An Optimized Power-Angle and Excitation Dual Loop Virtual Power System Stabilizer for Enhanced MMC-VSG Control and Low-Frequency Oscillation Suppression

Abstract

Modular Multilevel Converter Virtual Synchronous Generator (MMC-VSG) technology is gaining widespread attention for its ability to enhance the inertia and frequency stability of the power grid integrated with converter-interfaced renewable energy sources. However, the excitation voltage regulation in the MMC-VSG can generate equivalent negative damping torque and cause low-frequency oscillation problems similar to those in synchronous machines. This article aims to improve the system’s damping torque and minimize low-frequency oscillations by introducing a Virtual Power System Stabilizer (VPSS) into the power control loop. Building on the study of dynamic interactions between various control links of the MMC, this research establishes a reduced-order model (ROM) and a Phillips–Heffron state equation for the MMC-VSG single machine infinite bus system, using a hybrid modeling approach and a zero-pole truncation method. It also analyzes the mechanism of low-frequency oscillations in the MMC-VSG system through the damping torque method. The analysis reveals that the negative damping torque produced during the excitation voltage regulation process causes changes in the virtual power angle, which in turn increases the risk of low-frequency oscillation in the MMC-VSG. To address this issue, the article proposes an optimized control method for the MMC-VSG dual power loop architecture (power-angle/excitation) VPSS. This strategy compensates for the inadequate damping torque of a single loop VPSS and effectively suppresses low-frequency oscillations in the system.

1. Introduction

The increasing penetration of distributed renewable energy and power electronics in power systems has introduced challenges, such as weak damping and low inertia, affecting the safe and reliable operation of electricity systems [1,2]. Virtual synchronous generators (VSGs) are widely used to support grids with high levels of distributed energies [3,4]. However, if the negative damping torque generated by the VSG excitation controller exceeds the inherent positive damping torque, low-frequency oscillations may occur [5].

The existing studies of low-frequency oscillation mainly focus on two areas: mathematical models for assessing system stability and methods for suppressing low-frequency oscillations. Full-order models can provide accurate stability assessments but are computationally intensive. Reduced-order models (ROMs) can offer a practical alternative but could lead to larger modeling errors due to oversimplifying issues [6,7,8]. Traditional reduced-order models are implemented by ignoring inner control loop transients and using a single power loop, which is not suitable for power converters since their state switching timescale is different from traditional synchronous generators (SGs) [9,10,11]. A 5th-order ROM is developed in [12] by using hybrid modeling and zero-pole truncation and neglecting certain transients in inner voltage/current loops. A frequency-domain modeling method for analyzing low-frequency oscillations, considering VSG parameter coupling, is presented in [13], and a second-order VSG model is adopted to illustrate the impact of VSG on power grid oscillations. Other techniques, like Kron reduction, singular perturbation, and optimization algorithms are also used in model order reduction [14,15,16,17].

Many scholars also focus on the mechanism and suppression methods of low-frequency oscillation for VSG-based systems. References [18,19,20,21,22,23] explore the low-frequency oscillation mechanism of VSGs and reveal that the system damping of VSGs mainly comprises two parts: the inherent positive damping of the virtual rotor motion and the additional negative damping provided by the equivalent electromagnetic torque. When the additional negative damping exceeds the inherent positive damping, low-frequency oscillations occur due to insufficient system damping [18,20]. The work in [18] quantitatively investigates the power oscillation problem of VSG systems by constructing a grid-connected small-signal model, and the research results confirm that insufficient virtual damping torque is one of the main reasons for inducing system instability. Modeling results in [19] show that VSG subsystems can negatively affect small-signal angular stability. Research in [20] indicates that while VSGs impact low-frequency oscillations, a proper control strategy can enhance positive damping torque to mitigate low-frequency oscillations.

The increasing penetration of inverter-based resources (IBRs) can introduce new stability challenges, as discussed in [21]. The paralleled SG–VSG systems show lower stability margins as compared to paralleled VSG systems [22]. Suppressing low-frequency oscillations in VSGs is crucial [23,24,25,26,27,28], and methods are proposed to enhance positive damping of the virtual rotor [23,24,25] and reduce negative damping from the equivalent electromagnetic torque [26,27,28]. Reference [23] analyzes the oscillation mechanism in a VSG-infinite grid system and proposes a simplified damping adjustment method to suppress low-frequency oscillations. Additionally, an inner loop control algorithm for VSGs that balances the d–q axis components of voltage and current is proposed in [24] to reduce the impact of sub-synchronous oscillations of a power grid.

An optimization strategy to improve the dynamic response of the grid-connected active power (GCAP) in VSG is proposed in [25] by combining virtual negative impedance with an active power transient damping control algorithm to achieve oscillation suppression. In [26], the modeling of a two-machine interconnected power system with VSG and SG is realized by constructing state-space equations and using the Phillips–Heffron model, and a virtual power system stabilizer (VPSS) is introduced to reduce negative damping torque. In [27], a new control prototype based on a twin delayed deep deterministic policy gradient (TD3) adaptive controller is proposed by replacing the conventional VSG module in the MMC control subsystem to improve system stability. Modifications in [28] further enhance system stability and resilience while retaining the benefits of the traditional VSG control model.

However, effective VSG control methods to suppress low-frequency oscillations still require further research efforts. To address the limitations of conventional two-level VSGs, such as high switching harmonics, low fault tolerance, and high voltage stress, multilevel converters have been explored and applied. The cascaded H bridge multilevel converter (CHMC) system can be used to enhance power grid stability by optimizing the virtual inertia and damping in both island and grid-connected modes [28]. However, most of the research pays much less attention to the coupling effect between active and reactive power loops of MMC-VSG, which could degrade the response speed. Meanwhile, MMC-VSGs have attracted significant research attention due to their high fault tolerance, low switching harmonics, and greater control freedom.

In view of the above research, this article focuses on developing an effective reduced-order model (ROM) for the MMC-VSG and proposes a novel optimization control method, using a virtual power system stabilizer (VPSS) based on an improved power-angle and excitation dual loop architecture. This approach aims to enhance the stability of the MMC-VSG ROM and mitigate the risk of low-frequency oscillations. The main contributions are summarized as follows:

(1)

A comparative analysis is conducted between a full-order model and a 4th-order ROM of MMC-VSG. The 4th-order ROM is evaluated for accuracy by examining dominant eigenvalue distribution and trajectories relative to key parameter variations.

(2)

A new Phillips–Heffron model and an enhanced control strategy for the 4th-order ROM of MMC-VSG, featuring a dual VPSS, are proposed. This new controller significantly enhances system damping performance and mitigates low-frequency oscillations.

The rest of the paper is organized as follows: The reduced order MMC-VSG model and the low-frequency oscillation mechanism of MMC-VSG are analyzed in Section 2. In Section 3, the proposed power-angle and excitation dual loop virtual power system stabilizer are interpreted. Simulation results are presented and discussed in Section 4, followed by the conclusions in Section 5.

2. MMC-VSG Modeling and Low-Frequency Oscillation Suppression

2.1. Reduced Order MMC-VSG Model

Figure 1 shows the system topology and control scheme of a three-phase grid connected MMC-VSG, which is connected to the grid through a transformer at the point of common coupling (PCC). L_T represents the leakage inductance of the grid-connected transformer from MMC to the PCC, and R_g and L_g are the equivalent resistance and inductance of the power line, respectively.

To investigate the stability of VSGs, the existing research mainly considers the case of VSG modeling with a single power control loop, generally neglecting the inner voltage and current loops. However, MMC-VSGs have a narrower control bandwidth and complex nonlinear interactions between control loops [23]. Therefore, the modeling methods discussed in most existing literature are no longer suitable for MMC-VSGs. The conventional MMC-VSG system mainly includes an outer power control loop and inner voltage and current dual control loops. The voltage reference E*(U_dqref) and frequency reference ω^*(ω₀) for the inner voltage and current loop are provided by the output of the power control loop. The d-axis command of the voltage controller is outputted from the reactive power loop (U_dref = E*) and the q-axis command is 0 (U_qref = 0). According to Figure 1 and Appendix A, the small signal model of full-order MMC-VSG can be obtained as

$$\mathsf{\Delta }\dot{x}=A_{\mathrm{MMC}}\mathsf{\Delta }x+B_{\mathrm{MMC}}\mathsf{\Delta }u$$

(1)

$$\mathsf{\Delta }y=C_{\mathrm{MMC}}\mathsf{\Delta }x+D_{\mathrm{MMC}}\mathsf{\Delta }u,$$

(2)

where the input variables Δu = [ΔP_ref, ΔQ_ref]^T, ΔP_ref, and ΔQ_ref are reference active and reactive power; variables Δy = [ΔP_e, ΔQ_e]^T, ΔP_e, and ΔQ_e are output active and reactive power. The state variable Δx is equal to [Δδ, Δω, Δe_d, Δe_q, Δi_d, Δi_q, Δϕ_1d, Δϕ_1q, Δϕ_2d, and Δϕ_2q]^T. The terms Δδ and Δω stand for equivalent power angle and angular velocity; Δϕ_1d, Δϕ_1q, Δϕ_2d, and Δϕ_2q are intermediate variables obtained from inner voltage and current loops; terms Δi_d and Δi_q are equivalent state currents; and terms Δe_d and Δe_q are equivalent state voltages. The detailed derivation and expressions of (1) and (2) can be found in Appendix A. A complete state-space model of a single MMC-VSG can be obtained. In the full-order model of MMC-VSG, the power, voltage, and current loops are all described by using the state space model. The ROM in this paper is established by using a hybrid modeling approach and a zero-pole truncation method [12].

The hybrid modeling is realized based on a state-space model (Equations (1) and (2)) for the power loop and transfer functions for the inner voltage and current loops. The reduced-order model maintains the dominant slow response and key interactions between fast and slow states, as described in Appendix A.

$$\left\{\begin{array}{c}{\mathit{I}}_{\mathrm{ref}}=({\mathit{U}}_{\mathrm{ref}}-\mathit{U})G_{\nu }\\ {\mathit{e}}_{\mathrm{ref}}=\mathit{U}+G_i{\mathit{I}}_{\mathrm{ref}}+(\mathrm{j}\omega L-G_i)\mathit{I}\\ \mathit{e}-\mathit{U}=(\mathrm{s}L+\mathrm{j}\omega L)\mathit{I}\\ \mathit{e}={\mathit{e}}_{\mathrm{ref}}/(1+\mathrm{s}{T}_i)\end{array}\right.$$

(3)

where U_ref is the reference voltage outputted from the power loop, I_ref is the reference current outputted from the voltage loop, I is the output current, e_ref is the output voltage before time delay, e is the output voltage after time delay, U is the voltage of the PCC point, and G_i and G_v stand for current and voltage loops using the proportional integral (PI) control method, respectively, which can be expressed by

$$G_i=k_{p2}+{\displaystyle \frac{k_{i2}}{s}},G_{\mathrm{v}}=k_{p1}+{\displaystyle \frac{k_{i1}}{s}}.$$

Thus, the space vector equation can be obtained by

$$\mathit{U}={\displaystyle \frac{G_{\nu }{G}_i}{s{T}_i+G_{\nu }{G}_i}}\mathit{E}-{\displaystyle \frac{G_i-\mathrm{j}\omega L+(sL+\mathrm{j}\omega L)(1+s{T}_i)}{s{T}_i+G_{\nu }{G}_i}}\mathit{I}$$

(4)

(4) can be rewritten as

$$\mathit{U}={\displaystyle \frac{G_{\nu }{G}_i}{s{T}_i+G_{\nu }{G}_i}}\mathit{E}-{\displaystyle \frac{G_i-\mathrm{j}\omega L+(sL+\mathrm{j}\omega L)(1+s{T}_i)}{s{T}_i+G_{\nu }{G}_i}}\mathit{I}=G\mathit{E}-Z_{inner}\mathit{I}$$

(5)

where G is the control coefficient and Z_inner is the equivalent output impedance of the inner control loop. Considering T_i ≪ 1, we can get G ≈ 1, and

$$Z_{inner}\approx {\displaystyle \frac{G_i+sL}{G_{\nu }{G}_i}}={\displaystyle \frac{1}{k_{i1}}}s{\displaystyle \frac{\left(s^2\frac{L}{k_{i2}}+s\frac{k_{p2}}{k_{i2}}+1\right)}{(s\frac{k_{p1}}{k_{i1}}+1)(s\frac{k_{p2}}{k_{i2}}+1)}}.$$

By selecting T_c = T₀/2 as the cutoff time constant for the order reduction truncation, Z_inner can be reduced in order, namely

$$Z_{inner}\approx {\displaystyle \frac{G_i+sL}{G_{\nu }{G}_i}}={\displaystyle \frac{1}{k_{i1}}}s{\displaystyle \frac{\left(s^2\frac{L}{k_{i2}}+s\frac{k_{p2}}{k_{i2}}+1\right)}{(s\frac{k_{p1}}{k_{i1}}+1)(s\frac{k_{p2}}{k_{i2}}+1)}}\approx {\displaystyle \frac{1}{k_{i1}}}s.$$

(6)

Z_inner is pure resistance and can be written as R_inner. The voltage balance equation of the MMC-VSG reduced order model can be expressed as

$$\mathit{E}-{\mathit{U}}_{\mathrm{g}}=(Zinner+\mathrm{j}{X}_{\mathrm{g}}+R_{\mathrm{g}}){\mathit{I}}_{\mathrm{g}}=(Zinner+Z_{\mathrm{g}}){\mathit{I}}_{\mathrm{g}}=((Rinner+R_{\mathrm{g}})+\mathrm{j}{X}_{\mathrm{g}}){\mathit{I}}_{\mathrm{g}}$$

(7)

The structure of the mixed 4th-order ROM of MMC-VSG consists of a 2nd-order power loop, and a 2nd-order equivalent output state can be obtained from Appendix A and Equation (7), namely

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}\mathsf{\Delta }\omega }{\mathrm{d}t}}={\displaystyle \frac{\mathsf{\Delta }{P}_{\mathrm{ref}}}{J{\omega }_0}}-{\displaystyle \frac{\mathsf{\Delta }{P}_e}{J{\omega }_0}}-{\displaystyle \frac{D_{\mathrm{P}}(\mathsf{\Delta }\omega )}{J{\omega }_0}}\\ {\displaystyle \frac{\mathrm{d}\mathsf{\Delta }\delta }{\mathrm{d}t}}=\mathsf{\Delta }\omega \\ \left[\begin{array}{c}\mathsf{\Delta }Pe\\ \mathsf{\Delta }Qe\end{array}\right]=\left[\begin{array}{cc}{k}_{11mix}& k_{12mix}\\ k_{21mix}& k_{22mix}\end{array}\right]\left[\begin{array}{c}\mathsf{\Delta }\delta \\ \mathsf{\Delta }E\end{array}\right]\end{array}\right..$$

(8)

This model contains two inputs (P_ref and Q_ref), and two outputs (P_e and Q_e), where R’²_g = R²_g + R²_inner. The parameters k_mix are presented in

Table 1. Expression for parameters $k_{mix}$.

Parameters	Expression
k11mix	${\displaystyle \frac{3U_{\mathrm{g}}{E}_0}{R_{\mathrm{g}}^{\prime 2}+X_{\mathrm{g}}^2}}\left(X_{\mathrm{g}}\cos {\delta }_0+R_{\mathrm{g}}^{\prime }\sin {\delta }_0\right)$
k12mix	${\displaystyle \frac{3U_{\mathrm{g}}{E}_0}{R_{\mathrm{g}}^{\prime 2}+X_{\mathrm{g}}^2}}\left(X_{\mathrm{g}}\sin {\delta }_0-R_{\mathrm{g}}^{\prime }\cos {\delta }_0\right)$
k21mix	${\displaystyle \frac{3R_{\mathrm{g}}^{\prime }}{R_{\mathrm{g}}^{\prime 2}+X_{\mathrm{g}}^2}}\left(2E_0-U_{\mathrm{g}}\cos {\delta }_0\right)+{\displaystyle \frac{3U_{\mathrm{g}}{X}_{\mathrm{g}}}{R_{\mathrm{g}}^{\prime 2}+X_{\mathrm{g}}^2}}\sin {\delta }_0$
k22mix	${\displaystyle \frac{3X_{\mathrm{g}}}{R_{\mathrm{g}}^{\prime 2}+X_{\mathrm{g}}^2}}\left(2E_0-U_{\mathrm{g}}\cos {\delta }_0\right)-{\displaystyle \frac{3U_{\mathrm{g}}{R}_{\mathrm{g}}^{\prime }}{R_{\mathrm{g}}^{\prime 2}+X_{\mathrm{g}}^2}}\sin {\delta }_0$

.

The block diagram of the 4th-order ROM of the MMC-VSG can be obtained, as shown in Figure 2.

2.2. Low-Frequency Oscillation Mechanism of MMC-VSG

The excitation equation of MMC-VSG can be written as

$$E=E_0+\mathsf{\Delta }E=E_0+k_{\mathrm{q}}\mathsf{\Delta }{Q}_{\mathrm{ref}}-k_{\mathrm{q}}\mathsf{\Delta }{Q}_e.$$

(9)

By combining (9) with $\mathsf{\Delta }{Q}_e=k_{21mix}\mathsf{\Delta }\delta +k_{22mix}\mathsf{\Delta }E$ in (8), we have

$$\mathsf{\Delta }E=k_{\mathrm{q}}\mathsf{\Delta }{Q}_{\mathrm{ref}}-k_{\mathrm{q}}(k_{21mix}\mathsf{\Delta }\delta +k_{22mix}\mathsf{\Delta }E),$$

(10)

where k_q is the equivalent excitation coefficient. By combining (8) and (10), the Phillips–Heffron model for the 4th-order ROM MMC-VSG system can be obtained, as shown in Figure 3.

Figure 3 shows that while virtual excitation enhances the voltage regulation in the MMC-VSG system, it also introduces complex disturbances that increase the risk of virtual power-angle instability. When the equivalent damping torque provided by the MMC-VSG is negative, low-frequency oscillations dominated by the virtual power-angle δ could be triggered. Based on the instability mechanism, two strategies can be designed to avoid low-frequency oscillations: (1) compensating for the negative damping torque of the virtual excitation regulator; (2) directly compensating for the negative damping torque of the power-angle regulator. Let the transfer function from Δδ to ΔP_e in Figure 3 be F(s), and the following equation can be obtained,

$$F(s)=k_{11mix}+{\displaystyle \frac{k_{12mix}{k}_{21mix}{k}_{\mathrm{q}}}{1+k_{\mathrm{q}}{k}_{22mix}}}.$$

(11)

When there is no additional damping control, the electromagnetic torque provided by the excitation link is ΔT_e. As shown in Figure 3, ΔP_e is composed of two damping torques generated by k_11mix and k_21mix branches, which can be expressed by

$$\mathsf{\Delta }{P}_e=\mathsf{\Delta }{P}_{11}+\mathsf{\Delta }{P}_{12},$$

(12)

where

$$\mathsf{\Delta }{P}_{11}=k_{11mix}\mathsf{\Delta }\delta \mathsf{\Delta }{P}_{12}={\displaystyle \frac{k_{12mix}{k}_{21mix}{k}_{\mathrm{q}}}{1+k_{\mathrm{q}}{k}_{22mix}}}\mathsf{\Delta }\delta .$$

Due to the impact of parameters k_12mix and k_21mix on ΔP₁₂, the negative damping torque may be generated during unexpected disturbances.

In the complex frequency domain, electromagnetic torque can be expressed by

$$\mathsf{\Delta }{T}_e=F({\lambda }_S)\mathsf{\Delta }\delta ({\lambda }_S),$$

(13)

where λ_S = ζ_s ± jω_s are the typical eigenvalues for representing electromechanical oscillation modes of the system.

Meanwhile, in the complex frequency domain, Δω can be obtained by

$$\mathsf{\Delta }\omega ({\lambda }_{\mathrm{S}})={\displaystyle \frac{{\zeta }_{\mathrm{s}}}{{\omega }_0}}\mathsf{\Delta }\delta ({\lambda }_{\mathrm{S}})+{\displaystyle \frac{j{\omega }_{\mathrm{s}}}{{\omega }_0}}\mathsf{\Delta }\delta ({\lambda }_{\mathrm{S}}).$$

(14)

According to the damping torque analysis method, the input signal (electromagnetic torque) of the MMC-VSG regulators can be expressed as

$$\begin{array}{l}\mathsf{\Delta }{T}_e=F({\lambda }_S)\mathsf{\Delta }\delta ({\lambda }_S)=T_{\delta }\mathsf{\Delta }\delta ({\lambda }_S)+T_{\omega }\mathsf{\Delta }\omega ({\lambda }_S)=T_{\delta }\mathsf{\Delta }\delta ({\lambda }_S)+T_{\omega }({\displaystyle \frac{{\zeta }_s}{{\omega }_0}}\mathsf{\Delta }\delta ({\lambda }_S)+j{\displaystyle \frac{{\omega }_s}{{\omega }_0}}\mathsf{\Delta }\delta ({\lambda }_S))\\ =((T_{\delta }+T_{\omega }{\displaystyle \frac{{\zeta }_s}{{\omega }_0}})+j{\displaystyle \frac{{\omega }_s}{{\omega }_0}})\mathsf{\Delta }\delta ({\lambda }_{\mathrm{S}})=(\mathrm{Re}[F({\lambda }_{\mathrm{S}})]+j\mathrm{Im}[F({\lambda }_{\mathrm{S}})])\mathsf{\Delta }\delta ({\lambda }_{\mathrm{S}})\end{array}$$

(15)

where T_ωΔω (λ_S) is the damping torque, T_δΔδ (λ_S) is the synchronous torque of MMC-VSG, T_ω is the damping torque coefficient, T_δ is the synchronous torque coefficient, and F(λ_S) is the transfer function from Δδ to ΔT_e. Therefore,

$$\begin{array}{c}{T}_{\omega }={\displaystyle \frac{{\omega }_0}{{\omega }_s}}\mathrm{Im}[F({\lambda }_{\mathrm{S}})]\\ T_{\delta }=\mathrm{Re}[F({\lambda }_{\mathrm{S}})]-T_{\omega }{\displaystyle \frac{{\zeta }_{\mathrm{s}}}{{\omega }_0}}=\mathrm{Re}[F({\lambda }_{\mathrm{S}})]-\mathrm{Im}[F({\lambda }_{\mathrm{S}})]{\displaystyle \frac{{\zeta }_{\mathrm{s}}}{{\omega }_{\mathrm{s}}}}.\end{array}$$

(16)

Furthermore, the state equation of the MMC-VSG is derived by decomposing ΔT_e as

$$J{\omega }_0\Delta \ddot{\delta }+(D_p+T_{\omega })\Delta \dot{\delta }+T_{\delta }+k_{11mix}=0.$$

(17)

Equation (17) indicates that the magnitude of T_ω also influences the behaviors of low-frequency oscillations. In fact, increasing T_ω can enlarge the system stability margin. Therefore, it is necessary to design an MMC-VSG control strategy to effectively compensate for the negative damping torque of the virtual excitation regulator and avoid low-frequency oscillation in grid-connected systems.

3. The Proposed Dual VPSS of MMC-VSG

The structure of the proposed dual VPSS scheme is illustrated in Figure 4. The excitation VPSS and the power-angle VPSS are introduced to compensate for the negative damping torques of the virtual excitation regulator and the power-angle regulator, respectively, to improve the system stability, particularly under small signal disturbances.

3.1. The Improved Excitation VPSS

Traditional mechanical motions rely on the PSS to reduce the negative damping torque to suppress low-frequency oscillations. However, active support control strategies can also be adopted to reduce the negative damping torque by configuring the VPSS to guarantee the stability of the power system. With regards to the proposed VPSS control strategy, Δω is taken as a control variable and fed into the excitation regulator through the transfer function G_upss, and an additional electromagnetic torque ∆T_upss can be generated.

Assuming the transfer function of ΔE to ∆T_upss is F_upss (s), we have

$$F_{upss}(s)=k_{12mix}.$$

(18)

Therefore,

$$\begin{array}{c}\mathsf{\Delta }{T}_{upss}(s)=F_{upss}(s)G_{upss}(s)\mathsf{\Delta }\omega (s)\\ =k_{12mix}{G}_{upss}(s)\mathsf{\Delta }\omega (s)\end{array}.$$

(19)

The conventional PSS usually contains a leading-lag compensator, and the transfer function of the excitation VPSS can be expressed by

$$G_{upss}(s)=k_{upss}{\displaystyle \frac{1+T_{6}s}{1+T_{5}s}}{\displaystyle \frac{1+T_{8}s}{1+T_{7}s}},$$

(20)

where k_upss is the proportional coefficient of VPSS and T₅ to T₈ are the time constants of the excitation VPSS leading-lag compensator.

In the control diagram of MMC-VSG 4th-order ROM (Figure 3), k_21mix and k_12mix represent the relative gains of the E–P and δ–Q coupling channels. These gains, affected by the powerline impedance to the inductance ratio and power-angle, indicate the coupling strength between the active and reactive power loops. With a constant impedance-to-inductance ratio, increasing the power-angle value enlarges k_21mix and k_12mix, enlarging the coupling strength. The control strategy in Figure 3 can be adapted to a single input single output (SISO) model (Figure 5) for power-angle regulation. This approach provides two decoupling methods: compensating reactive power due to k_21mix (compensation method 1) and compensating active power due to k_12mix (compensation method 2).

The decoupling ratio can be expressed as

$${\displaystyle \frac{\partial Q/\partial \delta }{\partial P/\partial E}}={\displaystyle \frac{k21mix}{k12mix}}={\displaystyle \frac{\frac{3R_{\mathrm{g}}^{\prime }}{R_{\mathrm{g}}^{\prime 2}+X_{\mathrm{g}}^2}\left(2E_0-U_{\mathrm{g}}\cos {\delta }_0\right)+\frac{3U_{\mathrm{g}}{X}_{\mathrm{g}}}{R_{\mathrm{g}}^{\prime 2}+X_{\mathrm{g}}^2}\sin {\delta }_0}{\frac{3U_{\mathrm{g}}{E}_0}{R_{\mathrm{g}}^{\prime 2}+X_{\mathrm{g}}^2}\left(X_{\mathrm{g}}\sin {\delta }_0-R_{\mathrm{g}}^{\prime }\cos {\delta }_0\right)}}\approx -E_0.$$

(21)

It can be seen that the impact of phase angle δ on reactive power Q is much greater than the impact of output voltage on active power P. Hence, the reactive power Q is compensated to decouple active and reactive power to realize the negative torque compensation in this paper.

Therefore, the excitation VPSS transfer function with decoupled power loops can be written as

$$G_{upss}(s)={\displaystyle \frac{E_0\cot (\phi +{\delta }_0)}{s}}+k_{upss}{\displaystyle \frac{1+T_{6}s}{1+T_{5}s}}{\displaystyle \frac{1+T_{8}s}{1+T_{7}s}}.$$

(22)

The improved Phillips–Heffron state equation with the excitation VPSS can be written as

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}\mathsf{\Delta }x}{\mathrm{d}t}}={\mathit{A}}_{phu}\mathsf{\Delta }x+{\mathit{B}}_{phu}\mathsf{\Delta }u+{\mathit{B}}_{upss}\mathsf{\Delta }{u}_{upss}\\ \mathsf{\Delta }y={\mathit{c}}_{phu}\mathsf{\Delta }x\end{array}\right..$$

(23)

The details of (22) and (23) and the parameters of T₅ to T₈ are calculated and presented in the Appendix A.

3.2. The Improved Power-Angle VPSS

The Phillips–Heffron model of the MMC-VSG system reveals that the oscillation is a small disturbance power-angle stability issue caused by negative damping. Thus, suppressing low-frequency oscillations in the MMC-VSG system can be achieved by optimizing and controlling the power-angle.

To improve the system stability, an additional damping control variable, Δδ_pss, is introduced into the control loop. Its effect on system damping characteristics is analyzed using the Phillips–Heffron model, where the damping torque contributed by Δδ to the power oscillation is generated by the two control loops of P₁₁ and P₁₂ (Figure 4). The damping torque ΔT_δpss contributed by Δδ_pss can be expressed as

$$\begin{array}{c}\mathsf{\Delta }{T}_{\delta pss}(s)=F_{\delta pss}(s)G_{\delta pss}(s)\mathsf{\Delta }\omega (s)=\mathsf{\Delta }{T}_{\delta pss11}+\mathsf{\Delta }{T}_{\delta pss12}\\ =F_{\delta \mathrm{pss}11}(\mathrm{s})G_{\delta pss}(\mathrm{s})\mathsf{\Delta }\omega (\mathrm{s})+F_{\delta pss12}(\mathrm{s})G_{\delta pss}(\mathrm{s})\mathsf{\Delta }\omega (\mathrm{s})\end{array},$$

(24)

where

$$\mathsf{\Delta }{T}_{\delta pss11}(s)=k_{11mix}{G}_{\delta pss}(s)\mathsf{\Delta }\omega (s)$$

$$\mathsf{\Delta }{T}_{\delta pss12}(s)={\displaystyle \frac{k_{12mix}{k}_{21mix}{k}_{\mathrm{q}}^{\prime }}{1+k_{\mathrm{q}{k}_{22mix}}^{\prime }}}{G}_{\delta pss}(s)\mathsf{\Delta }\omega (s).$$

The transfer function of the power-angle VPSS can be expressed by

$$G_{\delta pss}(s)=k_{\delta pss}{\displaystyle \frac{1+T_{2}s}{1+T_{1}s}}{\displaystyle \frac{1+T_{4}s}{1+T_{3}s}}.$$

(25)

In the formula, k_δpss is the proportional coefficient of the power-angle VPSS, and T₁ to T₄ are the time constants of the leading and lagging compensators of the power-angle of the VPSS. The Heffron–Hills state equation with the power-angle VPSS can be written as

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}\mathsf{\Delta }x}{\mathrm{d}t}}=A_{ph\delta }\mathsf{\Delta }x+B_{ph\delta }\mathsf{\Delta }u+B_{\delta pss}\mathsf{\Delta }{u}_{\delta pss}\\ \mathsf{\Delta }y={\mathit{C}}_{\mathrm{ph}\delta }\mathsf{\Delta }x\end{array}\right..$$

(26)

The detailed derivation of (25) and (26), along with the parameters T₁ to T₄, is provided in Appendix A. Additionally, considering the coupling relationship between the dynamic and droop characteristics of MMC-VSG, optimizing the active power control loop (APCL) is essential for achieving dynamic and steady-state decoupling. By introducing the correction variable ∆δ_δpss of the power-angle into the feed-forward damping control and removing D from the original damping loop, the closed-loop transfer function expression for the power-angle VPSS loop can be obtained by

$$\mathsf{\Delta }{P}_e=G_r(s)\mathsf{\Delta }{P}_{\mathrm{ref}},$$

(27)

where

$$G_r(s)={\displaystyle \frac{k_P^{\prime }(G_{\delta pss}s+1)}{J{\omega }_0{s}^2+(k_P^{\prime }{G}_{\delta pss}+k)s+k_P^{\prime }}}.$$

(28)

According to (28), the natural oscillation angular frequency ω’ and damping coefficient ξ’ of the improved MMC-VSG can be calculated by

$$\begin{array}{c}{\xi }^{\prime }={\displaystyle \frac{k+k_p\prime G_{\delta pss}}{2}}\sqrt{{\displaystyle \frac{1}{k_P\prime J{\omega }_0}}}\\ \omega \prime =\sqrt{{\displaystyle \frac{k_P\prime }{J{\omega }_0}}}\end{array}.$$

(29)

From the above equations, we can see that the natural oscillation frequency of MMC-VSG remains unchanged after introducing the feedforward power-angle VPSS. The damping characteristics can be adjusted by changing the magnitude of the feedforward coefficient G_δpss.

At this point, the steady-state output power of MMC-VSG can be written as

$$\begin{array}{c}{P}_{\mathrm{out}}=\underset{s\to 0}{lim}\left(P_e\right)=\underset{s\to 0}{lim}(G_r(s)P_{\mathrm{ref}}+G_f(s)({\omega }_0-\omega ))\hfill \\ =\underset{s\to 0}{lim}(\frac{k_{\mathrm{P}}^{\prime }(G_{pvpss}s+1)}{J{\omega }_0{\mathrm{s}}^2+(k_{\mathrm{P}}^{\prime }{G}_{pvpss}+k)\mathrm{s}+k_{\mathrm{P}}^{\prime }}{P}_{\mathrm{ref}}+\frac{k_{\mathrm{P}}^{\prime }(J{\omega }_0\mathrm{s}+k)}{J{\omega }_0{\mathrm{s}}^2+(k_{\mathrm{P}}^{\prime }{G}_{pvpss}+k)\mathrm{s}+k_{\mathrm{p}}^{\prime }}({\omega }_0-\omega ))\hfill \end{array}$$

(30)

Then we have

$$P_{out}=P_{\mathrm{ref}}+k({\omega }_0-\omega ).$$

(31)

Equation (31) indicates that in the proposed MMC-VSG scheme, the mathematical expression of damping can be changed from D_p to k to realize the decoupling between the droop control and the damping control.

By integrating the power-angle and excitation VPSS into the MMC-VSG system, the low-frequency oscillation of the MMC-VSG system can be suppressed, and the system stability margin can be enlarged based on above theoretical calculation.

4. Simulation Verification

A simulation model is established to verify the effectiveness of the simplified MMC-VSG model and the power-angle and excitation dual VPSS control method.

4.1. The 4th-Order MMC-VSG ROM

The key parameters of the state matrix A in the small signal model of MMC-VSG are shown in

Table 2. MMC-VSG parameters.

MMC Parameters	Values	MMC Parameters	Values
Network side voltage	8 kV	DC side voltage Udc	20 kV
Transformer voltage Eabc	8.2 kV	Bridge arm inductance L0	3 mH
Line inductance Lg	5 mH	Inertia J	0.1
Submodule capacitance Csm	$5000 \mathsf{\mu }\mathrm{F}$	Damping coefficient Dp	200
Submodule number Nsm	10	Delay coefficient Ti	10−6
Proportional coefficient kp(u)	0.5	Proportional coefficient kp(i)	22
Integration coefficient ki(u)	0.1	Integration coefficient ki(i)	1700

. The relevant states of the eigenvalues can be calculated as shown in

Table 3. MMC-VSG eigenvalues and related status.

Mode	Eigenvalue	Oscillation Frequency	Damping Ratio	Main State Variables	Participation Factor
1	−7.8979 × 105 ± j2.3463 × 102	1.257 × 106	1	Δed/Δeq	0.9998/1
2	−1.4857 × 103 ± j3.2787 × 103	5.729 × 103	0.4127	Δϕ2d/Δϕ2q	0.9809/1
3	−1.4369 × 103 ± j3.2153 × 103	5.6051 × 103	0.4080	Δϕ2d/Δϕ2q	1/0.9813
4	−19.2171 ± j70.0938	13.0450	0.2594	Δϕ1d/Δϕ1q	1/0.9891
5	−2.1082 ± j20.49691	3.3723	0.0962	Δδ/Δω	0.9873/1

.

Table 3 shows the eigenvalue distribution of an individual MMC-VSG, with all eigenvalues located in the left half of the s-plane, confirming the system stability. Among the ten eigenvalues, five types of oscillation frequencies can be observed. The medium and high-frequency eigenvalues have larger absolute real parts and are far from the imaginary axis, indicating rapid attenuation and minimal impact on system stability. Demonstratively, the study focuses on the four eigenvalues corresponding to the oscillation frequencies of 13.045 Hz and 3.373 Hz.

The state matrix A of the MMC-VSG 4th-order ROM has four eigenvalues, and the related two oscillation modes are demonstrated with regards to two typical oscillation frequencies, 11.1988 Hz and 3.3069 Hz. The related states of the eigenvalues are shown in

Table 4. Two typical MMC-VSG eigenvalues and oscillation modes.

Mode	Values	MMC Parameters	Values
1	−20.3886 ± j67.3457	11.1988	0.2898
2	−1.9538 ± j20.6857	3.3069	0.094

.

The trajectories of all eigenvalues and the dominant eigenvalues between the reduced 4th-order ROM and the full-order model are presented in Figure 6 and Figure 7, respectively, in consideration of different parameter variation. As can be seen, the trajectories of the dominant eigenvalues of the 4th-order ROM and the full-order model are basically coincided, indicating that the reduced 4th-order ROM has enough accuracy to represent the full-order model.

4.2. Oscillation Modes of the Control System

Table 4 shows that, without the proposed dual VPSS control, the eigenvalues representing the system oscillation modes are −20.3886 ± j67.3457 and −1.9538 ± j20.6857, respectively. In addition, the damping ratio of Mode 2 is much smaller than that of Mode 1, which could lead to a high possibility of instability and oscillation. Therefore, the parameter settings are mainly focused on Mode 2 in this study. By setting the target eigenvalue of the oscillation mode to λ_2′ = −6.9538 ± j20.6857, the time constants T_1–T₄ and k_δpss, as well as T₅–T₈ and k_upss, can be configured according to (28) and (25), respectively, as depicted in

Table 5. VPSS parameters.

VPSS Parameters	Value	VPSS Parameters	Value
T1	0.09	T5	0.09
T2	0.085	T6	0.2
T3	0.09	T7	0.09
T4	0.085	T8	0.2
kδpss	2	kupss	1
λ2,δpss	−7.1694 ± j20.5935	λ2,upss	−7.0609 ± j14.127

.

Figure 8 shows that after activating the power-angle VPSS in the APCL and the excitation VPSS in the RPCL, the eigenvalues related to Mode 2 are shifted to the left side of the complex plane, and the system damping ratio is improved. Two negative real roots are introduced, and their impact on stability is negligible. However, as k_δpss and k_upss are increased, the damping ratios of Mode 2 and Mode 1 become closer, potentially leading to low system stability.

4.2.1. The Influence of the Excitation VPSS on System Oscillation

Based on the parameters listed in Table 5, the system eigenvalues can be calculated by incorporating the excitation VPSS. Figure 9 shows the eigenvalue trajectories with and without the excitation VPSS as k_upss increases. Initially, with a small k_upss, the dominant eigenvalues related to Mode 2 (s₁ and s₂) are close to the imaginary axis, while s₃ and s₄ have negative real parts independent of the system oscillation modes. The distance between s₅ and s₆ on the real axis is much greater than that of s₁ and s₂. As the gain coefficient k_upss is increased, s₁ and s₂ gradually depart from the imaginary axis, and the system damping is then enhanced. At k_upss = 6, s₁ and s₂ have negative real parts, and s₅ and s₆ show dominant influence on system stability. Further increasing k_upss causes s₅ and s₆ to gradually approach the imaginary axis, reducing damping and increasing the instability risk.

The oscillation modes and damping ratios corresponding to different k_upss are listed in

Table 6. The electromechanical oscillation modes and damping ratios at different k_upss.

kupss	Eigenvalue	Oscillation Frequency	Damping Ratio
0	−1.9538 ± j20.6857	3.3069	0.094
1	−6.3659 ± j13.9135	2.4352	0.4161
2	−7.6794 ± j10.0497	2.0130	0.6072
4	−8.7002 ± j4.7666	1.5789	0.8770

. According to Table 6, as the gain coefficient k_upss is increased, the imaginary part of the system eigenvalues with regards to the oscillation mode and the natural oscillation frequency remain unchanged, and the damping ratio is gradually increased. Corresponding to Figure 9, as k_upss is increased, the eigenvalue λ₂, corresponding to the dominant oscillation mode, is shifted to the left along the real axis, resulting in an enlargement in the system stability margin.

Nyquist plots of the MMC-VSG system with different k_upss are presented in Figure 10. With the proposed excitation VPSS, the active power Nyquist curve avoids surrounding the (−1, j0) point, ensuring system stability. Increasing the gain coefficient k_upss shifts the real axis point to the right, indicating a higher stability margin. A simulation model with the parameters in Table 2 is built for validation. The excitation VPSS parameter k_upss is set to 0, 2, and 5, respectively. The MMC-VSG’s reference active power is 10 MW, with reactive power equal to 0 Mvar. The grid frequency is 50 Hz, altered by 1 Hz at t = 1.0 s, and restored to 50 Hz at t = 2.0 s. Simulation results with and without the excitation VPSS control for different k_upss values are shown in Figure 11. The results show that the proposed excitation VPSS is capable of compensating for the active and reactive power and increasing the reactive power response speed. The MMC-VSG system’s stability margin can be enlarged by increasing k_upss.

4.2.2. The Influence of the Power-Angle VPSS on System Oscillation

The system eigenvalues with and without the power-angle VPSS can be calculated based on the system parameters listed in Table 5. The eigenvalue trajectories as k_δpss are changed as shown in Figure 12.

The root trajectories with regards to varying k_δpss and k_upss are similar. With small k_δpss, the dominant eigenvalues s₁ and s₂ (λ₂) are near the imaginary axis, while s₃ and s₄ have negative real roots independent of the system oscillation modes. The distance between s₅ and s₆ on the imaginary axis is much greater than that of s₁ and s₂. Since s₁ and s₂ are the dominant eigenvalues for system oscillation, the stability margin can be analyzed through λ₂. As the gain coefficient k_δpss increases, s₁ and s₂ move further from the imaginary axis, enhancing the system’s damping characteristics. Beyond a certain point, further increasing k_δpss can reduce the system stability margin. Eventually, s₅ and s₆ move past the imaginary axis into the right half-plane, leading to instability.

The oscillation modes and damping ratios corresponding to different k_δpss are listed in

Table 7. Electromechanical oscillation modes and damping ratios corresponding to different k_δpss.

kupss	Eigenvalue	Oscillation Frequency	Damping Ratio
0	−1.9538 ± j20.6857	3.3069	0.094
1	−4.1584 ± j20.6584	3.3538	0.1973
2	−6.7728 ± j20.2488	3.3982	0.3172
4	−12.0423 ± j17.8323	3.4246	0.5596

. As k_δpss increases, the imaginary part of the eigenvalues and the natural oscillation frequency remain nearly unchanged, while the damping ratio improves. Figure 13 shows that the Nyquist curve for both the excitation VPSS and the power-angle VPSS exhibit similar trends, with curves encircling the (−1, j0) point.

Simulations via MATLAB 2022B were conducted with k_δpss set to 0, 5, and 10. Figure 14 shows that different k_δpss values have minimal impact on steady-state operation, indicating that the power-angle VPSS can hardly affect the steady-state droop control. Without the power-angle VPSS, the MMC-VSG system exhibits evident oscillations, overshoots, and slow responses during dynamic power variation. With the power-angle VPSS, increasing k_δpss can reduce dynamic overshoots and improves the stability margin.

Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 show that both a power-angle VPSS and an excitation VPSS can improve the performance of oscillation suppression as compared to the cases without additional damping control. The power-angle VPSS can shorten the transient period and reduce oscillation amplitude. For the same real part of λ₂, an excitation VPSS requires a smaller gain coefficient, which is consistent with the Nyquist plot. The power-angle VPSS also provides better oscillation suppression, as the eigenvalues s₁ and s₂ are farther from the imaginary axis. In practical power systems, the rapid approach of s₅ and s₆ toward the imaginary axis, compared to s₁ and s₂ moving away, means that s₅ and s_6′s influence cannot be ignored when their distance from the imaginary axis is similar to that of s₁ and s₂. At this time, the system stability initially increases and then decreases with higher k_upss or k_δpss, leading to a reduced stability margin.

4.2.3. The Feasibility of the Proposed Dual Loop VPSS

The above case studies demonstrate that combining the power-angle and the excitation VPSSs can comprehensively improve the MMC-VSG system performance of oscillation suppression. To verify the proposed power-angle and excitation dual VPSS control strategy, a MATLAB/Simulink simulation model was conducted. The parameters of the MMC-VSG system are listed in Table 2, with k_δpss = 2 and k_upss = 1. The MMC-VSG system is configured with a reference active power of 10MW and a reactive power of 0 Mvar, corresponding to λ_2,δpss = −7.1694 ±j20.5935 and λ_2,upss = −7.0609 ± j14.127, respectively. The grid frequency is f₀ = 50 Hz. At t = 1.0s, the reactive power changes to 10 Mvar. The output active/reactive power are shown in Figure 15 and Figure 16.

The simulation results verify that the dual VPSS control significantly reduces output active power and oscillation amplitude compared to systems without it, resulting in shorter oscillation transients and better suppression. For the same oscillation mode (same real part of λ₂), the dual VPSS control achieves smaller active/reactive power, lower oscillation amplitude, and faster settling times than systems without it. In addition, compared to the single VPSS controls, the dual VPSS provides greater damping torque, smaller oscillation amplitude, and shorter transient times. Additionally, as shown in Figure 16, high-frequency oscillations with small magnitude take place after the occurrence of disturbance, and the phenomena can be attributed to the roots with a small damping ratio, which affect the MMC-VSG performance.

5. Conclusions

To simplify the MMC-VSG system and reduce calculation complexity, a new 4th-order ROM has been proposed and validated by analyzing the system’s pole positions and main eigenvalue trajectories. A Phillips–Heffron model was also developed by linearizing the MMC-VSG system, which is used to reveal the low-frequency oscillation mechanism. To better suppress these oscillations and achieve improved control performance, a power-angle/excitation dual VPSS control method has been introduced. Compared to existing MMC-VSG modeling methods, the 4th-order ROM can provide relatively accurate modeling with reduced complexity as compared to the full-order model. Furthermore, the power-angle/excitation dual VPSS control method can improve both systems damping performance and stability margin.

The proposed power-angle/excitation dual VPSS control method addresses only the small-signal stability of the MMC-VSG system. However, the transient stability mechanism is more complex. Future work will explore diverse approaches to address these issues, including studying multi-machine interactions, replacing traditional control modules with artificial intelligence algorithms, and developing an experimental prototype for validating the proposed solution.