First, the equivalent motion model is developed to clearly depict the VSG’s synchronizing behaviors, where TVCSM reshapes the P-δ curve and thus adjusts equivalent driving force in transient process. Further, the impact mechanism of TVCSM on the transient stability of VSG is clearly revealed based on the acceleration/deceleration area method. A novel analysis idea combining phase portrait method and inverse trajectory method is proposed to quantitatively assess the transient stability margin. Finally, the influences of power reference, maximum allowable current and grid impedance on transient stability is investigated. The valuable conclusions are as follows:

1) Based on the developed equivalent motion model, it is found that TVCSM introduces switching dynamics to VSG and significantly reshapes the equivalent power angle curve of virtual rotor motion.

2) TVCSM modifies the equivalent electromagnetic torque, resulting in a larger acceleration area and a smaller deceleration area. Both CCA and CCT of VSG are significantly reduced with TVCSM considered.

3) Decreasing power reference and increasing maximum allowable current will increase both CCA and CCT of VSG considering TVCSM. The effect of grid impedance on CCA and CCT is inconsistent, which is manifested as increasing CCA and decreasing CCT.

The works in this paper not only clearly reveal the impact mechanism of TVCSM on VSG’s transient stability, but also give a reference for improving control and relay protection design.