Transient Stability Enhancement Method for VSGs Based on Power Angle Deviation for Reactive Power Control Loop Modification

Abstract

Virtual synchronous generators (VSGs) simulate the operating characteristics of conventional synchronous generators to provide inertia, voltage and frequency support for new-type power systems dominated by power electronics. However, in the event of grid faults, VSGs inevitably experience transient angle instability, which leads to great challenges to the safe and stable operation of the power system. To address the problem of transient instability so that VSGs can continue to support the power system during a grid fault, this paper firstly analyzes the adverse effect of a reactive power control (RPC) loop on the transient stability of the system and proposes a method for adding the variation in the power angle into RPC to increase the voltage reference of a VSG during grid faults, which can solve the transient instability problem under both equilibrium point existence and nonexistence by increasing the active power output of the VSG. The effect of the additional coefficient on the transient characteristics of the system is then analyzed using a small-signal model, and it is found that this method also enhances the frequency stability of the system. Finally, the feasibility of the proposed method and the correctness of the theoretical analysis are confirmed by a simulation platform.

1. Introduction

With the increasing prominence of the fossil energy crisis, global warming and other problems, renewable energy has been rapidly developed [1]. New-energy sources such as photovoltaics and wind power are predominantly integrated into the power grid via power electronic devices [2]. This shift has progressively transformed the operational characteristics of power systems, leading to a system architecture characterized by high penetration of both renewable energy and power electronic devices. Consequently, the proportion of conventional synchronous generators in the power system continues to decline [3,4]. Although power electronic devices have the advantage of flexible control, power electronic systems lack sources of rotational inertia support such as traditional synchronous generators, which seriously threatens the safe and stable operation of power systems [5,6]. A virtual synchronous generator (VSG) provides inertia, voltage and frequency support for the power system by simulating the operating characteristics and excitation characteristics of a synchronous generator [7] and can maintain good operation under a weak grid; they have, therefore, become a hotspot in current research on new-energy grid-connected power generation technology [8,9].

Despite the many advantages of VSGs, they show various kinds of stability issues under different disturbances [10,11]. Existing studies on the stability of VSGs mainly focus on the analysis of the stability of small disturbances under normal conditions [12,13,14]; moreover, small-signal stability analysis usually linearizes near the equilibrium point (EP) to obtain a small-signal model. Similar to synchronous generators, VSGs also have problems such as transient instability when subjected to the action of large disturbances (e.g., grid voltage dips). Transient stability refers to the ability of the VSG to maintain synchronization with the grid under large disturbances, which can also be described as transient power angle stability or synchronization stability. Once VSG transient instability occurs, the VSG will lose its supporting function in the grid if no corresponding measures are taken, which will seriously damage the normal operation of the power system. Therefore, the problem of the transient stability of VSGs under large disturbances deserves to be investigated.

To enhance the transient stability of VSGs during grid faults, the following studies have been performed: In ref. [15], the rate of change in frequency (RoCoF) is fed back to the reactive power control (RPC) loop, which improves the transient stability of the system by increasing the active power output from the VSGs. The control parameters modification method can also be applied for improvement in large-disturbance stability [16,17]. In Ref. [16], a transient damping method is proposed. It introduces additional transient damping into the active power control (APC) loop of a VSG along with frequency deviation during grid faults. Through this method, it is possible to switch between normal and fault states to enhance transient stability. In ref. [17], adaptive control of inertia and damping is utilized. This method adjusts inertia and damping to their maximum values during a grid fault. It slows down the rate of change in the power angle and varies the reference value of the equivalent active power. The above methods only ensure that VSGs can regain stable operation in the presence of the EP, but they do not take the transient instability of VSGs into account when the EP does not exist.

Several studies have focused on the transient stability of the system when the EP does not exist after a severe fault in the grid. A method to reduce the active power reference value according to the degree of grid voltage dip during a grid fault, thereby reducing the active power imbalance, is proposed in ref. [18]. In ref. [19], the reactive power reference is adjusted to counteract the effects of RPC, thus enhancing the transient stability of the system. However, it is a challenge to quantify changes in reference values [20], and these two methods do not guarantee that the EP will always be restored. To enhance the transient stability of the system, ref. [21] proposes a controller that can adaptively reduce the power reference value of a VSG. Ref. [22] introduces a mode-adaptive power angle control method that automatically switches the gain of the APC loop after a grid fault, thereby preventing the positive feedback operation of the APC loop and eliminating the risk of transient instability. Nevertheless, it does not restore the EP under severe grid faults, and the power angle fluctuates considerably around the steady-state value when the faults are not cleared.

Therefore, in order to ensure the stable operation of the power system during grid faults, there is a need for a methodology that not only allows the VSG to maintain transient stability in the presence of the EP but also restores the EP and reaches a new stable state in the absence of the EP. The contributions of this paper are summed up as follows:

(1)

The adverse effects of RPC on VSG transient stability are analyzed, and the transient instability mechanism of a VSG under grid voltage dips is investigated.

(2)

A method to improve transient stability during voltage dips in the grid is proposed. It not only restores the transient stability of a VSG under slight voltage dips but also restores the EP of the system under severe voltage dips and maintains stable operation.

The rest of the paper is organized as follows: Section 2 describes the general topology and power control strategy of VSG grid-connected systems. Two types of power angle destabilization mechanisms and the effect of RPC on VSG transient stability are analyzed. Section 3 proposes a method to improve the transient stability of VSGs. Then the effect of the additional coefficient on the VSG system is analyzed, and the critical coefficients are calculated by an iterative method for different voltage dip levels. Section 4 verifies the effectiveness of the proposed methods through experiments. Section 5 concludes this paper.

2. VSG Transient Stability Analysis

2.1. Grid-Connected System and Control Strategy

The topology of a three-phase grid-connected VSG is shown in Figure 1. U_dc is the DC-side voltage, and the LC filter is used to filter out high harmonics. L_f, C_f and R_f are the filtering inductor, the filtering capacitor and the parasitic resistor of the filtering inductor, respectively. Z_g = R_g + jX_g denotes the line impedance, and u_gk (k = a, b, c) is the three-phase grid voltage. u_tk and i_k are the three-phase VSG output voltage and current, respectively. e_k and e_refk represent the three-phase bridge arm voltage and the three-phase modulation signal output by the voltage and current controller, respectively. The control strategy of a VSG consists of APC and RPC [23], and the control block diagram is shown in the dashed frame in Figure 1.

APC achieves the control of the system frequency by simulating the second-order swing equation of the synchronous generator rotor; the control equation can be expressed as

$$\mathit{J}{\displaystyle \frac{\mathit{d}\mathit{\omega }}{\mathit{dt}}}={\displaystyle \frac{{\mathit{P}}_{\mathit{ref}}-{\mathit{P}}_{\mathit{e}}}{{\mathit{\omega }}_0}}-{\mathit{K}}_{\mathit{D}}\left(\mathit{\omega }-{\mathit{\omega }}_0\right)$$

(1)

where J is the virtual inertia coefficient and the introduction of J enables the VSG to simulate the inertia characteristic of a physical synchronous generator. K_D is the damping coefficient, P_ref and P_e are the active power reference value and the active power output from the VSG, ω and ω₀ are the virtual rotor angular velocity and the rated angular velocity, respectively.

RPC adopts droop control, and its control equation can be expressed as

$${\mathit{U}}_{\mathit{ref}}{=\mathit{U}}_0{+\mathit{K}}_{\mathit{q}}\left({\mathit{Q}}_{\mathit{ref}}-{\mathit{Q}}_{\mathit{e}}\right)$$

(2)

where U_ref is the output voltage amplitude of the VSG; U₀ is the rated voltage magnitude; Q_ref and Q_e are the reactive power reference and the reactive power output from the VSG, respectively; and K_q is the droop coefficient.

2.2. Analysis of Transient Instability Mechanism

A grid-connected system using the VSG control strategy can be equated to a controlled voltage source connected to the point of common coupling (PCC) [24,25]. Generally, the transmission line resistance is ignored, and the transmission line is regarded as a purely inductive line; then the active and reactive power outputs from the VSG can be expressed as

$${\mathit{P}}_{\mathit{e}}={\displaystyle \frac{3{\mathit{U}}_{\mathit{ref}}{\mathit{U}}_{\mathit{g}}}{2{\mathit{X}}_{\mathit{g}}}}\sin \delta$$

(3)

$${\mathit{Q}}_{\mathit{e}}={\displaystyle \frac{3{\mathit{U}}_{\mathit{ref}}\left({\mathit{U}}_{\mathit{ref}}-{\mathit{U}}_{\mathit{g}}\cos \delta \right)}{2{\mathit{X}}_{\mathit{g}}}}$$

(4)

respectively, where U_g is the grid voltage magnitude and the power angle δ is defined as the phase angle difference between U_ref and U_g.

The normal operation of the power system requires good transient stability, i.e., the system can restore the original steady state or reach a new steady state when a large disturbance occurs. Furthermore, the power angle stability is an important index to measure the transient stability. There are two types of power angle instability: The first is that the system has an EP, but the operating point crosses the unstable equilibrium point (UEP), resulting in instability. The second type is characterized by the absence of an EP in the system [26].

Figure 2 demonstrates the P_e−δ curves for different levels of grid voltage dips. In the initial state, the system is operating at a stable equilibrium point of a, and the power angle at this point is the stable power angle δ₀. When the grid voltage drops to 0.8 p.u., the working point of the system changes to a′. The active power output from the VSG is less than the reference value, and the virtual rotor of the VSG accelerates under the influence of unbalanced power; then P_e increases. When P_e is greater than P_ref, the rotor decelerates until the speed is rated frequency ω₀. After a few cycles of oscillation, the system reaches point c. When the grid voltage drops to 0.6 p.u., the operating point changes abruptly to a′′, the virtual rotor accelerates, and it starts to decelerate when P_e is greater than P_ref. If the energy accumulated during acceleration is not completely consumed during deceleration, i.e., acceleration area S₁ is larger than maximum deceleration area S₂, the operating point will cross the UEP of point d′. Thereafter, P_ref will consistently be greater than P_e. The rotor continues to accelerate, δ continues to increase, and eventually, the power system is destabilized. When the grid voltage drops to 0.5 p.u., the operating point changes to a′′′. P_ref is greater than P_e, and the deceleration area does not exist. The virtual rotor keeps accelerating under the action of unbalanced active power, δ continues to increase, and the system loses stability.

The above analysis ignores the role of RPC by considering the VSG output voltage to be a constant value. In fact, RPC has an important impact on both the P_e−δ curves and the transient stability analysis. The adverse effect of reactive power control on the transient stability will be analyzed in the following.

By associating (2) and (4), the relationship between U_ref and δ can be obtained as

$${\mathit{U}}_{\mathit{ref}}={\displaystyle \frac{\mathit{m}}{3{\mathit{K}}_{\mathit{q}}}}+{\displaystyle \frac{\sqrt{{\mathit{m}}^2+\mathit{n}}}{3{\mathit{K}}_{\mathit{q}}}}$$

(5)

therefore, P_e can be expressed as

$${\mathit{P}}_{\mathit{e}}={\displaystyle \frac{\left(\mathit{m}+\sqrt{{\mathit{m}}^2+\mathit{n}}\right){\mathit{U}}_{\mathit{g}}}{2{\mathit{X}}_{\mathit{g}}{\mathit{K}}_{\mathit{q}}}}\sin \delta$$

(6)

where m and n can be expressed as

$$\mathit{m}=1.5{\mathit{K}}_{\mathit{q}}{\mathit{U}}_{\mathit{g}}\cos \delta -{\mathit{X}}_{\mathit{g}}$$

(7)

$$\mathit{n}=6{\mathit{K}}_{\mathit{q}}{\mathit{X}}_{\mathit{g}}\left({\mathit{K}}_{\mathit{q}}{\mathit{Q}}_{\mathit{ref}}{+\mathit{U}}_0\right)$$

(8)

Due to the coupling phenomenon between APC and RPC, the P_e−δ curves are shifted from the standard sinusoidal curve to lower left, as shown by the dashed lines in Figure 3. The main reason for the change can be explained as the rapid increase in δ leads to the decrease in Q_e, and Q_e causes the drop in terminal voltage through reactive voltage control, which finally leads to the decrease in P_e.

Figure 4 illustrates the phase plane diagrams under different grid voltage dips. The dashed lines represent the case where U_ref = U₀, and the solid lines represent the case where the effect of RPC is considered. The initial equilibrium point is (δ, ∆ω) = (δ₀, 0). Taking the grid voltage drop to 0.6 p.u. as an example, when U_ref =U₀, the curve finally converges, and the system reaches a new steady state. However, when the coupling between APC and RPC is considered, the curve eventually diverges, δ continues to increase, and the system loses stability. According to the above analysis, the VSG system considering RPC is more likely to lose stability due to the decrease in U_ref. Ignoring the effects of RPC will lead to an overly optimistic judgement on the stability of the system; therefore, U_ref cannot be simply regarded as a constant value.

3. Improvement Measures for RPC Based on Power Angle Deviation

3.1. The Proposed Method

According to the above analysis, the transient instability of the VSG system is caused by the reduction in active power due to grid voltage dips. The decrease in P_e results in the maximum deceleration area being less than the acceleration area or the maximum deceleration area not existing at all. Therefore, in this paper, a method is proposed to increase the output active power of the VSG after a grid fault to improve transient stability. The method adds the variation in δ in the RPC loop to increase P_e by increasing U_ref, which increases or restores the maximum deceleration area. Finally, the proposed method improves the transient stability of the system under two types of instability. The control block diagram is shown in Figure 5.

From Figure 5, RPC can be expressed as

$${\mathit{U}}_{\mathit{ref}}{=\mathit{U}}_0{+\mathit{K}}_{\mathit{q}}\left({\mathit{Q}}_{\mathit{ref}}-{\mathit{Q}}_{\mathit{e}}\right){+\mathit{K}}_{\delta }\left[{\displaystyle \int (\mathit{\omega }-{\mathit{\omega }}_0)}\mathit{dt}-{\delta }_0\right]$$

(9)

where K_δ is the additional compensation factor, and the steady power angle δ₀ can be calculated as

$${\delta }_0={\sin }^{-1}{\displaystyle \frac{2{\mathit{X}}_{\mathit{g}}{\mathit{P}}_{\mathit{ref}}}{3{\mathit{U}}_{\mathit{ref}}{\mathit{U}}_{\mathit{g}0}}}$$

(10)

Since δ can be expressed as (11), (9) can be rewritten as (12):

$$\delta ={\displaystyle \int (\mathit{\omega }-{\mathit{\omega }}_0)}\mathit{dt}$$

(11)

$${\mathit{U}}_{\mathit{ref}}{=\mathit{U}}_0{+\mathit{K}}_{\mathit{q}}\left({\mathit{Q}}_{\mathit{ref}}-{\mathit{Q}}_{\mathit{e}}\right){+\mathit{K}}_{\delta }\left(\delta -{\delta }_0\right)$$

(12)

Then the expression for U_ref can be obtained by combining (4) and (12) as

$${\mathit{U}}_{\mathit{ref}}={\displaystyle \frac{\mathit{m}}{3{\mathit{K}}_{\mathit{q}}}}+{\displaystyle \frac{\sqrt{{\mathit{m}}^2+\mathit{n}+6{\mathit{K}}_{\mathit{q}}{\mathit{X}}_{\mathit{g}}{\mathit{K}}_{\delta }(\delta -{\delta }_0)}}{3{\mathit{K}}_{\mathit{q}}}}$$

(13)

The proposed method introduces the term Δδ to dynamically adjust U_ref based on the transient state of the system, thereby enhancing the voltage level of the system (equivalent to excitation regulation in synchronous generators).

It can be seen that the amplitude of U_ref is related to both δ and U_g, where U_g is included in m (m = 1.5K_qU_gcosδ−X_g). The effects of different factors on U_ref are shown as follows.

The derivative of U_ref with respect to δ is

$${\displaystyle \frac{{\mathit{dU}}_{\mathit{ref}}}{\mathit{d}\delta }}={\displaystyle \frac{-{\mathit{U}}_{\mathit{g}}\sin \delta }{2}}+{\displaystyle \frac{1}{3{\mathit{K}}_{\mathit{q}}}}\times {\displaystyle \frac{2\mathit{m}(-1.5{\mathit{K}}_{\mathit{q}}{\mathit{U}}_{\mathit{g}}\sin \delta )+6{\mathit{K}}_{\mathit{q}}{\mathit{X}}_{\mathit{g}}{\mathit{K}}_{\delta }}{\sqrt{{\mathit{m}}^2+\mathit{n}+6{\mathit{K}}_{\mathit{q}}{\mathit{X}}_{\mathit{g}}{\mathit{K}}_{\delta }(\delta -{\delta }_0)}}}$$

(14)

It is difficult to quantify the relationship between U_ref and δ through (14). According to (13), when the grid voltage drops to a certain extent, the curves of the variation in U_ref with δ can be obtained as shown in Figure 6. When K_δ = 0, i.e., without adding the proposed method, it can be seen that U_ref decreases with the increase in δ. When K_δ = 300, the overall decreasing trend of U_ref decreases slightly, and as δ increases, U_ref first increases and then decreases. However, when δ exceeds π, the effect of the additional term is more pronounced, making U_ref increase all the time. The role of the additional term becomes more important when K_δ = 700, and U_ref increases monotonically with δ. In summary, when a fault occurs in the grid, the proposed method can achieve the increase in U_ref, thereby increasing the output active power of the VSG if K_δ is large enough. However, K_δ must not be excessively large; otherwise, it will lead to a large voltage difference between the PCC and the grid.

Since U_g is contained only in m, the relationship between U_ref and δ can be obtained by means of the intermediate variable m when K_δ is a defined value. The derivative of U_ref with respect to m is first found to be

$${\displaystyle \frac{{\mathit{dU}}_{\mathit{ref}}}{\mathit{dm}}}={\displaystyle \frac{1}{3{\mathit{K}}_{\mathit{q}}}}+{\displaystyle \frac{\mathit{m}}{3{\mathit{K}}_{\mathit{q}}\sqrt{{\mathit{m}}^2+\mathit{n}+6{\mathit{K}}_{\mathit{q}}{\mathit{X}}_{\mathit{g}}{\mathit{K}}_{\delta }(\delta -{\delta }_0)}}}>0$$

(15)

This implies that U_ref has the same monotonicity as m. Then the derivative of m with respect to U_g is shown in (16)

$${\displaystyle \frac{\mathit{dm}}{{\mathit{dU}}_{\mathit{g}}}}={\displaystyle \frac{1}{2}}\cos \delta \left\{\begin{array}{l}\ge 0,\delta \in [0,\pi /2]\\ < 0,\delta \in (\pi /2,\pi ]\end{array}\right.$$

(16)

According to (16), the monotonicity of m is determined by δ. When δ ∈ [0, π/2], m increases with the increase in U_g. When δ ∈ (π/2, π], m decreases with the increase in U_ref. The trend of U_ref with U_g is the same as that of m, as shown by the pink curves in Figure 6.

The power angle curves after adding the proposed control method are shown in Figure 7. Although the EP is present when the grid voltage drops to 0.7 p.u., the system still loses stability, as shown by the solid blue line in Figure 7. The operating point changes from a to a′. Under the effect of unbalanced power, δ starts to increase so that the compensation term starts to work, and U_ref increases with the increase in δ; finally, P_e also increases. Thus, as shown by the blue dashed line in Figure 7, the new P_e-δ curve is higher than the original P_e-δ curve. The acceleration area decreases, and the maximum deceleration area increases in the P_e-δ curve with the addition of the proposed method, where ∆S₁ is the decrease in acceleration area and ∆S₂ is the increase in deceleration area. Ultimately, the acceleration area is less than the maximum deceleration area, and the system also reaches a new steady state.

When the grid voltage drops to 0.5 p.u., the EP does not exist, and the system loses stability, as shown by the solid pink line in Figure 7. As shown by the pink dashed line, the effect of the additional term makes the new P_e-δ curve much higher than the original P_e-δ curve. The intersection between P_e and P_ref creates a new deceleration area that did not exist before, allowing the VSG system to reach a new stable operating state.

Based on the previous analysis, it can be obtained that the transient angle stability of the VSG in two types of instability cases can be enhanced with the proposed method.

3.2. Effect of Additional Coefficient on VSG

In order to analyze the impact of the proposed method on system transient stability, it should first be linearized around the stable equilibrium point δ_s. Subsequently, we define the intermediate function f(δ) as shown in (17).

$$\mathit{f}(\delta ){=\mathit{U}}_{\mathit{ref}}\cdot \sin \delta$$

(17)

Then ΔP_e can be expressed as

$${\Delta \mathit{P}}_{\mathit{e}}={\displaystyle \frac{1.5{\mathit{U}}_{\mathit{g}}}{{\mathit{X}}_{\mathit{g}}}}\mathit{f}\prime (\delta ){|}_{{\delta =\delta }_{\mathit{s}}}\cdot {\Delta \delta =\mathit{G}}_1\cdot \Delta \delta$$

(18)

$${\mathit{G}}_1={\displaystyle \frac{1.5{\mathit{U}}_{\mathit{g}}}{{\mathit{X}}_{\mathit{g}}}}\mathit{f}\prime (\delta ){|}_{{\delta =\delta }_{\mathit{s}}}$$

(19)

where G₁ is the gain coefficient between ΔP_e and Δδ, and f′(δ) is the derivative of f(δ). f′(δ) can expressed as

$$\mathit{f}\prime (\delta )={\displaystyle \frac{{\mathit{dU}}_{\mathit{ref}}}{\mathit{d}\delta }}\sin {\delta +\mathit{U}}_{\mathit{ref}}\cos \delta$$

(20)

U_ref increases with the increase in K_δ, as shown by the solid line in Figure 6. According to (14), dU_ref/dδ also increases with the increase in K_δ. Moreover, there must be δ_s ∈ (0, π/2) when the VSG system reaches a steady state; therefore, sinδ_s > 0, cosδ_s > 0. G₁ will increase with the increase in K_δ. The small-signal model of the VSG system is established as shown in Figure 8.

The small-signal transfer function of ∆δ and ∆ω can be expressed as

$$\left\{\begin{array}{l}{\displaystyle \frac{\Delta \delta }{{\Delta \mathit{P}}_{\mathit{ref}}}}={\displaystyle \frac{1}{{\mathit{J}\mathit{\omega }}_0{\mathit{s}}^2{+\mathit{K}}_{\mathit{D}}{\mathit{\omega }}_0{\mathit{s}+\mathit{G}}_1}}\\ {\displaystyle \frac{\Delta \mathit{\omega }}{{\Delta \mathit{P}}_{\mathit{ref}}}}={\displaystyle \frac{\mathit{s}}{{\mathit{J}\mathit{\omega }}_0{\mathit{s}}^2{+\mathit{K}}_{\mathit{D}}{\mathit{\omega }}_0{\mathit{s}+\mathit{G}}_1}}\end{array}\right.$$

(21)

Assuming a step change in the active power reference value, i.e., P_ref = 1/s, the expression for ∆ω in the frequency domain can be written as

$$\Delta \mathit{\omega }={\displaystyle \frac{1}{{\mathit{J}\mathit{\omega }}_0{\mathit{s}}^2{+\mathit{K}}_{\mathit{D}}{\mathit{\omega }}_0{\mathit{s}+\mathit{G}}_1}}$$

(22)

According to the standard form of second-order systems, the expression for ∆ω can be rewritten as

$$\Delta \mathit{\omega }={\displaystyle \frac{1}{{\mathit{G}}_1}}{\displaystyle \frac{{{\mathit{\omega }}_{\mathit{n}}}^2}{{\mathit{s}}^2+2{\mathit{\zeta }\mathit{\omega }}_{\mathit{n}}{{\mathit{s}+\mathit{\omega }}_{\mathit{n}}}^2}}$$

(23)

where ω_n is the natural oscillation angular frequency, and ζ is the damping ratio.

$${\mathit{\omega }}_{\mathit{n}}=\sqrt{{\displaystyle \frac{{\mathit{G}}_1}{{\mathit{J}\mathit{\omega }}_0}}}$$

(24)

$$\mathit{\zeta }={\displaystyle \frac{{\mathit{K}}_{\mathit{D}}}{2\mathit{J}}}\sqrt{{\displaystyle \frac{{\mathit{J}\mathit{\omega }}_0}{{\mathit{G}}_1}}}$$

(25)

In general, the VSG system operates in an underdamped state, i.e., 0 < ζ < 1; then the expressions for ∆δ and ∆ω in the time domain can be expressed as

$$\Delta \mathit{\omega }(\mathit{t})={\displaystyle \frac{1}{{\mathit{G}}_1}}{\displaystyle \frac{{\mathit{\omega }}_{\mathit{n}}}{\sqrt{1-{\mathit{\zeta }}^2}}}{\mathit{e}}^{-{\mathit{\zeta }\mathit{\omega }}_{\mathit{n}}\mathit{t}}\sin \left(\sqrt{1-{\mathit{\zeta }}^2}{\mathit{\omega }}_{\mathit{n}}\mathit{t}\right)$$

(26)

$$\Delta \delta (\mathit{t})={\displaystyle \frac{1}{{\mathit{G}}_1}}\left[1-{\displaystyle \frac{1}{\sqrt{1{-\mathit{\zeta }}^2}}}{\mathit{e}}^{-{\mathit{\zeta }\mathit{\omega }}_{\mathit{n}}\mathit{t}}\sin \left(\sqrt{1-{\mathit{\zeta }}^2}{\mathit{\omega }}_{\mathit{n}}\mathit{t}+\mathit{\phi }\right)\right]$$

(27)

where $\sin \mathit{\phi }=\sqrt{1-{\mathit{\zeta }}^2}$ and $\cos \mathit{\phi }=\mathit{\zeta }$.

Deriving ∆ω(t) with respect to time yields the expression for RoCoF as

$$\mathit{RoCoF}={\displaystyle \frac{\mathit{d}\Delta \mathit{\omega }(\mathit{t})}{\mathit{dt}}}={\displaystyle \frac{1}{{\mathit{G}}_1}}{\displaystyle \frac{{{\mathit{\omega }}_{\mathit{n}}}^2}{\sqrt{1-{\mathit{\zeta }}^2}}}{\mathit{e}}^{-{\mathit{\zeta }\mathit{\omega }}_{\mathit{n}}\mathit{t}}\sin \left(\mathit{\phi }-\sqrt{1-{\mathit{\zeta }}^2}{\mathit{\omega }}_{\mathit{n}}\mathit{t}\right)$$

(28)

RoCoF has a maximum value when t = 0. And the maximum value of RoCoF at this time is

$${\mathit{RoCoF}}_{\mathit{max}}={\left.{\displaystyle \frac{\mathit{d}\Delta \mathit{\omega }(\mathit{t})}{\mathit{dt}}}\right|}_{\mathit{t}=0}={\displaystyle \frac{1}{{\mathit{J}\mathit{\omega }}_0}}$$

(29)

As seen from (29), RoCoF_max is only related to J and is not affected by the additional term. Since RoCoF_max is inversely proportional to J, it is possible to reduce RoCoF_max by increasing J. Regarding the influence of J on transient stability, ref. [27] provides a pertinent explanation. For the instability case where an equilibrium point exists, a smaller J leads to a smaller change in δ, meaning that low inertia improves transient stability. For the instability case where the equilibrium point is absent, a larger J leads to a longer critical fault clearance time, indicating that high inertia enhances transient stability.

There is a maximum value of ∆ω(t) when RoCoF = 0, at which point the expression for t can be expressed as

$$\mathit{t}={\displaystyle \frac{\arcsin \sqrt{1-{\mathit{\zeta }}^2}}{\sqrt{1-{\mathit{\zeta }}^2}{\mathit{\omega }}_{\mathit{n}}}}$$

(30)

Bringing (30) into (26) can obtain the maximum value of ∆ω(t) as

$${\Delta \mathit{\omega }}_{\mathit{max}}={\displaystyle \frac{1}{{\mathit{G}}_1}}{\displaystyle \frac{{\mathit{\omega }}_{\mathit{n}}}{\sqrt{1-{\mathit{\zeta }}^2}}}{\mathit{e}}^{-{\mathit{\zeta }\mathit{\omega }}_{\mathit{n}}{\displaystyle \frac{\arcsin \sqrt{1-{\mathit{\zeta }}^2}}{\sqrt{1-{\mathit{\zeta }}^2}{\mathit{\omega }}_{\mathit{n}}}}}\sin \mathit{\phi }\approx \sqrt{{\displaystyle \frac{1}{{\mathit{J}\mathit{\omega }}_0{\mathit{G}}_1}}}{\mathit{e}}^{-\mathit{\zeta }}=\sqrt{{\displaystyle \frac{1}{{\mathit{J}\mathit{\omega }}_0{\mathit{G}}_1}}}{\mathit{e}}^{-{\displaystyle \frac{{\mathit{K}}_{\mathit{D}}}{2}}\sqrt{{\displaystyle \frac{{\mathit{\omega }}_0}{{\mathit{JG}}_1}}}}$$

(31)

It can be observed that ∆ω_max is correlated with J, K_D and the additional coefficient K_δ (K_δ is included in G₁). Among them, the relationship between ∆ω_max and K_D is more obvious, and increasing K_D can enlarge the damping ratio ζ and thus decrease ∆ω_max.

According to the above analysis, G₁ is proportional to K_δ, so the relationship between K_δ and ∆ω_max can be judged from the relationship between G₁ and ∆ω_max. In order to get the relationship between ∆ω_max and G₁, deriving ∆ω_max on G₁ can be obtained as

$${\displaystyle \frac{{\mathit{d}\Delta \mathit{\omega }}_{\mathit{max}}}{{\mathit{dG}}_1}}={\displaystyle \frac{1}{\sqrt{{\mathit{J}\mathit{\omega }}_0}}}{\displaystyle \frac{1}{2{{\mathit{G}}_1}^{2/3}}}{\mathit{e}}^{-\mathit{\zeta }}(-1{+\mathit{K}}_{\mathit{D}}\sqrt{{\displaystyle \frac{{\mathit{\omega }}_0}{\mathit{J}}}}{\displaystyle \frac{1}{\sqrt{{\mathit{G}}_1}}})$$

(32)

When G₁ ∈ (0,${\mathit{K}}_{\mathit{D}}^2\times {\mathit{\omega }}_0/\mathit{J}$), ∆ω_max increases with the increase in G₁; when G₁ ∈ (${\mathit{K}}_{\mathit{D}}^2\times {\mathit{\omega }}_0/\mathit{J}$, +∞), ∆ω_max decreases with the increase in G₁. The minimum value of G₁ is calculated to be greater than ${\mathit{K}}_{\mathit{D}}^2\times {\mathit{\omega }}_0/\mathit{J}$. Therefore, in the range of G₁, ∆ω_max decreases as G₁ increases; that is, ∆ω_max decreases as K_δ increases. Hence, the inclusion of the additional term not only enables the VSG system to restore stable operation after a grid fault but also enables the frequency deviation to be effectively reduced.

The curves of ∆ω with time for different K_δ are shown in Figure 9. The curves indicate that ∆ω_max decreases with the increase in K_δ and that RoCoF_max does not change with the change in K_δ. Moreover, ∆δ_max can be obtained by integrating ∆ω_max. Therefore, the influence of K_δ on ∆δ_max is identical to its influence on ∆ω_max. Figure 10 shows the phase plane curves of the VSG with different K_δ. It can be seen that both ∆δ_max and ∆ω_max decrease as K_δ increases.

3.3. Control Parameter Design

The analysis above has demonstrated the effectiveness of the proposed method for transient stability enhancement in VSGs. Moreover, the frequency stability of a VSG is also enhanced as K_δ increases. Nevertheless, it is not the case that a larger K_δ is the better, as too large a K_δ will lead to a greater voltage difference between the PCC and the grid. This leads to an increase in the fault current, which affects the safety of the system. Thus, it is necessary to calculate the critical value of K_δ.

Bringing (3) and (13) into (1), the second-order differential equation for the VSG with respect to δ after adding the proposed method is obtained as

$$\mathit{J}{\displaystyle \frac{{\mathit{d}}^2\delta }{{\mathit{dt}}^2}}={\displaystyle \frac{{\mathit{P}}_{\mathit{ref}}}{{\mathit{\omega }}_0}}-{\mathit{K}}_{\mathit{D}}{\displaystyle \frac{\mathit{d}\delta }{\mathit{dt}}}-{\displaystyle \frac{1}{3{\mathit{\omega }}_0{\mathit{K}}_{\mathit{q}}}}[\mathit{m}+\sqrt{{\mathit{m}}^2+\mathit{n}+6{\mathit{K}}_{\mathit{q}}{\mathit{X}}_{\mathit{g}}{\mathit{K}}_{\delta }(\delta -{\delta }_0)}]$$

(33)

Algorithm 1 can efficiently calculate parameter critical values, and the specific calculation steps are shown as the following:

Algorithm 1: Iterative algorithm
Step	Description
Initialization	Set Kδ = 0. Define the level of grid voltage dips.
2. Calculation	Solve the second-order differential equation (33) to obtain the power angle δ.
3. Judgement	Set the critical power angle δr at the UEP as π. If δ > δr (system is destabilized), then set Kδ = Kδ + 1 and go to Step 2. If δ < δr (system is stabilized), then terminate the loop and set Kδmin = Kδ.
4. Output	Return the minimum stabilizing value Kδmin.

Figure 11 shows the critical values of K_δ under different grid voltage dips, which form the stability boundary. The region where K_δ > K_δmin is the stable region, and the region where K_δ < K_δmin is the unstable area.

4. Verification

In order to verify the effectiveness of the transient stability enhancement control method proposed above, the model of the grid-connected system shown in Figure 1 is constructed based on MATLAB/Simulink. The experimental fault conditions are grid voltage sags to 0.7 p.u. and 0.5 p.u., respectively. Then, scenarios of both severe and slight drops are simulated to evaluate the performance of different control strategies under the two grid fault severity levels. The transient stability performance of the proposed method is compared with three other methods: conventional VSG control, the adaptive parameter method mentioned in ref. [17] and the mode-switching method proposed in ref. [22]. The main experimental parameters are shown in

Table 1. Main parameters used in experiment.

Parameters	Description	Value
Pref	Active power reference	300 kW
Qref	Reactive power reference	0 kVar
Udc	DC voltage	1000 V
U0	Rated voltage	563 V
Ug	Normal grid voltage	563 V
ω0	Rated angular frequency	100 π rad/s
Lg	Grid inductance	2 mH
J	Inertia coefficient	10 kg∙m2
KD	Damping coefficient	50 N∙m∙s/rad
Kq	Q-V droop gain	0.00125 V/Var
Lf	Filter inductance	1.5 mH
Cf	Filter capacitance	100 μF

.

• Case 1: Grid voltage dips to 0.7 p.u.

To verify the effectiveness of the proposed method under a slight grid voltage drop, the system is designed to drop the grid voltage to 0.7 p.u. after 3 s of stable operation. Figure 12, Figure 13, Figure 14 and Figure 15 show the response curves of active power output, grid voltage, angular frequency and power angle for the four different control methods in Case 1.

Figure 12 shows the experimental results of the conventional VSG control. It can be seen that without adding the additional control strategy, the EP of the system exists after the fault. However, the power angle gradually increases, VSG frequency generates oscillation, and the active power cannot be maintained near the reference value; finally, the VSG system loses stability.

The experimental results of the adaptive inertia and damping parameter method mentioned in ref. [17] are shown in Figure 13. Although the VSG system can regain stability, the regulation time is relatively long. It can be observed that the power angle takes at least 4 s to reach a new stable state. Figure 14 shows the results of using the mode-switching method proposed in ref. [22]. Although the active power output from the VSG can eventually be stabilized, the amplified experimental result shows that the frequency and power angle still oscillate after stabilization.

As a comparison, the method proposed above enables the system to restore stable operation after a very short period of regulation following a grid fault. As shown in Figure 15, the output active power of the VSG is maintained at the rated value, with stable power angle and frequency.

2.

Case 2: Grid voltage dips to 0.5 p.u.

To verify the effectiveness of the proposed method under a severe grid voltage drop, the system is designed to drop the grid voltage to 0.5 p.u. after 3 s of stable operation. Figure 16, Figure 17, Figure 18 and Figure 19 show the response curves of active power output, grid voltage, angular frequency and power angle for the four different control methods in Case 2.

Figure 16 shows the experimental results of conventional VSG control. It is apparent that P_e is always less than the reference value after a severe fault and the equilibrium point does not exist. Therefore, the VSG system loses its stability after a severe grid fault.

Nevertheless, as shown in Figure 17, the adaptive parameter method cannot restore the stable operation of an unstable VSG system under severe fault. This indicates that the adaptive parameter method only serves to restore system stability under a slight fault. With the mode-adaptive method mentioned in ref. [22], although it can restore the stability of the system after a serious fault, the oscillation of frequency and power angle is more obvious, as shown in Figure 18. In Case 1, it is difficult to observe the oscillation without zooming in on the waveforms, while in Case 2, the oscillation is more noticeable and can be observed without enlargement.

Figure 19 shows the waveforms with the proposed method. When the system experiences a severe power grid voltage drop, the system can reach a new stable operating state after a period of adjustment with the proposed method. Apparently, the proposed strategy exhibits excellent stability recovery capability under more severe fault conditions.

To investigate the effect of the compensation coefficient on stability, Figure 20 shows the frequency and power angle response waveforms with different values of the compensation coefficient K_δ. A grid voltage drop fault is set after 3 seconds of stable system operation. Figure 20a shows the waveforms when K_δ = K_δ1. Under such circumstances, the maximum angular frequency deviation ∆ω_max is 2.25 rad/s, and the maximum power angle deviation ∆δ_max is 0.94 rad. When K_δ = K_δ2, as shown in Figure 20b, along with the increase in K_δ, ∆ω_max decreases from 2.25 rad/s to 1.99 rad/s, and ∆δ_max decreases from 0.94 rad to 0.78 rad. It can be concluded that as the coefficient K_δ increases, the frequency stability of the system also improves.

Clearly, the experimental results demonstrate that the proposed method enables VSG systems to operate stably even under severe grid voltage dips. Moreover, the increase in K_δ enhances system frequency stability and power angle stability.

5. Conclusions

The transient power angle stability of VSGs under different grid voltage dips is investigated in this paper. Firstly, the causes of transient instability in VSG systems during grid voltage dips are analyzed. It is shown that RPC can adversely affect the transient stability of the system. Based on the analysis, a method is proposed to restore system stability in two instability cases by increasing the voltage of the VSG output. Meanwhile, the proposed method reduces the acceleration area and increases the maximum deceleration area. Then, to investigate the impact of different parameters on the transient characteristics of the VSG, theoretical analysis is carried out using a small-signal model. Moreover, the critical compensation coefficient is calculated, providing a theoretical basis for parameter selection. Finally, the effectiveness of the proposed method is verified by the MATLAB/Simulink platform.